Control of Nonlinear Distributed Parameter Systems (Lecture Notes in Pure and Applied Mathematics)

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Control of Nonlinear Distributed Parameter Systems edited by Goong Chen, Texas A&M University, College Station, Texas Irena Lasiecka, University of Virginia, Charlottesville, Virginia Jianxin Zhou, Texas A&M University, College Station, Texas

iii

Preface This volume is an outgrowth of the conference “Advances in Control of Nonlinear Distributed Parameter Systems”, held on October 22-23, 1999, at Texas A&M University, College Station, Texas. The conference was jointly sponsored by the National Science Foundation (NSF), The Institute of Mathematics and Its Applications (IMA) and Texas A&M University. Fifty-five researchers attended and twenty-six talks were delivered during the two-day event. Ten papers in this volume were written by those conference speakers. To further broaden the scope and appeal of this volume, we have invited seven additional papers from experts working in this field. Thus, a total of seventeen papers have constituted the volume. The mathematical theory of control is highly interdisciplinary—it is a part of applied mathematics serving perhaps the most important link between mathematics and technology: complex systems in aerospace, civil and mechanical engineering must be controlled in order to achieve designated mission or operational requirements. Many ultra-modern electronic and optical devices are also designed for and dedicated to the purpose of acting as control mechanisms and media, i.e., actuators and sensors. Most of those devices are inherently nonlinear. The strong interest in mathematical control problems among mathematicians and engineers alike can be witnessed in the large number of papers published in the various journals of IEEE and SIAM. Even though steady progress has been made in the overall study of the mathematics of control, and wider and wider applications to new problems have been found, the leading edge of the field, as a mathematical subject, is indisputably the area of control of distributed parameter systems (DPS). This area concerns investigation of the control laws, stability and optimization of systems and feedback syntheses for systems whose states are spatially and/or temporally distributed and whose governing equations are partial differential or functional (typically time delay) equations. Studies in the area also include the associated questions of modelling, identification and estimation, analysis and design, computation and visualization, etc., of DPS. Rapid progress has occurred in this area since its inception during the 1960’s and its initial burst of growth in the 1970’s. After nearly three decades of research, though many interesting questions remain open, control theory for linear DPS has attained a certain level of maturity. The momentum of DPS research is now visibly moving toward the study of control of nonlinear partial differential equations. Nonlinear DPS (NDPS) are very much model-dependent. Since comprehensive, unified theories are virtually nonexistent, research opportunities and challenges are

iv extraordinarily numerous. Very substantial payoffs from the study of control and optimization of wide-ranging, application-driven nonlinear DPS in various areas of high technology may be expected to yield a substantial payoff through operational economies and enhanced system performance. We hope the present volume will stimulate active development of the mathematical theory in this critically important area. Two major influences are driving the recent sharp surge of interest in control of nonlinear distributed parameter systems:

(A)

Advances in “smart” materials, active actuators and sensors, microelectromechanical systems (MEMS), etc.

Existing advanced, or “smart” materials largely consist of sophisticated laminates incorporating specialized layers in an overall matrix form; they are fabricated to achieve a variety of desirable properties. Actuators and/or sensors consisting of piezoelectric/piezoceramic, opto-thermo-electric materials or microprocessors can be bonded to external surfaces or embedded within the layered structure itself. The response of such individual components is totally nonlinear, resulting in an overall system of nonlinear partial differential equations as the operative mathematical model. For example, aircraft propellers and helicopter rotor blades may be designed so that varying pitch is achieved by torsional actuation within the blade itself rather than by a mechanically articulated mechanism at the point where the blades are mounted. Another example concerns active noise suppression in aircraft cabins. This is achieved by means of actuator panels in the cabin walls, acting to achieve cancellation of high amplitude noise signals propagated through the fuselage. Many more examples of ultra-modern micromachined elastic structures in diverse applications may be found in a large number of new technical journals. The complete list of nonlinear distributed parameter systems finding applications in the area of advanced materials is much too long for us to cover in any representative way here.

(B)

Advances in nonlinear PDEs and dynamical systems

The existing, now almost classical, theory of control of linear DPS is of rather limited use in the nonlinear arena. New nonlinear methodology for control, stabilization and optimization needs to be developed for such systems. During the past thirty years, dramatic breakthroughs in theory and methods for nonlinear PDEs have been made, including Lax’s entropy solution and Glimm’s method for hyperbolic conservation laws, The Mountain Pass Lemma of Ambrosetti and Rabinowitz, the method of viscosity solutions, Hopf bifurcation phenomena in infinite dimensional spaces studied by Crandall and Rabinowitz,. . . , enabling researchers to treat an increasing number of genuinely

v nonlinear PDEs with confidence. These equations, or systems of equations often have unstable, multiple solutions, depending on the geometry of the domain – a totally bewildering situation prior to recent developments. The emergence of the new field of dynamical systems and chaos has likewise shifted the focus of attention from the classical qualitative theory of ODEs and PDEs to that of fractals, strange attractors, randomness, and their manipulations, control and applications—these are some of the most intensively investigated topics in the general scientific community at the present time. Both exogenous and endogenous factors, i.e., (A) and (B), respectively, above, are simultaneously at work, enriching and propelling the study of control of nonlinear distributed parameter systems and cross-fertilizing other intimately allied disciplines. These synergistic effects amply testify to the timeliness of the publication of this volume. The chapters in this volume cover interests in various aspects of NDPS. For example, the paper by Seidman and Antman is related to Category (A) above. The two papers by Ding and by Li and Zhou involve the application of the Mountain Pass Lemma and are thus more associated with Category (B). The paper by Chen, Huang, Juang and Ma studying chaotic phenomena due to nonlinear boundary conditions has overlapping interests in both (A) and (B). We hope the wide range of topics in these, and the other papers not explicitly cited here, will provide a useful reference for the study of nonlinear distributed parameter systems and stimulate further interest and research in this important area. We thank all the contributing authors for their work and and their patience with repetitive revisions. We are grateful to Dr. Deborah Lockhart at NSF, Professor Willard Miller, Jr. of IMA, and Professor Richard E. Ewing, Dean of College of Science at Texas A&M University, for the financial support to the conference. Finally, we thank Ms. Maria Allegra and Helen Paisner at Marcel Dekker and Professor M. Zuhair Nashed of the University of Delaware for their kind assistance in expediting the editorial and publication process.

Goong Chen and Jianxin Zhou College Station, Texas Irena Lasiecka Charlottesville, Virginia

vi

Dedicated to Professor David L. Russell on the Occasion of his 60th Birthday

Contents Preface

vii iii

1. Shape Sensitivity Analysis in Hyperbolic Problems with non Smooth Domains John Cagnol and J. Paul Zolesio 1 2. Unbounded Growth of Total Variations of Snapshots of the 1D Linear Wave Equation due to the Chaotic Behavior of Iterates of Composite Nonlinear Boundary Reflection Relations Goong Chen, Tingwen. Huang, Jong Juang and Daowei Ma 15 3.

Velocity method and Courant metric topologies in shape analysis of partial differential equations Michel Delfour and J. Pual Zolesio 45

4. Nonlinear Periodic Oscillations In Suspension Bridges Zhonghai Ding

69

5. Canonical Dual Control for Nonconvex Distributed-Parameter Systems: Theory and Method David Y. Gao 85 6. Carleman estimate for a parabolic equation in a Sobolev space of negative order and their applications Oleg Imanuvilev and Masahiro Yamamoto 113 7.

Bilinear control for global controllability of the semilinear parabolic equations with superlinear terms Alexander Khapalov 139

8. A Nonoverlapping Domain Decomposition for Optimal Boundary Control of the Dynamic Maxwell System John E. Lagnese 157 9.

Boundary Stabilizibility of a Nonlinear Structural Acoustic Model Including Thermoelastic Effects Catherine Lebiedzik 177

10.

On Modelling, Analysis and Simulation of Optimal Control Problems for Dynamic Networks of Euler-Bernoulli-and Rayleigh-beams Guenter Leugering and Wigand Rathman 199

viii

Contents

11. Local Characterizations of Saddle Points and Their Morse Indices Yongxin Li and Jianxin Zhou 233 12. Static Buckling in a Supported Nonlinear Elastic Beam David Russell and Luther White

253

13. Optimal control of a nonlinearly viscoelastic rod Thomas Seidman and Stuart Antman

273

14. Mathematical Modeling and Analysis for Robotic Control Sze-Kai Tsui

285

15. Optimal Control and Synthesis of Nonlinear Infinite Dimensional Systems Yuncheng You 299 16. Forced Oscillation of The Korteweg-De Vries-Burgers Equation and Its Stability Bingyu Zhang 337

Shape Sensitivity Analysis in Hyperbolic Problems with non Smooth Domains

John Cagnol1 , Université Léonard de Vinci, FST, DER-CS, 92916 Paris La Défense Cedex, France, E-mail: [email protected] Jean-Paul Zol´ esio, CNRS, Ecole des Mines de Paris, 06902 Sophia Antipolis Cedex, France. E-mail: [email protected] Abstract The control with respect to the domain is inherently not linear due to the non linear structure of the set of domains. In this paper we investigate the weak shape differentiability of the solution to the generalized wave equation when the domain has a Lipschitz continuous boundary. By the means of the “hidden regularity”, a result for C 2 -boundary was obtained recently, when the right hand side is in L2 . To extend that result to Lipschitz continuous boundary, we first investigate the regularity of the solution at the boundary. We need an exact estimate of the L2 -norm of the normal derivative. Then, we build an increasing sequence of smooth domains, and we establish the shape differentiability result as a consequence of the situation for C 2 -boundary.

1

Introduction

The control with respect to the domain is inherently not linear due to the non linear structure of the set of domains. In this paper we investigate the sensitivity of the solution of an hyperbolic PDE with respect to the domain. This analysis is carried out with the wave equation with an homogeneous Dirichlet boundary condition. The novelty lies in the absence of regularity of the domain with respect to which the analysis is done. In a sense we extend the result presented in [6] to the case of Lipschitz-continuous domains. Let N ≥ 2 be an integer and D be a bounded domain of RN . Throughout this paper Ω will be an open domain, star-shaped, included in D whose boundary Γ is assumed to be Lipschitz continuous. Moreover we will assume Ω has a bounded perimeter. The family of such domains Ω shall be denoted O. 1 At the time this paper was presented, the first author was at the University of Virginia, Charlottesville, VA. Research supported by the INRIA under grant 1/99017.

1

2

Cagnol and Zol´ esio

Let T be a non negative real and I = [0, T ] be the time interval. We note Q =]0; T [×Ω the cylindrical evolution domain and Σ =]0, T [×Γ the lateral boundary associated to any element Ω of the family O.

1.1

Shape Differentiability

¯ RN )) with hV, n∂D i = 0 and free Let E be the set of V ∈ C([0, S]; C 1 (D, divergence. For any V ∈ E we consider the flow mapping Ts (V ). At the point x, V has the form as follows: ∂ V (s)(x) = (1) Ts ◦ Ts−1 (x) ∂s For each s ∈ [0, S[, Ts is a one-to-one mapping from D onto D such that i) T0 = I ¯ D)) ¯ with Ts (∂D) = ∂D ii) s 7→ Ts belongs to C 1 ([0; S[, C 1 (D; ¯ D)) ¯ iii) s 7→ Ts−1 belongs to C([0; S[, C 1 (D; We refer to [8] and [9] for further discussion on such mappings. The family O is stable under the perturbations Ω 7→ Ωs (V ) = Ts (V )(Ω). We denote by Qs the perturbed cylinder ]0; T [×Ωs (V ), Γs = ∂Ωs and Σs =]0, T [×Γs the perturbed lateral boundary. Let m ≥ 1 be an integer. Let f ∈ L1 (I, H m (D)) with its m-th timederivative in L1 (I, L2 (D)). Let ϕ ∈ H m+1 (D) and ψ ∈ H m (D). Let K be a coercive and symmetric N × N -matrix whose coefficients belong to W 2,∞ (D). To each element Ω ∈ O we associate the solution y = y(Ω) of the following problem  2 ∂ y − div (K∇y) = f on Q    t y = 0 on Σ (2) y(0) = ϕ on Ω    ∂t y(0) = ψ on Ω Throughout this paper we shall note P the operator ∂tt y − div (K∇). A Galerking method proves y ∈ H(I, Ω) = H 1 (I, L2 (Ω)) ∩ L2 (I, H01 (Ω)) For any V ∈ E and s ∈ [0; S] we set ys = y(Ωs ) ∈ L2 (Qs ). Following [5], [6], [13] the mapping Ω 7→ y(Ω) is said to be shape differentiable in L2 (I, H m (D)) (3)

∃Y ∈ C 1 ([0; S], L2 (I, H m (D)))

(4)

Y (s, ·, ·)|Qs = y(Ωs )

Sh. Sensitivity Analysis in Hyperbolic Pb. with non Smooth Domains then ∂s Y (0, ·, ·)|Q which is the restriction to Q of the derivative with respect to the perturbation parameter s at s = 0 is independent of the choice of Y verifying (3) and (4). (cf. [13]). Definition 1.1 (shape derivative). The shape derivative is that unique element ∂ 0 y (Ω; V ) = ∈ L2 (Q) Y ∂s s=0 (t,x)∈Q The weak shape differentiability can be defined analogously, replacing (3) by the existence of Y in C 1 ([0; S], L2σ (I, H m (D))).

1.2

Known Results for C 2 -boundary

When the boundary is C 2 it was proven in [11] that (2) has a unique solution in i m−i Z m (I, Ω) = ∩m (Ω)) i=0 C (I, H In [5], [6] the question of the shape differentiability is solved for various conditions of regularity of the data, but the domain Ω needs to be C 2 . The main result was Theorem 1.1 (Cagnol-Zolsio, 1997). Let m be a positive integer and let Ω be a domain with a C max{m,2} boundary. i) If m ≥ 1 then the solution to (2) is shape differentiable at Ω, strongly in L2 (I, H m−1 (D)). ii) If m = 0 hen the solution to (2) is shape differentiable at Ω, weakly in L1 (I, L2 (D)). the shape derivative y 0 ∈ Z m (I, Ω) and is solution to  2 0 ∂t y − div (K∇y 0 ) = 0 on Q    0 ∂y y = − ∂n hV (0), ni on Σ (5) 0  y (0) = 0 on Ω   ∂t y 0 (0) = 0 on Ω

1.3

Main Result

In this paper we extend the result of theorem 1.1 to the case of Lipschitz continuous domains Ω. Problem (2) is well-posed and, as we said earlier, the solution y lies in H(I, Ω). In [7] and [10] it is proven that the normal derivative belongs to L2 (Σ). That leads to the well-posedness of (5). Hence looking for the shape derivative in the case of Lipschitz continuous boundaries makes sense. In this paper we shall prove the following result Theorem 1.2. When m = 0, the solution to problem (2) is weakly shape differentiable at Ω in L1 (I, L2 (D)). The shape derivative y 0 belongs to H(I, Ω) and is solution to (5).

3

4


Remark 1.1. When m ≥ 1, the result can be improved to a weak differentiability in L∞ (I, L2 (D)).

2

Mollification of the Domain

Given a Lipschitz continuous domain Ω, we build an increasing sequence of smooth sub-domains converging to Ω with Haussdorf convergence of the boundaries. See also [12].

2.1

Properties of Lipschitz Continuous Domains

Definition 2.1. An open set Ω ⊂ RN is said to have the cone property if π ∃R > 0, ∃θ ∈]0, [, ∀x ∈ ∂Ω, ∃d, Cx (R, θ, d) ⊂ Ω 2 where Cx (R, θ, d) is the interior of a cone of revolution with the vertex at x, height R cos(θ/2) and the axis pointing toward the versor d. When Ω has a Lipschitz continuous boundary then Ω and RN r Ω have the cone property (cf. [1], [2]). Let R(Ω) and θ(Ω) be the parameters arising from the cone condition on Ω and R(RN r Ω) and θ(RN r Ω) be the parameters arising from the cone condition on RN rΩ. We note R = min(R(Ω), R(RN rΩ)) and θ = min(θ(Ω), θ(RN r Ω)). Remark 2.1. The reals R and θ do not depend on x. Lemma 2.1. Let |X| denote the measure of X, 1 1 M + M − − + ∃M > 0, ∃M < 1, ∀κ ≥ , ∀x ∈ ∂Ω, N ≤ Ω ∩ B x, ≤ N R κ κ κ Proof. Let x ∈ ∂Ω, the cone property yields the existence of a versor d such that Cx ( κ1 , θ, d) ⊂ Ω. Since κ1 < R we get Cx

1 1 , θ, d ⊂ Ω ∩ B x, κ κ

Let B(p) be the volume of the p-th dimensional ball of radius 1. We refer to [3, pp. 208–210] for an expression of B(p) as a function of p. The volume of the p-th dimensional ball of radius r is B(p)r p . Then, the volume of Cx ( κ1 , θ, d) is 1 1 1 N −1 hence N κ cos(θ/2)B(N − 1)( κ ) N −1 Ω ∩ B x, 1 ≥ 1 1 cos(θ/2)B(N − 1) 1 κ Nκ κ therefore

− Ω ∩ B x, 1 ≥ M κ κN

Sh. Sensitivity Analysis in Hyperbolic Pb. with non Smooth Domains with M − = N1 cos(θ/2)B(N − 1). Considering the cone property for RN r Ω yields the existence of M > 0 such that (RN r Ω) ∩ B(x, κ1 )| ≥ κMN . Let M + = 1 − M , we obtain + Ω ∩ B x, 1 ≤ M κ κN Let χ be the characteristic function of Ω and (ρκ ) be a mollifier. Let us note ξκ = χ ∗ ρκ . Proposition 2.1. There exists M − > 0 and M + < 1 such that 1 , ∀x ∈ ∂Ω, M − ≤ ξκ (x) ≤ M + R R R Proof. One has ξκ (x) = R2 χ(t)ρκ (t − x) dt hence ξκ (x) = Ω ρκ (t − x) dt. thus Z ξκ (x) = ρκ (t − x) dt ∀κ ≥

1 Ω∩B(x, κ )

Using the symmetry property of ρκ we get Ω ∩ B(x, 1 ) Z κ ξκ (x) = ρκ (t) dt B(x, 1 ) 1 B(0, κ ) κ Lemma 2.1 applies and gives the result. Lemma 2.2. supp ξκ = Ω + B(0, κ1 ) and supp (1 − ξκ ) = (RN r Ω) + B(0, κ1 ) Proof. The lemma is a consequence of supp (χ ∗ ρκ ) ⊂ supp χ + supp ρκ . We use χ ≥ 0 and ρκ ≥ 0 to prove the first equality. The second equality can be proven by the same techniques. Proposition 2.2. Let κ ≥ R1 and x ∈ RN then ξκ (x) > M + =⇒ x ∈ Ω ξκ (x) < M − =⇒ x 6∈ Ω Proof. From proposition 2.1 we have ξκ−1 (]M + , +∞[) ∩ ∂Ω = ∅ therefore ξκ−1 (]M + , +∞[) ⊂ Ω or ξκ−1 (]M + , +∞[) ⊂ RN r Ω. Elements x of Ω whose distance to the boundary is more than κ1 satisfy ξκ (x) = 1 thus ξκ−1 (]M + , +∞[) ⊂ Ω Analogous arguments show that ξκ−1 (] − ∞, M − [) ⊂ RN r Ω.

5

6


2.2

Definitions and Preliminary Results

Let Gκ ⊂ RN × R be the graph of ξκ . Since ξκ is C ∞ , the set Gκ is a C ∞ manifold. We note π1 : RN × R : (x, y) 7→ x π2 : RN × R : (x, y) 7→ y The restriction of π2 to Gκ is injective. We note −1 Γ(κ, t) = π1 ◦ π2|Gκ (t) ⊂ RN Lemma 2.3. Let κ be a positive integer and α and β be two reals such that 0 ≤ α < β ≤ 1. There exists t ∈]α, β[ such that Γ(κ, t) is C ∞ . Proof. From the Sard’s theorem, the image of the critical points of π2κ has measure 0 in R. Hence there exists t ∈]α, β[ such that (π2 |Gκ )−1 is not critical, therefore (Γ(κ, t), t) is regular and Γ(κ, t) is C ∞ . For a real t provided by lemma 2.3, let Ω(κ, t) = (π1 ◦ (π2 |Gκ )−1 )(]t, +∞[) ⊂ be the level set, then ∂Ω(κ, t) = Γ(κ, t). Corollary 2.1. Under the hypothesis of lemma 2.3, there exists t ∈]α, β[ such that Ω(κ, t) is C ∞ . RN

2.3

Construction of a Sequence

The purpose of this section is to build an isotonic sequence of domains (Ωk )k≥0 , whose projective limit is Ω. Let α > M + . Construction of the first term: Let κ0 be an integer larger that R1 . Let β0 = 1, from lemma 2.3 there exists t ∈]α, β0 [ such that Ω(κ0 , t) is C ∞ . Let us note Ω0 = Ω(κ0 , t) and β1 = t. The set Ω0 built that way satisfies Ω0 ⊂ Ω moreover the distance d0 = d(ξκ−1 (M + ), ξκ−1 (t)) > 0. 0 0 Construction of the next terms: Let κ1 ≥ max(κ0 + 1, d10 ). There exists t ∈]α, β1 [ such that Ω(k1 , t) is C ∞ . Let Ω1 = Ω(κ1 , t) and β2 = t. We have Ω1 ⊂ Ω Since ξκ1 (x) = 1 for all x whose distance to the boundary of Ω is less than d0 we have ξκ1 (x) = 1 for all x ∈ Ω0 hence Ω0 ⊂ Ω1 Let d1 = d(ξκ−1 (M + ), ξκ−1 (t)) > 0. Then we build Ω2 and so on so forth. 0 1 For each k ≥ 0, Γk = Γ(kκ , βκ ) which is also the boundary of Ωk .

Sh. Sensitivity Analysis in Hyperbolic Pb. with non Smooth Domains

2.4

Properties

Proposition 2.3. The sequence (Ωk )k≥0 has the subsequent properties i) It is an increasing sequence of domains k ii) The limit ∪+∞ k=0 Ω is equal to Ω

Proof. i) This is a consequence of the construction k ii) Since Ωk ⊂ Ω it is obvious that ∪+∞ k=0 Ω ⊂ Ω. Let x ∈ Ω, since Ω is open there exists r > 0 such that B(x, r) ⊂ Ω. Let k be such that k κk ≥ max( 1r , k0 ) then ξk (x) = 1 hence x ∈ Ωk . It follows Ω ⊂ ∪+∞ k=0 Ω .

Proposition 2.4. Let K be a compact subset of Ω, there exists k ≥ k0 such that K ⊂ Ωk . Proof. Let r be the distance between K and Ω. Let k be such that κk ≥ max( 1r , κ0 ) then for all x ∈ K we have ξκk (x) = 1 hence x ∈ Ωk .

2.5

Mollification of the Transported Domain

Transported domains Ωs = Ts (Ω) were considered in the introduction. They are Lipschitz continuous so the construction which has been performed with Ω can be repeated for those domains. That yields an isotonic sequence of domains (Ωks )k≥0 which tends to Ωs . Property 2.4 holds when replacing Ω by Ωs . Once s is given, all subsequent properties on Ω will hold for Ωs as well. Remark 2.2. There is no reason to have Ts (Ωk ) = Ωks . Let Qks = I × Ωks , Γks = ∂Ωks and Σks = I × Γks . We shall note ysk the solution of the problem  2 ∂ y − div (K∇ysk ) = f on Qks    kt ys = 0 on Σks (6) y k (0) = ϕ on Ωks    s k ∂t ys (0) = ψ on Ωks

3

Continuity Result for the Wave Equation

In this section and the next one, we suppose m = 0, that is f ∈ L1 (I, L2 (D)),

ϕ ∈ H 1 (D),

ψ ∈ L2 (D)

The aim of this section is to prove the solution to the wave equation in the mollified domain tends to the solution of the wave equation in the Lipschitz continuous domain. It is not a general continuity result (see [4]) since it only works with the sequence of domain built in the previous section. In the next section, that convergence will turn out to be enough to prove the shape differentiability result that we are looking for.

7

8


3.1

Weak Convergence

For k ≥ 0 we note Qk = I × Ωk and Σk = I × Γk . Let us consider  P y k = f on Qk    k y = 0 on Σk y k (0) = ϕ on Ωk    ∂t y k (0) = ψ on Ωk

(7)

That problem has a unique solution in Z 1 (I, Ωk ). The energy estimate gives the subsequent lemma (see [6, lemma 5]) Lemma 3.1. Let O be an open C 2 domain in D and µ ∈ Z 2 (I, O). We note Z a(µ) = kP µkL1 (I,L2 (O)) and b(µ) =

(K∇µ(0).∇µ(0) + (∂t µ(0))2 )dx O

then (8)

k∂t µkL∞ (I,L2 (O)) ≤ 2a(µ) +

(9)

kµkL∞ (I,H01 (O)) ≤ 2a(µ) +

p

p

b(µ)

b(µ)

Proposition 3.1. Let O be an open C 2 domain in D and µ ∈ Z 1 (I, O) with P µ ∈ L1 (I, L2 (Ω)). With the notations of lemma 3.1, identities (8) and (9) hold. Proof. This proposition is a consequence of lemma 3.1, the the density of Z 2 (I, O) in Z 1 (I, O) and the continuity of the wave equation with respect to the data. The hypothesis of that proposition are satisfied for O = Ωk and µ = y k . Let us note Z k k a = kf kL1 (I,L2 (Ωk )) and b = (K∇ϕ.∇ϕ + ψ 2 )dx Ωk

then k∂t y k kL∞ (I,L2 (Ωk )) ≤ 2ak +

√

bk

√ ky k kL∞ (I,H01 (Ωk )) ≤ 2ak + bk R Let a∗ = kf kL1 (I,L2 (D)) and b∗ = D (K∇ϕ.∇ϕ + ψ 2 )dx then for all k we have ak ≤ a∗ and bk ≤ b∗ . Moreover y k can be extended by 0 on RN r Ωk . hence √ ky k kW 1,∞ (I,L2 (Ω))∩L∞ (I,H01 (Ω)) ≤ 2a∗ + b∗

Sh. Sensitivity Analysis in Hyperbolic Pb. with non Smooth Domains that yields ky k kH(I,Ω) is bounded, hence there exists a converging subsequence weakly in H 1 (Q) Let us note y ∗ such an2 element. We have y ∗ ∈ H(I, Ω)

(10)

Remark 3.1. As a corollary of (10) we have y ∗ = 0 on Σ. Proposition 3.2. One has P y ∗ = f on Q Proof. Let θ ∈ C0∞ (Q), since P y k = f on Qk we get Z ∞ ∀θ ∈ C0 (Q), (P y k )θ − f θ = 0 Qk

using proposition 2.4 we obtain the subsequent identity, when k is large enough Z ∞ ∀θ ∈ C0 (Q), (P y k )θ − f θ = 0 C0∞ (Q),

R

Q

∗ ∗ lemma 3.4 yields ∀θ ∈ Q (P y )θ − f θ = 0 therefore P y = f on Q. Proposition 3.3. One has y k (0) = ϕ and ∂t y k (0) = ψ on Ω. The proof of that lat proposition is analogous to the proof of proposition 3.2. Then we obtain  P y ∗ = f on Q    ∗ y = 0 on Σ (11) y  ∗ (0) = ϕ on Ω   ∂t y ∗ (0) = ψ on Ω

Since that problem is well-posed we have y ∗ = y. The subsequent lemma follows: Proposition 3.4. y k * y weakly in H(I, Ω) as Qk → Q

3.2

Strong Convergence

We consider 1 Ek (t) = 2

Z Ωk

D E K∇y k (t), ∇y k (t) + (∂t y k (t))2

and E∞ (t) the corresponding energy when replacing Ωk by Ω and y k by y. Lemma 3.2. Ek (0) → E∞ (0) when k → +∞. Proof. One has Z 1 Ek (0) = hKϕ, ϕi + ψ 2 2 Ωk since supp ϕ b Ω and supp ψ b Ω, proposition 2.4 gives Z 1 Ek (0) = hKϕ, ϕi + ψ 2 2 Ω when k is large enough. 2

At this point we do not know it is unique

9

10

Cagnol and Zol´ esio Lemma 3.3. One has Z kykH(I,Ω) = T E∞ (0) +

Z (T − τ )f ∂t y

0

Z ky kH(I,Ωk ) = T Ek (0) +

T

T

Ω

Z

k

0

Ωk

(T − τ )f ∂t y k

Proof. We shall do the proof for the second identity, the proof of the first one is analogous. The energy estimates gives Z ∂t Ek (t) = f ∂t y Ωk

this gives Ek (τ ) = Ek (0) + Z ky kH(I,Ωk ) =

Rτ R 0

Ωk

P y k ∂t y k we get Z

T

k

T

Z

Z

τ

Ek (τ ) dτ = T Ek (0) + 0

0

Ωk

0

P y k ∂t y k

Proposition 3.5. When k tends to +∞ we have ky k kH(I,Ω) → kykH(I,Ω) Proof. Lemma 3.2 and proposition 3.4 give Z TZ Z T Ek (0) + (T − τ )f ∂t y → E∞ (0) + 0

Ωk

0

T

Z (T − τ )f ∂t y Ω

lemma 3.3 yields ky k kH(I,Ωk ) → kykH(I,Ω) since (Ωk ) is an increasing sequence of domains and y k is extend by 0 out of Ωk we get ky k kH(I,Ωk ) = ky k kH(I,Ω) Corollary 3.1. We have y k → y strongly in H(I, Ω) as Qk → Q Remark 3.2. The same proof gives ysk → ys strongly in H(I, Ωs ) as Qks → Qs for all s ∈ [0, S].

Sh. Sensitivity Analysis in Hyperbolic Pb. with non Smooth Domains

4 Shape Differentiability 4.1 Absolute Continuity Let θ ∈ L1 (I, L2 (D)), we note Z hk (s) =

Qks

ysk θ dx dt

Z h(s) =

yθ dx dt Qs

the shape differentiability for smooth domains gives Z 0k h (s) = ys0 θ dx dt Qks

Let y¯ be the solution to the subsequent well-posed problem  P (¯ y ) = 0 on Q    ∂y y¯ = − ∂n hV (0), ni on Σ (12)  y¯(0) = 0 on Ω   ∂t y¯(0) = 0 on Ω at this point we do not know that y¯ is the shape derivative of the state function y, and it is precisely what we are going to prove. Let us note Z ¯ h(s) = y¯s θ dx dt Qs

The absolute continuity of hk gives (13)

∀k ∈ N∗ , ∀s ∈ [0, S], hk (s) = hk (0) +

Z

s

h0k (σ) dσ

0

From proposition 3.4, the left hand side and the first term of the right hand side converge to h(s) and h(0) Rrespectively. To prove the Rabsolute continuity of s s¯ h it is sufficient to prove that 0 h0k (σ) dσ converges to 0 h(σ) dσ as k tends to +∞. To achieve that goal let us introduce the following adjoint problem  P (Λk ) = θ on Qks    k s Λs = 0 on Σks (14) Λk (T ) = 0 on Ωks    s k ∂t Λs (T ) = 0 on Ωks From proposition 3.1 we get Λks → Λs strongly in H(I, Ωs ) as Qks → Qs

11

12


Following [6] we have Lemma 4.1. Let Ks = (DTs )−1 (K ◦ Ts )(∗ DTs−1 ) then 2 2 ! D Z ED E ∂(ysk + Λks ) 1 ∂(ysk − Λks ) 0k k k k h (s) = K V (s), n − n , n s s s s dΓ dt 4 Σks ∂nks ∂nks For the sake of shortness we suppose K = I. Following [6] we have Lemma 4.2. Let µks ∈ L2 (I, H01 (Ωks )) ∩ H1 (I, L2 (Ωks )) such that P µks ∈ L2 (Qks ) then Z k 2 D E ∂µs k V (s), n s = ∂nks Σks Z E D E Z D −2 (div (V (s) − 2ε(V ))∇µks , ∇µks ∂t µks (0) ∇µks (0), V (s)(0) − Ωks

Z

−2

Qks

D E P µks ∇µks , V (s) +

Z Qks

Qks

D E 2 ∂t µks div (V − s) − 2∂t µks ∇µks , V (s)

Let µks,α = ysk + αΛks with α ∈ {−1; 1}. It satisfies the hypothesis of lemma 4.2 since P ysk = f and P Λks = θ, Moreover ysk and Λks as well as their time derivative and its gradient vanish on Ω s r Ωks , therefore the integrals on Ωks and Qks of lemma 4.2 can be replaced by integrals on Ω s and Qs respectively. It ¯ k (s) converges to follows that h ! Z ∂(ys + Λs ) 2 ∂(ys − Λs ) 2 hKs ns , ns i hV (s), ns i dΓ dt − ∂ns ∂ns Σs it follows the Proposition 4.1. One has ¯ k (s) = h0 (s) lim h

k→+∞

From lemma 3.1, the real hk (s) is dominated by a constant independent of s and k. Since aks and bks are bounded by a∗ and b∗ respectively. Corollary 4.1. The function h is absolutely continuous.

4.2

Differentiability

Lemma 4.3. When s → 0 one has ys * y in H 1 (I × D) Proof. Proposition 3.1 works with O = Ωs and µ = ys . Since we extend ys by zero out of Ωs we get √ kys kW 1,∞ (I,L2 (D))∩L∞ (I,H 1 (D)) ≤ 2a∗ + b∗

Sh. Sensitivity Analysis in Hyperbolic Pb. with non Smooth Domains Since a∗ and b∗ depend only on the hold-all D, we extract a subsequence converging to an element y∗ . The last point to be proven is that y∗ is solution of (2). Even though we do not have an isotonic sequence, property 2.4 holds when replacing Ωk by Ωs and “k large enough” by “s small enough”. The only problem is to prove y∗ vanishes on the lateral boundary Σ. Let dX denote the distance to the set X. The convergence of the boundary gives dΩs → dΩ strongly in L2 (I × D), hence dΩs ys → dΩ y ¯ s and dΩs = 0 on Ωs , we get in L2 (I × D) as s → 0. Since ys = 0 on D r Ω dΩ y = 0 hence y = 0 almost everywhere in D r Ω. As Ω is Lipschitz continuous, ¯ it has the Keldysh stability property, therefore y = 0 quasi everywhere in Dr Ω. ¯ That yields y = 0 on ∂(D r Ω). We end up with y∗ = 0 on Σ. Proposition 4.2. When s → 0 one has ys → y strongly in H 1 (I × D) Proof. The proof is based on the ideas of section 3.2. The energy to be considered is Z 1 hK∇ys (t), ∇ys (t)i + (∂t ys (t))2 2 Ωs Again, we do not have an isotonic sequence, however because Ωs is the image of Ω by the flow mapping Ts , each compact of Ω is included in Ωs for s small enough. We derive kys kH(I,Ωs ) → kykH(I,Ω) when s → 0. That leads to kys kH(I,D) → kykH(I,D) . Since a∗ and b∗ depend only on D, the domination of kys kH(I,D) is straightforward. That proves ¯ ¯ lim h(s) = h(0)

s→0

That gives the weak shape differentiability in L1 (I, L2 (D)) of the state function. Remark 4.1. When m = 0, taking θ ∈ L∞ (I, L2 (Ω)) is required because the weak shape differentiability for C 2 -boundary takes place in L1 (I, L2 (Ωk )). When m ≥ 1, the test θ can be taken L1 (I, L2 (θ)), that leads to the shape differentiability in L∞ (I, L2 (D)) of the state function.

References [1] S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand Mathematical Studies, 1965.

13

14


[2] C. Baiocchi and A. Capelo, Variational and quasivariational inequalities, John Wiley & Sons Inc., New York, 1984. [3] M. Berger and B. Gostiaux, Differential geometry: manifolds, curves, and surfaces, Springer-Verlag, New York, 1988. [4] D. Bucur, Contrˆ ole par rapport au domaine dans les EDP, PhD thesis, Ecole des Mines de Paris, 1995. [5] J. Cagnol and J.-P. Zolésio, Hidden shape derivative in the wave equation, in Systems modelling and optimization (Detroit, MI, 1997), Chapman & Hall/CRC, Boca Raton, FL, 1999, pp. 42–52. [6] , Shape derivative in the wave equation with Dirichlet boundary conditions, J. Differential Equations, 158 (1999), pp. 175–210. [7] A. Chaïra, Equation des ondes et régularité sur un ouvert lipschitzien, Comptes Rendus de l’Académie des Sciences, Paris, series I, Partial Differential Equations, 316 (1993), pp. 33–36. [8] M. C. Delfour and J.-P. Zolésio, Structure of shape derivatives for non smooth domains, Journal of Functional Analysis, 104 (1992), pp. 1–33. [9] , Shape analysis via oriented distance functions, Journal of Functional Analysis, 123 (1994), pp. 129–201. [10] , Hidden boundary smoothness in hyperbolic tangential problems on nonsmooth domains, in Systems modelling and optimization (Detroit, MI, 1997), Chapman & Hall/CRC, Boca Raton, FL, 1999, pp. 53–61. [11] I. Lasiecka, J.-L. Lions, and R. Triggiani, Non homogeneous boundary value problems for second order hyperbolic operators, Journal de Mathématiques pures et Appliquées, 65 (1986), pp. 149–192. [12] J. Neˇcas, Sur les domaines de type N, Czechoslovak Math, 12 (1962), pp. 274–287. (Russian with a French summary). [13] J.-P. Zolésio, Introduction to shape optimization and free boundary problems, in Shape Optimization and Free Boundaries, M. C. Delfour, ed., vol. 380 of NATO ASI, Series C: Mathematical and Physical Sciences, Kluwer Academic Publishers, 1992, pp. 397–457.

Unbounded Growth of Total Variations of Snapshots of the 1D Linear Wave Equation due to the Chaotic Behavior of Iterates of Composite Nonlinear Boundary Reflection Relations

Goong Chen(1),(2) , Texas A&M University, College Station, Texas Tingwen Huang(1) , Texas A&M University, College Station, Texas Jonq Juang(3) , National Chiao Tung University, Hsinchu, Taiwan, ROC Daowei Ma(4) , Wichita State University, Wichita, Kansas Abstract Consider a linear one-dimensional wave equation on an interval. If the boundary conditions are also linear, then the total variation of the gradient (wx (·, t), wt (·, t)) on the spatial interval remains bounded as t → ∞, provided that the initial condition (w(·, 0), wt (·, 0)) has finite total variation. However, if we let the left-end boundary condition pump energy into the system linearly, while the right-end boundary condition be selfregulating of the van der Pol type with a cubic nonlinearity, then chaotic vibrations occur when the parameters enter a certain regime. In this paper, we characterize the chaotic behavior of the gradient (wx (·, t), wt (·, t)) by proving that its total variation grows unbounded (with generically given initial conditions) as t → ∞, even though the initial condition has a finite total variation. The proofs are obtained by the technique of interval covering sequences based on Stefan cycles and homoclinic orbits of the composite nonlinear boundary reflection map.

(1) E-mails: [email protected] and [email protected]. (2) Supported in part by Texas A&M University Interdisciplinary Research Initiative IRI 99-22. (3) Work completed while on sabbatical at Texas A&M University. Supported in part by a grant from NSC of R.O.C. E-mail: [email protected]. (4) E-mail: [email protected].

15

16

1

Chen et al.

Introduction

In this paper, we study a special property of chaotic vibration of the wave equation, that of unbounded growth of total variations of snapshots (wx (·, t), wt (·, t)) on the spatial interval of the one-dimensional (1D) wave equation as t → ∞. Earlier, in a series of papers [3–6], we have studied chaotic vibration of the 1D wave equation (1.1)

wxx (x, t) − wtt (x, t) = 0,

0 < x < 1,

t > 0,

subject to the following boundary conditions (1.2) left-end x = 0 : wt (0, t) = −ηwx (0, t),

η > 0,

(1.3) right-end x = 1 : wx (1, t) = αwt (1, t) − βwt3 (1, t),

η 6= 1,

t > 0;

0 < α ≤ 1,

β > 0,

where the boundary condition (1.2) signifies energy injection or pumping into the system, while (1.3) signifies a feedback with cubic nonlinearity of the van der Pol type. Note that in (1.1) we have set the spatial domain to be the unit interval I ≡ (0, 1) just for convenience. Two initial conditions (1.4)

w(x, 0) = w0 (x),

wt (x, 0) = w1 (x),

0 < x < 1,

are also prescribed. Then it was established in [5] that for fixed α, β, the composite reflection map Gη ◦ Fα,β : I¯ → I¯ is chaotic (cf. [5, (9)–(12)] or (1.9)–(1.10) below for Gη and Fα,β ) and, therefore, for initial conditions (1.4) of generic type, (wx (x, t), wt (x, t)) displays chaotic behavior. Here, we follow Devaney’s definition of chaos [8]; see also [2]. To make this paper sufficiently self-contained, let us repeat the solution procedure for (1.1)–(1.4) from [4] using the method of characteristics. Define (1.5)

u(x, t) =

1 [wx (x, t) + wt (x, t)], 2

v(x, t) =

1 [wx (x, t) − wt (x, t)]. 2

Then (u, v) satisfies the following initial-boundary value problem (IBVP), a first-order diagonalized symmetric hyperbolic system ∂ u(x, t) ∂ u(x, t) 1 0 (1.6) = , 0 < x < 1, t > 0, 0 −1 ∂x v(x, t) ∂t v(x, t) with boundary conditions (1.7) (1.8)

[u(0, t) − v(0, t)] = −η[u(0, t) + v(0, t)], u(1, t) + v(1, t) = α[u(1, t) − v(1, t)] + β[u(1, t) − v(1, t)]3 .

Unbounded Growth of Total Variations

17

The algebraic equations (1.7) and (1.8) define the reflection relations (1.9)

v(0, t) = Gη (u(0, t)) ≡

(1.10)

u(1, t) = Fα,β (v(1, t)),

1+η u(0, t), 1−η

at, respectively, the left-end x = 0 and the right-end x = 1, where in (1.10), Fα,β : R → R is a nonlinear mapping such that for each given v ∈ R, u ≡ Fα,β (v) is the unique real solution of the cubic equation (1.11)

β(u − v)3 + (1 − α)(u − v) + 2v = 0.

The initial conditions are now (1.12)

u(x, 0) = u0 (x) = 12 [w00 (x) + w1 (x)], 0 < x < 1. v(x, 0) = v0 (x) = 12 [w00 (x) − w1 (x)],

From time to time, we also need that u0 and v0 satisfy the compatibility conditions (1.13)

v0 (0) = Gη (u0 (0)),

u0 (1) = Fα,β (v0 (1)).

Using the maps Fα,β and Gη , we can represent the solution (u, v) of (1.6) explicitly as follows [5, (13), (14), p. 425]: for t = 2k + τ , k = 0, 1, 2, . . . , 0 ≤ τ < 2 and 0 ≤ x ≤ 1, (1.14)

 τ ≤ 1 − x,  (Fα,β ◦ Gη )k (u0 (x + τ )), −1 k+1 G ◦ (Gη ◦ Fα,β ) (v0 (2 − x − τ )), 1 − x < τ ≤ 2 − x, u(x, t) =  η k+1 2 − x < τ < 2;  (Fα,β ◦ Gη )k (u0 (τ + x − 2)), τ ≤ x,  (Gη ◦ Fα,β ) (v0 (x − τ )), v(x, t) = Gη ◦ (Fα,β ◦ Gη )k (u0 (τ − x)), x < τ ≤ 1 + x,  (Gη ◦ Fα,β )k+1 (v0 (2 + x − τ )), 1 + x < τ < 2,

              

where (Gη ◦Fα,β )n = (Gη ◦Fα,β )◦(Gη ◦Fα,β )◦· · ·◦(Gη ◦Fα,β ), the n-times iterative composition of Gη ◦ Fα,β . Since the solution representation (1.14) depend on (Gη ◦ Fα,β )n , it constitutes a natural Poincaré section for the solution of (1.6). We say that the solution of (1.6) is chaotic if the map Gη ◦ Fα,β : I¯ → I¯ (or ¯ Gη ◦ Fα,β : I → I, where I is an invariant subset of Gη ◦ Fα,β contained in I) is chaotic. Since (wx , wt ) is topologically conjugate to (u, v) in the sense of [4, §5], we also say that the gradient w of the system (1.1)–(1.4) is chaotic. The orbit diagram of the map Gη ◦ Fα,β , where α and β are held fixed, say α = 1/2, β = 1, and only η is varying, can be seen from [5, Fig. 3, p. 433] (for 0 < η < 1) and [5, Fig. 4, p. 434] (for η > 1), wherein period-doubling cascades are manifest. For the purpose of making this paper sufficiently self-contained,

18

Chen et al.

we reproduce these two figures in Figs. 1 and 2, respectively. Furthermore, the existence of homoclinic orbits has been established for the parameter range (1.15) 1+α . 1+α 1− √ 1+ √ ≤ η < 1, 3 3 3 3

1 11.

20

Chen et al.

Fig. 3. The profile of u(x, t) at t = 52 52, with α = 0.5 0.5, β = 1, η = 0.525 0.525, for the system (1.6), (1.7), (1.8) and (1.17). (Reprinted from [5, p. 436], courtesy of World Scientific, Singapore.)

Fig. 4. The profile of v(x, t) at t = 52 52, with α = 0.5, β = 1, η = 0.525 0.525, for the system (1.6) (1.7), (1.8) and (1.17). (Reprinted from [5, p. 346], courtesy of World Scientific, Singapore.)


21

Fig. 5. The profile of u(x, t) at t = 102 102, with α = 0.5, β = 1, η = 0.525 0.525, for the system (1.6) (1.7), (1.8) and (1.17). (Reprinted from [5, p. 437], courtesy of World Scientific, Singapore.) The chaos here is due to the period-doubling cascade.

Fig. 6. The profile of v(x, t) at t = 102 102, with α = 0.5, β = 1, η = 0.525 0.525, for the system (1.6) (1.7), (1.8) and (1.17). (Reprinted from [5, p. 437], courtesy of World Scientific, Singapore.) As with Fig. 5, the chaos here is due to the period-doubling cascade.

22

Chen et al.

Fig. 7. The profile of u(x, t) at t = 52 52, with α = 0.5, β = 1, η = 1.52 1.52, for the system (1.6) (1.7), (1.8) and (1.17). (Reprinted from [5, p. 442], courtesy of World Scientific, Singapore.) The chaos here is due to a homoclinic orbit of Gη ◦ Fα,β .

Fig. 8. The profile of v(x, t) at t = 52 52, with α = 0.5, β = 1, η = 1.52 1.52, for the system (1.6) (1.7), (1.8) and (1.17). (Reprinted from [5, p. 442], courtesy of World Scientific, Singapore.) As with Fig. 7, the chaos here is due to a homoclinic orbit of Gη ◦ Fα,β .


23

In this paper, we give some informative answers to the question [Q] posed above. The rest of the paper is divided into three parts. In §2, we present a few facts about linear vibration in order to show the contrasts between linearity and nonlinearity. The main theorems are established in §3. In §4, miscellaneous discussions are given. A useful proposition, which was used in §2, is given separately in the Appendix near the end of the paper.

2

Bounds on the Total Variation of (u(·, t), v(·, t)) of the Linear Wave Equation as t → ∞ Consider the wave equation (1.1), but with linear boundary conditions such

as (2.1)

left-end x = 0 : w(0, t) = 0,

t > 0,

(2.2)

right-end x = 1 : wx (1, t) = 0,

t > 0,

in lieu of (1.2) and (1.3). Let the initial conditions satisfy (2.3) where

w(x, 0) = w0 (x) ∈ H01 (0, 1),

wt (x, 0) = w1 (x) ∈ L2 (0, 1),

H01 (0, 1) = {f : (0, 1) → R | f (0) = 0; f, f 0 ∈ L2 (0, 1)}

is a Sobolev space with norm Z kf kH01 (0,1) =

1

1/2 (f 2 + f 02 )dx .

0

Then for the system (1.1), (2.1)–(2.3), the energy Z 1 1 2 E(t) = (2.4) [w (x, t) + wt2 (x, t)]dx 2 0 x is conserved and, therefore, we have the estimate (2.5)

kw(·, t)kH01 (0,1) + kwt (·, t)kL2 (0,1) ≤ C(kw0 kH01 (0,1) + kw1 kL2 (0,1) )

for some constant C > 0 independent of (w0 , w1 ). This type of Sobolev estimates is quite well known for the IBVP of the linear wave equation. Not so well known are similar types of estimates in terms of total variations for the linear wave equation. Let us convert (1.1), and (2.1)–(2.3) into a firstorder diagonalized symmetric hyperbolic system (1.6) through (1.5). Then (2.1) and (2.2) lead to the reflection relations (2.6) v(0, t) = G(u(0, t)) ≡ u(0, t),

u(1, t) = F (v(1, t)) ≡ −v(1, t),

t > 0.

24

Chen et al.

Assume that the initial conditions u0 and v0 (cf. (1.12)) are continuous on I¯ and satisfy the compatibility conditions v0 (0) = G(u0 (0)),

(2.7)

u0 (1) = F (v0 (1)).

Then from the representation formula (1.14) we easily obtain ∀ t > 0,

VI¯(u(·, t)) + VI¯(v(·, t)) = VI¯(u0 ) + VI¯(v0 ),

(2.8)

i.e., the sum of the total variations of u and v at any t on I¯ is conserved. If u0 and v0 are continuous on I¯ but the compatibility conditions in (2.7) are violated, then discontinuities can propagate along characteristics x − t = −k, x + t = 1 + k for k = 0, 1, 2, . . . so (2.8) needs to be modified to VI¯(u(·, t)) + VI¯(v(·, t)) ≤ VI¯(u0 ) + VI¯(v0 ) + C,

(2.9)

∀ t > 0,

for some constant C: C ≡ |u0 (0) − v0 (0)| + |u0 (1) + v0 (1)|.

(2.10)

Using (1.5) and (2.9), we further deduce that e VI¯(wx (·, t)) + VI¯(wt (·, t)) ≤ 2[VI¯(w00 ) + VI¯(w1 ) + C]

(2.11)

e > 0. From for some constant C

Z

x

w(x, t) =

wx (ξ, t)dξ 0

and for any {xk ∈ [0, 1] | k = 0, 1, . . . , n} satisfying 0 = x0 < x1 < x2 < · · · < xn = 1, n−1 X k=0

(2.12)

|w(xk+1 , t) − w(xk , t)| =

n−1 X Z xk+1

k=0 n−1 X

xk

Z

wx (ξ, t)dξ

xk+1 1 ≤ dξ + 2 xk k=0 Z 1 1 = + wx2 (x, t)dx, 2 0

we obtain VI¯(w(·, t)) ≤

1 + 2

Z

1

Z

xk+1

wx2 (ξ, t)dξ

xk

wx2 (x, t)dx.

0

Therefore (2.11) can be furthered strengthened. We summarize the above in the following.


25

Theorem 2.1. Consider the system (1.1), (2.1), (2.2) and (2.3), with w0 ∈ C 1 ([0, 1]) and w1 ∈ C 0 ([0, 1]). Then (2.13) ee VI¯(w(·, t)) + VI¯(wx (·, t)) + VI¯(wt (·, t)) ≤ 2[VI¯(w00 ) + VI¯(w1 ) + E(0) + C], ee for some C > 0 depending only on C in (2.10). ¯ and w1 ∈ C 0 (I), ¯ then From the estimate (2.13) we see that if w0 ∈ C 1 (I) as t → ∞, the left-hand side (LHS) of (2.13) remains bounded, provided that initially w00 and w1 have bounded total variations. The LHS of (2.13) can grow unbounded (when and) only when initially (at least one of) w00 and w1 have unbounded total variations. This is possible, as shown in the following example. Example 2.1. Choose Z x 1 w0 (x) = ξ(ξ − 1) sin dξ, w1 (x) = 0; 0 < x < 1. ξ 0 Then (w0 , w1 ) ∈ H01 (0, 1) × L2 (0, 1). The compatibility conditions (2.7)1 and (2.7)2 are satisfied. Therefore the solution (u, v) of (1.6), (2.6)–(1.12) is continuous for any (x, t) ∈ [0, 1] × [0, ∞). Here 1 w00 (x) = x(x − 1) sin , x is easily verified to have VI¯(w00 ) = ∞.

0 < x < 1,

We see that the LHS of (2.13) is ∞ for any t > 0. Next, let us consider, again, linear boundary conditions but somewhat different from the ones in (2.1) and (2.2). The IBVP system is  wxx (x, t) − wtt (x, t) = 0, x ∈ (0, 1), t > 0,    wt (0, t) + γw(0, t) = 0, t > 0, (2.14) w (1, t) = 0, t > 0,    x w(x, 0) = w0 (x), wt (x, 0) = w1 (x), x ∈ (0, 1). Note that the boundary condition (2.14)2 is integrable along the t-direction: (2.15)

w(0, t) = w(0, 0)e−γt ,

t ≥ 0.

Again, converting (2.14) into a first-order diagonalized symmetric hyperbolic system using (1.5) and utilizing (2.15), we obtain the following snapshots at t = 1, 2, . . . , inductively: (2.16)  u0 (x), v0 (x) are given (according to (1.12)); and w(0, 0) is also known,    u(x, k + 1) = −v(1 − x, k), v(x, k + 1) = u(1 − x, k) + γak e−γ e−γ(1−x) ,    ak+1 = w(0, k + 1) = e−γ w(0, k), w(0, 0) ≡ a0 .

26

Chen et al.

If γ < 0, then (2.15) implies that w(x, t) can grow unbounded in general and, therefore, the total variations of w, wx , wt , u and v can not be expected to remain bounded even if the initial condition w0 and w1 (or u0 , v0 ) have bounded total variations at t = 0. This is a classical instability case. So let us only consider the case γ > 0. (The case γ = 0 is already covered in Theorem 2.1.) Theorem 2.2. For (2.14), let u and v be defined by (1.5). Then there exist a constant C > 0, depending only on γ > 0 and the right side of (2.10), such that (2.17)

VI¯(u(·, t)) + VI¯(v(·, t)) ≤ VI¯(u0 ) + VI¯(v0 ) + C.

Proof. We need only establish (2.17) for integral values of t. Let us use (2.16) to verify that the following holds:  VI¯(u(·, 2k)) ≤ VI¯(u0 ) + γ|a0 |Ak ,    VI¯(u(·, 2k + 1)) ≤ VI¯(v0 ) + γ|a0 |Bk , (2.18) VI¯(v(·, 2k)) ≤ VI¯(v0 ) + γ|a0 |Bk ,    VI¯(v(·, 2k + 1)) ≤ VI¯(u0 ) + γ|a0 |Ak+1 , for k = 1, 2, . . . , where Ak =

e−γ − e−(2k−1)γ , 1 − e−2γ

Bk =

e−2γ − e−2(k+1)γ . 1 − e−2γ

We prove by induction. When k = 1, (2.16) gives   u(x, 1) = −v0 (1 − x), v(x, 1) = u0 (1 − x) + γa0 e−γ e−γ(1−x) ,  a1 = a0 e−γ ;   u(x, 2) = −u0 (x) − γa0 e−γ e−γx , v(x, 2) = −v0 (x) + γa0 e−2γ e−γ(1−x) ,  a2 = a0 e−2γ ;   u(x, 3) = v0 (1 − x) − γa0 e−2γ e−γx , v(x, 3) = −u0 (1 − x) − γa0 e−γ e−γ(1−x) + γa0 e−3γ e−γ(1−x) ,  a3 = a0 e−3γ . Therefore, (2.18) is clear for k = 1. Suppose (2.18) is valid for k ≤ n. We now prove (2.18) for k = n + 1. By (2.16), u(x, 2(n + 1)) = −v(1 − x, 2n + 1); (2.19)

VI¯(u(·, 2(n + 1))) = VI¯(v(·, 2n + 1)) ≤ VI¯(u0 ) + γ|a0 |An+1 ; v(x, 2(n + 1)) = u(1 − x, 2n + 1) + γa2n+1 e−γ e−γ(1−x) = u(1 − x, 2n + 1) + γa0 e−(2n+2)γ e−γ(1−x) .


27

Thus VI¯(v(·, 2(n + 1))) ≤ VI¯(u(·, 2n + 1)) + VI¯(γa0 e−2(n+1)γ e−γ(1−x) ) ≤ VI¯(u(·, 2n + 1)) + γ|a0 |e−2(n+1)γ ≤ VI¯(v0 ) + γ|a0 |B2n + γ|a0 |e−2(n+1)γ # " e−2γ − e−2(n+1)γ −2(n+1)γ = VI¯(v0 ) + γ|a0 | · +e 1 − e−2γ = VI¯(v0 ) + γ|a0 | · (2.20)

e−2γ − e−2(n+2)γ 1 − e−2γ

= VI¯(v0 ) + Bn+1 .

Therefore, (2.18)1 and (2.18)3 are verified by (2.19) and (2.20), respectively. The proof of (2.18)2 and (2.18)4 can be done in the same way. Therefore we have proved (2.17). Corollary 2.1. For the IBVP (2.14), assuming that w0 ∈ C 1 ([0, 1]) and w1 ∈ C 0 ([0, 1]). Then the estimate (2.13) holds for t > 0. If, instead of the linear boundary conditions pair (2.14)2 and (2.14)3 , we consider (2.21)

wx (0, t) − γw(0, t) = 0, wt (1, t) = 0, t > 0,

γ > 0,

t > 0,

where one of them is of Robin type [7, §1.2 and 1.3], then the treatment becomes much more challenging. After some extra efforts, we have succeeded in establishing an estimate similar to (2.13), as shown below. Theorem 2.3. Consider the system (1.1), (2.21), with initial conditions w0 ∈ C 1 ([0, 1]) and w1 ∈ C 0 ([0, 1]) in (1.4). Then there exist two positive constants C1 , C2 such that (2.22)

VI (w(·, t)) + VI (wx (·, t)) + VI (wt (·, t)) ≤ C1 [VI (w00 ) + VI (w1 )] + C2

for all t > 0. Proof. First, we will establish the inequality (2.23)

e1 [V (u0 ) + V (v0 )] + C e2 VI (u(·, t)) + VI (v(·, t)) ≤ C I I

e1 and C e2 , for all t > 0, from which (2.22) will for some positive constants C naturally follow. Here, as before, u, v, u0 and v0 are defined through (1.5) and (1.12).

28

Chen et al. From (2.21)1 , we have Z

[u(0, t) + v(0, t)] − γ [u(0, τ ) − v(0, τ )]dτ + w(0, 0) = 0, 0 Z t Z t v(0, t) + γ v(0, τ )dτ = − u(0, t) − γ u(0, τ )dτ + γw(0, 0), 0 0 Z t Z t −γt d γt γt d −γt e v(0, τ )dτ = −e u(0, τ )dτ + γw(0, 0), e e dt dt 0 0 t

which, after further simplification and integrations by parts, leads to Z t (2.24) v(0, t) = −u(0, t) + 2γ e−γ(t−τ ) u(0, τ )dτ + γw(0, 0)e−γt , t > 0. 0

This is the reflection relation at the left end x = 0. At the right end x = 1, the reflection relation is (2.25)

u(0, t) = v(0, t),

t > 0.

From (2.24) and (2.25), it is clear (based on a representation similar to (1.14)) b1 that (2.23) will hold if we can prove that there exist two positive constants C b and C2 such that (2.26)

b1 · V[0,T ] (u(0, ·)) + C b1 , V[0,T ] (v(0, ·)) ≤ C

for all

T > 0,

under the assumption of (2.24). (The reflection relation (2.25) is easy and simple so it does not require any separate consideration.) But (2.24) implies (2.26), following the application of a technical Proposition A in the Appendix near the end of the paper. We leave out the details that (2.23) yields (2.22). Note that if (2.21)2 is replaced by wx (1, t) = 0, then (2.25) correspondingly will be changed to u(0, t) = −v(0, t) and the arguments from (2.24) through (2.26) also need to be adapted accordingly. Nevertheless, such modifications are straightforward and the estimate (2.22) remains valid. As a summary of this section, we have successfully shown that for major typical homogeneous linear boundary conditions of the wave equation, the total variations of the snapshot of the gradient as well as the state of the wave equation at any time t on I will remain uniformly bounded in time, if the initial total variation on I is finite.


3

29

Unbounded Growth of the Total Variations of (u(·, t), v(·, t)) as t → ∞ When Chaotic Vibration Occurs

This is the main section of the paper where we will treat question [Q]. Let us first estimate the growth of total variations of (u(·, t), v(·, t)) due to the post period-doubling of the reflection map Gη ◦ Fα,β . We will utilize special properties of the Stefan Cycle as given in Robinson [11, pp. 67–70]. Let fµ : J → J be continuous on a finite closed interval J ⊂ R. Assume that fµ has completed a period-doubling cascade as the parameter µ crosses a value µ0 . Therefore, we may now assume that fµ has a prime periodic point with period n = m ·2k , where m is an odd integer greater than or equal to 3, as such an n is above the doubling cascade portion · · · 2j 2j−1 2j−2 · · · 2 in Sharkovskii’s ladder. From now on, to simplify notation, we just write fµ as f. k Define g = f 2 . Then g has a periodic orbit O = {xj | j = 1, 2, . . . , m}

(3.1)

of prime odd period m such that g(xj ) = xj+1 ,

(3.2)

j = 1, 2, . . . , m − 1,

for

satisfying either (3.3)

xm < xm−2 < · · · < x3 < x1 < x2 < x4 < · · · < xm−1

or (3.4)

xm−1 < xm−3 < · · · < x4 < x2 < x1 < x3 < · · · < xm−2 < xm .

Let us just treat the case (3.3) below because relation (3.4) is just a reflection of (3.3) centered at x1 and all the arguments for (3.3) will also go through for (3.4) after a straightforward modification. Now define ([11, p. 67]) m − 1 closed intervals (3.5)

I1 = [x1 , x2 ], I2 = [x3 , x1 ], I3 = [x2 , x4 ], . . . , I2j−1 = [x2j−2 , x2j ], I2j = [x2j+1 , x2j−1 ], . . . ,

for j = 2, . . . , (m − 1)/2.

For two closed intervals J1 and J2 , we say that J1 f -covers J2 , in notation f

J1 −→ J2 , if J2 ⊆ f (J1 ). Then we have the following. Proposition 3.1. ([11, p. 68]) Assume that J is a finite closed interval and g : J → J is continuous with a prime periodic orbit of odd period m. Then there exist m − 1 closed subintervals I1 , I2 , . . . , Im−1 of J defined through (3.1)–(3.3) and (3.5) that overlap at most at endpoints such that (3.6)

g

g

g

g

g

g

I1 −→ I2 −→ I3 −→ · · · −→ Im−1 −→ I1 −→ I1 ∪ I2 .

30

Chen et al.

Theorem 3.1. Assume that J is a closed interval and g : J → J is continuous with a prime periodic orbit of odd period m. Then lim VJ (gn ) = ∞.

(3.7)

n→∞

Proof. We first show that (3.8)

lim VI1 (gkm ) = ∞;

see I1 defined in (3.5)).

k→∞

Utilizing (3.6), we have I1 gm -covers I1 ∪ I2 . Therefore VI1 (gm ) ≥ `(I1 ) + `(I2 ) = (x2 − x1 ) + (x1 − x3 ) = x2 − x3 ,

(3.9)

by (3.5),

where `(Ij ) denotes the length of the interval Ij . Also, since I1 gm -covers I1 ∪I2 , we can find two subintervals I1,1 and I1,2 of I1 , overlapping at most at endpoints, such that gm (I1,1 ) = I1 ,

(3.10)

gm (I1,2 ) = I2 .

Next, from (3.6) and (3.10), we have gm

g

g

g

g

g

g

g

g

g

g

(3.11)

I1,1 −→ I1 −→ I2 −→ I3 −→ · · · Im−1 −→ I1 −→ I1 ∪ I2 ,

(3.12)

I1,2 −→ I2 −→ I3 −→ I4 −→ · · · Im−1 −→ I1 −→ I1 −→ I1 ∪ I2 ,

gm

g

and, therefore, I1,1 has two subintervals I1,11 and I1,12 such that (3.13)

g2m (I1,11 ) = I1

and g2m (I1,12 ) = I2 ,

with I1,11 and I1,12 overlapping at most at endpoints. Similarly, from (3.12), we obtain two closed subintervals I1,21 and I1,22 of I1,2 , overlapping at most at endpoints, such that g2m (I1,21 ) = I1 ,

g2m (I1,22 ) = I2 .

We therefore obtain VI1 (g2m ) ≥ VI1,11 (g2m ) + VI1,12 (g2m ) + VI1,21 (gm ) + VI1,22 (gm ) ≥ `(I1 ) + `(I2 ) + `(I1 ) + `(I2 ) = 2(x2 − x3 ). This process can be continued indefinitely. In general, from a subinterval I1,a1 a2 ...ak where aj = 1 or 2 for j = 1, 2, . . . , k, we have g km

g

g

g

g

g

g km

g

g

g

g

g

I1,a1 a2 ...ak−1 ,1 −→ I1 −→ I2 −→ I3 −→ · · · −→ Im−1 −→ I1 ∪ I2 , g

I1,a1 a2 ...ak−1 ,2 −→ I2 −→ I3 −→ I4 −→ · · · −→ Im−1 −→ I1 −→ I1 ∪ I2 .

31


From either of the above two sequences we can find two subintervals I1,a1 ...ak 1 and I1,a1 ...ak 2 of I1,a1 ...ak , overlapping at most at endpoints, such that g(k+1)m (I1,a1 ...ak 1 ) = I1

g(k+1)m (I1,a1 ... ,ak 2 ) = I2 ,

and

and because the collection of subintervals {I1,a1 a2 ...ak ak+1 | aj = 1, 2; j = 1, 2, . . . , k + 1} has non-overlapping interior, X VI1 (g(k+1)m ) ≥ VI1 ,a1 a2 ...ak ak+1 (g(k+1)m) ) aj =1,2 j=1,... ,k+1

≥ (k + 1)(x2 − x3 ) → ∞ as k → ∞.

(3.14)

Therefore, we have established (3.8). To show (3.7), it is sufficient to show lim VI1 (gkm+j ) = ∞,

k→∞

j = 1, 2, . . . , m − 1.

for

We utilize the covering sequence gj

g

g

g

g

g

g

g

g

I1 −→ Ij+1 −→ Ij+2 −→ · · · −→ Im−1 −→ I1 −→ I1 −→ · · · −→ I1 −→ I1 ∪ I2 | {z } I1 appearing j+1 times (j)

(j)

to deduce that I1 has two closed subintervals I1,1 and I1,2 , overlapping at most at endpoints, such that (j)

gm+j (I1,1 ) = I1 ,

(j)

gm+j (I1,2 ) = I2 .

(j)

Inductively, if I1,a1 ... ,ak is constructed, satisfying either g km+j

(j)

g

g

g

g

g

g

I1,a1 ... ,ak−1 1 −−−−−−−→ I1 −→ I2 −→ I3 −→ · · · −→ Im−1 −→ I1 −→ I1 ∪ I2 , or (j)

g km+j

g

g

g

g

g

g

g

I1,a1 ...ak−1 2 −−−−−−−→ I2 −→ I3 −→ I4 −→ · · · −→ Im−1 −→ I1 −→ I1 −→ I1 ∪I2 , (j)

depending, respectively, on ak = 1 or 2, then we can find I1,a1 ... ,ak ’s two (j)

(j)

subintervals I1,a1 ... ,ak 1 and I1,a1 ... ,ak 2 , overlapping at most at endpoints, such that (j) (j) g(k+1)m+j (I1,a1 ...ak 1 ) = I1 , g(k+1)m+j (I1,a1 ...ak 2 ) = I2 . Again, we have VI1 (g(k+1)m+j ) ≥

X

VI1 ,a1 ...ak+1 (g(k+1)m+j) ≥ (k +1)(x2 −x3 ) → ∞ as k → ∞.

aj =1,2 j=1,... ,k+1

The proof of (3.7) is now complete.

32

Chen et al.

Corollary 3.1. Assume that J is a closed interval f : J → J is continuous with a prime periodic orbit of period m · 2k , where m is odd. Then lim VJ (f n ) = ∞.

(3.15)

n→∞

Proof. Let O1 = {f ` (ξ) | ` = 0, 1, . . . , m · 2k − 1} be an orbit of f with prime k k period m · 2k . Then O2 = {f j·2 (ξ) | j = 1, 2, . . . , m} is an orbit of g ≡ f 2 with prime period m. Write O2 in the form of (3.1) such that (3.2), (3.3) and (3.5) are satisfied. Therefore, we have O2 = {xj | j = 1, 2, . . . , m} and for some integer j1 : 0 < j1 ≤ m, k

k

x1 = f j1·2 (ξ),

k

x2 = f (j1 +1)·2 (ξ), . . . , xm = f (j1 +m)·2 (ξ).

The main idea of the proof is to show that k )+`)

lim VIe0 (f j·(m·2

(3.16)

j→∞

=∞

for any ` = 0, 1, 2, . . . , m · 2k − 1, for some subinterval Ie0 ⊆ J (where Ie0 depends on given `). Given any such ` ∈ {0, 1, 2, . . . , m · 2k − 1}, we can find a positive integer `ˆ > 0 such that ` + `ˆ = j1 · 2k

(modm · 2k ).

Define (3.17)

`ˆ

y1 = f (ξ),

y2 = f

ˆ k `+2

(ξ),

( [y1 , y2 ], if y1 < y2 , Ie0` = [y2 , y1 ], if y1 > y2 .

Then k

ˆ

f ` (y1 ) = f `+` (ξ) = f j1 ·2 (ξ) = x1 ,

ˆ

k

k

f ` (y2 ) = f `+`+2 (ξ) = f (j1 +1)·2 (ξ) = x2 ,

and we have the covering sequence `

g g g g g g f Ie0` −→ I1 −→ I2 −→ I3 −→ · · · −→ Im−1 −→ I1 −→ I1 ∪ I2 , k ` and Ie` , overlapping at where g = f 2 . So Ie0` contains two subintervals Ie0,1 0,2 most at endpoints, such that k +`

f m·2

` (Ie0,1 ) = I1 ,

k +`

f m·2

` (Ie0,2 ) = I2 .

` In general, if Ie0,a is constructed, for aj = 1, 2, j = 1, 2, . . . , p, satisfying the 1 ...ap following covering sequences


33

(i) if ap = 1, then (3.18) pm·2k +`

g g g g g f ` Ie0,a −−−−−−−→ I1 −→ I2 −→ · · · −→ Im−1 −→ I1 −→ I1 ∪ I2 ; 1 ... ,ap−1 1

(ii) if ap = 2, then (3.19) pm·2k +`

g g g g g g f ` Ie0,a −−−−−−−→ I2 −→ I3 −→ · · · −→ Im−1 −→ I1 −→ I1 −→ I1 ∪ I2 . 1 ...ap−1 2 ` ` From (3.18) and (3.19), we have two subintervals Ie0,a and Ie0,a of 1 ...ap 1 1 ...ap 2 Ie0,a ...ap , such that 1

k +`

f (p+1)·m·2

` (Ie0,a ) = I1 , 1 ...ap 1

k +`

f (p+1)·m·2

` (Ie0,a ) = I2 . 1 ...ap 2

The rest of the arguments follows in the same way as in the proof of Theorem 3.1. Therefore (3.16) follows from each ` ∈ {0, 1, . . . , m·2k −1}. Hence (3.15) follows. Theorem 3.2. Consider the IBVP (1.6)–(1.8), and (1.12). Assume that the composite reflection map f = Gη ◦ Fα,β has a periodic orbit O = {f ` (ξ) | ` = 0, 1, . . . , m · 2k − 1}, with prime period m · 2k , where m is odd. Further, assume that the initial conditions u0 and v0 in (1.12) are continuous and satisfy the compatibility conditions in (1.13) such that for some integer j0 : 0 ≤ j0 ≤ m · 2k − 1, (3.20)

f j0 (ξ),

k

f j0 +2 (ξ) ∈ Range z,

z ≡ u0

or

z ≡ v0 .

Then lim [VI¯(u(·, t)) + VI¯(v(·, t))] = ∞.

(3.21)

t→∞

k

Proof. Let us assume that {f j0 (ξ), f j0 +2 (ξ)} ⊆ Range u0 . Then we can construct a subinterval Ie0` as in (3.17) by letting ` = j0 therein. From the proof of Cor. 3.1 and (1.14), we easily deduce that lim [VI¯(u(·, n)) + VI¯(v(·, n))] = ∞.

n→∞

It is also easy to show that for any τ : 0 < τ < 1, by using the continuity of the total variations with respect to τ , that lim [VI¯(u(·, n + τ )) + VI¯(v(·, n + τ ))] = ∞.

n→∞

Therefore (3.21) follows.

34

Chen et al.

Remark 3.1. It seems natural to believe that Theorem 3.2 remains valid even if condition (3.20) is weakened to the following: “there exist integers j1 and j2 : 0 ≤ j1 < j2 ≤ m · 2k − 1, such that f j2 (ξ) ∈ (Range u0 ) ∪ (Range v0 ).”

f j1 (ξ),

(3.22)

However, in order to establish (3.21) under condition (3.22), the arguments used in the proof of Cor. 3.1 also need to be considerably strengthened in order to take care of the laborious “bookkeeping” details of finer interval covering sequences, which we do not yet have an elegant way to handle so far. Next, we study the growth of total variations of snapshots (u(·, t), v(·, t)) when the composite reflection map Gη ◦ Fα,β has homoclinic orbits. There are two cases to be considered: (i) η > 1, and (ii) 0 < η < 1. Write f = Gη ◦ Fα,β . Here we only consider the case that f has a bounded invariant interval J such that f : J → J. For this to hold, we need [5, Lemma 2.5] either r r 1+η1+α 1+α 1 + η 1 + αη (i) M ≡ (3.23) ≤ , if 0 < η < 1, 1−η 3 3β 2η βη or (3.24)

(ii)

1+η1+α M ≡− 1−η 3

r

1+α 1+η ≤ 3β 2

r

α+η , β

if η > 1,

in addition to (1.15), with (3.25)

J = [−M, M ].

Two graphs of f are provided in Figs. 9 and 10, where η = 0.552, 1.812, respectively. Theorem 3.3. Assume that 0 < α ≤ 1, β > 0, η > 0 and η 6= 1. Assume also that (1.15), (3.23)–(3.25) are also satisfied so that J = [−M, M ] is a bounded invariant interval of Gη ◦ Fα,β . Then lim VJ ((Gη ◦ Fα,β )n ) = ∞.

n→∞

Proof. We first consider the case η > 1. {xi ∈ J | i = 0, 1, 2, . . . } as follows. Let (3.26)

q

1+α β , the −1 min{f (x0 )}, f −1 (x1 ),

x0 = vI = x1 = x2 = .. .

xn+1 = .. .

f −1 (x

n ),

Define a sequence of points

 positive v-axis intercept of f [5, (32), p. 428],                    


35

Then for n = 0, 1, 2, . . . , xn ∈ J and xn ↓ 0 as n → ∞. Also, define subintervals I0 = [x1 , x0 ],

(3.27)

I1 = [x2 , x1 ], . . . , In = [xn+1 , xn ], . . . .

Then because f (I0 ) = [0, x1 ] we have the following covering sequence f

f

f

f

f

f

(3.28) In −→ In−1 −→ In−2 −→ · · · −→ I1 −→ I0 −→

n [

Ij

j=1 f

f

f

f

f

f

−→ Ik −→ Ik−1 −→ · · · −→ I1 −→ I0 −→

(3.29)

n [

I` , for k = 1, . . . , n.

`=1

Therefore from (3.28), In has n subintervals In,j , j = 1, 2, . . . , n, overlapping at most at endpoints, such that f n (In,j ) = Ij ,

(3.30)

j = 1, 2, . . . , n.

Fig. 9. A degenerate homoclinic orbit of the map f = Gη ◦ Fα,β , where α = 0.5, β = 1 and η = 0.552 0.552. (Reprinted from [5, p. 426], courtesy of World Scientific, Singapore.)

Using the second part of the covering sequence in (3.29) f

f

f

f

f

f

f

f

f

Ik −→ Ik−1 −→ · · · −→ I0 −→ In−k −→ · · · −→ I2 −→ I1 −→ I0 −→

n [ `=1

I` ,

36

Chen et al.

Fig. 10. A degenerate homoclinic orbit of the map f = Gη ◦ Fα,β , where α = 0.5 0.5, β = 1 and η = 1.812 1.812. (Reprinted from [5 p. 426], courtesy of World Scientific, Singapore.)

we also obtain n subintervals Ik,1 , . . . , Ik,n of Ik , overlapping at most at endpoints, such that (3.31)

j = 1, . . . , n; for k = 1, 2, . . . , n − 1.

f n (Ik,j ) = Ij ,

From (3.30) and (3.31), we obtain V[0,x0] (f n ) ≥ V[xn+1 ,x0 ] (f n ) n n X X ≥ VIk,j (f n ) ≥ [(x0 − x1 ) + (x1 − x2 ) + · · · + (xn − xn+1 )] k,j=1

(3.32)

k=1

= n(x0 − xn+1 ) → ∞,

as

n → ∞.

Next, we consider the case 0 < η < 1. Let us modify (3.26) only slightly by redefining (3.26)2 as x1 = max{f −1 (x0 )},

(3.33)

x1 < 0.

The rest of (3.26) remains unchanged. Now, define intervals I0 = [x2 , x0 ], (3.34)

I1 = [x1 , x3 ],

I2n = [x2n+2 , x2n ],

I2 = [x4 , x2 ],

I3 = [x3 , x5 ], . . . ,

I2n+1 = [x2n+1 , x2n+3 ], . . . ,

using the fact that x1 < x3 < x5 < · · · < x2n+1 < · · · < 0 < · · · < x2n < x2n−2 < · · · < x4 < x2 < x0 .


37

Then because f (I0 ) = [x1 , 0], we have the following covering sequence f

f

f

f

f

f

I2n+1 −→ I2n −→ I2n−1 −→ · · · −→ I1 −→ I0 −→

n [

I2j+1 .

j=0

The rest of the proof can be done in the same way as in (3.28)–(3.32). Theorem 3.4. Consider the IBVP (1.6)–(1.8), (1.12). Assume that η satisfies either (3.23) or (3.24) so that the composite reflection map f = Gη ◦ Fα,β has a bounded invariant interval J = [−M, M ] and a homoclinic orbit in J. Further, assume that the initial conditions u0 and v0 in (1.12) satisfy the compatibility conditions in (1.13) such that (3.35) Range z ⊇ In , z ≡ u0 or z ≡ v0 for some n ∈ {0, 1, 2, . . . , }, cf. (3.27) or (3.34). Then lim [VI¯(u(·, t)) + VI¯(v(·, t))] = ∞.

t→∞

Proof. Same as that of Theorem 3.2. Remark 3.2. We believe that (3.35) can be weakened at least to Range u0 ∪ Range v0 ⊇ In , for some n ∈ {0, 1, 2, . . . }.

Remark 3.3. The proof of Theorem 3.3 is essentially similar to those of Theorem 3.1 and Corollary 3.1, and is based on the following fact: “Let J be a finite closed interval and f : J → J is continuous. (3.36)

If f has a homoclinic orbit in J, then lim VJ (f n ) = ∞”. n→∞

Actually, (3.36) above stands alone as a theorem itself and can also be proved by quoting the proofs of Theorem 3.1 and Corollary 3.1, provided that the homoclinic orbit in (3.36) is nondegenerate, because by Theorem 1.16.5 in Devaney P [8, p. 124], the map f is then topologically conjugate to the shift map k σ on 2 and, therefore, f has some periodic orbits of prime period m · 2 , with m being odd and k ∈ {0, 1, 2, . . . }. Hence the proofs of Theorem 3.1 and Corollary 3.1 apply. When the homoclinic orbit in (3.36) is degenerate, then f is “more chaotic” (than the case when the homoclinic orbit is nondegenerate) and has homoclinic bifurcations. The renormalization procedure for the “model case” quadratic map fµ (x) = µx(1−x) as µ → 4 (the degenerate homoclinic orbit case) as mentioned in [8, §1.16] suggests that for µ = 4, f4 should be in the “post period doubling era” and therefore, f4 has many period-m · 2k orbits, with m being odd. It is quite obvious that our f in (3.36) ought to also have this kind of period-m · 2k orbits (when the homoclinic orbit in (3.36) is degenerate) and, therefore, the

38

Chen et al.

proofs of Theorem 3.1 and Corollary 3.1 again apply. Nevertheless, we could not locate a precise reference to this effect. In passing, we may also note that the condition (3.35) is stated quite differently from (3.20), in the sense that the end-points of intervals In in (3.35) are not periodic points. (Or rather, the end-points of In have an “infinite periodicity”.)

4

Miscellaneous Remarks

In this paper, we have successfully shown that when chaotic vibration occurs for the wave equation caused by the nonlinear boundary condition specified here, the total variations of snapshots tend to infinity as t → ∞ for a large class of initial data, even though the total variation of any such initial data is finite at time t = 0. Our theorems in §3 have covered the case of “stable” chaos on a bounded invariant interval. A different type of “unstable” chaos, discussed in [5, §5], happens on an invariant Cantor set (rather than a bounded invariant interval, because the map Gη ◦ Fα,β does not have one for that set of α, β and η values). In that case, it is trivial to show that the total variations of snapshots also tend to infinity as t → ∞ for a large class of initial data, even though initially, the total variation of the state is finite. One may ask a converse question to [Q]: “[−Q] Assume that there exist initial conditions (u0 , v0 ) for (1.6), (1.9), (1.10) and (1.12) and an invariant interval I¯ of Gη ◦ Fα,β such that (4.1)

VI¯(u0 , v0 ) < ∞,

VI¯(u(·, t)) + VI¯(v(·, t)) → ∞

as

t → ∞.

Is the map Gη ◦ Fα,β necessarily chaotic?” The answer is negative, as the following counterexample has shown. Example 4.1. Let α = 0.5, β = 1, either η ∈ (0, 0.433) or η ∈ (2.312, ∞). Then as Figs. 1 and 2 ([5, Figs. 3 and 4, pp. 433–434]) have shown, the map Gη ◦ Fα,β has a locally attracting period-2 orbit near 0, which is repelling. Let g(x) = x2 . For x ∈ [0, 1], (4.2)

   x2 ,         0, u0 (x) = 2(n + 1) 2n + 1   , x−   n n2    2n 2n + 1   , x+  − n+1 (n + 1)2

1 , n = 1, 2, . . . , n 2n + 1 if x = , n = 1, 2, . . . , 2n(n + 1) h 2n + 1 1 i if x ∈ , + 1) n h 2n(n 1 2n + 1 if x ∈ , . n + 1 2n(n + 1)

if x =

Then u0 (x) is continuous. Choose any v0 , continuous of bounded total h variation i 1 satisfying the compatibility condition (1.13). On each subinterval n+1 , n1 , the


39

total variation of u0 is 1/n2 + 1/(n + 1)2 . Therefore u0 has bounded total variation on the interval [0,1]. Let the period-2 orbit of Gη ◦ Fα,β be {p, −p}, where p > 0. Then for each y ∈ [0, 1], y 6= n1 for n = 1, 2, . . . , we have lim |(Gη ◦ Fα,β )n (u0 (y))| = p.

n→∞

h Therefore, on each subinterval to p, and

1 1 n+1 , n

i

, the total variation of (Gη ◦Fα,β )n tends

lim [VI¯(u(·, t)) + VI¯(v(·, t))] = ∞, but VI¯(u0 ) + VI¯(v0 ) < ∞.

(4.3)

t→∞

The above negative result seems to have weakened the connection between chaotic vibration and unbounded growth of total variations of snapshots. However, we may take note of the following recent result in [10]. Let f : J → J be chaotic (according to Devaney [8, p. 50]), on the finite closed interval J. Then f has sensitive dependence on initial data [2]. This sensitive dependence on initial data is regarded as a major feature of chaotic maps; [10] has proved the following: “(i) Let f : J → J has sensitive dependence on initial data. lim VJ 0 (f n ) = ∞ for every closed subinterval J 0 of J.

Then

n→∞

(ii) Let f : J → J be continuous with finitely many extremal points, satisfying lim VJ 0 (f n ) = ∞ for every closed subinterval J 0 of J. Then f n→∞ has sensitive dependence on initial data.” The theorems in [10] actually explains why the breakdown (4.3) happens: the initial data in (4.2) has infinitely many extremal points, i.e., there are infinitely many oscillations on a finite closed interval and, thus, it is “too oscillatory”. The growth rate of VI¯(f n ) as estimated in (3.14) and (3.32) are linear with respect to n. Sharper estimates may also be possible, at least for certain special cases. In [9], examples of exponential growth have been found. Related issues such as Remarks 3.1 and 3.2 and others are also being investigated in [9].

Appendix A Key Proposition In this section, we prove the following. Proposition A. Assume that u and v are related through Z (A.1)

v(t) = αu(t) + β 0

t

e−γ(t−τ ) u(τ )dτ + f (t),

t ≥ 0,

40

Chen et al.

where α, β ∈ R, γ > 0. Then |β| |β| V[0,T ] (v) ≤ |α| + (A.2) V[0,T ] (u) + |u(0+)| + V[0,T ] (f ), γ γ for all T > 0, where u(0+) = lim u(t). t→0+

We first establish the following technical lemma. Lemma B. Let γ > 0 and

Z

t

g(t) = [Qu](t) ≡

e−γ(t−τ ) u(τ )dτ,

∀ t ≥ 0.

0

Then V[0,T ] (g) ≤

(A.3)

1 [|u(0+)| + V[0,T ] (u)], γ

∀ T > 0.

Proof. (1) We first assume that u is increasing and continuous. Then Z t 0 g (t) = −γ e−γ(t−s) u(s)ds + u(t) 0 s=t Z t = −e−γ(t−s) u(s) + e−γ(t−s) du(s) + u(t) s=0 0 Z t = e−γt u(0) + (A.4) e−γ(t−s) du(s). 0

Note that the integral in (A.4) is a Stieltjes integral [1, Chap 9]. Since g is absolutely continuous, we see that Z T Z T Z TZ t 0 −γt V[0,T ] (g) = |g (t)|dt ≤ |u(0)| e dt + e−γ(t−s) du(s)dt 0 0 0 0 Z T Z T 1 ≤ |u(0)| + e−γt dt eγs du(s) γ 0 s Z 1 1 T −γs = |u(0)| + (e − e−γT )eγs du(s) γ γ 0 Z 1 1 T ≤ |u(0)| + du(s) γ γ 0 1 1 = |u(0) + [u(T ) − u(0)] γ γ 1 = [|u(0)| + V[0,T ] (u)]. γ Therefore (A.3) is true when u is increasing and continuous.


41

(2) We now assume that u is increasing and left-continuous. Then u = u0 +

(A.5)

X

rc H c ,

c∈J

where u0 is increasing and continuous, u0 (0) = u(0+), J is a (possibly empty) countable set of nonnegative real numbers, rc > 0, and Hc (t) =

0, 0 ≤ t ≤ c, 1, t > c,

is the Heaviside function. Then V[0,T ] (u) = V[0,T ] (u0 ) +

X

rc ,

c∈J,c 0 there exists a constant c(r, s) > 0 with the following property: if the sequence f1 , . . . , fn in C r (RN ) is such that n X (2.1) kfi kC r < α, 0 < α < s, i=1

then for the map F = (I + fn ) ◦ · · · ◦ (I + f1 ) kF − IkC r ≤ α c(r, s).

(2.2)

Associate with a fixed open or closed subset4 Ω0 of RN the following family of images of Ω0 by the elements of F0k def

X (Ω0 ) =

n

F (Ω0 ) ⊂ RN : ∀F ∈ F0k

o

This induces a bijection (2.3)

[F ] 7→ F (Ω0 ) : F0k /G(Ω0 ) → X (Ω0 ).

between X (Ω0 ) and the quotient group of F0k by the subgroup o n def G(Ω0 ) = F ∈ F0k : F (Ω0 ) = Ω0 (2.4) 4 In her paper [5] A.M. Micheletti assumes that Ω0 is a bounded connected open domain of class C k in order to make all the images F (Ω0 ) bounded connected open domains of class C k . However the construction of the Courant metric only requires that Ω0 be closed or open.

48

M.C. Delfour and Jean-Paul Zol´ esio

of transformations which map Ω0 onto Ω0 . The next step is to construct a complete metric space topology on F0k which induces a complete metric space topology on the quotient group F0k /G(Ω0 ). The space X (Ω0 ) is then identified with the topological quotient group F0k /G(Ω0 ). This quotient metric is called the Courant metric by A.M. Micheletti. The construction of the metric topology on F0k /G(Ω0 ) is not as straightforward as it might appear at first sight. The obvious candidates for the metric do not usually satisfy the triangle inequality and only yield a pseudo-metric on the quotient group. In order to get around this difficulty Micheletti introduced the following construction. Given F ∈ F0k , consider finite factorizations of F and F −1 of the form F = (I + fn ) ◦ · · · ◦ (I + f1 ) and F −1 = (I + gm ) ◦ · · · ◦ (I + g1 ) Define (2.5)

def

d(I, F ) =

n X

inf

(f1 ,...,fn )

kfi kC k +

i=1

inf

(g1 ,...,gm )

m X

kgi kC k

i=1

where the infima are taken with respect to all finite factorizations of F and F −1 in F0k . Extend this definition to all pairs F and G in F0k def

d(F, G) = d(I, G ◦ F −1 )

(2.6)

By definition d is right-invariant since for all F , G and H in F0k (2.7)

d(F, G) = d(F ◦ H, G ◦ H)

and the three axioms which define a metric on F0k are verified. Theorem 2.2. F0k is a complete metric group. Corollary 2.1. The topology induced by the metric d on the topological group F0k coincides with the topology which has as a basis of neighborhoods of the identity in F0k the sets n o def E(ε) = F ∈ F0k : kF − IkC k + kF −1 − IkC k < ε . It is readily seen that we have the following properties. Lemma 2.3. Given an open or a closed5 subset Ω0 of RN , the family def

G(Ω0 ) =

n

F ∈ F0k : F (Ω0 ) = Ω0

o

is a closed subgroup of F0k . 5

As noted earlier it is not necessary that Ω0 be a bounded connected open C k domain.

Velocity method and Courant metric topologies

49

By definition for each Ω ∈ X (Ω0 ) there exists F ∈ F0k such that Ω = F (Ω0 ). Therefore the following map is well-defined and bijective (2.8)

Ω 7→ χ(Ω) = F ◦ G(Ω0 ) : X (Ω0 ) → F0k /G(Ω0 ).

Using χ we now introduce the following complete metric on X (Ω0 ). Theorem 2.3. Given an open or a closed subset Ω0 of RN , the function (2.9)

def

δ(F ◦ G(Ω0 ), H ◦ G(Ω0 )) =

inf

˜ G,G∈G(Ω 0)

˜ d(F ◦ G, H ◦ G)

is a metric on F0k /G(Ω0 ). The topology induced by δ coincides with the quotient topology of F0k /G(Ω0 ) and the space F0k /G(Ω0 ) is complete. Finally it is natural to define on X (Ω0 ) the following metric def

(2.10)

ρ(Ω1 , Ω2 ) = δ(χ(Ω1 ), χ(Ω2 ))

induced by the bijection χ on X (Ω0 ). With that metric, X (Ω0 ) is a complete metric space. This metric is called the Courant metric (metrica di Courant) in [5].

3

The generic framework of Micheletti

The construction of a complete metric topology on the group and a Courant metric on the quotient group naturally extends to other underlying spaces Θ than the Banach space C0k (RN ). Given a Banach space Θ of transformations of RN , define the space def

F(Θ) =

F : RN → RN : F − I ∈ Θ and F −1 − I ∈ Θ .

Associate with F ∈ F(Θ) the distance (3.1)

def

d(I, F ) =

inf

(f1 ,...,fn )

n X

kfi kΘ +

i=1

inf

(g1 ,...,gm )

m X

kgi kΘ ,

i=1

where the infima are taken over all finite factorizations in F(Θ) of the form F = (I + fn ) ◦ · · · ◦ (I + f1 ) and F −1 = (I + gm ) ◦ · · · ◦ (I + g1 ), Extend this definition to all pairs F and G in F(Θ) (3.2)

def

d(F, G) = d(I, G ◦ F −1 )

fi , gi ∈ Θ.

50


Define for some fixed open or closed subset Ω0 of RN the subgroup def

G(Ω0 ) = {F ∈ F(Θ) : F (Ω0 ) = Ω0 } and the Courant metric on the equivalence classes of F(Θ)/G(Ω0 ) def

∀F, G ∈ F(Θ),

(3.3)

δ([F ], [H)] =

∀F1 , F2 ∈ F(Θ),

(3.4)

inf

˜ G,G∈G(Ω 0)

˜ d(F ◦ G, H ◦ G)

def

ρ(F1 (Ω0 ), F2 (Ω0 )) = δ([F1 ], [F2 ])

on the images F (Ω0 ) of Ω0 by elements F of F(Θ). Of course F(Θ)/G(Ω0 ) will be a well-defined complete metric space for the Courant metric only for appropriate choices of spaces Θ. As an illustration consider the spaces of transformations of RN introduced by Murat and Simon [6] in 1976 in the construction of metric spaces of domains. For integers k ≥ 0, they correspond to the following choices of the space Θ:

def

=

W k+1,c(RN , RN ) n o f ∈ W k+1,∞(RN , RN ) : ∀ 0 ≤ |α| ≤ k + 1, ∂ α f ∈ C(RN , RN )

and W k+1,∞(RN , RN ).

The first space W k+1,c (RN , RN ) algebraically and def

topologically coincides with the space C k+1 (RN ) = C k+1 (RN , RN ). corresponding space of transformations def

F k+1 (RN ) =

The

F : RN → RN :

F − I ∈ C k+1 (RN ) and F −1 − I ∈ C k+1 (RN )

is a topological group for the metric d as in the case of F0k+1 in § 2. The def

second space W k+1,∞ (RN , RN ) coincides with C k,1 (RN ) = C k,1 (RN , RN ). The associated space of transformations def

(3.5)

F k,1 (RN ) =

F : RN → RN :

F − I ∈ C k,1 (RN ) and F −1 − I ∈ C k,1 (RN )

is complete for the topology induced by the metric d, but is not a topological group. In both cases the Courant metric defines a complete metric topology on the corresponding quotient space. Also recall from [6] that F 1 (RN ) transports locally Lipschitzian (graph) domains onto locally Lipschitzian (graph) domains but that F 0,1 (RN ) does not.


3.1

51

Approach of Murat and Simon

The construction introduced by Murat and Simon [6] to obtain a complete metric topology on the quotient spaces are different from the ones of Micheletti [5] which were seemingly not known to them. They worked with the pseudometric def

dp (F2 , F1 ) = kF2 ◦ F1−1 − IkW k+1,∞ + kF1 ◦ F2−1 − IkW k+1,∞ rather than the metric defined by the infima over finite factorizations of F2 ◦F1−1 and F1 ◦ F2−1 . They recover a metric from the pseudo-metric by using an auxiliary construction which depends on the third property of a pseudo-metric. We briefly recall the definition and the result. Definition 3.1. A pseudo-metric on a space E is a function δ : E × E → R+ with the following properties (i) δ(F2 , F1 ) = 0 ⇐⇒ F2 = F1 (ii) δ(F2 , F1 ) = δ(F1 , F2 ) for all F1 and F2 (iii) δ(F1 , F3 ) ≤ δ(F1 , F2 ) + δ(F2 , F3 ) + δ(F1 , F2 ) δ(F2 , F3 ) P (δ(F1 , F2 ) + δ(F2 , F3 )) for all Fi ’s, where P : R+ → R+ is a continuous increasing function. Proposition 3.1. Let δ be a pseudo-metric on E. For all α, 0 < α < 1, there exists a constant ηα > 0 such that the function δ(α) : E × E → R+ defined as def

δ(α) (F1 , F2 ) = inf{δ(F1 , F2 ), ηα }α is a metric on E.

3.2

Approach of Micheletti

The pseudo-metric can be completely by-passed. By combining the construction of [5] with the properties established in [6], we readily get the completeness of the (Micheletti) metric for the group of transformations and of the Courant metric for the quotient group. In both cases the results can also be obtained directly by adapting with obvious technical changes in the case C k,1 ((RN ) the sequence of lemmas and theorems of § 2. The first case is technically analogous to F0k by choosing Θ = C k (RN ). Theorem 3.1. Let k ≥ 1 be an integer. (i) The topology induced by d on F k (RN ) makes it a complete metric topological group. Moreover around the identity I for all 0 < ε < 1 def

E(ε) =

n

E(ε) ⊂ S(ε) ⊂ E(2c(k, 1)ε)

o N ) : kF − Ik k + kF −1 − Ik k < ε . F ∈ F k (R C n o C def

S(ε) =

F ∈ F k (RN ) : d(I, F ) < ε .

52

M.C. Delfour and Jean-Paul Zol´ esio and the topology coincides with the topology which has as a basis of neighborhoods of the identity in F k (RN ) the sets E(ε) (the constant c(k, 1) is as specified in Lemma 2.2).

(ii) Given an open or closed subset Ω0 of RN , def

δ(F ◦ G, H ◦ G) =

(3.6)

inf

˜ G,G∈G(Ω 0)

˜ d(F ◦ G, H ◦ G)

is a metric on F k (RN )/G(Ω0 ). The topology induced by δ is complete and coincides with the quotient topology of F k (RN )/G(Ω0 ). In the other case Θ = W k+1,∞ (RN , RN ) is equal to C k,1 (RN ) which is a Banach space when endowed with the norm def

(3.7)

kf kC k,1 = kf kC k + ck (f ) |f (y) − f (x)| |∂ α f (y) − ∂ α f (x)| def def c(f ) = sup ck (f ) = max sup . |y − x| |x − y| |α|=k x6=y x6=y

We get a similar result for F k,1 (RN ) except that it is not a topological group. Theorem 3.2. Let k ≥ 1 be an integer. (i) F k,1 (RN ) is a group under composition. The function d induces a complete metric topology on F k,1 (RN ). Moreover around the identity I for all 0 < ε < 1 E(ε) ⊂ S(ε) ⊂ E(c(k) ε) n o k,1 N ) : kF − Ik k,1 + kF −1 − Ik k,1 < ε . E(ε) = F ∈ F (R C n o C def

def

S(ε) =

F ∈ F k,1 (RN ) : d(I, F ) < ε

for some constant c(k) independent of ε. (ii) Given an open or closed subset Ω0 of RN , (3.8)

def

δ(F ◦ G, H ◦ G) =

inf

˜ G,G∈G(Ω 0)

˜ d(F ◦ G, H ◦ G)

is a metric on F k,1 (RN )/G(Ω0 ). The topology induced by δ is complete and coincides with the quotient topology of F k,1 (RN )/G(Ω0 ).

4

Unconstrained families of domains

In this section we study equivalences between the Velocity Method (cf. [9], [10]) and methods using a family of transformations. In § 4.1 we give some general conditions to construct a family of transformations of RN from a non-autonomous velocity field. Conversely we show how to construct a nonautonomous velocity field from a family of transformations of RN . In § 4.2 the equivalences of § 4.1 are specialized to velocities in C0k+1 (RN ), C k+1 (RN ), and C k,1 (RN ), k ≥ 0.


4.1

53

Equivalence between velocities and transformations

Let the real number τ > 0 and the map V : [0, τ ] × RN → RN be given. The map V can be viewed as a family {V (t) : 0 ≤ t ≤ τ } of non-autonomous velocity fields on RN defined by def

x 7→ V (t)(x) = V (t, x) : RN 7→ RN .

(4.1) Assume that (4.2)

∀x ∈ RN ,

(V)

∃c > 0, ∀x, y ∈ RN ,

V (·, x) ∈ C [0, τ ]; RN

kV (·, y) − V (·, x)kC([0,τ ];RN ) ≤ c|y − x|

where V (·, x) is the function t 7→ V (t, x). Associate with V the solution x(t; V ) of the ordinary differential equation dx (t) = V t, x(t) , dt

(4.3)

t ∈ [0, τ ],

x(0) = X ∈ RN ,

and introduce the homeomorphism def

X 7→ Tt (V )(X) = xV (t; X) : RN → RN .

(4.4) and the maps

def

(4.5)

(t, X) 7→ TV (t, X) = Tt (V )(X) : [0, τ ] × RN → RN ,

(4.6)

(t, x) 7→ TV−1 (t, x) = Tt−1 (V )(x) : [0, τ ] × RN → RN .

def

In the sequel we shall drop the V in TV (t, X), TV−1 (t, x) and Tt (V ) whenever no confusion arises. Theorem 4.1. (i) Under assumption (V) the map T has the following properties: ∀X ∈ RN , T (·, X) ∈ C 1 [0, τ ]; RN and ∃c > 0, (T1) ∀X, Y ∈ RN , kT (·, Y ) − T (·, X)kC 1 ([0,τ ];RN ) ≤ c|Y − X|, (4.7) (T2) (T3)

∀t ∈ [0, τ ], X 7→ Tt (X) = T (t, X) : RN → RN is bijective, ∀x ∈ RN , T −1 (·, x) ∈ C [0, τ ]; RN and ∃c > 0, kT −1 (·, y) − T −1 (·, x)kC([0,τ ];RN ) ≤ c|y − x|.

∀x, y ∈ RN ,

(ii) Given a real number τ > 0 and a map T : [0, τ ] × RN → RN verifying assumptions (T1), (T2) and (T3), then the map (4.8)

def

(t, x) 7→ V (t, x) =

∂T t, Tt−1 (x) : [0, τ ] × RN → RN ∂t

verifies assumption (V), where Tt−1 is the inverse of X 7→ Tt (X) = T (t, X). If, in addition, T (0, ·) = I, then T (·, X) is the solution of (4.3).

54


(iii) Given a real number τ > 0 and a map T : [0, τ ] × RN → RN satisfying assumptions (T1), (T2) and T (0, ·) = I, then there exists τ 0 > 0 such that the conclusions of part (ii) hold on [0, τ 0 ]. This equivalence theorem says that we can either start from a family of velocity fields {V (t)} on RN or a family of transformations {Tt } of RN provided that the map V , V (t, x) = V (t)(x), verifies (V) or the map T , T (t, X) = Tt (X), verifies (T1) to (T3). Starting from V , the family of homeomorphisms {Tt (V )} generates the family Ωt = Tt (V )(Ω) = {Tt (V )(X) : X ∈ Ω}.

(4.9)

of perturbations of the initial domain Ω. Interior (resp. boundary) points of Ω are mapped onto interior (resp. boundary) points of Ωt . This is the basis of the Velocity method.

4.2

Equivalence for special families of velocities

In this section we specialize Theorem 4.1 to velocities in C k,1 (RN ), C0k+1 (RN ), and C k+1 (RN ), k ≥ 0. The following notation will be helpful def

f (t) = Tt − I,

f 0 (t) =

dTt , dt

def

g(t) = Tt−1 − I,

whenever Tt−1 exists and the identities g(t) = −f (t) ◦ Tt−1 = −f (t) ◦ [I + g(t)] dTt V (t) = ◦ Tt−1 = f 0 (t) ◦ Tt−1 = f 0 (t) ◦ [I + g(t)]. dt Recall also the notation c(f ) and ck (f ) in (3.7). Theorem 4.2. Let k ≥ 0 be an integer. (i) Given τ > 0 and a velocity field V such that (4.10)

V ∈ C([0, τ ]; C k (RN , RN )) and ck (V (t)) ≤ c

for some constant c > 0 independent of t, the map T given by (4.3)-(4.5) satisfies conditions (T1), (T2), and (4.11) f ∈ C 1 ([0, τ ]; C k (RN , RN )) ∩ C([0, τ ]; C k,1 (RN , RN )),

ck (f 0 (t)) ≤ c

for some constant c > 0 independent of t. Moreover condition (T3) is satisfied and there exists τ 0 > 0 such that (4.12)

g ∈ C([0, τ 0 ]; C k (RN , RN )),

for some constant c independent of t.

ck (g(t)) ≤ ct


55

(ii) Given τ > 0 and T : [0, τ ] × RN → RN satisfying conditions (4.11) and T (0, ·) = I, there exists τ 0 > 0 such that the velocity field V (t) = f 0 (t)◦Tt−1 satisfies conditions (V) and (4.10) in [0, τ 0 ]. Proof. We prove the theorem for k = 0. The general case is obtained by induction over k. (i) By assumption on V , the conditions (V) given by (4.2) are satisfied and by Theorem 4.1 the corresponding family T satisfies conditions (T1) to (T3). (Conditions (4.11) on f ). For any x and s ≤ t Z t Tt (x) − Ts (x) = V (r) ◦ Tr (x) dr s Z t |Tt (x) − Ts (x)| ≤ c|Tr (x) − Ts (x)| + |V (r) ◦ Ts (x)| dr s Z t |f (t)(x) − f (s)(x)| ≤ c|f (r)(x) − f (s)(x)| + kV (r)kC dr. s

By assumption on V and Gronwall’s inequality ∀t, s ∈ [0, τ ],

kf (t) − f (s)kC ≤ c |t − s|

for another constant c independent of t. Moreover |(f (t) − f (s))(y) − (f (t) − f (s))(x)| = |(Tt − Ts )(y) − (Tt − Ts )(x)| Z t ≤ |V (r) ◦ Tr (y) − V (r) ◦ Tr (x)| dr s Z t ≤ c|(Tr − Ts )(y) − (Tr − Ts )(x)| + c|Ts (y) − Ts (x)| dr s Z t ≤ c|(f (r) − f (s))(y) − (f (r) − f (s))(x)| + cc0 |y − x| dr s

for some other constant c0 by the second condition (T1). Again by Gronwall’s inequality’s there exists another constant c such that |(f (t) − f (s))(y) − (f (t) − f (s))(x)| ≤ c|t − s| |y − x| ⇒ c(f (t) − f (s)) ≤ c |t − s| (4.13)

⇒ f ∈ C([0, τ ]; C 0,1 (RN , RN )) and kf (t) − f (s)kC 0,1 ≤ c |t − s|.

Moreover f 0 (t) = V (t) ◦ Tt and |f 0 (t)(x) − f 0 (s)(x)| ≤ |V (t)(Tt (x)) − V (s)(Tt (x))| + |V (s)(Tt (x)) − V (s)(Ts (x))| ≤ kV (t) − V (s)kC + c(V (s)) kTt − Ts kC ≤ kV (t) − V (s)kC + c kf (t) − f (s)kC .

56


Finally |f 0 (t)(y) − f 0 (t)(x)| ≤ |V (t)(Tt (y)) − V (t)(Tt (x))| ≤ c(V (t)) |Tt (y) − Ts (x)| ≤ c c(Tt ) |y − x| and c(f 0 (t)) ≤ c for some new constant c independent of t. Therefore f ∈ C 1 ([0, τ ]; C(RN , RN )) and c(f 0 (t)) ≤ c. (Conditions (4.12) on g). Since conditions (T1) and (T2) are satisfied there exists τ 0 > 0 such that conditions (T3) are satisfied by Theorem 4.1 (iii). Moreover from conditions (4.11) |g(t)(y) − g(t)(x)| ≤ |f (t)(Tt−1 (y)) − f (t)(Tt−1 (x))| ≤ c(f (t))|Tt−1 (y) − Tt−1 (x)| ≤ c(f (t)) (|g(t)(y) − g(t)(x)| + |y − x|) ⇒ (1 − c(f (t))) |g(t)(y) − g(t)(x)| ≤ c(f (t)) |y − x| ≤ ct |y − x|. Choose a new τ 00 = min{τ 0 , 1/(2c)}. Then for 0 ≤ t ≤ τ 00 , c(g(t)) ≤ 2ct. Now g(t) − g(s) = −f (t) ◦ [I + g(t)] + f (s) ◦ [I + g(s)] kg(t) − g(s)kC ≤kf (t) ◦ [I + g(t)] − f (t) ◦ [I + g(s)]kC + kf (t) ◦ [I + g(s)] − f (s) ◦ [I + g(s)]kC ≤c(f (t))kg(t)] − g(s)kC + kf (t) − f (s)kC ≤ct kg(t) − g(s)kC + kf (t) − f (s)kC . For t in [0, τ 00 ], ct ≤ 1/2, and kg(t) − g(s)kC ≤ 2kf (t) − f (s)kC ⇒ g ∈ C([0, τ 00 ]; C(RN , RN )) and c(g(t)) ≤ 2ct. Thus conditions (4.11) on f are satisfied for k = 0. For k = 1 we start from the equation Z t DTt − DTs = DV (r) ◦ Tr DTr dr s

and use the fact that DTt−1 = [DTt ]−1 ◦ Tt−1 in connection with the identity Dg(t) = −Df (t) ◦ Tt−1 DTt−1 = −(Df (t)[DTt ]−1 ) ◦ Tt−1 . (ii) From conditions (4.11) on f the transformation T satisfies conditions (T1). To check condition (T2) we consider two cases: k ≥ 1 and k = 0. For k ≥ 1


57

the function t 7→ Df (t) = DTt − I : [0, τ ] → C k−1 (RN , RN )N is continuous. Hence t 7→ det DTt : [0, τ ] → R is continuous and det DT0 = 1. So there exists τ 0 > 0 such that Tt is invertible for all t in [0, τ 0 ] and (T2) is satisfied in [0, τ 0 ]. In the case k = 0 consider for any Y the map h(X) = Y − f (t)(X). For any X1 and X2 , |h(X2 ) − h(X1 )| ≤ c(f (t)) |X2 − X1 |. But by assumption f ∈ C([0, τ ; C 0,1 (RN )) and c(f (0)) = 0 since f (0) = 0. Hence there exists τ 0 > 0 such that c(f (t)) ≤ 1/2 for all t in [0, τ 0 ] and h is a contraction. So for all Y in RN there exists a unique X such that Y − [Tt (X) − X] = h(X) = X ⇐⇒ Tt (X) = Y, Tt is bijective, and (T2) is satisfied in [0, τ 0 ]. By Theorem 4.1 (iii) from (T1) and (T2), there exists another τ 0 > 0 for which conditions (T3) on g and (V) on V (t) = f 0 (t) ◦ Tt−1 are also satisfied. Moreover we have seen in the proof of part (i) that conditions (4.12) on g follow from (T2) and (4.11). Using conditions (4.11) and (4.12) |V (t)(y) − V (t)(x)| ≤|f 0 (t)(Tt−1 (y)) − f 0 (t)(Tt−1 (x))| ≤c(f 0 (t)) |Tt−1 (y) − Tt−1 (x)|

≤c(f 0 (t)) [1 + c(g(t))] |y − x| ≤ c0 |y − x| and c(V (t)) ≤ c0 . Also |V (t)(x) − V (s)(x)| =|f 0 (t)(Tt−1 (x)) − f 0 (s)(Ts−1 (x))| ≤|f 0 (t)(Tt−1 (x)) − f 0 (t)(Ts−1 (x))| + |f 0 (t)(Ts−1 (x)) − f 0 (s)(Ts−1 (x))| ≤c(f 0 (t)) |Tt−1 (x) − Ts−1 (x)| + kf 0 (t) − f 0 (s)kC

≤c kg(t) − g(s)kC + kf 0 (t) − f 0 (s)kC . Therefore since both g and f 0 are continuous V ∈ C([0, τ 0 ]; C(RN , RN )) and c(V (t)) ≤ c

for some constant c independent of t. This completes the proof for k = 0. As in part (i) for k = 1 we use the identity DV (t) = Df 0 (t) ◦ Tt−1 DTt−1 = (Df 0 (t)[DTt ]−1 ) ◦ Tt−1 and proceed in the sane way. The general case is obtained by induction over k. We now turn to the case of velocities in C0k (RN ). Theorem 4.3. Let k ≥ 1 be an integer.

58


(i) Given τ > 0 and a velocity field V such that V ∈ C([0, τ ]; C0k (RN , RN )),

(4.14)

the map T given by (4.3)–(4.5) satisfies conditions (T1), (T2), and f ∈ C 1 ([0, τ ]; C0k (RN , RN ))

(4.15)

Moreover conditions (T3) is satisfied and there exists τ 0 > 0 such that g ∈ C([0, τ 0 ]; C0k (RN , RN )).

(4.16)

(ii) Given τ > 0 and T : [0, τ ] × RN → RN satisfying conditions (4.15) and T (0, ·) = I, there exists τ 0 > 0 such that the velocity field V (t) = f 0 (t)◦Tt−1 satisfies conditions (V) and (4.14) on [0, τ 0 ]. Proof. It will be convenient to use the notation C0k for the space C0k (RN ). As in the proof of Theorem 4.2 we only prove the theorem for k = 1. The general case is obtained by induction on k, the various identities on f , g, f 0 and V , and the techniques of Theorem 2.1 and Lemmas 2.1 and 2.2. (i) By the embedding C01 (RN , RN ) ⊂ C 1 (RN , RN ) ⊂ C 0,1 (RN , RN ), it follows from (4.14) that V ∈ C([0, τ ]; C 0,1 (RN , RN )) and condition (4.10) of Theorem 4.2 are satisfied. Therefore conditions (4.11) and (4.12) of Theorem 4.2 are also satisfied in some interval [0, τ 0 ]. (Conditions (4.15) on f ). It remains to show that f (t) and f 0 (t) belong to the subspace C0 (RN , RN ) of C(RN , RN ) and prove the appropriate properties for Df (t) and Df 0 (t). Recall from the proof of the previous theorems that there exists c > 0 such that Z t Z t |f (t)(x)| ≤ c |V (r)(x)| dr ≤ c |(V (r) − V (0))(x)| dr + c t |V (0)(x)|. 0

0

By assumption on V (0), for ε > 0 there exists a compact set K such that ∀x ∈ {K,

|V (0)(x)| ≤ ε/(2c)

and there exists δ, 0 < δ < 1, such that ∀0 ≤ t ≤ δ,

kV (r) − V (0)kC ≤ ε/(2c)

⇒ ∀0 ≤ t ≤ δ, ∀x ∈ {K,

|f (t)(x)| ≤ ε

⇒ f (t) ∈ C0 .

Proceeding in this fashion from the interval [0, δ] to the next interval [δ, 2δ] using the inequality Z t |(f (t) − f (s))(x)| ≤ c |(V (r) − V (δ))(x)| dr + c |t − δ| |V (δ)(x)|, s


59

the uniform continuity of V ∀t, s,

|t − s| < δ,

kV (t) − V (s)kC ≤ ε/(2c)

and the fact that V (δ) ∈ C0 , that is, there exists a compact set K(δ) such that ∀x ∈ {K(δ),

|V (δ)(x)| ≤ ε/(2c),

we get f (t) ∈ C0 , δ ≤ t ≤ 2δ, and hence f ∈ C([0, τ ]; C0 ). For f 0 (t) we make use of the identity f 0 (t) = V (t) ◦ Tt . Again by assumption for any ε > 0 there exists a compact set K(t) such that |V (t)(x)| ≤ ε on {K(t). Thus by choosing the compact Kt0 = Tt−1 (K(t)), |f 0 (t)(x)| ≤ ε on {Kt0 and f 0 ∈ C([0, τ ]; C0 ). In order to complete the proof, it remains to establish the same properties for Df (t) and Df 0 (t). The matrix Df (t) is solution of the equations

(4.17)

d Df (t) = DV (t) ◦ Tt DTt , Df (0) = 0 dt ⇒ Df 0 (t) = DV (t) ◦ Tt Df (t) + DV (t) ◦ Tt .

From the proof of Theorem 2.1 in [5] for each t the elements of the matrix def

A(t) = DV (t) ◦ Tt = DV (t) ◦ [I + f (t)] belong to C0 since DV (t) and f (t) do. By assumption V ∈ C([0, τ ]; C0k ) and V and all its derivatives ∂ α V are uniformly continuous in [0, τ ] × RN . Therefore for each ε > 0 there exists δ > 0 such that ∀ |t − s| < δ, ∀ |y 0 − x0 | < δ,

|DV (t)(y 0 ) − DV (s)(x0 )| < ε.

Pick 0 < δ0 < δ such that ∀ |t − s| < δ0 ,

kTt − Ts kC = kf (t) − f (s)kC < δ

⇒ ∀x, ∀ |t − s| < δ0 ,

|Tt (x) − Ts (x)| < δ

⇒ |DV (t)(Tt (x)) − DV (s)(Ts (x))| < ε ⇒ kA(t) − A(s)kC < ε

⇒ A ∈ C([0, τ ]; (C0 )N ).

For each x, Df (t)(x) is the unique solution of the linear matrix equation (4.17). To show that Df (t) ∈ (C0 )N we first show that Df (t)(x) is uniformly continuous for x in RN . For any x and y Z t |Df (t)(y) − Df (t)(x)| ≤ |V (r, Tr (y)) − V (r, Tr (x))| dr 0 Z t ≤ c|Tr (y) − Tr (x)| dr 0 Z t ≤c |f (r)(y) − f (r)(x)| + |y − x| dr. 0

60


But f ∈ C([0, τ ]; C0 ) is uniformly continuous in (t, x): for each ε > 0 there exists δ, 0 < δ < ε/(2cτ ) such that ∀ |t − s| < δ, ∀ |y − x| < δ,

|f (t)(y) − f (s)(x)| < ε/(2cτ ).

Substituting in the previous inequality for each ε > 0, there exists δ > 0 such that ∀ t, ∀ |y − x| < δ,

|Df (t)(y) − Df (t)(x)| < ε.

Hence Df (t) is uniformly continuous in RN . Furthermore from the equation (4.17) we have the following inequality Z t |Df (t)(x)| ≤ |DV (r)(Tr (x)| |Df (r)(x)| + |DV (r)(Tr (x))| dr Z t 0 (4.18) ≤ c (|Df (r)(x)| + 1) dr 0

since V ∈ C([0, τ ]; C01 ). By Gronwall’s inequality |Df (t)(x)| ≤ ct for some other constant c independent of t. Thence Df (t) ∈ C(RN , RN )N . Finally to show that Df (t) vanishes at infinity we start from the integral form of (4.17) Z t Df (t)(x) = DV (r)(Tr (x)) DTr (x) dr Z t 0 |Df (t)(x)| ≤c |DV (r)(Tr (x)) − DV (r)(x)| + |DV (r)(x)| dr Z t 0 0 ≤c |f (r)(x)| + |DV (r)(x)| dr. 0

By the same technique as before for f (t), it follows that the elements of Df (t) belong to C0 since both f (s) and DV (r) do. Finally for the continuity with respect to t Z t Df (t) − Df (s) = A(r)Df (r) + A(r) dr s Z t kDf (t) − Df (s)kC ≤ kA(r)kC kDf (r) − Df (s)kC s

+ kA(r)kC (1 + kDf (r)kC ) dr.

Again, by Gronwall’s inequality, there exists another constant c such that kDf (t) − Df (s)kC ≤ c |t − s|.


61

Therefore Df ∈ C([0, τ ]; (C0 )N ) and f ∈ C([0, τ ]; C01 ). For Df 0 we repeat the proof for f 0 using the identity Df 0 (t) = DV (t) ◦ Tt Df (t) + DV (t) ◦ Tt to get Df 0 ∈ C([0, τ ]; (C0 )N )

⇒ f 0 ∈ C([0, τ ]; C01 ).

(Conditions (4.16) on g). From the remark at the beginning of part (i) of the proof, the conclusion of Theorem 4.2 are true for g and it remains to check the other conditions on g and Dg using the identities g(t) = −f (t) ◦ [I + g(t)],

Dg(t) = −Df (t) ◦ [I + g(t)] (I + Dg(t)).

By the proof of Theorem 2.1 in [5], g(t) ∈ C0 since Df (t) and g(t) do. Therefore g(t) ∈ C01 . The continuity follows by the same argument as for f 0 and hence g ∈ C([0, τ ]; C01 ). (ii) By assumption from conditions (4.15) conditions (T1) are satisfied. For (T2) observe that for k ≥ 1 the function t 7→ Df (t) = DTt − I : [0, τ ] → C k−1 (RN , RN )N is continuous. Hence t 7→ det DTt : [0, τ ] → R is continuous and det DT0 = 1. So there exists τ 0 > 0 such that Tt is invertible for all t in [0, τ 0 ] and (T2) is satisfied in [0, τ 0 ]. Furthermore we have seen in the proof of part (i) that conditions (T1), (T2) and (4.15) imply conditions (T3) and (4.16) on g in [0, τ 0 ], τ 0 > 0. Therefore the velocity field V (t) = f 0 (t) ◦ Tt−1 = f 0 (t) ◦ [I + g(t)] satisfies the conditions (V) specified by (4.2). From the proof of Theorem 2.1 in [5] V (t) ∈ C0k since f 0 (t) and g(t) belong to C0k . By assumption f ∈ C 1 ([0, τ ]; C0k ). Hence f 0 and all its derivatives ∂ α f 0 , |α| ≤ k, are uniformly continuous on [0, τ ] × RN , that is, given ε > 0, there exists δ > 0 such that ∀ |t − s| < δ, ∀ |y 0 − x0 | < δ,

|∂ α f 0 (t)(y 0 ) − ∂ α f 0 (s)(x0 )| < ε.

Similarly g ∈ C([0, τ 0 ]; C0k ) and there exists 0 < δ0 ≤ δ such that ∀ |t − s| < δ0 , ∀ |y − x| < δ0 ,

|∂ α g(t)(y) − ∂ α g(s)(x)| < δ.

Therefore for |t − s| < δ0 kTt−1 − Ts−1 kC = kg(t) − g(s)kC < δ and since δ0 < δ ∀x,

|f 0 (t)(Tt−1 (x)) − f 0 (t)(Ts−1 (x))| < ε

⇒ kV (t) − V (s)kC < ε

⇒ V ∈ C([0, τ 0 ]; C0 ).

62


We then proceed to the first derivative of V DV (t) = Df 0 (t) ◦ Tt−1 DTt−1 = Df 0 (t) ◦ [I + g(t)] [I + Dg(t)] and by uniform continuity of the right-hand side V ∈ C([0, τ 0 ]; C01 ). By induction on k we finally get V ∈ C([0, τ 0 ]; C0k ). The proof of the last theorem is based on the fact that the vector functions involved are uniformly continuous. The fact that they vanish at infinity is not an essential element of the proof. Therefore the theorem is valid with C k (RN , RN ) in place of C0k (RN , RN ). Theorem 4.4. Let k ≥ 1 be an integer. (i) Given τ > 0 and a velocity field V such that (4.19)

V ∈ C([0, τ ]; C k (RN , RN )),

the map T given by (4.3)–(4.5) satisfies conditions (T1), (T2), and (4.20)

f ∈ C 1 ([0, τ ]; C k (RN , RN )).

Moreover conditions (T3) are satisfied and there exists τ 0 > 0 such that (4.21)

g ∈ C([0, τ 0 ]; C k (RN , RN )).

(ii) Given τ > 0 and T : [0, τ ] × RN → RN satisfying conditions (4.20) and T (0, ·) = I, there exists τ 0 > 0 such that the velocity field V (t) = f 0 (t)◦Tt−1 satisfies conditions (V) and (4.19) on [0, τ 0 ].

5

Continuity of shape functions

In this section we give a characterization of the continuity of a shape function6 (5.1)

Ω 7→ J(Ω) : A ⊂ P(RN ) → B.

defined on a family A in P(RN ) with values in a Banach space B with respect to the Courant metric in terms of its continuity along the flows generated by a family of velocity fields using the equivalence Theorems 4.2, 4.3 and 4.4. Checking the continuity along flows is usually easier and more natural. We specifically consider the continuity of shape functions with respect to the Courant metric associated with the quotient spaces of transformations F0k /G(Ω) of § 2 and F k (RN )/G(Ω) and F k,1 (RN )/G(Ω) of § 3 corresponding to the families of velocity fields C0k (RN ), C k (RN ), and C k,1 (RN ). 6 To be well-defined on the quotient spaces the shape function J must be independent of the choice of the representative in the equivalence class.


5.1

63

Courant metrics and flows of velocities

We start with the space C0k (RN ) used in [5]. Theorem 5.1. Let k ≥ 1 be an integer, B a Banach space, and Ω a non-empty open subset of RN . Consider a shape function J : NΩ ([I]) → B defined in a neighborhood NΩ ([I]) of [I] in F0k /G(Ω). Then J is continuous at Ω for the Courant metric if and only if lim J(Tt (Ω)) = J(Ω)

(5.2)

t&0

for all families of velocity fields {V (t) : 0 ≤ t ≤ τ } satisfying the condition V ∈ C([0, τ ]; C0k (RN , RN )).

(5.3)

Proof. It is sufficient to prove the theorem for a real valued function J. The Banach space case is readily obtained by considering the new real valued function j(T ) = |J(T (Ω)) − J(Ω)|. (i) If J is δ-continuous at Ω, then for all ε > 0 there exists δ > 0 such that ∀T, [T ] ∈ NΩ ([I]),

δ([T ], [I]) < δ,

|J(T (Ω)) − J(Ω)| < ε.

Condition (5.3) on V coincides with condition (4.14) of Theorem 4.3 which implies conditions (4.15) and (4.16): f ∈ C 1 ([0, τ ]; C0k (RN , RN )) and g ∈ C([0, τ ]; C0k (RN , RN )) ⇒ kTt − IkC k (RN ) → 0 and kTt−1 − IkC k (RN ) → 0 as t → 0. But by definition of δ δ([Tt ], [I]) ≤ kTt−1 − IkC k + kTt − IkC k → 0 as t → 0. and we get the convergence (5.2) of the function J(Tt (Ω)) to J(Ω) as t goes to zero for all V satisfying (5.3). (ii) Conversely it is sufficient to prove that for any sequence {[Tn ]} such that δ([Tn ], [I]) goes to zero there exists a subsequence such that J(Tnk (Ω)) → J(I(Ω)) = J(Ω) as k → ∞. Indeed let ` = lim inf J(Tn (Ω)) and L = lim sup J(Tn (Ω)). n→∞

n→∞

By definition of the liminf, there is a subsequence, still indexed by n, such that ` = lim inf n→∞ J(Tn (Ω)). But since there exists a subsequence {Tnk } of {Tn } such that J(Tnk (Ω)) → J(Ω), then necessarily ` = J(Ω). The same reasoning

64


applies to the limsup and hence the whole sequence J(Tn (Ω)) converges to J(Ω) and we have the continuity of J at Ω. We now prove that we can construct a velocity V associated with a subsequence of {Tn } verifying conditions (4.14) of Theorem 4.3 and hence conditions (5.3). By Corollary 2.1 to Theorem 2.2 and the same technique as in the proof of Theorem 2.3 and Theorem 2.2 in [5] associate with a sequence {Tn } such that δ([Tn ], [I]) → 0 a subsequence, still denoted {Tn }, such that kfn kC k + kgn kC k = kTn−1 − IkC k + kTn − IkC k ≤ 2−2(n+2) . For n ≥ 1 set tn = 2−n and observe that tn − tn+1 = −2−(n+1) . Define the following C 1 -interpolation in (0, 1/2]: for t in [tn+1 , tn ]

def

Tt (X) = Tn (X) + p

tn+1 − t tn+1 − tn

(Tn+1 (X) − Tn (X)),

def

T0 (X) = X

where p ∈ P 3 [0, 1] is the polynomial of order 3 on [0, 1] such that p(0) = 1 and p(1) = 0 and p(1) (0) = 0 = p(1) (1). (Conditions on f .) By definition for all t, 0 ≤ t ≤ 1/2, f (t) = Tt − I ∈ C0k (RN ). Moreover for 0 < t ≤ 1/2 ∂T ∂T Ttn+1 (X) = Tn+1 (X), (tn , X) = 0 = (tn+1 , X) ∂t ∂t ∂T ∂T Tn+1 (X) − Tn (X) (1) tn+1 − t ⇒ f 0 (t) = (t, X) = p (t, ·) ∈ C0k (RN ) ∂t |tn − tn+1 | tn+1 − tn ∂t Ttn (X) = Tn (X),

and f (·)(X) = T (·, X) − I ∈ C 1 ((0, 1/2]; RN ). By definition f (0) = 0. For each 0 < t ≤ 1/2 there exists n ≥ N such that tn+1 ≤ t ≤ tn and tn+1 − t kf (t) − f (0)kC k = kf (t)kC k = kfn + p (fn+1 − fn )kC k tn+1 − tn ≤ 2kfn kC k + kfn+1 kC k ≤ 2 2−2(n+2) + 2−2(n+3) ≤ 2−(n+1) ≤ t. Define at t = 0 f 0 (t) = 0. By the same technique there exists a constant c > 0, and for each 0 < t ≤ 1/2 there exists n ≥ N such that tn+1 ≤ t ≤ tn and kf 0 (t) − f 0 (0)kC k = kf 0 (t)kC k

∂T

kTn+1 − Tn kC k kfn+1 − fn kC k

=

∂t (t, ·) k ≤ c |tn+1 − tn | = c 2−(n+1) C −2(n+2)

≤ c22

/2−(n+1) ≤ c 2−1 2−(n+1) ≤ c 2−(n+1) ≤ c t ⇒ kf 0 (tkC k ≤ ct.

So for each X the functions t 7→ f (t)(X) and t 7→ Tt (X) belong to C 1 ([0, 1/2]; RN ). By uniform C k -continuity of the Tn ’s and the continuity


65

with respect to t for each X, it follows that f ∈ C 1 ([0, 1/2]; C0k (RN )) and the condition (4.15) of Theorem 4.3 is satisfied. Hence the corresponding velocity V satisfies conditions (4.14). Finally V satisfies conditions (5.3) and by (5.2) for all ε > 0 there exists δ > 0 such that ∀0 ≤ t ≤ δ,

|J(Tt (Ω)) − J(Ω)| < ε.

In particular there exists N > 0 such that for all n ≥ N , tn ≤ δ and ∀n ≥ N,

|J(Tn (Ω)) − J(Ω)| = |J(Ttn (Ω)) − J(Ω)| < ε

and this proves the δ-continuity for the subsequence {Tn }. The case of the Courant metric for the space C k (RN ) is a corollary to Theorem 5.1. Theorem 5.2. Let k ≥ 1 be an integer, B a Banach space, and Ω a non-empty open subset of RN . Consider a shape function J : NΩ ([I]) → B defined in a neighborhood NΩ ([I]) of [I] in F k (RN )/G(Ω). Then J is continuous at Ω for the Courant metric if and only if lim J(Tt (Ω)) = J(Ω)

(5.4)

t&0

for all families of velocity fields {V (t) : 0 ≤ t ≤ τ } satisfying the condition (5.5)

V ∈ C([0, τ ]; C k (RN , RN )).

The proof of the theorem for the Courant metric topology associated with the space C k,1 (RN ) is similar to the proof of the first theorem with obvious changes. Theorem 5.3. Let k ≥ 0 be an integer, Ω a non-empty open subset of RN , and B a Banach space. Consider a shape function J : NΩ ([I]) → B defined in a neighborhood NΩ ([I]) of [I] in F k,1 (RN )/G(Ω). Then J is continuous at Ω for the Courant metric if and only if lim J(Tt (Ω)) = J(Ω)

(5.6)

t&0

for all families {V (t) : 0 ≤ t ≤ τ } of velocity fields in C k,1 (RN , RN ) satisfying the conditions (5.7)

V ∈ C([0, τ ]; C k (RN , RN )) and ck (V (t)) ≤ c

for some constant c independent of t. Proof. As in the proof of Theorem 5.1, it is sufficient to prove the theorem for a real valued function J. (i) If J is δ-continuous at Ω, then for all ε > 0 there exists δ > 0 such that ∀T, [T ] ∈ NΩ ([I]),

δ([T ], [I]) < δ,

|J(T (Ω)) − J(Ω)| < ε.

66


Under condition (5.7) from Theorem 4.2 f = T· − I ∈ C([0, τ ]; C k,1 (RN )) and kTt − IkC k,1 → 0 as t → 0 g(t) = Tt−1 − I ∈ C k,1 (RN ) and kTt−1 − IkC k,1 ≤ ct → 0 as t → 0. But by definition of δ δ([Tt ], [I]) ≤ kTt−1 − IkC k,1 + kTt − IkC k,1 → 0 as t → 0. and we get the convergence (5.6) of the function J(Tt (Ω)) to J(Ω) as t goes to zero for all V satisfying (5.7). (ii) Conversely, as in the proof of Theorem 5.1, it is sufficient to prove that given any sequence {[Tn ]} such that δ([Tn ], [I]) → 0 there exists a subsequence such that J(Tnk (Ω)) → J(I(Ω)) = J(Ω) as k → ∞. By the same technique as in the proof of Theorem 2.3 and Theorem 2.2 in [5] associate with a sequence {Tn } such that δ([Tn ], [I]) → 0 a subsequence, still denoted {Tn }, such that kfn kC k,1 + kgn kC k,1 = kTn−1 − IkC k,1 + kTn − IkC k,1 ≤ 2−2(n+2) . For n ≥ 1 set tn = 2−n and observe that tn − tn+1 = −2−(n+1) . Define the following C 1 -interpolation in (0, 1/2]: for t in [tn+1 , tn ] def

Tt (X) = Tn (X) + p

tn+1 − t tn+1 − tn

(Tn+1 (X) − Tn (X)),

def

T0 (X) = X

where p ∈ P 3 [0, 1] is the polynomial of order 3 on [0, 1] such that p(0) = 1 and p(1) = 0 and p(1) (0) = 0 = p(1) (1). (Conditions on f .) By definition for all t, 0 ≤ t ≤ 1/2, f (t) = Tt −I ∈ C k,1 (RN ). Moreover for 0 < t ≤ 1/2 ∂T ∂T Ttn+1 (X) = Tn+1 (X), (tn , X) = 0 = (tn+1 , X) ∂t ∂t ∂T ∂T Tn+1 (X) − Tn (X) (1) tn+1 − t ⇒ f 0 (t) = (t, X) = p (t, ·) ∈ C k,1 (RN ) ∂t |tn − tn+1 | tn+1 − tn ∂t Ttn (X) = Tn (X),

and f (·)(X) = T (·, X) − I ∈ C 1 ((0, 1/2]; RN ). By definition f (0) = 0. For each 0 < t ≤ 1/2 there exists n ≥ N such that tn+1 ≤ t ≤ tn and tn+1 − t kf (t) − f (0)kC k,1 = kf (t)kC k,1 = kfn + p (fn+1 − fn )kC k,1 tn+1 − tn ≤ 2kfn kC k,1 + kfn+1 kC k,1 ≤ 2 2−2(n+2) + 2−2(n+3) ≤ 2−(n+1) ≤ t.


67

Define at t = 0 f 0 (t) = 0. By the same technique there exists a constant c > 0, and for each 0 < t ≤ 1/2 there exists n ≥ N such that tn+1 ≤ t ≤ tn and kf 0 (t) − f 0 (0)kC k,1 = kf 0 (t)kC k,1

∂T

kTn+1 − Tn kC k,1 kfn+1 − fn kC k,1

= ≤c (t, ·) =c

∂t |tn+1 − tn | 2−(n+1) k,1 C

≤ c 2 2−2(n+2) /2−(n+1) ≤ c 2−1 2−(n+1) ≤ c 2−(n+1) ≤ c t ⇒ kf 0 (tkC k,1 ≤ ct and ck (f 0 (t)) ≤ ct. and for each X the functions t 7→ f (t)(X) and t 7→ Tt (X) belong to C 1 ([0, 1/2]; RN ). By uniform C k -continuity of the Tn ’s and the continuity with respect to t for each X, it follows that f ∈ C 1 ([0, 1/2]; C k (RN )). Moreover it can be shown that ck (f 0 (t)) ≤ ct

⇒ ∀t, s ∈ [0, τ ],

ck (f (t) − f (s)) ≤ c0 |t − s|

for some c0 > 0. The result is straightforward for k = 0 and then the general case follows by induction on k. As a result f ∈ C([0, 1/2]; C k,1 (RN )) and the condition (4.11) of Theorem 4.2 is satisfied. Hence the corresponding velocity V satisfies conditions (4.10). Finally the velocity field V satisfies conditions (5.7) and by (5.6) for all ε > 0 there exists δ > 0 such that ∀t ≤ δ,

|J(Tt (Ω)) − J(Ω)| < ε.

In particular there exists N > 0 such that for all n ≥ N , tn ≤ δ and ∀n ≥ N,

|J(Tn (Ω)) − J(Ω)| = |J(Ttn (Ω)) − J(Ω)| < ε

and we have the δ-continuity for the subsequence {Tn }. Remark 5.1. The conclusions of Theorems 5.1, 5.2 and 5.3 are generic. They also have their counterpart in the constrained case. For instance a generalization of Theorem 5.1 in the constrained case has been announced by [2] for an open subset D of RN of class C 2 . The difficulty lies in the second part of the theorem which requires a special construction to make sure that the family of transformations {Tt : 0 ≤ t ≤ τ } constructed from the sequence {Tn } are homeomorphisms of D.

References [1] C. Bardos and G. Chen Control and stabilization for the wave equation. III. Domain with moving boundary, SIAM J. Control Optim. 19 (1981), no. 1, 123–138.

68


[2] S. Boisgerault and N. Gomez, personal communication. [3] M.C. Delfour and J.-P. Zolésio, Velocity method and Lagrangian Formulation for the computation of the shape Hessian, SIAM J. Control Optim. 29 (1991), no. 6, 1414–1442. [4] M.C. Delfour and J.-P. Zolésio, Structure of shape derivatives for nonsmooth domains, J. Funct. Anal. 104 (1992), no. 1, 1–33. [5] A.M. Micheletti, Metrica per famiglie di domini limitati e propriet` a generiche degli autovalori, Ann. Scuola Norm. Sup. Pisa Ser. III, 26 (1972), 683–694. [6] F. Murat and J. Simon, Sur le contrˆ ole par un domaine géométrique, Rapport 76015, Université Pierre et Marie Curie, Paris, 1976. [7] D.L. Russell, Introduction to Formation Theory of linear elastic materials, Lecture notes, Ecole CEA-EDF-INRIA, Pole Universitaire Leonard de Vinci, Paris, 1997. [8] J.-P.- Zolésio and C.Truchi, Shape stabilization of wave equation, in “Boundary control and boundary variations” (Nice, 1986), 372–398, Lecture Notes in Comput. Sci., 100, Springer, Berlin-New York, 1988. [9] J. Sokolowski and J.-P. Zolésio, Introduction to shape optimization. Shape sensitivity analysis, Springer Series in Computational Mathematics, 16. Springer-Verlag, Berlin, 1992. [10] J.-P. Zolésio, Identification de domaines par déformation, thèse de doctorat d’état, Université de Nice, France, 1979.

Nonlinear Periodic Oscillations In Suspension Bridges

Zhonghai Ding1 , University of Nevada, Las Vegas, Nevada Abstract In this paper, we investigate nonlinear periodic oscillations in a suspension bridge system which is described by the nonlinearly coupled wave and beam equations. By applying the Mountain Pass Theorem to a dual variational formulation of the problem, it is proved that the suspension bridge system has at least two periodic oscillation solutions.

1

Introduction

The suspension bridge is a common type of civil engineering structures. It is well known that a suspension bridge may display certain oscillations under external aerodynamic forces. Under the action of a strong wind, in particular, a narrow and very flexible suspension bridge can undergo dangerous oscillations. The collapse of the Tacoma Narrows suspension bridge caused by a wind blowing at a speed of 42 miles per hour in the State of Washington on November 7, 1940, is one of the most striking examples. The Federal Works Agency Report [3] on the collapse has created a widespread demand for a comprehensive investigation of dynamic oscillation problems in suspension bridges in order to understand the causes of such destructive oscillations, and to develop design techniques to prevent their recurrence in future. A systematic study of the mathematical theory of suspension bridges appears to be initiated by Bleich, McCullough, Rosecrans and Vincent [5] in 1950. Since then, the extensive studies on dynamics of suspension bridges were carried out by many researchers (see, for example, [1], [15], [17]-[20] and the references therein), and more recently by Lazer and McKenna [12], [13]. Based upon the observation of the fundamental nonlinearity in suspension bridges that the stays connecting the supporting cables and the roadbed resist expansion, but do not resist compression, new models describing oscillations in suspension bridges have been developed by Lazer and McKenna in [12], [13]. The new models are described by systems 1

This research is supported in part by NSF Grant DMS 96-22910.

69

70

Ding

of coupled nonlinear partial differential equations. Multiple large amplitude periodic oscillations have been found theoretically and numerically in the single one-dimensional Lazer-McKenna suspension bridge equation (see [6], [7], [9], [10], [12]-[14] and references therein). However, there is very little discussion on multiple large amplitude periodic oscillations in suspension bridge systems of coupled nonlinear partial differential equations in the existing literature. The objective of this paper is to study nonlinear large amplitude periodic oscillations in a suspension bridge model described by two coupled nonlinear partial differential equations. Consider a simplified suspension bridge configuration: the roadbed of length L is modeled by a horizontal vibrating beam with both ends being simply supported; the supporting cable is modeled by a horizontal vibrating string with both ends being fixed; and the vertical stays connecting the roadbed to the supporting cable are modeled by one-sided springs which resist expansion but do not resist compression. Let u(x, t) and w(x, t) denote the downward deflections of the cable and the roadbed, respectively. The following suspension bridge model has been proposed by Lazer-McKenna [12]: (1.1)  mc utt − Quxx − K(w − u)+ = mc g + f1 (x, t), 0 < x < L, t > 0,    mb wtt + EIwxxxx + K(w − u)+ = mb g + f2 (x, t), 0 < x < L, t > 0, u(0, t) = u(L, t) = 0,    w(0, t) = w(L, t) = 0, wxx (0, t) = wxx (L, t) = 0, where (w − u)+ = max{w − u, 0}; mc and mb are the mass densities of the cable and the roadbed, respectively; Q is the coefficient of cable tensile strength; EI is the roadbed flexural rigidity; K is the Hooke’s constant of the stays; f1 and f2 represent the external aerodynamic forces. The total energy E(t), including all kinetic and potential energies, is given by

E(t) =

1 2

Z

L

2 [mc u2t + Qu2x ] + [mb wt2 + EIwxx ] + K((w − u)+ )2 dx.

0

In the absence of nonconservative external forces, i.e., f1 (x, t) = 0 and f2 (x, t) = 0, system (1.1) is conservative, i.e., E 0 (t) = 0. Except a recent paper [2] by Ahmed and Harbi who investigated the asymptotic stability of (1.1) with nonlinear damping, the model (1.1) has not yet received in-depth study in the literature. In this paper, we study nonlinear periodic oscillations in (1.1). We are interested in periodic oscillation being symmetric about x = L/2,

(1.2)

u(x, t + T ) = u(x, t), u(x, t) = u(L − x, t),

w(x, t + T ) = w(x, t), 0 ≤ x ≤ L, t > 0 w(x, t) = w(L − x, t), 0 ≤ x ≤ L/2, t > 0,

Nonlinear Oscillations In Suspension Bridges

71

where T is the period of periodic oscillations. By rescaling and translating x and t, system (1.1) with (1.2) can be written in an equivalent form: (1.3)  mc utt − Quxx − K(w − u)+ = mc g + f1 (x, t), −π/2 < x < π/2, t > 0,     mb wtt + EIwxxxx + K(w − u)+ = mb g + f2 (x, t), −π/2 < x < π/2, t > 0,    u(−π/2, t) = u(π/2, t) = 0, t > 0, w(−π/2, t) = w(π/2, t) = 0, wxx (−π/2, t) = wxx (π/2, t) = 0, t > 0,      u(−x, t) = u(x, t), w(−x, t) = w(x, t), 0 ≤ x ≤ π/2, t > 0,   u(x, t + π) = u(x, t), w(x, t + π) = w(x, t), −π/2 ≤ x ≤ π/2, t > 0. By applying the Mountain Pass Theorem to a dual variational formulation of the problem, we prove that, under some periodic external forces, the above problem has at least two periodic solutions. One is a near-equilibrium periodic oscillation, and the other is a nonlinear periodic oscillation. The organization of this paper is as follows. In Section 2, we study the near-equilibrium periodic oscillation. In Section 3, we formulate an equivalent dual problem. In Section 4, we prove that the dual problem has at least two periodic solutions by applying the Mountain Pass Theorem.

2

Near-equilibrium periodic oscillation

To investigate the suspension bridge system (1.3), we assume throughout this paper that (2.1)

Q ≤ mc ,

EI ≤ mb .

These assumptions hold naturally for suspension bridges in civil engineering applications. The equation for equilibrium oscillation (we , ue ) of (1.3) is

(2.2)

 −Quxx − K(w − u)+ = mc g, −π/2 < x < π/2,      EIwxxxx + K(w − u)+ = mb g, −π/2 < x < π/2, u(−π/2) = u(π/2) = 0,   w(−π/2) = w(π/2) = 0, wxx (−π/2) = wxx (π/2) = 0,    u(−x) = u(x), w(−x) = w(x), 0 ≤ x ≤ π/2.

The following three propositions follow from a direct and tedious calculation. Let (mb + mc )g π 2 EI(mb + mc )g 2 h(x) = −x − . 2Q 2 Q2

72

Ding Proposition 2.1. If K >

4Q2 , and if EI

π π ω1 tanh ω2 − ω2 tanh ω1 mc ω1 ω2 2 2, < 2 · mb ω1 + ω22 ω1 tanh ω1 π − ω2 tanh ω2 π 2 2 where ω12

" # r 2K 1 4Q = K + K2 − > 0, 2Q EI

ω22

" # r 2K 1 4Q = K − K2 − > 0, 2Q EI

then the equilibrium solution of (2.2) is given by  EI   A1 ω12 cosh ω1 x + A2 ω22 cosh ω2 x + h(x),  ue (x) =  Q   mg   we (x) = A1 cosh ω1 x + A2 cosh ω2 x + b + h(x), K where A1 and A2 are determined from the boundary conditions in (2.2). Furthermore, w(x) − u(x) > 0 for |x| < π/2. Proposition 2.2. If K =

4Q2 , and if EI

mc 1 sinh ω0 π − ω0 π < · mb 2 sinh ω0 π + ω0 π s where ω0 =

K , then the equilibrium solution of (2.2) is given by 2Q

 EI     ue (x) = 2B1 + 2B2 Q ω0 cosh ω0 x + 2B2 x sinh ω0 x + h(x),     we (x) = B1 cosh ω0 x + B2 x sinh ω0 x + mb g + h(x), K where B1 and B2 are determined from the boundary conditions in (2.2). Furthermore, w(x) − u(x) > 0 for |x| < π/2. Proposition 2.3. If 0 < K
0 for |x| < π/2. For the purpose of our investigation, let (2.3)

f1 (x, t) = 0,

f2 (x, t) = ε sin 2t cos x,

where ε is a small parameter to be specified later. Let λ10 = Q − 4mc < 0,

µ10 = EI − 4mb < 0,

σ10 =

λ10 µ10 . λ10 + µ10

The near-equilibrium solution of system (1.3) with (2.3) can be obtained also from a direct and careful calculation. Proposition 2.4. Assume K > −σ10 and (ue , we ) being the equilibrium solution of (2.1) given in Propositions 2.1-2.3. If |ε| < ε0 , where ε0 > 0 is a constant determined by K, mc , mb , Q and EI, then the suspension bridge system (1.3) with the external forces (2.3) admits a near-equilibrium solution (u0 , w0 ) given by  εK   u0 (x, t) = ue (x) + cos x sin 2t,   (K + σ10 )(λ10 + µ10 )  (2.4)

     w0 (x, t) = we (x) +

ε(K + λ10 ) cos x sin 2t. (K + σ10 )(λ10 + µ10 )

Furthermore, w0 (x, t) − u0 (x, t) > 0 for |x| < π/2 and t > 0.

74

3

Ding

A duality formulation

The objective of the rest of this paper is to show that, in addition to the near-equilibrium solution given in (2.4), the suspension bridge system (1.3) with (2.3) has at least one nonlinear periodic solution. To prove the existence of such a solution, we derive first an equivalent duality formulation of system (1.3) with (2.3). Define the wave operator L1 by   L1 u = mc utt − Quxx , u(−π/2, t) = u(π/2, t) = 0,  u(x, t) = u(−x, t), u(x, t + π) = u(x, t). Define the beam operator L2 by  L2 w = mb wtt + EIwxxxx ,    w(−π/2, t) = w(π/2, t) = 0, w (−π/2, t) = wxx (π/2, t) = 0,    xx w(x, t) = w(−x, t), w(x, t + π) = w(x, t). Denote by {λmn } the eigenvalues of L1 and by {µmn } the eigenvalues of L2 . Then it follows from a direct calculation that (3.1)

λmn = Q(2n + 1)2 − 4mc m2 , m, n = 0, 1, 2, · · · , µmn = EI(2n + 1)4 − 4mb m2 , m, n = 0, 1, 2, · · · .

The eigenfunctions of L1 corresponding to eigenvalue λmn are the same as that of L2 corresponding to eigenvalue µmn , which are given by ϕ0n (x, t) = cos(2n + 1)x, n ≥ 0, ϕmn (x, t) = cos(2n + 1)x cos 2mt, m ≥ 1, n ≥ 0, ψmn (x, t) = cos(2n + 1)x sin 2mt, m ≥ 1, n ≥ 0. Assume throughout the rest of this paper that the material parameters mc , mb , Q and EI are chosen such that (3.2)  r r Q EI    both and are rational numbers; mc mb λ = Q(2n + 1)2 − 4mc m2 6= 0, µmn = EI(2n + 1)4 − 4mb m2 6= 0,    mn λmn + µmn 6= 0, for m ≥ 1, n ≥ 1. Let Ω = [−π/2, π/2] × [−π/2, π/2], and H be a Hilbert space defined by H = {u ∈ L2 (Ω) | u(−x, t) = u(x, t)}.


75

It is easy to check that the set of eigenfunctions {ϕmn , ψmn } is an orthogonal basis of H. Under the assumption (3.2), L1 , L2 and L1 + L2 have p compact inverses from H to H. Noting that the assumption of both Q/mc and p EI/mb being rational is necessary due to the well-known fact that certain number theoretical difficulties may be encountered. Under the above notations, system (1.3) with (2.3) can be written as (3.3)

L1 u − K(w − u)+ = mc g, L2 w + K(w − u)+ = mb g + ε sin 2t cos x.

Let u ¯ = u − u0 and w ¯ = w − w0 where (u0 , w0 ) is the near-equilibrium solution given by (2.4), then (3.3) becomes (3.4)

L1 u ¯ − K [((w ¯−u ¯) + (w0 − u0 ))+ − (w0 − u0 )] = 0, L2 w ¯ + K [((w¯ − u ¯) + (w0 − u0 ))+ − (w0 − u0 )] = 0.

Since system (1.3) with (2.3) is equivalent to system (3.4), the problem of finding a nonlinear periodic solution in system (1.3) with (2.3) becomes the problem of finding a nontrivial periodic solution of system (3.4). From (3.4), one has L1 u ¯ + L2 w ¯ = 0. −1 By applying L−1 1 L2 to both sides of this equation, we have

L−1 ¯ + L−1 ¯ = 0. 2 u 1 w Let w ˜ = L−1 ¯ and u ˜ = L−1 ¯, then w ˜+u ˜ = 0, u ¯ = L2 u ˜ and w ¯ = L1 w. ˜ 1 w 2 u Substituting them into the second equation of (3.4), we obtain L2 L 1 w ˜ + K ((L1 + L2 )w ˜ + (w0 − u0 ))+ − (w0 − u0 ) = 0. Let w ˆ = (L1 + L2 )w ˜ and g0 = w0 − u0 , then the above equation can be written as L2 L1 (L1 + L2 )−1 w (3.5) ˆ + K (w ˆ + g0 )+ − g0 = 0. Let Aw ˆ = L2 L1 (L1 + L2 )−1 w. ˆ The eigenvalues of A are given by

σmn =

λmn µmn , λmn + µmn

where the corresponding eigenfunctions are {ϕmn , ψmn }. Under assumption (2.1), it is easy to check that

76

Ding

σ20 < σ10 < 0 < σ00 . Assume throughout this paper that the only eigenvalue of A in the interval (σ20 , σ00 ) is σ10 . Let β be a given constant satisfying −σ10 < K < β < −σ20 .

(3.6) Let

Aβ w ˆ = Aw ˆ + β w, ˆ Fβ (w) ˆ = βw ˆ − K [(w ˆ + g0 )+ − g0 ] . Under the assumption (3.6), Aβ has a compact inverse from H to H. Equation (3.5) can be written as −Aβ w ˆ + Fβ (w) ˆ = 0.

(3.7)

By (3.6), Fβ : < −→ < is a monotone increasing function. Thus Fβ has a monotone increasing inverse given by

Fβ−1 (w) ˆ =

=

 1  w, ˆ    β−K

if w ˆ > −(β − K)g0

  1   (w ˆ ≤ −(β − K)g0 ˆ − Kg0 ), if w β 1 1 [w ˆ + (β − K)g0 ]+ − [w ˆ + (β − K)g0 ]− − g0 , β−K β

where u− = max{−u, 0}. Let v = Aβ w, ˆ then equation (3.7) can be written as (3.8)

−1 −A−1 β v + Fβ (v) = 0.

Note that v = 0 is a trivial solution of (3.8). Proposition 3.1. Let condition (3.6) be satisfied. If equation (3.8) has a nontrivial solution v in H, then the suspension bridge system (3.4) admits a nontrivial periodic solution (¯ u, w) ¯ such that (¯ u, w) ¯ ∈ H 2 (Ω) × H 3 (Ω). The proof of this proposition is based on analyzing u ¯ = −L2 (L1 + L2 )−1 A−1 β v, and bootstrapping regularities.

w ¯ = L1 (L1 + L2 )−1 A−1 β v,


4

77

Nonlinear periodic oscillation

In this section, we prove that (3.8) admits a nontrivial solution in H by applying the Mountain Pass Theorem. Define a functional I(v) : H −→ < by Z

1 −1 − Aβ v · v + F(v) dxdt, 2

I(v) = Ω

where

F(v) 2 2 1 1 β−K 2 = [v + (β − K)g0 ]+ + [v + (β − K)g0 ]− − g0 v − g0 . 2(β − K) 2β 2 Lemma 4.1. I(v) is continuous Frechet differentiable with I 0 (v)ϕ =

Z h Ω

i −1 −A−1 v + F (v) ϕdxdt, β β

for any v, ϕ ∈ H. Consequently, the solutions of (3.8) in H correspond to critical points of I(v) in H. The Mountain Pass Theorem due to Ambrosetti and Rabinowitz [4] has been used to prove the existence of critical points of functionals satisfying a condition called the Palais-Smale (PS) condition, which occurs repeatedly in the critical point theory. We say that a functional J satisfies the (PS) condition if any sequence {vn } ⊂ H for which J(vn ) is bounded and J 0 (vn ) → 0 possesses a convergent subsequence. Mountain Pass Theorem. Let E be a real Banach space. J ∈ C 1 (E, 0 such that J|∂Bρ ≥ α, (b). there is an e ∈ E\B ρ such that J(e) ≤ 0. Then J possesses a critical value c ≥ α. Moreover c can be characterized as c = inf

max J(u),

g∈Γ u∈g([0,1])

where Γ = {g ∈ C([0, 1], E) | g(0) = 0, g(1) = e}. In the next several lemmas, we show that I(v) satisfies all conditions in the Mountain Pass Theorem. K − σ10 Lemma 4.2. If (3.6) is satisfied, β < , and |ε| < ε0 , where ε0 is 2 defined in Proposition 2.4, then v = 0 is a strict local minimum of I(v) in H.

78

Ding

Proof. It is easy to verify that I(0) = 0 and I 0 (0) = 0. We follow an approach used in [8] to prove v = 0 is a strict local minimum of I(v). For s > 0 and ϕ ∈ H with kϕkL2 = 1, we have I 0 (sϕ)ϕ Z h

1 (sϕ + (β − K)g0 )+ ϕ β − K Ω i 1 − (sϕ + (β − K)g0 )− ϕ − g0 ϕ dxdt β Z Z s = −s A−1 ϕ · ϕdxdt + ϕ2 dxdt β β − K Ω Ω Z 1 1 + (sϕ + (β − K)g0 )− ϕdxdt, − β −K β Ω

=

− sA−1 β ϕ·ϕ+

where we have used the relation u = u+ − u− in the last equality. Let 1 −1 H1 = span ϕmn , ψmn |β + σmn | > , 2β which is finite dimensional. Let H = H1 ⊕ H2 where H2 is the orthogonal complement of H1 in H. In fact, 1 −1 H2 = the closure of span ϕmn , ψmn |β + σmn | ≤ . 2β For any ψ ∈ H2 , we have Z Z A−1 ψ · ψdxdt ≤ 1 ψ 2 dxdt. β 2β Ω Ω Thus, by letting ϕ = ϕ1 + ϕ2 where ϕ1 ∈ H1 and ϕ2 ∈ H2 , we have I 0 (sϕ)ϕ = −s

Z h

+

Ω

A−1 β ϕ1

· ϕ1 +

1 1 − β−K β

Z

A−1 β ϕ2

· ϕ2

i

s dxdt + β−K

Z Ω

ϕ21 + ϕ22 dxdt

(s(ϕ1 + ϕ2 ) + (β − K)g0 )− (ϕ1 + ϕ2 )dxdt Ω

Z Z 1 1 1 1 2 ≥ s ϕ1 dxdt + s ϕ22 dxdt − − β−K |β + σ20 | β−K 2β Ω Ω Z 1 1 − (s(ϕ1 + ϕ2 ) + (β − K)g0 )− (|ϕ1 | + |ϕ2 |)dxdt. − β−K β Ω


79

Under the assumption (3.6) and |ε| < ε0 , we know from Proposition 2.4 that g0 (x, t) = w0 (x, t) − u0 (x, t) > 0 for |x| < π/2, and gx (−π/2, t) > 0 and gx (π/2, t) < 0. Since kϕ1 kL2 ≤ kϕkL2 = 1, there is a small s0 > 0, which is dependent of β, K and g0 only, such that for any 0 < s < s0 , 1 π π s|ϕ1 (x, t)| ≤ (β − K)g0 (x, t), − ≤ x ≤ . 2 2 2 Let Ωs = {(x, t) ∈ Ω | sϕ2 ≤ − 12 (β − K)g0 }. Thus I 0 (sϕ)ϕ

Z Z 1 1 1 1 2 ≥ s ϕ1 dxdt + s ϕ22 dxdt − − β−K |β + σ20 | β − K 2β Ω Ω Z 1 1 1 − (sϕ2 + (β − K)g0 )− (|ϕ1 | + |ϕ2 |)dxdt − β−K β 2 Ω Z Z 1 1 1 1 ≥ s ϕ21 dxdt + s ϕ22 dxdt − − β−K |β + σ20 | β − K 2β Ω Ω Z 1 1 −s ϕ22 dxdt − β−K β Z Ωs 1 1 1 − (tϕ2 + (β − K)g0 )− |ϕ1 |dxdt − β−K β 2 Ωs Z 1 ≥ C1 s − C2 (sϕ2 + (β − K)g0 )− |ϕ1 |dxdt, 2 Ωs where C1 = min

1 1 1 − , β−K |β + σ20 | 2β

For any δ T > 0, define Ωδ = {(x, t) ∈ Ω | (x, t) ∈ Ωt (Ω \ Ωδ ), −sϕ2 ≤ −δ. Thus, Z

C2 =

> 0, 1 2 (β

− K)g0 ≤ δ}. Hence for any

Z

1= Ω

(ϕ21

+

ϕ22 )dxdt

≥ Ωs

T (Ω\Ωδ )

ϕ22 dxdt ≥

\ δ2 (Ω \ Ωδ )). mes(Ω s s2

Then

mes(Ωs

\

(Ω \ Ωδ )) ≤

1 1 − > 0. β−K β

s2 . δ2

80

Ding

Since kϕ1 kL2 ≤ 1, it is easy to check that kϕ1 kL∞ ≤ C3 where C3 is a positive constant independent of ϕ1 . From Propositions 2.1-2.3, we know that g0 (x, t) = w0 (x, t) − u0 (x, t) = 0 when x = ±π/2, g0 (x, t) > 0 for |x| < π/2, and g0 (x, t) is continuous on Ω. Then for any δ1 > 0, one can choose δ > 0 sufficiently small such that kϕ1 kL∞ (Ωs T Ωδ ) ≤ δ1 . Thus I 0 (sϕ)ϕ Z

!

Z

1 (sϕ2 + (β − K)g0 )− |ϕ1 |dxdt 2 Ωs Ωδ Ωs (Ω\Ωδ ) ! Z Z 1 ≥ C1 s − C2 δ1 (sϕ2 + (β − K)g0 )− dxdt +C2 C3 T T 2 Ωs Ωδ Ωs (Ω\Ωδ ) q \ ≥ C1 s − C2 δ1 skϕ2 kL2 (Ωs T Ωδ ) mes(Ωs Ωδ )

≥ C1 s − C2

T

+

T

−C2 C3 skϕ2 kL2 (Ωs T(Ω\Ωδ )

q mes(Ωs

\

(Ω \ Ωδ )

C 2 C3 s 2 . δ By choosing sufficiently small δ > 0, one has C1 − C2 δ1 π > C1 /2. Then fix δ, and choose s1 > 0 small enough such that ≥ C1 s − C2 δ1 sπ −

C 1 C 2 C3 s C1 − ≥ , for 0 ≤ s ≤ s1 . 2 δ 4 Thus for 0 < s ≤ min{s0 , s1 }, we have Z s Z s C1 C1 2 I(sϕ) = I 0 (τ ϕ)ϕdτ ≥ τ dτ = s . 4 8 0 0 Thus v = 0 is a strictly local minimum of I(v) in H. Lemma 4.3. If −σ10 < K < −σ20 , the following equation (4.1)

Av + Kv + = 0

admits only the trivial solution v = 0 in H. Proof. Under the assumption −σ10 < K < −σ20 and by applying an abstract symmetry theorem due to Lazer and McKenna [11], all solutions of (4.1) can be expressed as v(x, t) = g(t) cos x. It is straightforward to check that if v ∈ H is a solution of (4.1), then u = −L2 (L1 + L2 )−1 v ∈ H and w = L1 (L1 + L2 )−1 v ∈ H is a solution to the following system L1 u − K(w − u)+ = 0, L2 w + K(w − u)+ = 0.


81

Since v = g(t) cos x, one has u = g1 (t) cos x and w = g2 (t) cos x. Thus (g1 , g2 ) satisfies   mc g100 + Qg1 − K(g2 − g1 )+ = 0, m g00 + EIg2 + K(g2 − g1 )+ = 0, (4.2)  c 2 g1 (t + π) = g1 (t), g2 (t + π) = g2 (t). A simple phase plane analysis of (4.2) shows that it admits only the trivial solution g1 (t) = g2 (t) = 0. Lemma 4.4. If (3.6) is satisfied, then I(v) satisfies the (PS) conditions. Proof. Assuming {vn } ⊂ H such that I(vn ) is bounded and I 0 (vn ) −→ 0 strongly in L2 (Ω), we need to show that {vn } has a convergent subsequence. We claim that {vn } is bounded in L2 (Ω). Assume the contrary, kvn kL2 −→ vn ∞ as n −→ ∞. Let wn = . Since I 0 (vn ) −→ 0, i.e., kvn kL2 −1 −A−1 β vn + Fβ (vn ) −→ 0,

we have −A−1 β wn

+ − 1 β−K 1 β−K 1 + g0 − g0 − g0 −→ 0. wn + wn + β−K kvn kL2 β kvn kL2 kvn kL2

Since kwn kL2 = 1, there exists a subsequence of {wn }, denoted by itself, such that wn −→ w weakly in L2 (Ω). Since A−1 β is compact from H to H, we have −1 lim A−1 β wn = Aβ w

n→∞

in L2 (Ω).

Then + − 1 β−K 1 β−K g0 − g0 −→ A−1 wn + wn + β w. β−K kvn kL2 β kvn kL2 By applying Proposition B.1 in [16], we have β−K −1 − + lim wn + = (β − K)[A−1 β w] − β[Aβ w] n→∞ kvn kL2

strongly in L2 (Ω).

Since the weak limit of {wn } is unique, we have lim wn = w strongly in L2 (Ω) n→∞ and −1 − + w = (β − K)[A−1 β w] − β[Aβ w] .

Let u = A−1 β w, then u statisfies Au + Ku+ = 0.

82

Ding

By Lemma 4.3, we have u = 0. Hence w = 0. Thus lim wn = 0 strongly in n→∞

L2 (Ω), which contradicts to kwn kL2 = 1.

Since {vn } is bounded, there exists a subsequence of {vn }, denoted by itself, such that lim vn = v weakly in L2 (Ω). By repeating the same argument as n→∞

above, we have lim vn = v strongly in L2 (Ω). n→∞

Lemma 4.5. Assume (3.6) is satisfied, and |ε| < ε0 , where ε0 is defined in Proposition 2.4. If β satisfies further −

(4.3)

2 1 1 + + < 0, β + σ10 β − K β

then lim I(sϕ10 ) = −∞. s→∞

Proof. It is easy to check that Z Z Z 1 π2 + 2 − 2 (ϕ10 ) dxdt = (ϕ10 ) dxdt = (ϕ10 )2 dxdt = . 2 Ω 8 Ω Ω Thus for any η > 0 and s being sufficiently large, we have I(sϕ10 )

=

 " + #2 Z  1 1 β − K − ϕ2 + ϕ10 + g0 2 Ω  β + σ10 10 β − K s

s2

1 β

"

β−K ϕ10 + g0 s

 − #2 

Z

β−K dxdt − s ϕ21 g0 dxdt −  2 Ω

Z −

Z Ω

g02 dxdt

i 1h i 1 1 h + 2 − 2 2 ≤ ϕ + ϕ10 + η + ϕ10 + η dxdt β + σ10 10 β − K β Ω Z Z β−K −s ϕ21 g0 dxdt − g02 dxdt 2 Ω Ω s2 π 2 1 2 1 1 1 = − + + 8η + 16 β + σ10 β − K β β−K β Z Z β−K −s ϕ21 g0 dxdt − g02 dxdt. 2 Ω Ω s2 2

Since η > 0 is arbitrary and (4.3) holds, we have lim I(sϕ10 ) = −∞.

s→∞


83

By applying the Mountain Pass Theorem, we obtain from Lemmas 4.1-4.5 the following theorem. K − σ10 , and |ε| < ε0 , 2 where ε0 is defined in Proposition 2.4, then I(v) admits at least one nontrivial critical point v ∈ H. Theorem 4.1. If (3.6) and (4.3) are satisfied, β
0 such that I(s0 ϕ10 ) ≤ 0. Since β is introduced to formulate the dual problem, one can derive easily from (3.6) and (4.3) that the only condition on K needed is −σ10 < K < min{−σ20 , −2σ10 }. Hence, as a consequence of Theorem 4.1 and Lemma 3.1, we obtain the following main result of this paper. Theorem 4.2. If −σ10 < K < min{−σ20 , −2σ10 }, and |ε| < ε0 , where ε0 is defined in Proposition 2.4, then the suspension bridge system (1.3) with (2.3) admits, in addition to an explicit near-equilibrium periodic oscillation T given in Proposition 2.4, at least one nonlinear periodic oscillation in H 2 (Ω) H 3 (Ω).

5

Conclusion

In this paper, we have studied nonlinear periodic oscillations in a suspension bridge system governed by the nonlinearly coupled wave and beam equations. It is shown that the suspension bridge system has at least two periodic oscillations: one is an explicit near-equilibrium oscillation, and the other is a nonlinear periodic oscillation. More theoretical and numerical results on multiple nonlinear periodic oscillations in suspension bridges are being reported in several other papers.

References [1] A. M. Abdel-Ghaffer, Suspension bridge vibration: continuum formulation, J. Engrg. Mech., 108(1982), pp. 1215-1232. [2] N. U. Ahmed and H. Harbi, Mathematical analysis of dynamic models of suspension bridges, SIAM J. Appl. Math., 58(1998), pp. 853-874. [3] O. H. Amann, T. Von Karman and G. B. Woodruff, The failure of the Tacoma Narrows Bridge, Federal Works Agency, Washington D. C., 1941. [4] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14(1973), pp. 349-381.

84

Ding

[5] F. Bleich, C. B. McCullough, R. Rosecrans and G. S. Vincent, The mathematical theory of suspension bridges, Bureau of Public Roads, U. S. Department of Commence, Washington D. C., 1950. [6] Q. H. Choi, T. Jung and P. J. McKenna, The study of a nonlinear suspension bridge equation by a variational reduction method, Appl. Analysis, 50(1993), pp. 73-92. [7] Y. S. Choi, K. C. Jen and P. J. McKenna, The structures of the solution set for periodic oscillations in a suspension bridge model, IMA J. Appl. Math., 47(1991), pp. 283-306. [8] Y. S. Choi, P. J. McKenna and M Romano, A mountain pass method for the numerical solution of semilinear wave equations, Numer. Math., 64(1993), pp. 487-509. [9] J. Glover, A. C. Lazer and P. J. McKenna, Existence and stability of large scale nonlinear oscillations in suspension bridges, Z. Angew. Math. Phys., 40(1989), pp. 172-200. [10] L. D. Humphreys, Numerical mountain pass solutions of a suspension bridge equation, Nonlinear Analysis, Vol. 28(1997), pp. 1811-1826. [11] A. C. Lazer and P. J. McKenna, A symmetry theorem and applications to nonlinear partial differential equations, J. Diff. Equations, 72(1988), pp. 95-106. [12] , Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis, SIAM Review, 32(1990), pp. 537-578. [13] , Large scale oscillation behavior in loaded asymmetric systems, Ann. Inst. H. Poincare Anal. Non Lineaire, 4(1987), pp. 243-274. [14] P. J. McKenna and W. Walter, Nonlinear oscillations in a suspension bridge, Arch. Rational Mech. Anal., 98(1987), pp. 167-177. [15] B. Pittel and V. Jakubovic, A mathematical analysis of the stability of suspension bridges based on the example of the Tacoma bridge, Vestnik Leningrad. Univ., 24 (1969), pp. 80-91. [16] P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, Regional Conference Series in Mathematics, No. 65, AMS, 1986. [17] R. H. Scanlan, The action of flexible bridges under wind. Part I: Flutter theory, J. Sound Vibration, 60(1978), pp. 187-199. [18] , The action of flexible bridges under wind. Part II: Buffeting theory, J. Sound Vibration, 60(1978), pp. 201-211. [19] A. Selberg, Oscillation and aerodynamic stability of suspension bridges, Acta Polytech. Scand., 13(1961), pp. 308-377. [20] E. G. Wiles, Report of aerodynamic studies on proposed San Pedro-Terminal Island suspension bridge, California, Research, Bureau of Public Roads, U. S. Department of Commerce, Washington, D. C., 1960.

Canonical Dual Control for Nonconvex Distributed-Parameter Systems: Theory and Method

David Y. Gao, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061. E-mail: [email protected] Abstract This paper presents a potentially powerful canonical dual transformation method and associated duality theory for solving fully nonlinear distributed-parameter control problems. The extended Lagrange duality and the interesting triality theory proposed recently in finite deformation theory are generalized into nonconvex dynamical systems. A bifurcation criterion is proposed, which leads to an effective dual feedback control against the chaotic vibration in Duffing system.

1

Problems and Motivations

We shall study a duality approach for solving the following very general abstract distributed parameter problem ((P) for short), (1.1)

(P) :

ρu,tt + A(u, µ) = 0 ∀u ∈ Uk ,

where the feasible space Uk is a convex, non-empty subset of a reflexive Banach space U over an open space-time domain Ωt = Ω × (0, tc ) ⊂ Rn × R+ , in which, certain essential boundary-initial conditions are prescribed. We assume that for a given distributed parameter control field µ(x, t) over Ωt , the mapping A(u, µ) is a potential operator from Uk into its dual space U ∗ , i.e., there exists a Gˆ ateaux differentiable potential functional Pµ (u) = P (u; µ), such that the directional derivative of P at u ¯ ∈ Uk in the direction δu can be written as δPµ (¯ u; δu) = hDPµ (¯ u), δui ∀δu ∈ Uk , where the operator DPµ (¯ u) = A(¯ u, µ) is the Gâteaux derivative of Pµ at the point u ¯; the bilinear form h·, ·i : U × U ∗ → R places U and U ∗ in duality. By nonlinear operator theory we know that the mapping A : Uk → U ∗ is monotone if and only if P is convex on Uk . 85

86

Gao

The problem (P) is said to be exactly controllable if for certain given initial data (u0 (x), v0 (x)) in Uk and the final state (¯ uc (x), v¯c (x)) there exists suitable control function µ(x, t) such that the solution u(x, t) of the problem (P) satisfies (1.2)

u(x, tc ) = u ¯c (x), u,t (x, tc ) = v¯c (x) ∀x ∈ Ω.

Dually, the problem (P) is said to be observable if for certain given input control µ(x, t), there exists an output function h(u) such that the initial state (uo (x), vo (x)) can be uniquely determined from the output z = h(u(x, t)) over any interval 0 < t < tc . The abstract form of problem (P) covers a great variety of situations. Very often, the total potential Pµ (u) can be written as Pµ (u) = Φµ (u, Λ(u)) = Wµ (Λ(u)) − Fµ (u), where Λ is a Gˆ ateaux differentiable operator from U into another Banach space E; the functional Wµ (ξ) is the so-called stored (or internal) potential; while the functional Fµ (u) represents the external potential of the system. In convex Hamilton systems, the total potential Pµ (u) is convex and its Gâteaux derivative A(u; µ) = DPµ (u) is usually an elliptic operator in conservative problems. In linear field theory of mathematical physics, Λ is usually a gradient-like operator, say Λ = grad, and Wµ (ξ) is a quadratic functional, for example, Z 1 Pµ (u) = a(x)|∇u|2 dΩ − Fµ (u), 2 Ω where a(x) > 0 ∀x ∈ Ω. In this case, the governing equation (1.1) reads (1.3)

ρu,tt = ∇ · (a(x)∇u) + DFµ (u) ∀(x, t) ∈ Ωt .

It is a linear wave equation if Fµ (u) is a linear functional, say Fµ (µ) = hu , u∗ (µ)i, where u∗ (µ) is a given function of the input control field µ(x, t). If Fµ (u) is nonlinear, then the governing equation (1.3) is semi-linear. In boundary control problems, the distributed-parameter µ also appears in the feasible set Uk . In applications of engineering mechanics, the state variable u could be also a vector-valued function and Λ is a tensor type operator. For example, in the shear control of extended beam structures, the actuators are filaments attached to the upper and lower beam surfaces (y = ±h). The external signals effect a change of the properties of these filaments in such way that they produce shear forces µ± (x, t). Thus, µ± (x, t) is, in effect, the applied distributed-control, and the composite beam/actuator system is then an instance of an active, or “smart” structure. Since the repeated operation of these actuator devices results large shear deformations, the traditional Timoshenko beam model can not be used to the study of these phenomena because it assumes that the shear deformation is

Canonical Dual Control for Nonconvex Systems

87

a function of x and t alone and does not vary in the lateral beam direction. In order to study the control problems of smart structures, several extended beams models have been proposed recently by Gao and Russell (1994, 1996), where the state variable space U = C 1 (Ωt ; R2 ) is a displacement space over the space time domain Ωt = (0, `) × (−h, h) × (0, tc ). The element u = {χ(x, y, t), w(x, t)} ∈ U is a continuous, differentiable vector in R2 with domain Ωt , where χ(x, y, t) measures the shear deformation of the beam at the point (x, y), while w(x, t) is the deflection of the beam. In the case that the elastic beam subjected to the transverse load f (x, t) undergone infinitesmall deformation, the total potential is a quadratic functional Z 1 Pµ (χ, w) = [χ2 + β(χ,y + w,x )2 ] dΩ 2 Ω ,x Z ` − (µ+ (x, t)χ(x, h, t) + µ− (x, t)χ(x, −h, t) + f (x, t)w) dx. 0

If the beam is clamped at x = 0 and simply supported at x = `, and subjected to a compressive load at x = `, the kinematical admissible space Uk ⊂ U can be definded as   χ(x, −y, t) = −χ(x, y, t), w(0, t) = w(`, t) = 0,       χ(0, y, t) = χ,x (`, y, t) = 0 ∀y ∈ [−h, h], χ Uk = ∈ U . (χ, w) = (χ0 , w0 ), (χ,t , w,t ) = (χ˙ 0 , w˙ 0 ) w       ∀(x, y) ∈ Ω, t = 0 In this case, the abstract governing equation (1) is a linear coupled partial differential system ρχ χ,tt = χ,xx + βχ,yy , (1.4)

ρw w,tt = βw,xx +

β 2h [χ,x (x, h, t)

− χ,x (x, −h, t)] + f (x, t),

χ,y (x, ±h, t) + w,x (x, t) = ±µ± (x, t). Since the total potential of this system is strictly convex, for the given input control function µ± (x, t), this system possesses a unique stable solution. Due to the efforts of more than thirty years research by many wellknown mathematicians and scientists, the mathematical theory for distributedparameter control systems have been well-established for convex Hamilton systems governed by partial differential equations (cf. e.g., Russell, 1973, 1978, 1986, 1996; Chen et al, 1991; Komornik, 1994; Lasiecka and Triggiani, 1999) with substantial applications in mechanics and structures (see, for examples, Lagnese and Lions, 1988; Lasiecka, 1998a; Lasiecka and Triggiani, 1987, 1999; Zuazua, 1996). In linear systems, there exists a very elegant duality relationship between the controllability and observability (see Dolecki and Russell, 1977).

88

Gao

If the system reversible, the well-known Russell principle states that the stabilizability implies its exact controllability. The celebrated review articles by Russell (1978) and Lions (1988) still serve the excellent introductions to the mathematical aspects of controllability, stabilization and perturbations for distributed-parameter systems. Duality is a fundamental concept that underlies almost all natural phenomena. In classical optimization and calculus of variation, duality methods possess beautiful theoretical properties, potentially powerful alternative performances and pleasing relationships to many other fields. The associated theory and extremality principles have been well studied for convex static and Hamilton systems (cf. e.g., Toland, 1978, 1979; Auchmuty, 1983, 1989, 1997; Strang, 1986; Rockafellar and Wets, 1997). There is a growing interest in studying and applications of convex duality theory in optimal control (cf., e.g., Mossino (1975), Chan and Ho (1979), Chan (1985), Chan and Yung (1987), Barron (1990), Tanimoto (1992), Lee and Yung (1997), Bergounioux et al (1999), Arada and Raymond (1999) and many others). The interesting one-to-one analogy between the optimal control and engineering structural mechanics was discovered by Zhong et al (1993, 1999). Recently, the so-called primal-dual interior-point (PDIP) method has been considered as a revolution in linear constrained optimization problems (cf. e.g., Gay et al, 1998; Wright, 1998). It was shown by Helton et al (1998) that the fundamental H ∞ optimization problem of control can be naturally treated with the PDIP methods. However, the beautiful duality relationship in convex systems is broken in nonconvex problems. In many applications of engineering and sciences, the total potential of system is usually nonconvex, and even nonsmooth. The exact controllability and stability for nonconvex/nonsmooth systems are much more difficult. For example, in the well-known von Kármán thin plate model, the state variable u is a vector-valued function u = {χ(x, t), w(x, t)} over Ωt ⊂ R2 × R, where χ = {χα } (α = 1, 2) is an in-plane displacement vector, while w(x, t) stands for the deflection of the plate at (x, t) ∈ Ωt . The total potential is a nonlinear functional Z 1 1 P (χ, w) = a(w, w) + b(ξ(χ, w), ξ(χ, w)) − (1.5) f w dΩ, 2 2 Ω where a(w, w) and b(ξ, ξ) are two bilinear forms, defined respectively by Z a(w, w) = K [(1 − ν)(∇∇w)(∇∇w) + ν∆w∆w] dΩ, Z Ω b(ξ, ξ) = hξαβ Cαβγθ ξγθ dΩ, Ω

and ξ is a Cauchy-Green type strain tensor, defined by 1 ξαβ = (χα,β + χβ,α + w,α w,β ), α, β = 1, 2. 2


89

The governing equations for dynamical von Kármán plate are coupled nonlinear partial differential system (1.6)

ρw w,tt = h∇ · (σ · ∇w) − K0 ∆∆w + f, ρχ χ,tt = ∇ · σ, σ = Cξ(χ, w).

This coupled nonlinear partial differential system is a typical example in finite deformation mechanics. The mathematical control theory for large deformation plates and shells has emerged as the most challenging and active research field in recent years. In a series of papers by Lasiecka and her colleagues (see, for examples, Horn and Lasiecka, 1994, 1995; Favini et al, 1996; Lasiecka, 1998, 1999), many important contributions and open questions have been addressed for stabilizability of the so-called full von K´ arm´ an system with nonlinear boundary feedback (see Lasiecka, 1998). A detailed documentation on mathematical control theory of coupled nonlinear PDE’s has been given in a lecture note by Lasiecka (1999). Since the von K´ arm´ an model is valid only for plates subjected to the moderately large deflections, only the second-order nonlinear term w,α w,β is considered, and the governing equation is linearly dependent on the in-plane deformation χ. In many engineering applications, the acceleration term ρχ,tt can usually be ignored. Thus, the second equation in (1.6) reads ∇ · σ(χ, w) = 0. If the plate is subjected to compressive load on the boundary, the plate will be in the post-buckling state when the compressive load reaches its critical point. In this case, the total potential is nonconvex (i.e. the so-called double-well energy) (see Gao, 1995). In one-dimensional problems, the in-plane equilibrium condition σ,x = 0 leads to a constant stress σ = −λ everywhere in the domain Ω = (0, `) ⊂ R. In this case, the nonlinear von K´ armán model (1.6) in R2 reduces to a linear equation in one-dimensional “beam” problem, i.e., ρw w,tt = hλw,xx − K0 w,xxxx + f. The main reason behind this von K´ arm´ an “paradox” is that the seconder order nonlinear term w,α w,β is considered for in-plane strain ξ, but it is ignored in the thickness direction. It may be appropriate for thin plates, but for onedimensional beam models, this is wrong! It is shown in Gao (1996) that the strain in the thickness direction of the beam is proportional to the second-order 2 , and cannot be ignored when the beam is subjected to moderately large term w,x rotations. Thus, an extended large deformation beam model was proposed as 1 2 (1.7) ρw w,tt + K0 w,xxxx − k0 (λ − w,x )w,xx − f = 0 in Ωt = (0, `) × (0, tc ), 2 where k0 > 0 is a positive material constant. The total potential energy associated with this nonlinear beam theory is a nonlinear functional Z Z 1 1 2 2 2 P (w) = dx − f w dx. (1.8) K0 w,xx + k0 ( w,x − λ) 2 I 2 I

90

Gao

In static problem, if the beam is clamped at x = 0, simply supported at x = `, the kinematically admissible space Uk can be written as Uk = {w ∈ C 2 (0, `)| w(0) = w,x (0) = 0, w(`) = w,xx (`) = 0}. It is clear that for the given Euler pre-buckling load λc > 0, Z Z 2 2 K0 w,xx dx ≥ λc w,x dx, ∀w ∈ Uk . I

I

Thus, on Uk , Z P (w) ≥ I

1 2

2 λc w,x

1 2 + k0 ( w,x − λ)2 2

Z dx −

f w dx I

= Pµ (w) + λ`λc /k0 − `λ2c /(2k02 ), where Pµ (w) is a nonlinear functional Z (1.9)

Pµ (w) = I

1 1 2 − µ)2 dx − k0 ( w,x 2 2

Z f w dx, I

and µ = λ − λc /k0 ∈ R is Clearly, when the parameter µ > 0, the R a1 constant. 1 2 stored energy Wµ (ε) = I 2 k0 ( 2 ε − µ)2 dx is the well-known van der Waals double-well function (see Figure 1a) of the linear strain ε = w,x , the beam is in a post-buckled (bifurcation) state. In this case, the total potential Pµ is nonconvex. It has three critical points: two local minimizers, corresponding to two possible stable buckled states, and one local maximizer, corresponding to an unstable buckled state. The global minimizer of Pµ depends on the lateral load f (see Figure 1b).

(a) Graph of Wµ (ε)

(b) Graphs of Pµ (u) (f > 0 solid, f < 0 dashed)

Fig. 1. Double-well energy and nonconvex potential


91

If the beam is subjected to a periodic dynamical load f (x, t), the two local minimizers of Pµ become extremely unstable, and the beam is in dynamical post-buckling state. In this case, the governing equation (1.7) is replaced by (1.10)

3 2 ρw w,tt = k0 ( w,x − µ)w,xx + f (x, t) in Ωt = (0, `) × (0, tc ), 2

If the deflection w(x, t) can be separated into w(x, t) = u(t)v(x), this postbuckling dynamical beam model is equivalent to the well-known Duffing equation: 1 u,tt = au(µo − u2 ) + µ(t). 2

(1.11)

where a > 0 and µo ∈ R are constants. This equation is extremely sensitive to the initial data. It is known that for certain give parameter µo and the driving input µ(t), this equation may produce the so-called chaotic solutions. The problem of controlling chaotic systems is of significant practical importance and has attracted considerable attention during the last years. Mathematically speaking, the total potential of the chaotic system is usually nonconvex or even nonsmooth. Very small perturbations of the system’s initial conditions and parameters may lead the system to different operating points with significantly different performance characteristics. This is the one of main reasons why the traditional perturbation analysis, the direct approaches and many standard control techniques cannot successfully be applied to chaotic systems. Based upon these observations and in order to handle the nonlinear problem, a school of new techniques has been developed (see, e.g., Fowler, 1989; Ott et al, 1990; Chen and Dong, 1992, 1993; Ogorzalek, 1993; Antoniou et al, 1996; Ghezzi and Piccardi, 1997; Mertzios and Koumboulis, 1996; Koumboulis and Mertzios, 2000). In the shear control of large deformation extended beam model, the equation (1.4) can be replaced by (see Gao, 2000a) (1.12) χ,xx + βχ,yy = 0, 2 2 + β − λα w ρw w,tt = 3α2 w,x ,xx +

β 2h [χ,x (x, h, t)

− χ,x (x, −h, t)] + f (x, t),

χ,y (x, ±h, t) + w,x (x, t) = ±µ± (x, t), where α > 0 is a given material constant and λ > 0 represents the axial load. The total potential associated with this model is a nonconvex functional Z 1 1 2 Pµ (χ, w) = [(χ2,x + αw,x − λ)2 + β(χ,y + w,x )2 ] dΩ 2 Ω 2 Z ` − (µ+ (x, t)χ(x, h, t) + µ− (x, t)χ(x, −h, t) + f (x, t)w) dx. 0

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In order to control the chaotic vibration of this nonconvex dynamical beam system, an efficient canonical dual feedback control method has been proposed recently by the author (Gao, 2000e). The duality theory in fully nonlinear variational problems was originally studied by Gao and Strang (1989) for large deformation nonsmooth mechanics. In order to recover the broken symmetry in fully nonlinear systems (see Definition 2), a so-called complementary gap function was introduced. It was realized recently in post-buckling analysis of nonlinear beam theory (Gao, 1996) that this function recovered the duality gap between the nonconvex primal problems and the Fenchel-Rockafellar dual problems. A self-contained comprehensive presentation of the mathematical theory for general nonconvex systems was given recently by Gao (1999). A so-called canonical dual transformation method and associated triality theory have been proposed for solving nonconvex/nonsmooth variational-boundary value problems. Compared with the traditional analytic methods and direct approaches, the main advantages of this canonical dual transformation method are (1) converting nonconvex/nonsmooth constrained variational problems into smooth unconstrained dual problems; (2) transforming certain fully nonlinear partial differential equations into algebraic systems; (3) providing powerful and efficient primal-dual alternative approaches. The aim of this article is to generalize the author’s previous results on nonconvex variational problems into distributed-parameter control systems. The rest of this paper is divided into four main sections. The next section set up the notation used in the paper. A general framework in fully nonlinear systems are discussed. Section 3 presents an extended Lagrangian critical point theorem and associated triality theory in general nonconvex dynamical systems. The critical points in fully nonlinear systems are classified. Section 4 is devoted mainly to the construction of dual action in fully nonlinear systems. The nice tri-duality proposed in static boundary value problems is generalized into control problems. Section 5 discusses the application in Duffing system. A bifurcation criterion is proposed which can be used for feedback controlling against chaotic vibrations.

2

Framework for Canonical Systems and Classification

Let U and U ∗ be two locally convex topological real linear spaces, placed in separating duality by a bilinear form h·, ·i : U × U ∗ → R. Let P : Us → R be a given functional, well-defined on a convex domain Us ⊂ U such that for any given u ∈ Us , P (u) is Gˆ ateaux differentiable. Thus, the Gˆ ateaux derivative DP of P at u ∈ Us is a mapping from Us into U ∗ . Let Us∗ ⊂ U ∗ be the range of the mapping DP : Us → U ∗ . If the relation u∗ = DP (u) is reversible on Us , then, for any given u∗ ∈ Us∗ , the classical Legendre conjugate functional P ∗ : Us∗ → R


93

of P (u) is defined by P ∗ (u∗ ) = hu(u∗ ), u∗ i − P (u(u∗ )). The conjugate pair (u, u∗ ) is called the Legendre duality pair on Us ×Us∗ ⊂ U ×U ∗ if and only if the equivelant relations (2.1)

u∗ = DP (u) ⇔ u = DP ∗ (u∗ ) ⇔ P (u) + P ∗ (u∗ ) = hu, u∗ i.

hold on Us × Us∗ . The following notations and definitions, used in Gao (1999), will be of convenience in nonconvex control problems. Definition 2.1. The set of functionals P : U → R which are either convex ˇ or concave is denoted by Γ(U ). In particular, let Γ(U) denote the subset of ˆ functionals P ∈ Γ(U ) which are convex and Γ(U ) the subset of P ∈ Γ(U ) which are concave. The canonical functional space ΓG (Us ) is a subset of functionals P ∈ Γ(Us ) which are Gˆ ateaux differentiable on Us ⊂ U , such that the relation u∗ = DP (u) is reversible for any given u ∈ Us . ♦ Clearly, if P ∈ ΓG (Us ) and Us∗ is the range of the mapping DP : Us → U ∗ , then the Legendre duality relations (2.1) hold on Us × Us∗ . Let (E, E ∗ ) be an another pair of locally convex topological real linear spaces paired in separating duality by the second bilinear form h· ; ·i : E ×E ∗ → R. The so-called geometrical operator Λ : U → E is a continuous, Gâteaux differentiable operator such that for any given u ∈ Ua ⊂ U, there exists an element ξ ∈ Ea ⊂ E satisfying the geometrical equation ξ = Λ(u). The directional derivative of ξ at u ¯ in the direction u ∈ U is then definded by (2.2)

δξ(¯ u; u) := lim

θ→0+

ξ(¯ u + θu) − ξ(¯ u) u)u, = Λt (¯ θ

where Λt (¯ u) = DΛ(¯ u) : U → E denotes the Gˆ ateaux derivative of the operator ∗ ∗ Λ at u ¯. For a given ξ ∈ E , GΛ (u) = hΛ(u) ; ξ ∗ i is a real-valued functional of u on U . Its Gˆ ateaux derivative at u ¯ ∈ U in the direction u ∈ U reads δGΛ (¯ u; u) = hΛt (¯ u)u ; ξ ∗ i = hu , Λ∗t (¯ u)ξ ∗ i, where Λ∗t (¯ u) : E ∗ → U ∗ is the adjoint operator of Λt associated with the two bilinear forms. Let V and V ∗ be the velocity and momentum spaces, respectively, placed in duality by the third bilinear form h∗ , ∗i : V ×V ∗ → R. For Newtonian systems,

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the kinetic energy K : V → R and its Legendre conjugate K ∗ : V ∗ → R are quadratic forms Z Z 1 2 1 −1 2 ∗ K(v) = ρv dΩ, K (p) = ρ p dΩ. Ω 2 Ω 2 Thus the canonical physical relations between V and V ∗ are linear: p = DK(v) = ρv ⇔ v = DK ∗ (p) = ρ−1 p. Let Va ⊂ V be a subspace defined by (2.3)

Va = {v ∈ V| v(x, 0) = v0 ∀x ∈ Ω}.

Finally, we let M be an admissible control space over Ωt . For any given µ ∈ M, we assume that there exists a Gâteaux differentiable functional Φµ : Ua × Ea ⊂ U × E → R, such that the total potential P (u; µ) of the system can be written as (2.4)

Pµ (u) = P (u; µ) = Φµ (u, Λ(u)),

and the total action of the system Z (2.5)

Πµ (u) =

tc

[K(u,t ) − Φµ (u, Λ(u))] dt

0

is well-defined on the feasible space Uk given by (2.6)

Uk = {u ∈ Ua | Λ(u) ∈ Ea , u,t ∈ Va }.

The following classification for distributed parameter control systems was originally introduced in nonlinear variational/boundary value problems by Gao (1998, 1999). Definition 2.2. Suppose that for the problem (P) given in (1), the associated total potential Pµ (u) is well-defined on its domain Us ⊂ U . If the geometrical operator Λ : U → E can be chosen in such a way that Pµ (u) = Φµ (u, Λ(u)), Φµ ∈ ΓG (Ua ) × ΓG (Ea ) and Us = {u ∈ Ua | Λ(u) ∈ Ea }. Then (1) the transformation {P ; Us } → {Φµ ; Ua × Ea } is called the canonical transformation, and Φµ : Ua × Ea → R is called the canonical functional associated with Λ; (2) the problem (P) is called geometrically nonlinear (or linear) if Λ : U → E is nonlinear (or linear); it is called physically nonlinear (resp. linear) if the duality mapping DΦµ : Ua × Ea → Ua∗ × Ea∗ is nonlinear (resp. linear); it is called fully nonlinear if it is both geometrically and physically nonlinear. ♦

Canonical Dual Control for Nonconvex Systems v∈ V d dt

6

u∈ U Λt + Λc = Λ

?

ξ∈ E

hv , pi

- V∗ 3 p −d dt

hu , u∗ i

? - U ∗ 3 u∗ 6Λ∗ = (Λ − Λ )∗ t

95

hξ ; ξ ∗ i

c

- E ∗ 3 ξ∗

Fig. 2. Framework in fully nonlinear Newtonian systems

The canonical transformation plays a fundamental role in duality theory of nonconvex systems. Clearly, if Φµ ∈ ΓG (Ua ) × ΓG (Ea ) is a canonical functional, the Gâteaux derivative DΦµ : Ua × Ea → Ua∗ × Ea∗ ⊂ U ∗ × E ∗ is a monotone mapping, i.e., the duality relations (2.7)

u∗ = Du Φµ (u, ξ),

ξ ∗ = Dξ Φµ (u, ξ)

are reversible between the paired spaces (Ua , Ua∗ ) and (Ea , Ea∗ ), where Du Φµ and Dξ Φµ denote the partial Gˆ ateaux derivatives of Φµ with respect to u and ξ, respectively. Thus, on Uk , the directional derivative of Pµ at u ¯ in the direction u ∈ Uk can be written as δPµ (¯ u; u) = hu , Du Φµ (¯ u, Λ(¯ u))i + hΛt (¯ u)u ; Dξ Φµ (¯ u, Λ(¯ u))i ∗ ∗ ∗ ¯ = hu , u ¯ i + hu ; Λ (¯ u)ξ i ∀u ∈ Uk . t

In terms of canonical variables, the governing equation (1) for the fully nonlinear problems can be written in the tri-canonical forms, namely, (2.8)

(1) geometrical equations: v = u,t , ξ = Λ(u), (2) physical relations: p = ρv, (u∗ , ξ ∗ ) = DΦµ (u, ξ), (3) balance equation: p,t + u∗ + Λ∗t (u)ξ ∗ = 0.

The framework for the fully nonlinear system is shown in Figure 2. Extensive illustrations of the canonical transformation and the tri-canonical forms in mathematical physics and variational analysis were given in the monograph by Gao (1999). In geometrically linear systems, where Λ : U → E is linear, we have Λ = Λt . For dynamical problems, if the total potential Pµ is convex, the total action

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associated with the problem (P) is a d.c. functional, i.e., the difference of convex functionals: Z tc Πµ (u) = [K(u,t ) − Pµ (u)] dt. 0

It was shown by Gao (1999) that the critical point of Πµ either minimizes or maximizes Πµ over the kinetically admissible space. The classical Hamiltonian associated with this d.c. functional Πµ is a convex functional on the phase space U × V ∗ , i.e. (2.9)

H(u, p) = K ∗ (p) + Pµ (u),

The classical canonical forms for convex Hamilton systems are well-known d u = Dp H(u, p), dt

−

d p = Du H(u, p). dt

Furthermore, if Φµ (u, ξ) = 12 hξ ; Cξi − hu , µi is a quadratic quadratic functional, where C : E → E ∗ is a linear operator, then the governing equations for linear system can be written as ρ¯ u,tt + Λ∗ CΛu = µ. For conservative systems, the operator Λ∗ CΛ is usually symmetric. In geometrically nonlinear systems, Λ 6= Λt , and the total potential Pµ (u) is usually a nonconvex functional. In this case, we have the following operator decomposition (2.10)

Λ(u) = Λt (u)u + Λc (u),

where Λc : U → E is called the complementary operator of the Gâteaux derivative operator Λt . By this decomposition, we have (2.11)

hΛ(¯ u) ; ξ¯∗ i = h¯ u , Λ∗t (¯ u)ξ¯∗ i − G(¯ u, ξ¯∗ )

where G : U × E ∗ → R is so-called complementary gap functional, defined by (2.12)

G(u, ξ ∗ ) = h−Λc (u) ; ξ ∗ i : U × E ∗ → R.

This functional was first introduced by Gao and Strang (1989) in finite deformation theory, which plays a key role in nonconvex variational problems. As a typical example in nonconvex dynamical systems, let us consider the post-buckling dynamical beam model (1.10) discussed in section 1. For a given feasible space Uk , we consider the following nonconvex variational problem over the domain Ωt = (0, `) × (0, tc ) Z 1 2 1 1 2 2 (2.13) Πµ (u) = ρu,t − a( u,x − µ) + uf dx dt → sta ∀u ∈ Uk , 2 2 Ωt 2


97

where a, µ are given positive constants. This nonconvex problem also appears very often in phase transitions and hysteresis. First, we let Λ = d/ dx be a linear operator, and Pµ (u) = Wµ (Λu) − Fµ (u) with Z ` Z ` 1 1 2 2 Wµ (ε) = uf dx, a( ε − µ) dx, F (u) = 2 0 2 0 Thus, Wµ (ε) is the so-called van der Waals’ double-well function of the linear “strain” ε = u,x . Since Wµ (ε) is not a canonical functional, the constitutive equation ε∗ = DWµ (ε) is not one-to-one. Thus, the Legendre conjugate of Wµ (ε) does not have a simple algebraic expression. The Fenchel conjugate Wµ∗ (ε∗ ) of the double-well energy Wµ (ε), defined by Wµ∗ (ε∗ ) = sup{hε ; ε∗ i − Wµ (ε)}, ε

is always a convex, lower semi-continuous functional. However, the well-known Fenchel-Young inequality Wµ (u,x ) ≥ hu,x ; ε∗ i − Wµ∗ (ε∗ ) leads to a so-called duality gap between the primal problem and the FenchelRockafellar dual problem (see Gao, 1999). This nonzero duality gap indicates that the well-established Fenchel-Rockafellar duality theory can only be used for solving convex variational problems. ¿From the theory of continuum mechanics we know that in finite deformation problems, ε = u,x is not a strain measure (it does not satisfy the axiom of material frame-indifference (cf. e.g., Gao, 1999). In order to recover this duality gap, we need to choose a suitable geometrical operator Λ, say, Λ(u) = 12 u2,x − µ, so that the nonconvex problem (2.13) can be put in our framework. In continuum mechanics, this quadratic measure ξ = Λ(u) is a Cauchy-Green type strain. Thus, in terms of u and ξ, Φµ (u, ξ) = Wµ (ξ) − Fµ (u) = 12 hξ ; aξi − hu , f i is a canonical functional. The Legendre conjugate of the quadratic functional Wµ (ξ) = 12 hξ ; aξi is simply defined by W ∗ (ξ ∗ ) = 12 ha−1 ξ ∗ ; ξ ∗ i. The operator decomposition (2.10) for this quadratic operator reads 1 Λ(u) = Λt (u)u + Λc (u), Λt (u)u = u,x u,x , Λc (u) = − u2,x − µ. 2 The complementary gap functional associated with this quadratic operator is a quadratic functional of u ∗

∗

Z

`

G(u, ξ ) = h−Λc (u) ; ξ i = 0

1 2 ∗ u ξ dx. 2 ,x

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For homogeneous boundary conditions, we have ∗

Z

hΛt (u)u ; ξ i =

`

∗

Z

u,x u,x ξ dx = −

0

`

u(u,x ξ ∗ ),x dx = hu , Λ∗t (u)ξ ∗ i,

0

which leads to the adjoint operator Λ∗t of Λt . Thus, the tri-canonical equations for this nonconvex problem can be listed as the following. 1 v = u,t , ξ = au2,x − µ, 2 p = ρv, ξ ∗ = DWµ (ξ) = aξ, u∗ = DFµ (u) = f p,t = −Λ∗t (u)ξ ∗ + u∗ = (u,x ξ ∗ ),x + f. Since the geometrical operator Λ is nonlinear, and the canonical constitutive equations are linear, the nonconvex problem (2.13) is a geometrically nonlinear system.

3

Extended Lagrangian and Triality Theory

The triality theory in nonconvex problems was originally proposed by the author (Gao, 1996, 1997, 1999, 2000) in static finite deformation theory and global optimization. In this section, we will generalize this interesting result into fully nonlinear dynamical systems. We assume that for a given fully nonlinear system, there exists a Gˆ ateaux differentiable operator Λ : Ua → Ea such that the total potential of the system can be written as Pµ (u) = Wµ (Λ(u)) − Fµ (u),

(3.1)

ˇ G (Ea ) is a convex canonical functional, while Fµ : Ua → R is a where Wµ ∈ Γ linear functional. Thus, the primal problem (P) can be reformulated as the following. Problem 1 (Primal Distributed-Parameter Control Problem). For any given primal feasible space Uk = {u ∈ Ua | u,t ∈ Va , Λ(u) ∈ Ea } and the final state (¯ uc (x), v¯c (x)), find the control field µ(x, t) ∈ M such that the solution u ¯(x, t) of the variational problem Z (3.2)

(P) :

tc

Πµ (u) =

[K(u,t ) − Wµ (Λ(u)) + Fµ (u)] dt → sta ∀u ∈ Uk

0

satisfying the controllability condition (¯ u(x, tc ), u ¯,t (x, tc )) = (¯ uc (x), v¯c (x)) ∀x ∈ Ω.


99

It is easy to check that the critical point condition DΠµ (¯ u) = 0 leads to the the canonical governing equation (3.3)

ρ¯ u,tt = DFµ (¯ u) − Λ∗t (¯ u)DWµ (Λ(¯ u)).

By the Legendre-Fenchel transformation, the conjugate of Wµ (ξ) is defined by Wµ∗ (ξ ∗ ) = sup{hξ ; ξ ∗ i − Wµ (ξ)}. ξ∈E

Since Wµ : Ea → R is a convex canonical functional, Wµ∗ (ξ ∗ ) is well-defined on the range Ea∗ of the duality mapping DWµ∗ : Ea → E ∗ , the Legendre duality relation ξ ∗ = DWµ (ξ) ⇔ ξ = DWµ∗ (ξ ∗ ) ⇔ Wµ (ξ) + Wµ∗ (ξ) = hξ ; ξ ∗ i holds on Ea × Ea∗ . Moreover, we have Wµ∗∗ (ξ) = Wµ (ξ) for all ξ ∈ Ea . Let Z = U × V ∗ × E ∗ be the so-called extended canonical phase space. Definition 3.1. Suppose that for a given problem (P), there exists a Gˆ ateaux differentiable operator Λ : U → E and canonical functionals Wµ ∈ Γ(E), Fµ ∈ Γ(U ) such that Pµ (u) = Wµ (Λ(u)) − Fµ (u). Then (1) the functional Hµ : Z → R defined by (3.4)

Hµ (u, p, ξ ∗ ) = K ∗ (p) − Wµ∗ (ξ ∗ ) + Fµ (u) ∈ Γ(U ) × Γ(V ∗ ) × Γ(E ∗ )

is called the extended canonical Hamiltonian density associated with Πµ ; (2) the functional Lµ : Z → R definded by (3.5)

Lµ (u, p, ξ ∗ ) = hu,t , pi − hΛ(u) ; ξ ∗ i − Hµ (u, p, ξ ∗ )

is called the extended Lagrangian density of (P) associated with Λ; (3) the functional Ξµ : Z → R definded by Z tc ∗ Ξµ (u, p, ξ ) = (3.6) Lµ (u, p, ξ ∗ ) dt 0

is called the extended Lagrangian form of (P). It is called the canonical Lagrangian form if Ξµ ∈ Γ(U) × Γ(V ∗ ) × Γ(E ∗ ). ♦ ∗ ¯ A point (¯ u, p¯, ξ ) ∈ Z is said to be a critical point of Ξµ if Ξµ is Gâteauxdifferentiable at (¯ u, p¯, ξ¯∗ ) and DΞµ (¯ u, p¯, ξ¯∗ ) = 0. It is easy to find out that the criticality condition DΞµ (¯ u, p¯, ξ¯∗ ) = 0 leads to the following canonical Lagrange equations ¯,t = DK ∗ (¯ p), Λ(¯ u) = Dξ ∗ Wµ∗ (ξ¯∗ ), u DΞµ (¯ u, p¯, ξ¯∗ ) = 0 ⇒ (3.7) ∗ ¯ p¯,t = DFµ (¯ u) − Λt (¯ u)ξ ∗ .

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Since Wµ and Fµ are canonical functionals, we know that, by the Legendre duality theory, any critical point of Ξ µ solves the variational problem (P). Since Fµ (u) : Ua → R is a linear functional, by the Riesz representation theory, there exists an element u ¯∗ (µ) ∈ U ∗ such that Fµ (u) = hu , u ¯∗ (µ)i. Thus, the extended Lagrangian associated with (P) can be written as (3.8) Ξµ (u, p, ξ ∗ ) =

Z

tc

[hu,t , pi − hΛ(u) ; ξ ∗ i − K ∗ (p) + W ∗ (ξ ∗ ) + hu , u ¯∗ (µ)i] dt.

0

Note that Ξµ : Va∗ × Ea∗ → R is a saddle functional for any given u ∈ Ua , we have always the equality inf sup Ξµ (u, p, ξ ∗ ) = sup ∗inf ∗ Ξµ (u, p, ξ ∗ ) ∀u ∈ Ua .

(3.9)

ξ ∗ ∈Ea∗ p∈V ∗

p∈Va∗ ξ ∈Ea

a

However, for any given (p, ξ ∗ ) ∈ Va∗ × Ea∗ , the convexity of Ξµ (·, p, ξ ∗ ) → R depends on the operator Λ. Let Lc ⊂ Za = Ua × Va∗ × Ea∗ be a critical point set of Ξµ , i.e. Lc = {(¯ u, p¯, ξ¯∗ ) ∈ Za | δΞ(¯ u, p¯, ξ¯∗ ; u, p, ξ ∗ ) = 0 ∀(u, p, ξ ∗ ) ∈ Za }. For any given critical point (¯ u, p¯, ξ¯∗ ) ∈ Lc , we let Zr = Ur × Vr∗ × Er∗ ⊂ Za be its neighborhood such that on Zr , (¯ u, p¯, ξ¯∗ ) is the only critical point of Ξµ . The following extremum results are of fundamental importance in the stability analysis of nonlinear dynamical systems. Theorem 3.1 (Triality Theorem). Suppose that (¯ u, p¯, ξ¯∗ ) ∈ Lc , and Zr ∗ ¯ is a neighborhood of (¯ u, p¯, ξ ). If hΛ(u) ; ξ¯∗ i is concave on Ur , then on Zr , (3.10)

Ξµ (¯ u, p¯, ξ¯∗ ) = min max min Ξµ (u, p, ξ ∗ ) = max min min Ξµ (u, p, ξ ∗ ). ∗ ∗ u

p

p

ξ

u

ξ

However, if hΛ(u) ; ξ¯∗ i is convex on Ur , then on Zr we have either Ξµ (¯ u, p¯, ξ¯∗ ) = min max min Ξµ (u, p, ξ ∗ ) = min max min Ξµ (u, p, ξ ∗ ) ∗ ∗ u

(3.11)

p

p

ξ

u

ξ

∗

= min max Ξµ (u, p, ξ ) = min max Ξµ (u, p, ξ ∗ ). ∗ ∗ ξ ,u

p

p,ξ

u

or Ξµ (¯ u, p¯, ξ¯∗ ) = max min max Ξµ (u, p, ξ ∗ ) = max min max Ξµ (u, p, ξ ∗ ) ∗ ∗ u

(3.12)

ξ

p

p

∗

ξ

u

= min max Ξµ (u, p, ξ ) = max min Ξµ (u, p, ξ ∗ ). ∗ ∗ ξ

u,p

u,p

ξ


101

ˇ a∗ ), K ∗ ∈ Γ(V ˇ a∗ ), if hΛ(u) ; ξ¯∗ i is concave on Ur , Proof. Since Wµ∗ ∈ Γ(E ˇ r ) × Γ(V ˆ a∗ ) is a saddle functional. Thus the then for the given ξ¯∗ , Ξµ ∈ Γ(U equality (3.10) follows from the saddle-Lagrangian duality theorem (cf. e.g., Gao, 1999). However, if hΛ(u) ; ξ¯∗ i is convex on Ur , then for any given ξ ∗ ∈ Er∗ , ˆ r ) × Γ(V ˆ ∗ ) is a super-critical functional (see the extended Lagrangian Ξµ ∈ Γ(U a Gao, 1999). By the super-Lagrangian duality theorem proved in Gao (1999), we have either (3.11) or (3.12).

4

Dual Action and Tri-Duality Theory

The goal of this section is to develop a dual approach for solving the distributed parameter control problem (P). For any given u ∈ Uk , the extended Lagrangian density Ξµ (u, p, ξ ∗ ) is a saddle functional on V ∗ × E ∗ , and we have Πµ (u) = sup ∗inf ∗ Ξµ (u, p, ξ ∗ ) ∀u ∈ Uk .

(4.1)

p∈V ∗ ξ ∈E

On the other hand, the dual action Πdµ : Va∗ × Ea∗ → R can be defined by Πdµ (p, ξ ∗ ) = sta{Ξµ (u, p, ξ ∗ )| ∀u ∈ Ua } Z tc = FµΛ (p, ξ ∗ ) − (4.2) [K ∗ (p) − Wµ∗ (ξ ∗ )] dt, ∀(p, ξ ∗ ) ∈ Va∗ × Ea∗ . 0

where FµΛ (p, ξ ∗ ) is the so-called Λ-dual functional of Fµ (u) defined by (4.3)

FµΛ (p, ξ ∗ )

Z = sta

u∈Ua

tc

[hu,t , pi − hΛ(u) ; ξ ∗ i + Fµ (u)] dt.

0

Since Fµ (u) = hu , u ¯∗ (µ)i is a linear functional, for any given (p, ξ ∗ ) ∈ Va∗ × Ea∗ and the applied control µ ∈ M, the solution u ¯ of this stationary problem (4.3) satisfies the balance equation (4.4)

p,t + Λ∗t (¯ u)ξ ∗ = u ¯∗ (µ) in Ωt .

For geometrically linear systems, where Λ is a linear operator, we have (4.5)

∗ ∗ c FµΛ (p, ξ ∗ ) = up|t=t ¯∗ (µ). t=0 , s.t. Λ ξ + p,t = u

In this case, (4.6)

Πdµ (p, ξ ∗ )

Z =

c up|t=t t=0

+

tc

[Wµ∗ (ξ ∗ ) − K ∗ (p)] dt

0

is the classical complementary action in linear engineering dynamical systems (see Tabarrok and Rimrott, 1994) defined on the dual feasible space Ts = {(p, ξ ∗ ) ∈ Va × Ea∗ | p,t + Λ∗ ξ ∗ = u ¯∗ (µ)}.

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In fully nonlinear systems, we let Ts ⊂ Va∗ × Ea∗ be a subspace such that for any given (p, ξ ∗ ) ∈ Ts , the critical point u ¯ can be determined by (4.4) as ∗ d u ¯ = u ¯(p, ξ ) and the dual action Πµ is well defined by (4.2). Thus, by the operator decomposition Λ = Λt + Λc , we have Z tc t=tc Λ ∗ Fµ (p, ξ ) = up|t=0 + (4.7) Gd (p, ξ ∗ ) dt, s.t. Λ∗t (¯ u)ξ ∗ + p,t = u∗ (µ), 0

Gd (p, ξ ∗ )

where = h−Λc (¯ u) ; ξ ∗ i is the so-called pure complementary gap functional. Then, the problem dual to the primal control problem (P) can be proposed as the following. Problem 2 (Dual Distributed-Parameter Control Problem). For a given dual feasible space Ts and the final state (uc (x), vc (x)), find the control field µ(x, t) ∈ M such that the dual solution (¯ p(x, t), ξ¯∗ (x, t)) of the dual variational problem (4.8)

(P d ) :

Πdµ (p, ξ ∗ ) → sta ∀(p, ξ ∗ ) ∈ Ts

and the associated state u ¯(x, t) satisfying the controllability condition (4.9)

(¯ u(x, tc ), ρ−1 p¯(x, tc )) = (uc (x), vc (x)) ∀x ∈ Ω.

The following lemma plays a key role in duality theory for nonlinear dynamical systems. Lemma 4.1. Let Ξµ (u, p, ξ ∗ ) be a given extended Lagrangian associated with (P) and Πdµ (p, ξ ∗ ) the dual action defined by (4.2). Suppose that Zr = Ur × Vr∗ × Er∗ is an open subset of Za and (¯ u, p¯, ξ¯∗ ) ∈ Zr is a critical point of Ξµ on Zr , Πµ is Gˆ ateaux differentiable at u ¯, and Πdµ is Gˆ ateaux differentiable ∗ d ∗ ¯ ¯ at (¯ p, ξ ). Then DΠµ (¯ u) = 0, DΠµ (¯ p, ξ ) = 0, and Πµ (¯ u) = Ξµ (¯ u, p¯, ξ¯∗ ) = Πdµ (¯ p, ξ¯∗ ).

(4.10)

The proof of this lemma can be found in Gao (1998) in parametrical variational analysis. Lemma 4 shows that the critical points of the extended Lagrangian are also the critical points for both the primal and dual variational problems. Theorem 4.1 (Tri-Duality Theorem). Suppose that (¯ u, p¯, ξ¯∗ ) ∈ Lc is ∗ ∗ a critical point of Ξµ and Zr = Ur × Vr × Er is a neighborhood of (¯ u, p¯, ξ¯∗ ) such that Vr∗ × Er∗ ⊂ Ts . If hΛ(u) ; ξ¯∗ i is concave on Ur , then (4.11)

Πµ (¯ u) = min Πµ (u) u∈Ur

iff

Πdµ (¯ p, ξ¯∗ ) = max∗ min Πdµ (p, ξ ∗ ). ∗ ∗ p∈Vr ξ ∈Er

However, if hΛ(u) ; ξ¯∗ i is convex on Ur , then (4.12)

Πµ (¯ u) = min Πµ (u)

iff

Πdµ (¯ p, ξ¯∗ ) =

(4.13)

Πµ (¯ u) = max Πµ (u)

iff

Πdµ (¯ p, ξ¯∗ ) = max∗ min Πdµ (p, ξ ∗ ). ∗ ∗

u∈Ur

u∈Ur

min Πdµ (p, ξ ∗ );

(p,ξ ∗ )∈Ts

p∈Vr ξ ∈Er


103

Proof. This theorem can be proved by combining Lemma 1 and the triality theorem.

5

Feedback Control Against Chaotic Duffing System

As a typical example, let us consider the very simple nonconvex dynamical problem over the time domain I = (0, tc ) Z 1 1 0 Πµ (u) = [ρu 2 − a( u2 − µo )2 + µu] dt → sta ∀u ∈ Uk . (5.1) 2 2 I The kinematically admissible space Uk for the initial-value problem of this onedimensional dynamical system is given simply as Uk = {u ∈ L4 (0, tc )| u0 ∈ L2 (0, tc ), u(0) = u0 , u0 (0) = v0 }. The criticality condition for Πµ leads to the well-known Duffing equation (5.2)

1 ρu00 = au(µo − u2 ) + µ(t), ∀t ∈ I, u ∈ Uk . 2

In terms of the nonlinear canonical measure ξ = Λ(u) = 12 u2 , the energy density Wµ (ξ) and its conjugate Wµ∗ (ς) are convex functions: 1 1 2 Wµ (ξ) = a(ξ − µo )2 , Wµ∗ (ς) = ς + µo ς. 2 2a The extended Lagrangian for this nonconvex system is Z Z 1 2 1 2 1 2 0 Ξµ (u, p, ς) = pu − ς( u − µo ) − p + ς dt + µu dt. (5.3) 2 2ρ 2a I I The criticality condition Du Ξµ (¯ u, p, ς) = 0 leads to the equilibrium equation p0 + u ¯ς = µ ∀t ∈ I. Clearly, the critical point u ¯ = (µ − p0 )/ς is well-defined for any nonzero ς. Thus, the dual feasible space can be defined as p(0) = ρv0 , −µo a ≤ ς(t) < +∞, 1 Ts = (p, ς) ∈ C (I) . ς(t) 6= 0 ∀t ∈ I, ς(0) = a( 12 u20 − µo ) Substituting u ¯ = (µ − p0 )/ς into Ξdµ , the dual action is obtained as Πdµ (p, ς) = sta Ξµ (u, p, ς) u∈Ua

(5.4)

Z

= p(tc )u(tc ) − ρv0 u0 +

[ I

1 2 (p0 − µ)2 1 ς + µo ς + − p2 ] dt, 2a 2ς 2ρ

104

Gao

which is well defined on Ts . The criticality condition for Πdµ leads to the dual Duffing system in the time domain I ⊂ R 0 1 0 1 (5.5) (p − µ) + p = 0, ς ρ 1 1 ς 2 ( ς + µo ) = (µ − p0 )2 . a 2

(5.6)

This system consists of the so-called differential-algebraic equations (DAE’s), which arise naturally in many applications (cf., e.g., Brenan et al, 1996; Beardmore and Song, 1998). Although the numerical solution of these types of systems has been the subject of intense research activity in the past few years, the solvability of each problem depends mainly on the so-called index of the system. Clearly, the algebraic equation (5.6) has zero solution ς = 0 if and only if σ = (µ − p0 ) = 0. Otherwise, for any nonzero σ(t) = µ(t) − p0 (t), the algebraic equation (5.6) has at most three real roots ςi (t) (i = 1, 2, 3), each of them leads to the state solution ui (t) = (µ(t) − p0i (t))/ςi (t). Theorem 5.1 (Stability and Bifurcation Criteria). For a given parameter µo > 0, initial data (u0 , v0 ) and the input control µ(t), if at a certain time period Is ⊂ I = (0, tc ), (5.7)

3 µc (t) = 2

µ(t) − p0 (t) a

2/3 > µ o , t ∈ Is

then the Duffing system possesses only one solution set (¯ u(t), p¯(t), ς¯(t)) satisfying ς¯(t) > 0 ∀t ∈ Is , and over the period Is , (5.8)

Πµ (¯ u) = min Πµ (u) iff Πdµ (¯ p, ς¯) = min Πdµ (p, ς),

(5.9)

Πµ (¯ u) = max Πµ (u) iff Πdµ (¯ p, ς¯) = max min Πdµ (p, ς). p

ς

However, if at a certain time period Ib ⊂ I = (0, tc ) we have µc (t) < µo , then, the system possesses three sets of different solutions (¯ ui , p¯i (t), ς¯i (t)), i = 1, 2, 3. In the case that the three solutions ςi (t) are in the following ordering (5.10)

−aµo ≤ ς¯3 (t) ≤ ς¯2 (t) ≤ 0 ≤ ς¯1 (t) ∀t ∈ Ib ,

then over the period Ib , the solution set (¯ u1 (t), p¯1 (t), ς¯1 (t)) satisfies either (5.8) or (5.9); while the solution sets (¯ ui (t), p¯i (t), ς¯i (t)) (i = 2, 3) satisfy (5.11)

Πµ (¯ ui ) = min Πµ (u) = max min Πdµ (p, ς) = Πdµ (¯ pi , ς¯i ) i = 2, 3. u

p

ς


105

This theorem can be proved by combining the multi-solution theorem given by Gao (1999, Theorem 3.4.4) and the triality theorem. Remark. By Theorem 3.4.4 proved by the author (Gao, 1999), for any given continuous function σ(t), if ς¯i (t) (i = 1, 2, 3) are the three solutions of the dual Euler-Lagrange equation (5.6) in the order of (5.10), then the associated u ¯1 (t) is a global minimizer of the total potential Z Z 1 1 2 2 Pµ (u) = a( u − µo ) dt − σ(t)u dt; 2 I 2 I while u ¯2 (t) is a local minimizer of Pµ and u ¯3 (t) is a local maximizer of Pµ . In algebraic geometry, the dual Euler-Lagrange equation (5.6) is the socalled singular algebraic curve in (ς, σ)-space, i.e. ς = 0 is on the curve (see Silverman & Tate, 1992, p. 99). With a change of variables, the singular cubic curve (5.6) can be given by the well-known Weierstrass equation y 2 = x3 + αx2 + βx + γ, where α, β, γ ∈ R are constants. If we let Cns be a set consisting of nonsingular points on the curve, then Cns is an Abelian group. This fact in algebraic geometry is very important in understanding the stability of the nonconvex dynamical systems. Actually, from Figure 3 we can see clearly that for a given input control, if µc (t) < µo , the cubic algebraic equation (5.6) possesses three different real solutions for ς(t). The two negative solutions ς¯(t) are the sources that lead to the chaotic motion of the system. Thus, the inequality (5.7) provides a bifurcation (or chaotic) criterion for the Duffing system. Fig. 3 also shows that if the continuous function σ(t) = µ(t) − p0 (t) is one-signed on certain time interval Ib ⊂ I = (0, tc ), each root ς¯(t) of (5.6) is also one-signed on Ii . σ

µo < µc

0.4

µo = µc 0.2

-1

-0.8 -0.6 -0.4 -0.2

µo > µc 0.2

0.4

ς

-0.2

-0.4

Fig. 3. Singular algebraic curve for the dual Duffing equation (5.6)

Theoretically speaking, for the same initial conditions, the Duffing equation (5.2) and its dual system (5.5-5.6) should have the same solution set. Numerically, the primal and dual Duffing problems give complementary bounding

106

Gao

approaches to the real solution. For the given data a = 1, µo = 1.5, u0 = 2, v0 = 1.4 and µ = 0, Figures 4 and 5 show the numerical primal (solid line) and dual (dashed line) solutions. From the dual trajectories in the dual phase space ς-p-p,t (Fig. 5(c-d)) we can see that at the point ς3 (t) = −aµo , if the function σ(t) = µ(t) − p,t (t) changes its sign, the state u(t) crosses the origin goes to another potential well in the phase space Z = U × V ∗ , and the bifurcation is then occurred. Thus, based on the canonical dual transformation method and theorems developed in this paper, the dual feedback control against the chaotic vibration of the Duffing system can be suggested as the following. 1. Periodic vibration on the whole phase plane. Choosing the controller µ(t) such that the function σ(t) = µ(t) − p0 (t) changes its sign at the point ς¯3 (t) = −aµo . 2. Unilateral vibrations on half phase planes (either u(t) > 0 or u(t) < 0). There are two methods: (1) choosing the controller µ(t) such that the function σ(t) = µ(t) − p0 (t) does not change its sign at the point ς¯3 (t) = −aµo ; (2) choosing µ(t) such that either (5.12)

µ(t) > p0 (t) + a(2µo /3)3

1/2

∀t ∈ I,

or (5.13)

µ(t) < p0 (t) − a(2µo /3)3

1/2

∀t ∈ I.

Detailed study on the exact controllability and stability for the Duffing system will be given in other papers (cf. e.g., Gao, 2000d).

6

Concluding Remarks

The concept of duality is one of the most successful ideas in modern mathematics and science. The inner beauty of duality theory owes much to the fact that the nature was originally created in a duality way. By the fact that the canonical physical variables appear always in pairs, the canonical dual transformation method can be used to solve many problems in natural systems. The associated extended Lagrange duality and triality theories have profound computational impacts. For any given nonlinear problem, as long as there exists a geometrical operator Λ such that the tri-canonical forms can be characterized correctly, the canonical dual transformation method and the associated triality principles can be used to establish nice theories and to develop powerful alternative algorithms for robust feedback control of chaotic systems. For static three-dimensional finite deformation problems, a general analytic solution form and associated extremality theory have been proposed (Gao, 1999, 1999b). A general canonical dual transformation method for solving nonsmooth global optimization is given recently (Gao, 2000c). In general n-dimensional distributed parameter systems, the dual algebraic equation (5.6) will be a tensor equation

107

Canonical Dual Control for Nonconvex Systems (a) Primal and dual solutions

(b) Primal and dual actions

3

1.5

2

1

1

0.5

0

0

−1

−0.5

−2

−1

−3

0

10

20

30

40

(c) Trajectories in phase space u−p

−1.5 −4

−2

0

2

4

(d) Primal and dual actions in phase space u−p

2 2 1 0

0

−2 2

−1

5

0 −2 −4

−2

0

2

4

0 −2 −5

Fig. 4. Primal and dual solutions in primal phase space

and the stability of the nonconvex system will depend on the eigenvalues of symmetrical canonical stress tensor field ς(x, t) (see Gao, 2000d). The triality theory can be used for studying the controllability, observability and stability of distributed parameter control problems. Acknowledgement. The author would like to thank the referee for the valuable suggestions and comments.

108

Gao (a) Trajectores in phase space ς−p,t

(b) Trajectores in ς−p space

4

2

2

1

0

0

−2

−1

−4 −2

−1

0

1

2

−2 −2

−1

0

1

2

(d) Dual solution in phase space ς−p−p,t

(c) Dual solution in dual phase space p−p,t 4 5 2 0

0

−5 2

−2

2

0 −4 −2

−1

0

1

2

0 −2 −2

Fig. 5. Duffing solutions in dual phase spaces

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Carleman estimate for a parabolic equation in a Sobolev space of negative order and their applications

O. Yu. Imanuvilov, Iowa State University, Ames IA 50011-2064 USA. E-mail: [email protected] M. Yamamoto, The University of Tokyo, Tokyo, Japan. E-mail: [email protected] Abstract We obtain a Carleman estimate for a second order parabolic equation when the coefficients are not bounded and the right hand side is taken in the Sobolev space L2 (0, T ; H −1(Ω)) and we apply it to • the global exact null-controllability of a semilinear parabolic equation whose semilinear term contains also derivatives of first order • conditional stability in continuation of the solution • inverse problem of determining f ∈ H −` (Ω) with ` < 1 at the right hand side.

1

Introduction

In this paper we first formulate Carleman estimates for a parabolic equation in a Sobolev spaces of negative order and apply them in order to establish the exact null-controllability for a semilinear parabolic equation, conditional stability in the continuation and the uniqueness in determining the source term. A weighted estimate for a partial differential equation Ly = g, which is called a Carleman estimate, was used by Carleman [3] for proving the unique continuation for an elliptic equation, and since then, it has been recognized as an important technique in the theory of partial differential equations. In particular, the Carleman estimate is very helpful for • the unique continuation (e.g. Hörmander [11], Isakov [16], [17] and the references therein). • observability inequality and exact controllability (Cheng, Isakov, Yamamoto and Zhou [5], Isakov and Yamamoto [18], Kazemi and Klibanov 113

114

O. Yu. Imanuvilov and M. Yamamoto [19], Lasiecka and Triggiani [22], Lasiecka, Triggiani and Zhang [23], Tataru [32]). • inverse problem of determining coefficients or non-homogeneous terms in partial differential equations (Bukhgeim and Klibanov [2], Imanuvilov and Yamamoto [13], Isakov [15], [17], Kha˘ıdarov [20] Klibanov [21], Yamamoto [34]).

Moreover for Carleman estimates and the applications to the hyperbolic equation, the reader can consult Lavrent’ev, Romanov and Shishat·ski˘ı[24], Ruiz [28], and for the parabolic case, Chae, Imanuvilov and Kim [4], Fursikov and Imanuvilov [10], Imanuvilov [12], Saut and Scheurer [29], Sogge [30], Tataru [33]. Except for Ruiz [28], those papers take L2 -spaces as the space of g = Ly (the right hand side of the partial differential equation under consideration), and such L2 -spaces make us assume more regularity in the applications. On the other hand, by our Carleman estimate, we can reduce the regularity assumptions on the right hand side of the parabolic equation. Because of the page limitation, however, we will omit the proofs of the Carleman estimate and the exact controllability (see Imanuvilov and Yamamoto [14]), and concentrate on the conditional stability in the continuation and the inverse problem This paper is composed of five sections: Section 2. Carleman estimate Section 3. Application to the exact null-controllability Section 4. Application to the continuation problem Section 5. Application to the inverse problem.

Acknowledgements The authors thank the referee for useful comments. The second named author is partly supported by Sanwa Systems Developement Co. Ltd. (Tokyo, Japan).

2

Carleman estimate

Let (t, x) ∈ Q ≡ (0, T ) × Ω, Σ ≡ (0, T ) × ∂Ω, where Ω ⊂ Rn is a connected bounded domain whose boundary ∂Ω is of class C 2 , ν(x) is the outward unit ∂ ∂ normal to ∂Ω, T ∈ (0, +∞) is an arbitrary moment of time, ∂t = ∂t , ∂i = ∂x , i 0

1 ≤ i ≤ n, Dβ = D β0 D β = ∂tβ0 ∂1β1 . . . ∂nβn , β = (β0 , β 0 ) = (β0 , β1 , . . . , βn ), |β| = 2β0 + β1 + · · · + βn , ∇ = (∂1 , ...., ∂n ). Let ω ⊂ Ω be an arbitrarily fixed subdomain and let us set Qω = (0, T ) × ω. Throughout this paper, Wpµ (Ω) = W µ,p (Ω), W2µ (Ω) = H µ (Ω), W0µ,p (Ω), p ≥ 1, µ ≥ 0, H0µ (Ω), µ ≥ 0, denote usual Sobolev spaces (e.g. Adams [1]), and we set L2 (Ω) = W20 (Ω). 0 Moreover Wp−µ (Ω) = (W0µ,p (Ω))0 , H −µ (Ω) = (H0µ (Ω))0 : the dual, where 1 1 p + p0 = 1.

115

Carleman estimate and applications Let us consider a parabolic equation (2.1) Ly ≡ ∂t y −

n X

∂i (aij (x)∂j y) +

i,j=1

n X

∂i (bi (t, x)y) + c(t, x)y = g

in

Q,

i=1

with the boundary condition

y Σ = 0.

(2.2)

Assume that  1 1 ≤ i, j ≤ n,   aij ∈ W∞ (Ω), aij = aji , ∞ r bi ∈ L (0, T ; L (Ω)), r > 2n, 1 ≤ i ≤ n, (2.3) n   c ∈ L∞ (0, T ; Wr−µ (Ω)), 0 ≤ µ < 1 , r1 > max 2

1

o

2n 3−2µ , 1

    , 

and the coefficients aij satisfy the uniform ellipticity: There exists β > 0 such that n X (2.4) aij (x)ζi ζj ≥ β|ζ|2 , ζ = (ζ1 , ...., ζn ) ∈ Rn , (t, x) ∈ Q. i,j=1

For the weak solution, we can show Lemma 2.1. Let y0 ∈ L2 (Ω), g ∈ L2 (0, T ; H −1 (Ω)) and let us assume the conditions (2.3) and (2.4). Then there exists a solution y ∈ L2 (0, T ; H01 (Ω)) to (2.1) and (2.2) with y(0, ·) = y0 . Moreover the solution is unique in L2 (Q), and we have an estimate: (2.5)

kykL2 (0,T ;H01 (Ω))∩C([0,T ];L2 (Ω)) ≤ C0 (ky(0, ·)kL2 (Ω) + kgkL2 (0,T ;H −1 (Ω)) ),

where the constant C0 > 0 depends continuously only on n X

kaij kW∞ 1 (Ω) +

i,j=1

n X

kbi kL∞ (0,T ;Lr (Ω)) + kckL∞ (0,T ;Wr−µ (Ω)) .

i=1

1

This lemma can be proved by a usual energy method and for completeness we will give a sketch of the proof in Appendix A. In view of Lemma 2.1, in the succeeding arguments, we can assume the smoothness of solutions which admit calculations such as integration by parts. More precisely, we can use a usual density argument, i.e., we can do everything for sufficiently smooth solutions, and then pass to the limit in the final inequality. In order to formulate our Carleman estimate, we need a special weight function. Lemma 2.2 ([4], [12]). Let ω0 ⊂ ω be an arbitrarily fixed subdomain of Ω such that ω0 ⊂ ω. Then there exists a function ψ ∈ C 2 (Ω) such that (2.6)

ψ(x) > 0, x ∈ Ω,

ψ|∂Ω = 0,

|∇ψ(x)| > 0, x ∈ Ω \ ω0 .

116

O. Yu. Imanuvilov and M. Yamamoto

Now, using the function ψ constructed in Lemma 2.2, we introduce weight functions: (2.7)

ϕ(t, x) = exp(λψ(x)) t(T −t) , 1 ϕ0 (t) = t(T −t)

α(t, x) =

exp(λψ(x))−exp(2λkψkC(Ω) ) , t(T −t)

where λ > 0 is a parameter. We are ready to state a Carleman estimate in a Sobolev space of negative order: Theorem 2.1. Let (2.3) - (2.4) be fulfilled and Pnthe functions ϕ, α be defined by (2.7). Moreover let g(t, x) = g0 (t, x) + i=1 ∂i gi (t, x) with g0 ∈ L2 (0, T ; H −1 (Ω)) and gi ∈ L2 (Q), 1 ≤ i ≤ n. Then there exists a number b > 0 such that for an arbitrary λ ≥ λ, b we can choose s0 (λ) > 0 satisfying: λ there exists a constant C1 > 0 such that for each s ≥ s0 (λ) and any y ∈ L2 (Q) satisfying (2.1) and (2.2), we have Z 1 2 2 (2.8) |∇y| + sϕy e2sα dx dt sϕ Q ! Z n X ≤ C1 kg0 esα k2L2 (0,T ;H −1 (Ω)) + kgi esα k2L2 (Q) + sϕy 2 e2sα dxdt , Qω

i=1

where the constant C1 > 0 is dependent continuously on aij , bi , 1 ≤ i, j ≤ n, c, λ, and independent of s. Moreover, if bi = 0, 1 ≤ i ≤ n, and y|Qω = 0, then we have Z 1 − 12 1 sα 2 2 2 (2.9) |∇y| + sϕ0 y e2sα dx dt kϕ ∂t (ye )kL2 (0,T ;H −1 (Ω)) + s 0 sϕ 0 Q ≤ C2 kgesα k2L2 (0,T ;H −1 (Ω))

∀ s ≥ s0 (λ),

where the constant C2 > 0 is dependent continuously on aij , 1 ≤ i, j ≤ n, c, λ, and independent of s. Proof. For the Carleman estimate (2.8), we refer to [14]. We will prove the second Carleman estimate (2.9). It is sufficient to prove (2.9) in the case of c = 0. Indeed, Lemma 2.3. Let c satisfy (2.3). Then we can take δ ∈ 0, 12 such that kcyesα kL2 (0,T ;H −1 (Ω)) ≤ C3 s−δ (k(sϕ)− 2 (∇y)esα kL2 (Q) + k(sϕ) 2 yesα kL2 (Q) ) 1

1

for y ∈ L2 (0, T ; H01 (Ω)) and large s > 0. Here a constant C3 > 0 is independent of y, s, but dependent on aij , c, Ω, T . The proof of the lemma is given in Appendix B. By taking s > 0 sufficiently large, Lemma 2.3 and the Carleman estimate (2.9) in the case of c = 0, yield (2.9) with c satisfying (2.3).

Carleman estimate and applications

117

Now we proceed to the proof of (2.9) in the case of c = 0. Instead of the b operator L, it suffices to prove (2.9) for the operator L: b = ∂t y − Ly

n X

aij (x)∂i ∂j y.

i,j=1

b s = esα Le−sα , z = yesα . We notice that the operator L b s can be Denote L written explicitly as follows n X

b s z =∂t z − L

aij ∂i ∂j z + 2sλϕ

i,j=1

n X

aij (∂i ψ)∂j z + sλ2 ϕa(x, ∇ψ, ∇ψ)z

i,j=1

−s λ ϕ a(x, ∇ψ, ∇ψ)z + sλϕz 2 2 2

n X

aij ∂i ∂j ψ − s(∂t α)z,

i,j=1

P where a(x, ζ, ζ) = ni,j=1 aij ζi ζj for ζ = (ζ1 , ..., ζn ). We note that the function z is the solution to the initial value problem bs z = gesα in Q, L

(2.10)

z|Σ = 0, z(0, ·) = 0.

Using the partition of unity and a standard argument (see e.g., [11, p. 191]), one can reduce the proof of the estimate (2.9) to the case when Ω ⊂ {x; |x| < δ} where the parameter δ can be chosen arbitrarily small. For each t ∈ [0, T ], let w(t, x) be the solution to the boundary value problem: −

n X

∂i (aij (x)∂j w(t, x)) = z(t, x),

x ∈ Ω,

w|∂Ω = 0.

i,j=1

Taking the scalar product of (2.10) with the function by parts and a priori estimates, we can obtain

sw t(T −t) ,

after integration

3

(2.11)

s3 kϕ 2 yesα k2L2 (0,T ;H −1 (Ω)) 1 1 1 ≤ C(skϕ 2 zk2L2 (Q) + kϕ− 2 ∇zk2L2 (Q) + kgesα k2L2 (0,T ;H −1 (Ω)) ). s

Therefore, by (2.8) and (2.11), we have (2.12)

3

s3 kϕ 2 yesα k2L2 (0,T ;H −1 (Ω)) ≤ Ckgesα k2L2 (0,T ;H −1 (Ω)) .

118


Next we have −1

ϕ0 2 ∂t z =

n X

− 12

aij (∂i ∂j z)ϕ0

n X

−1

− 2sλϕ0 2 ϕ

i,j=1

aij (∂i ψ)(∂j z)

i,j=1

−1

−1

−sλ2 ϕ0 2 ϕa(x, ∇ψ, ∇ψ)z + s2 λ2 ϕ0 2 ϕ2 a(x, ∇ψ, ∇ψ)z n X −1 −1 −1 −sλϕ0 2 ϕz aij ∂i ∂j ψ + s(∂t α)ϕ0 2 z + gesα ϕ0 2 . i,j=1

Therefore (2.13) ≤

1 − 12 kϕ ∂t zk2L2 (0,T ;H −1 (Ω)) s 0 n C X −1 kaij ϕ0 2 (∂i ∂j z)k2L2 (0,T ;H −1 (Ω)) s i,j=1

+Cs

n X

2 2 2 kaij (ϕϕ−1 0 ) ϕ (∂i ψ)∂j zkL2 (0,T ;H −1 (Ω)) 1

1

i,j=1 −1

2 2 2 2 2 +Cs3 k(ϕϕ−1 0 ) ϕ a(x, ∇ψ, ∇ψ)zkL2 (0,T ;H −1 (Ω)) + Cskϕ0 ϕzkL2 (Q) n o 3 1 2λ(kψkC(Ω) −ψ(x)) −λψ(x) 2 e ϕ 2 zk2L2 (0,T ;H −1 (Ω)) +Csk(2t − T )(ϕϕ−1 ) − e 0 1

3

1 −1 + kgesα ϕ0 2 k2L2 (0,T ;H −1 (Ω)) . s Noting that kqwkL2 (0,T ;H −1 (Ω)) ≤ CkwkL2 (0,T ;H −1 (Ω))

(2.14)

1 (Ω)), we have for q ∈ L∞ (0, T ; W∞

[the third, the fifth and the sixth terms at the right hand side of (2.13)] 3

≤Cs3 kϕ 2 zk2L2 (0,T ;H −1 (Ω)) + Ckgesα k2L2 (0,T ;H −1 (Ω)) . Moreover, by (2.3) and (2.14), we see −1

−1

kaij ϕ0 2 (t)(∂i ∂j z)(t, ·)kH −1 (Ω) ≤ Ckϕ0 2 (t)(∂i ∂j z)(t, ·)kH −1 (Ω) Z − 21 ≤Cϕ0 (t) sup (∂j z)(t, x)∂i µ(x)dx µ∈H01 (Ω),k∇µkL2 (Ω) =1

Ω

− 21

≤Cϕ0 (t)k(∂j z)(t, ·)kL2 (Ω) , so that we have C [the first term at the right hand side of (2.13)] ≤ s

Z Q

1 |∇z|2 dxdt. ϕ0


119

Finally we have 2 2 2 kaij (∂i ψ)(ϕϕ−1 0 ) (t, ·)ϕ(t, ·) (∂j z)(t, ·)kH −1 (Ω) ≤ kϕ(t, ·) (∂j z)(t, ·)kH −1 (Ω) Z ϕ 12 (t, x)(∂j z)(t, x)µ(x)dx ≤C sup 1 1

µ∈H0 (Ω),k∇µkL2 (Ω) =1

1

1

Ω

1 2

≤Ckϕ (t, ·)z(t, ·)kL2 (Ω) , so that

Z [the second term at the right hand side of (2.13)] ≤ C

sϕ0 |z|2 dxdt. Q

Hence (2.8) and (2.12) yield 1 − 12 kϕ ∂t zk2L2 (0,T ;H −1 (Ω)) ≤ Ckgesα k2L2 (0,T ;H −1 (Ω)) . s 0 Thus the proof of (2.9) is complete.

3

Exact null-controllability of semilinear parabolic equations

Henceforth aij , bi , 1 ≤ i, j ≤ n and c are assumed to satisfy (2.3) and we consider the semilinear parabolic equation (3.1)

G(y) = ∂t y −

n X

∂i (aij (x)∂j y) +

i,j=1

n X

bi (t, x)∂i y(t, x)

i=1

+c(t, x)y + f (t, x, ∇y, y) in Q with u ∈ U (ω),

= u+g and (3.2)

y Σ = 0,

y(0, x) = v0 (x),

x ∈ Ω,

where v0 and g are given, and u(t, x) is a locally distributed control in the space (3.3)

U(ω) = {u(t, x) ∈ L2 (Q); supp u ⊂ Qω }.

By the exact null-controllability we mean a problem of finding a control u ∈ U (ω) such that (3.4)

y(T, x) = 0,

x ∈ Ω.

For a semilinear term f , let us assume that (3.5)

f (t, x, ζ 0 , ζ0 ) ∈ C 1 (Q × Rn+1 ),

f (t, x, 0, 0) = 0,

(t, x) ∈ Q

120


and (3.6) ∂f (t, x, ζ 0 , ζ0 ) ≤ K, ∂ζi

(t, x) ∈ Q,

ζ ≡ (ζ 0 , ζ0 ) = (ζ1 , . . . , ζn , ζ0 ) ∈ Rn+1

for 0 ≤ i ≤ n with some constant K > 0. Set η(t, x) =

(3.7)

−eλψ(x) + e2λkψkC(Ω) , (T − t)`(t)

where (3.8)

` ∈ C ∞ [0, T ],

We set

( L2 (Q, ρ) =

T `(t) = t, t ∈ [ , T ]. 2

`(t) > 0, `(t) ≥ t, t ∈ [0, T ], Z

)

1

y = y(t, x); kykL2 (Q,ρ) ≡

y 2 ρdxdt

2

0 such that for λ ≥ λ b there exists and (3.6) be fulfilled. Then there exists λ λ b a constant s0 (λ) so that for g ∈ Xs (Q) with λ ≥ λ and s ≥ s0 (λ), there exists a solution pair (y, u) ∈ Y (Q) × U (ω) to (3.1), (3.2) and (3.4).


121

As for the proof, we refer to Imanuvilov and Yamamoto [14]. This is the exact null-controllability for a parabolic equation whose semilinear term depends also on ∇y. The main achievement by our Carleman estimate (Theorem 2.1) is that we can include the first order derivatives in the semilinear term. For the approximate controllability for a parabolic equation with semilinear term including ∇y, see Fernández and Zuazua [7]. We further refer to Fabre, Puel and Zuazua [6] and Fernández-Cara [8].

4

Conditional stability in the continuation

Let L be defined by (2.1) and let (2.3)–(2.4) hold. Let ω ⊂ Ω be an arbitrary subdomain. In this section, we discuss conditional stability in continuation of solutions to a parabolic equation. By the uniqueness in the continuation, we mean that y|(0,T )×ω = 0 implies y|Q = 0. We can refer to Isakov [15]–[17], Lavrent’ev, Romanov and Shishat·ski˘ı[24], Saut and Scheurer [29], Sogge [30]. In fact, for the uniqueness, we can have even sharper results: Lin [25], Poon [27]. The continuation problem is known to be ill-posed. In other words, we cannot expect the continuity of the map y|(0,T )×ω −→ y|Q . However, under a priori boundedness assumptions for y in Q, we can restore stability, which is called conditional stability. In this section, we will prove conditional stability by our Carleman estimate (Theorem 2.1). For the proof, we follow a scheme: ”Carleman estimate” −→ ”conditional stability”, which has been used by Hörmander [11], Isakov [15]. Thus the new ingredient in this section is the Carleman estimate in a Sobolev space of negative order. In comparison with conditional stability (e.g. [11], [15]) shown by traditional Carleman estimates, the advantages of our result are: • less regularity of coefficients • weaker norms of the right hand side, thanks to our Carleman estimate. Let us recall that ψ is defined in Lemma 2.2. We set (4.1)

Ωδ = {x ∈ Ω; ψ(x) > δ}

for sufficiently small δ > 0. Then we note that Ω0 = Ω by (2.6). We assume (4.2)

ω ⊂ Ω4δ .

This is true if δ > 0 is sufficiently small. Theorem 4.1. Let (2.3)–(2.4) be fulfilled and y ∈ L2 (Q) satisfy (4.3)

Ly = g ∈ L2 (0, T ; H −1 (Ω)).

122


Then for a given κ > 0, there exists θ ∈ (0, 1) depending on Ω, T , ω, κ, δ such that (4.4)

kykL2 (2κ,T −2κ;H 1 (Ωδ )) ≤ C(kgkL2 (κ,T −κ;H −1(Ω)) + kykL2 (Qω ) )θ kyk1−θ . L2 (Q)

Here limδ→0 θ = 0 and limκ→0 θ = 0. This estimate asserts the stability under a condition that kykL2 (Q) is bounded, and becomes trivial as δ → 0 or κ → 0. Proof. Let us take the cut off function χ ∈ C0∞ (Ω), 0 ≤ χ ≤ 1 on Ω such that ( 1, x ∈ Ω2δ χ(t, x) = 0, x ∈ Ω \ Ωδ .

(4.5)

We note that χ = 0 in a neighbourhood of ∂Ω. We further set (4.6)

v = χy.

Then v|∂Ω = 0 and n X

Lv =χLy −

aij (∂j y)∂i χ −

i,j=1

−

n X

∂i (aij y)∂j χ

i,j=1

aij (∂i ∂j χ)y +

i,j=1

n X

n X

bi y∂i χ in

Q.

i=1

Therefore, noting that aij = aji , we have

(4.7)

   n n  X  X Lv =χg +  (∂j aij )∂i χ + aij ∂i ∂j χ y − 2 ∂j (aij y∂i χ)   i,j=1

+

n X

bi y∂i χ

in

i,j=1

Q.

i=1

We can assume that T ≤ 1. For 0 < t0 < t1 < T , we set (4.8) Qt0 t1

= (t0 , t1 ) × Ω,

ϕ = ϕt0 t1 (t, x) =

exp(λψ(x)) , (t − t0 )(t1 − t)

(4.9) α = αt0 t1 (t, x) =

exp(λψ(x)) − exp(2λkψkC(Ω) ) (t − t0 )(t1 − t)

,

(t, x) ∈ Qt0 t1 .

123

Carleman estimate and applications Therefore, applying (2.8) in Theorem 2.1 to (4.7), we have Z ((sϕ)−1 |∇v|2 + sϕv 2 )e2sα dx dt Q t0 t1

≤C1 kχgesα k2L2 (t0 ,t1 ;H −1 (Ω)) +C1 +C1

n X

k((∂j aij )∂i χ + aij ∂i ∂j χ)yesα k2L2 (t0 ,t1 ;H −1 (Ω))

i,j=1 n X

n X

i=1

i,j=1

kbi (∂i χ)yesα k2L2 (t0 ,t1 ;H −1 (Ω)) + C1

Z

kaij y(∂i χ)esα k2L2 (Qt

0 t1 )

sϕv 2 e2sα dxdt.

+C1 Qω

Here we note that C1 is independent of t0 and t1 , but dependent on t1 − t0 . Consequently, by (4.2) and (4.5), we have Z t1 Z (4.10) ((sϕ)−1 |∇y|2 + sϕy 2 )e2sα dx dt t0

Ω2δ

≤ C1 kgesα k2L2 (0,T ;H −1 (Ω)) +C1 +C1 +C1

n X

k((∂j aij )∂i χ + aij ∂i ∂j χ)yesα k2L2 (t0 ,t1 ;L2 (Ωδ \Ω2δ ))

i,j=1 n X

kbi (∂i χ)yesα k2L2 (t0 ,t1 ;H −1 (Ω))

i=1 n X

Z kaij y(∂i χ)esα k2L2 (t0 ,t1 ;L2 (Ωδ \Ω2δ ))

sϕy 2 e2sα dxdt.

+ C1 Qω

i,j=1

On the other hand, in terms of bi ∈ L∞ (0, T ; Lr (Ω)) with r > 2n, we can rewrite (4.10) as follows. Z sα sα bi (t, x)ye (∂i χ)µdx kbi (t, ·)ye ∂i χkH −1 (Ω) ≤ sup µ∈H01 (Ω),k∇µkL2 (Ω) =1

≤

sup µ∈H01 (Ω),k∇µkL2 (Ω) =1

Ω

kbi (t, ·)kLn (Ω) kyesα ∂i χkL2 (Ω) kµk

2n

L n−2 (Ω)

≤Ckbi (t, ·)kLr (Ω) kyesα ∂i χkL2 (Ω) ≤ Ckbi (t, ·)kLr (Ω) kyesα kL2 (Ωδ \Ω2δ ) by the Hölder inequality, r > 2n and the Sobolev embedding. Moreover we directly see that sup t0 0 is independent of s > 0. Therefore from (4.10) and (2.3), we obtain Z t1 Z ((sϕ)−1 |∇y|2 + sϕy 2 )e2sα dx dt t0 Ω2δ Z (4.11) ≤ C4 kgk2L2 (t0 ,t1 ;H −1 (Ω)) + C4 y 2 dxdt + C4 kyesα k2L2 (t0 ,t1 ;L2 (Ωδ \Ω2δ )) . Qω

We set (4.12)

2λkψkC(Ω)

h(r) = e3δλ − e

o n 2λkψkC(Ω) − (1 + 4r 2 − 4r) e2δλ − e

for 0 ≤ r ≤ 12 . Then h(0) = exp(3δλ) − exp(2δλ) > 0 and we can take r = r(λ, δ) > 0 sufficiently small so that 0 < r < 12 and ε ≡ h(r) > 0.

(4.13)

Here we notice that λ > 0 and δ > 0 are fixed, and r can be determined independently of t0 and t1 , so that ε > 0 is independent of t0 and t1 and dependent on λ, δ and r. Let t0 + t1 t0 + t1 t∈ − r(t1 − t0 ), + r(t1 − t0 ) ≡ It0 t1 . 2 2 Then, since r > 0 is sufficiently small, It0 t1 ⊂ (t0 , t1 ), and, by (4.13), we can verify   exp(3δλ) − exp(2λkψkC(Ω) )       ≤ αt0 t1 (t, x),   2   1 2   − r (t − t ) 1 0 2 (4.14)       exp(3δλ)     ≤ ϕ (t, x), t ∈ I , x ∈ Ω  1  t t t t 3δ 0 1 0 1 2 (t − t ) 1 0 4 and (4.15) αt0 t1 (t, x) ≤

exp(2δλ) − exp(2λkψkC(Ω) ) 1 4 (t1

− t0 )2

,

t0 < t < t1 , x ∈ Ωδ \ Ω2δ .

125


2 1 In fact, for t ∈ It0 t1 , we have (t−t0 )(t1 −t) ≥ 12 − r (t1 −t0 )2 , and (t−t0 )(t ≤ 1 −t) 1 . Therefore, noting that exp(λψ(x)) − exp(2λkψkC(Ω) ) < 0, we 2 ( 12 −r) (t1 −t0 )2 obtain αt0 t1 (t, x) ≡

exp(λψ(x)) − exp(2λkψkC(Ω) )

(t − t0 )(t1 − t) exp(λψ(x)) − exp(2λkψkC(Ω) ) ≥ , t ∈ It0 t1 . 2 1 2 2 − r (t1 − t0 ) Since x ∈ Ω3δ implies that ψ(x) > 3δ, we see the first inequality in (4.14). The second inequality in (4.14) is straightforward. For (4.15) we have 1 (t1 − t0 )2 ≥ (t − t0 )(t1 − t) 4

(4.16) for t0 ≤ t ≤ t1 and

αt0 t1 (t, x) ≤

exp(λψ(x)) − exp(2λkψkC(Ω) ) 1 4 (t1

− t0 )2

by exp(λψ(x)) − exp(2λkψkC(Ω) ) < 0. Since x ∈ Ωδ \ Ω2δ , we obtain ψ(x) ≤ 2δ, which implies (4.15). Therefore by means of (4.11), (4.14) and (4.15), we have ! Z exp(3δλ) − exp(2λkψkC(Ω) ) s exp(3δλ) × exp 2s y 2 dxdt 2 1 2 1 2 (t − t ) 0 It0 t1 ×Ω3δ 4 1 2 − r (t1 − t0 ) ≤C4 kgk2L2 (t0 ,t1 ;H −1 (Ω)) + C4 kyk2L2 (Qω ) +C4 exp 2s

exp(2δλ) − exp(2λkψkC(Ω) ) 1 4 (t1

− t0 )2

! kyk2L2 (t0 ,t1 ;L2 (Ω)) .

Consequently, taking s ≥ 1 and noting that 1 1 1 (t1 − t0 )2 ≤ T 2, 4s exp(3δλ) 4 we obtain Z

exp(2λkψkC(Ω) ) − exp(3δλ) C4 T 2 y dxdt ≤ exp 2s 2 1 2 4 It0 t1 ×Ω3δ 2 − r (t1 − t0 ) ! C4 T 2 −2sε + M. exp 2 1 4 − r (t1 − t0 )2 2

2

! F

126


Here and henceforth we set F = kgk2L2 (t0 ,t1 ;H −1 (Ω)) + kyk2L2 (Qω ) ,

M = kyk2L2 (Q) ,

and C > 0 denotes a generic constant which dependent on λ > 0 and δ > 0, but independent of s > 0. Taking κ > 0 sufficiently small, we assume |t1 − t0 | ≥ κ > 0.

(4.17) Therefore we see

Z

(4.18) It0 t1 ×Ω3δ

y 2 dxdt ≤ C(e2sC F + e−2sCε M ).

We can choose s > 0 so that s = max (4.19)

1 M log , s0 (λ) + 1 , 2C(1 + ε) F

in order that the Carleman estimate (2.8) holds. Thus we obtain Z 1 ε (4.20) y 2 dxdt ≤ CM 1+ε F 1+ε . It0 t1 ×Ω3δ

Now we will complete the proof. Since (4.20) holds true provided that κ 3κ |t1 − t0 | ≥ κ, we can apply (4.20) in the time interval (t0 , t1 ) = 2 , 2 , so that Z (1+r)κ Z 2 ε 1+ε y 2 dxdt ≤ CkykL1+ε F . 2 (Q) 0 (1−r)κ

Here we set F0 =

Ω3δ

kgk2L2 κ ,T − κ ;H −1 (Ω) (2 ) 2

+ kyk2L2 (Qω ) . Similarly for any t ∈

(κ, T − κ), the estimate (4.20) yields Z Z 2 ε 1+ε y 2 dxdt ≤ CkykL1+ε 2 (Q) F0 It

Ω3δ

where It is an interval including t with the length ≥ 2rκ. Since a finite number of such intervals It cover (κ, T − κ), we obtain Z T −κ Z 2 ε 1+ε (4.21) y 2 dxdt ≤ CkykL1+ε . 2 (Q) F0 κ

Ω3δ

Now we will prove the estimate for ∇y. Set Q = Qκ,T −κ = (κ, T − κ) × Ω, exp(λψ(x)) ϕ = ϕκ,T −κ (t, x) = , (t − κ)(T − κ − t) (4.22) exp(λψ(x)) − exp(2λkψkC(Ω) ) α = ακ,T −κ (t, x) = , (t − κ)(T − κ − t)

(t, x) ∈ Qκ,T −κ .

127

Carleman estimate and applications Take χ1 (t, x) =

1, x ∈ Ω4δ 0, x ∈ Ω \ Ω3δ ,

in place of χ defined by (4.5). Then, similarly to (4.11), by |e2sα | ≤ 1, we obtain Z

T −κ Z

−1

(sϕ)

|∇y| e

2 2sα

dxdt ≤ C

κ

Ω4δ sα 2 +Ckye kL2 (κ,T −κ;L2(Ω3δ \Ω4δ ))

Z kgk2L2 (κ,T −κ;H −1(Ω))

2

+

y dxdt Qω

≤ CF0 + Ckyk2L2 (κ,T −κ;L2 (Ω3δ )) .

Hence (4.21) yields Z

T −2κ Z 2κ

2

Ω4δ

ε

1+ε (sϕ)−1 |∇y|2 e2sα dxdt ≤ CF0 + CkykL1+ε . 2 (Q) F0

Then we fix such large s > 0. Since (sϕ)−1 e2sα exp(λψ(x)) − exp(2λkψkC(Ω) ) 1 = (t − κ)(T − κ − t) exp(−λψ(x)) exp 2s s (t − κ)(T − κ − t)

! >0

for (t, x) ∈ [2κ, T − 2κ] × Ω4δ , the proof is complete.

5

Inverse Source Problem

In this section, we will consider a parabolic equation with a source term: (5.1)

∂t u + Au = f (x)R(t, x)

in Q

u|Σ = 0.

(5.2)

For simplicity we consider only a t-independent elliptic operator A (5.3)

(Au)(x) = −

n X

∂i (aij (x)∂j u) + c(x)u,

i,j=1

with the homogeneous Dirichlet boundary condition, where c ∈ C ∞ (Ω), ≥ 0, aij = aji ∈ C ∞ (Ω), 1 ≤ i, j ≤ n and there exists β > 0 such that P n 2 n i,j=1 aij (x)ζi ζj ≥ β|ζ| for ζ = (ζ1 , ...., ζn ) ∈ R and x ∈ Ω. We note that the condition c ≥ 0 is not essential, because we choose large M > 0 and can consider ∞ eM t u in place of u, if necessary. Let {ei (x)}∞ i=1 and {λi }i=1 be the sequences

128


of eigenvectors and eigenvalues of the operator A. In order to formulate our results, we introduce the following spaces ( ) ∞ ∞ X X Xα = u = ai ei ∈ L2 (Ω); λαi a2i < ∞ i=1

i=1

and we set kukX α =

∞ X

!1 2

λαi a2i

.

i=1

We note that = = ∩ H01 (Ω) ⊃ H02 (Ω), X −2 ⊂ H −2 (Ω) and X −` = H −` (Ω) for 0 ≤ ` < 32 (e.g. Fujiwara [9]). Let us recall that ω ⊂ Ω is an arbitrarily fixed subdomain. Assuming that R = R(t, x) is given, we discuss Inverse Source Problem. Let θ ∈ (0, T ) and 0 ≤ ` < 1 be arbitrarily fixed. Determine f ∈ H −` (Ω) and u(0, ·) ∈ X −2 (Ω) from X0

(5.4)

L2 (Ω),

X2

H 2 (Ω)

u(θ, x),

x∈Ω

and (5.5)

u|(0,T )×ω .

This inverse problem is closely related to the determination of the coefficient p in ∂t u + Au = p(x)u. In fact, let ∂t v + Av = q(x)v. Setting y = u − v, f = p − q and R = v, we have ∂t y + Ay = p(x)y + f (x)R(t, x) in Q. In this paper, in order to concisely show the essence in applying our Carleman estimate to the inverse problem, we assume that R is smooth: (5.6)

∂t R, R ∈ C 0,1 (Q).

Here we set C 0,1 (Q) = {y; y, ∇y ∈ C(Q)}. Although we can establish the stability in our inverse problem, we concentrate on the uniqueness for the conciseness. Our main result is stated as follows: Theorem 5.1. We assume (5.6) and (5.7)

R(θ, ·) > 0

on Ω.

Let u ∈ L2 (0, T ; X −2 ) be the weak solution to (5.1), (5.2) and u(0, ·) ∈ X −2 with f ∈ H −` (Ω) where ` < 1. If u(θ, ·) = 0 in Ω and u = 0 in Qω , then u(0, ·) = 0 and f = 0 in Q. As f , we can consider δS which is a delta function concentrated on an R (n − 1)-dimensional smooth hypersurface S ⊂ Ω: < δS , h >= S h(x)dSx for h ∈ C0∞ (Ω). Remark 5.1. For u(0, ·) ∈ X −2 and f ∈ H −` (Ω), we can prove that there exists a unique solution u ∈ L2 (0, T ; X −2 ) to (5.1) and (5.2) (e.g. Tanabe [31]).


129

Remark 5.2. In ususal Carleman estimates within L2 -spaces, we need the regularity y, ∂t y, ∂i y, ∂i ∂j y ∈ L2 (Q). However, for f ∈ H −` (Ω), 0 < ` < 1, we have A∂t u 6∈ L2 (Q), and, in general, we cannot expect that ∂i ∂j y ∈ L2 (Q). Proof. We regard A as an operator in L2 (Ω) with the homogeneous Dirichlet boundary condition: D(A) = H 2 (Ω) ∩ H01 (Ω). Then the fractional power Aγ , γ ∈ R, can be defined and we have ( 0 ≤ γ < 14 H 2γ (Ω), D(Aγ ) = (5.8) {u ∈ H 2γ (Ω); u|∂Ω = 0}, 14 < γ ≤ 1, γ 6= 34 and there exists a constant Cγ > 0 such that (5.9)

Cγ−1 kukH 2γ (Ω) ≤ kAγ ukL2 (Ω) ≤ Cγ kukH 2γ (Ω) ,

u ∈ D(Aγ )

(e.g. Fujiwara [9]). Moreover we see C −1 kukH −1 (Ω) ≤ kA− 2 ukL2 (Ω) ≤ CkukH −1 (Ω) , 1

u ∈ H −1 (Ω),

(5.10) C −1 kukX −2

≤ kA−1 ukL2 (Ω) ≤ CkukX −2 ,

u ∈ X −2 .

Henceforth, without loss of generality, we can translate the time variable and we discuss the whole problem in t ∈ (−δ, T ) with δ > 0 and we set θ=

(5.11) (5.12)

T : 2

∂t u + Au = f (x)R(t, x), x ∈ Ω, −δ < t < T, u(−δ, ·) ∈ X −2 , u|(−δ,T )×∂Ω = 0.

We note that −δ is considered as the initial time. We set Q = (0, T ) × Ω, not Q = (−δ, T ) × Ω. Moreover we can take sufficiently small T > 0 if necessary, so that we can assume that (5.13)

R(t, x) > 0,

x ∈ Ω, 0 ≤ t ≤ T

by means of (5.7). Henceforth C, C0 , Cγ , etc. denote positive constants which are independent of s, f , (t, x) ∈ Q. Next we examine the regularity of u in t ∈ (0, T ). By the theory of semigroup (e.g. Tanabe [31]), we can prove Lemma 5.1. Let f ∈ H −` (Ω) with ` < 1, R satisfy (5.6), and a ≡ u(−δ, ·) ∈ −2 X . Then ∂t u ∈ C([0, T ]; H01 (Ω)).

130

O. Yu. Imanuvilov and M. Yamamoto Proof of Lemma 5.1. By the semigroup theory, we can represent u by Z t u(t) = e−tA (e−δA a) + e−A(t−ξ) R(ξ)f dξ = e−A(t+δ) a +

−δ t+δ

Z

e−Aη R(t − η)f dη.

0

Here we write u(t) = u(t, ·), f = f (·) and R(t) = R(t, ·). In view of (5.6), we have −A(t+δ)

∂t u(t) = −Ae

−A(t+δ)

a+e

Z

t+δ

R(−δ)f +

e−Aη (∂t R(t − η)f )dη,

t > 0.

0

For γ > 0, there exists a constant Cγ > 0 such that kAγ e−tA gkL2 (Ω) ≤ Cγ t−γ kgkL2 (Ω) ,

t>0

(e.g. [31]). Therefore, by (5.9), (5.10) and γ < 1, we have 1

k∂t u(t)kH01 (Ω) ≤ CkA 2 ∂t u(t)kL2 (Ω)

5 3

≤C A 2 e−A(t+δ) A−1 a + A 2 e−A(t+δ) A−1 (R(−δ)f )

Z t+δ

`+1 `

−Aη − + A 2 e (A 2 (∂t R(t − η)f )dη

2 0 L (Ω)

− 52

≤C(t + δ) kA akL2 (Ω) + C(t + δ) kA−1 (R(−δ)f )kL2 (Ω) Z t+δ `+1 ` +C η− 2 sup kA− 2 (∂t R(t − η)f )kL2 (Ω) dη 0 − 52

−1

− 32

0≤η≤t+δ

≤Cδ kA−1 akL2 (Ω) + Cδ− 2 kA−1 (R(−δ)f )kL2 (Ω) 1−` ` 2C + (t + δ) 2 sup kA− 2 (∂t R(s)f )kL2 (Ω) . 1−` −δ≤s≤t 3

Thus the proof of Lemma 5.1 is complete. In view of Lemma 5.1, we can justify the following calculations: Setting z = ∂t u, we have ∂t z + Az = (∂t R)f in (0, T ) × Ω, (5.14) z(θ, ·) = R(θ, ·)f, z|(0,T )×ω = 0, z|(0,T )×∂Ω = 0. By (2.9) we have 1 − 12 kϕ ∂t (yesα )k2L2 (0,T ;H −1 (Ω)) + s 0

Z Q

1 2 2 |∇y| + sϕ0 y e2sα dxdt sϕ0

(5.15) ≤ C2 k(∂t R)f esαk2L2 (0,T ;H −1 (Ω)) ,

s > s0 (λ).


131

On the other hand, by (5.10) and z(0, ·)esα(0,·) = 0, we have kz(θ, ·)esα(θ,·) k2H −1 (Ω) ≤ CkA− 2 (z(θ, ·)esα(θ,·) )k2L2 (Ω) Z Z θ ∂ − 12 sα 2 =C |A (ze )| dx dt 0 ∂t Ω Z θ Z − 12 sα − 12 sα =2C {A ∂t (ze )}{A (ze )}dx dt 0 Ω Z θ Z 1 − 12 − 12 sα − 12 sα =2C {A (sϕ0 ) ∂t (ze )}{A ((sϕ0 ) 2 ze )}dx dt 1

0 T

Z ≤C 0

Ω −1

1

{s−1 kA− 2 (ϕ0 2 ∂t (zesα ))k2L2 (Ω) + skA− 2 (ϕ02 zesα )k2L2 (Ω) }dt 1

1

1 C −1 ≤ kϕ0 2 ∂t (zesα )k2L2 (0,T ;H −1 (Ω)) + Cskϕ02 zesα k2L2 (Q) . s

Hence (5.15) yields (5.16)

kz(θ, ·)esα(θ,·) kH −1 (Ω) ≤ Ck∂t RkC 0,1 (Q) kf esα kL2 (0,T ;H −1 (Ω)) .

Next we will prove (5.17)

kf esα kL2 (0,T ;H −1 (Ω)) ≤ C(s − 1)− 4 kf esα(θ,·) kH −1 (Ω) 1

where C > 0 is independent of s > 0. Proof of (5.17). By the mean value theorem, we can take κ = κ(t, x) such that 1 1 q(t, x) ≡ α(t, x) − α(θ, x) = (∂t2 α)(t, x)(t − θ)2 + (∂t3 α)(κ, x)(t − θ)3 . 2 6 By direct calculations, we have (5.18)

(∂t2 α)(t, x) ≤

−γ0 , − t)3

t3 (T

(∂t3 α)(t, x)(t − θ)3 ≤ 0,

(t, x) ∈ Q,

where γ0 > 0 is a constant independent of (t, x) ∈ Q. Hence we directly verify sγ0 sq(t,x) 2 |e (t, x) ∈ Q (5.19) | ≤ exp − 3 (t − θ) , 2t (T − t)3 and

(5.20)

|(∇q)(t, x)esq(t,x) | (t − θ)2 sγ0 (t − θ)3 2 ≤ C 3 × exp − 3 + (t − θ) t (T − t)3 t4 (T − t)4 2t (T − t)3 (s − 1)γ0 ≤ C(t − θ)2 exp − 3 (t − θ)2 2t (T − t)3 ≤ C(t − θ)2 exp(−C0 (s − 1)(t − θ)2 )

132


for large s > 1. We have kf e

sα(t,·)

kH −1 (Ω) =


≤ kf esα(θ,·) kH −1 (Ω)

Z f esα(θ,x) esq(t,x) µ(x)dx Ω


k∇(esq(t,·) µ)kL2 (Ω) .

By (5.19) and (5.20), we obtain (5.21)

kf esα(t,·) k2H −1 (Ω) n o 2 2 ≤ C e−2C0 s(t−θ) + s2 (t − θ)4 e−2C0 (s−1)(t−θ) kf esα(θ,·) k2H −1 (Ω) .

Since Z Z

0

≤

T

{exp(−2C0 s(t − θ)2 ) + s2 (t − θ)4 exp(−2C0 (s − 1)(t − θ)2 )}dt

∞

−∞

(e−2C0 (s−1)η + s2 η 4 e−2C0 (s−1)η )dη ≤ √ 2

2

C , s−1

the inequality (5.21) yields (5.17). Thanks to (5.17), from (5.14) and (5.16), we obtain (5.22)

kR(θ, ·)f esα(θ,·) kH −1 (Ω) ≤ C(s − 1)− 4 kf esα(θ,·) kH −1 (Ω) , 1

∀ s ≥ s0 .

Noting the definition of the H −1 (Ω)-norm, by (5.6) and (5.7), we can prove kf esα(θ,·) kH −1 (Ω) ≤ CkR(θ, ·)f esα(θ,·) kH −1 (Ω) .

(5.23)

In (5.22), taking s > 0 sufficiently large, we see f = 0. Thus the proof of Theorem 5.1 is complete.

A

Appendix A. Sketch of Proof of Lemma 2.1

1 (Ω), 1 ≤ i, j ≤ n, the unique existence of the solution in Since aij ∈ W∞ L2 (0, T ; H01 (Ω)) ∩ C([0, T ]; L2 (Ω)) is seen in the case of bi = 0, 1 ≤ i ≤ n and c = 0, for example, by Chapter 3, §1 and §4 in Lions and Magenes [26]. In the general case with non-zero bi and c satisfying (2.3), as for the unique existence of solution and the a priori estimate (2.5), in view of the general theorem in Chapter 3, §4 in [26], it is sufficient to prove: for any ε > 0, there exists a constant C = C(ε) > 0 such that Z (A.1) bi u∂i udx ≤ εkuk2H 1 (Ω) + C(ε)kuk2L2 (Ω) , 1 ≤ i ≤ n, u ∈ H 1 (Ω) Ω

Carleman estimate and applications and (A.2)

Z 2 cu dx ≤ εkuk2 1 + C(ε)kuk2 2 , H (Ω) L (Ω)

133

u ∈ H 1 (Ω),

Ω

Henceforth C > 0 denotes a generic constant independent of functions to be estimated. First we prove (A.1). By the Hölder inequality, we have Z bi u∂i udx ≤ kbi (t, ·)kLr (Ω) kuk 2r k∂i ukL2 (Ω) . L r−2 (Ω) Ω

Since r > 2n, the Sobolev imbedding theorem (e.g. [1]) implies H 2 −δ (Ω) ⊂ 2r L r−2 (Ω) for sufficiently small δ > 0. Hence with small ε > 0 we have Z bi u∂i udx ≤ kuk 1 −δ kukH 1 (Ω) H 2 (Ω) 1

Ω

C kuk2 1 −δ . H 2 (Ω) ε By the interpolation inequality (e.g. [1]), we see ≤εkuk2H 1 (Ω) +

kuk2

(A.3)

1

H 2 −δ (Ω)

≤ δkuk2H 1 (Ω) + C(δ)kuk2L2 (Ω)

for small δ > 0. We choose sufficiently small ε > 0 and δ > 0 such that also small, so that n Z X bi u∂i udx ≤ εkuk2 1 + C(ε)kuk2 2 . (A.4) H (Ω) L (Ω) i=1

δ ε

is

Ω

Next we will prove (A.2). For this, we show n o 1 2n Lemma A.1. Let 0 < µ < 2 and r1 > max 3−2µ , 1 , there exist constants 0 < δ
0 such that

kzvkW µ0 (Ω) ≤ CkvkH 1 (Ω) kzk r1

1

H 2 −δ (Ω)

.

As for the proof, we can refer to Lemma 2.2 in [14] for example. By Lemma A.1, we have Z cu2 dx ≤ kc(t, ·)k −µ ku2 kW µ (Ω) Wr1 (Ω) r0 Ω

1

≤Ckuk

H

1 −δ 2 (Ω)

kukH 1 (Ω) ≤ Cεkuk2H 1 (Ω) +

1 2.

C kuk2 1 −δ H 2 (Ω) ε

with 0 < δ < In view of the interpolation inequality (A.3), taking ε > 0 and δ > 0 so small that δε is also small, we obtain Z 2 cu dx ≤ εk∇uk2 2 + C(ε)kuk2 2 . L (Ω) L (Ω) Ω

Thus the proof of (A.1) and (A.2) is complete.

134

B


Appendix B. Proof of Lemma 2.3

Henceforth C > 0 denotes a generic constant independent of y, and dependent on c. First Z kcyesα k2L2 (0,T ;H −1 (Ω)) Z

T

≤C Z

{kck2L∞ (0,T ;W −µ (Ω)) r1

0 T

≤C

sup 0

≤C

kµkH 1 (Ω) =1

T

sup kµkH 1 (Ω) =1

0

sup kµkH 1 (Ω) =1

Z 2 cyesα µdx dt Ω

ky(t, ·)esα µk2W µ (Ω) }dt 0 r1

ky(t, ·)µesα k2W µ (Ω) dt. r0 1

Then by Lemma A.1 in Appendix A, we obtain ky(t, ·)µesα k2W µ (Ω) ≤ Ckµk2H 1 (Ω) kyesα k2

1

H 2 −δ (Ω)

r0 1

and so we see Z kcyesα k2L2 (0,T ;H −1 (Ω))

T

≤C

kyesα k2

1

H 2 −δ (Ω)

0

dt.

Thus the proof of Lemma 2.3 is complete, when we will have proved Z (B.1)

s

2δ

kyesα k2 2 1 L (0,T ;H 2 −δ (Ω))

≤C

(sϕy 2 + Q

1 |∇y|2 )e2sα dxdt sϕ

whenever δ ∈ (0, 12 ). In the rest part of the appendix, we will verify (B.1). We note that (B.2)

|(∂i ϕ)(t, x)|, |(∂i α)(t, x)| ≤ Cϕ(t, x),

(t, x) ∈ Q

and C −1 ϕ0 (t) ≤ ϕ(t, x) ≤ Cϕ0 (t), where ϕ0 (t) = and (B.3)

1 t(T −t) .

(t, x) ∈ Q

Therefore, by the interpolation inequality (e.g. [1], [26])

|ab| ≤

2 2 1 − 2δ 1−2δ 1 + 2δ 1+2δ + , |a| |b| 2 2


135

we have kyesα k

H

1

1 −δ 2 (Ω)

−δ

1

+δ

sα 2 2 ≤ Ckyesα kH kL2 (Ω) 1 (Ω) kye 1

−δ

δ

1

+δ

δ

− sα 2 2 4 2 ky(sϕ ) 2 e ≤Cky(sϕ0 )− 2 esα kH kL2 (Ω) (sϕ0 )− 4 − 2 0 1 (Ω) (sϕ0 ) 1

1

1

1

−δ

1

1

+δ

≤Cs−δ k(sϕ0 )− 2 ∇(yesα )kL2 2 (Ω) k(sϕ0 ) 2 esα ykL2 2 (Ω) 1

1

−δ

1

1

+δ

≤Cs−δ k(sϕ)− 2 esα ∇ykL2 2 (Ω) k(sϕ) 2 esα ykL2 2 (Ω) + Cs−δ k(sϕ) 2 esα ykL2 (Ω) 2 1 1−2δ −δ −δ 1 − 2δ − 12 sα 2 ≤Cs k(sϕ) e ∇ykL2 (Ω) 2 2 1 1+2δ 1 1 +δ −δ 1 + 2δ sα 2 2 +Cs + Cs−δ k(sϕ) 2 esα ykL2 (Ω) . k(sϕ) e ykL2 (Ω) 2 1

1

1

Thus the verification of (B.1) is complete.

References [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] A. L. Bukhgeim and M. V. Klibanov, Global uniqueness of a class of multidimensional inverse problems, Soviet Math. Dokl., 24 (1981), pp. 244–247. [3] T. Carleman, Sur un problème d’unicité pour les systèmes d’équations aux derivées partielles ` a deux variables independentes, Ark. Mat. Astr. Fys., 2B (1939), pp. 1–9. [4] D. Chae, O. Y. Imanuvilov, and S. M. Kim, Exact controllability for semilinear parabolic equations with Neumann boundary conditions, J. of Dynamical and Control Syst., 2 (1996), pp. 449–483. [5] J. Cheng, V. Isakov, M. Yamamoto, and Q. Zhou, Lipschitz stability in the lateral Cauchy problem for elasticity system, Preprint Series UTMS 99-33, 1999, Graduate School of Mathematical Sciences, The University of Tokyo. 1999. [6] C. Fabre, J.-P. Puel, and E. Zuazua, Approximate controllability of the semilinear heat equation, Proc. Roy. Soc. Edinburgh, 125 Sect. A (1995), pp. 31–61. [7] L. Fern´ andez and E. Zuazua, Approximate controllability for semilinear heat equation involving gradient terms, J. Optim. Theory Appl., 101 (1999), pp. 307– 328. [8] E. Fern´ andez-Cara, Null controllability of the semilinear heat equation, ESAIM:Control, Optimization and Calculus of Variations, 2 (1997), pp. 87–103. [9] D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order, Proc. Japan Acad., 43 (1967), pp. 82–86. [10] A. V. Fursikov and O. Y. Imanuvilov, Controllability of evolution equations, vol. 34 of Lecture notes series, Seoul National University, Seoul, South Korea, 1996. [11] L. Hörmander, Linear Partial Differential Operators, Springer-Verlag, Berlin, 1963. [12] O. Y. Imanuvilov, Boundary controllability of parabolic equations, Sbornik Mathematics, 186 (1995), pp. 879–900.

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[13] O. Y. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate, Inverse Problems, 14 (1998), pp. 1229–1245. [14] , On Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Preprint Series UTMS 98-46, 1998, Graduate School of Mathematical Sciences, The University of Tokyo. 1998. [15] V. Isakov, Inverse Source Problems, American Mathematical Society, Providence, Rhode Island, 1990. [16] , Carleman type estimates in an anisotropic case and applications, Journal of Differential Equations, 105 (1993), pp. 217–238. , Inverse Problems for Partial Differential Equations, Springer-Verlag, [17] Berlin, 1998. [18] V. Isakov and M. Yamamoto, Carleman estimate with the Neumann boundary condition and its applications to the observability inequality and inverse hyperbolic problems, to appear in Contemporary Mathematics. [19] M. A. Kazemi and M. V. Klibanov, Stability estimates for ill-posed Cauchy problems involving hyperbolic equations and inequalities, Appl. Math., 50 (1993), pp. 93–102. [20] A. Kha˘ıdarov, Carleman estimates and inverse problems for second order hyperbolic equations, Math. USSR Sbornik, 58 (198), pp. 267–277. [21] M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 8 (1992), pp. 575–596. [22] I. Lasiecka and R. Triggiani, Carleman estimates and exact boundary controllability for a system of coupled, nonconservative second-order hyperbolic equations, in Partial Differential Equation Methods in Control and Shape Analysis, vol. 188, Marcel Dekker, Inc., New York, 1997, pp. 215–243. [23] I. Lasiecka, R. Triggiani, and X. Zhang, Nonconservative wave equations with unobserved Neumann B.C.: global uniqueness and observability in one shot, to appear in Contemporary Mathematics. [24] M. M. Lavrent’ev, V. G. Romanov, and S. P. Shishat·ski˘ı, Ill-posed Problems of Mathematical Physics and Analysis, American Mathematical Society, Providence, Rhode Island, 1986. [25] F. H. Lin, A uniqueness theorem for parabolic equations, Comm. Pure Appl. Math., 43 (1990), pp. 127–136. [26] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, Berlin, 1972. [27] C.-C. Poon, Unique continuation for parabolic equations, Comm. Partial Differential Equations, 21 (1996), pp. 521–539. [28] A. Ruiz, Unique continuation for weak solutions of the wave equation plus a potential, J. Math. Pures Appl., 71 (1992), pp. 455–467. [29] J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations, 66 (1987), pp. 118–139. [30] C. D. Sogge, Unique continuation theorem for second order parabolic differential operators, Ark. Mat., 28 (1990), pp. 159–182. [31] H. Tanabe, Equations of Evolution, Pitman, London, 1979. [32] D. Tataru, Boundary controllability for conservative PDEs, Appl. Math. Optim. (1995), pp. 257–295. [33] , Carleman estimates, unique continuation and controllability for anizotropic


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PDE’s, Contemporary Math., 209 (1997), pp. 267–279. [34] M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems, J. Math. Pures Appl., 78 (1999), pp. 65–98.

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Bilinear control for global controllability of the semilinear parabolic equations with superlinear terms

Alexander Khapalov, Washington State University, Pullman, Washington

Abstract In this paper we establish the global approximate controllability of the semilinear heat equation with superlinear term, governed in a bounded domain by a pair of controls: (a) the traditional internal either locally distributed or lumped control and (b) the lumped control entering the system as a time-dependent coefficient. The motivation for the latter is due to the well known lack of global controllability properties for this class of pde’s when they are steered solely by the former controls. Our approach involves an asymptotic technique allowing us to “separate and combine” the impacts generated by the above-mentioned two types of controls. In particular, the addition of bilinear control allows us to reduce the use of the additive one to the local controllability technique only.

Key words: semilinear heat equation, global approximate controllability, bilinear controls, lumped control, asymptotic analysis. AMS(MOS) subject classifications. 93, 35. 1. Introduction. 1.1. Problem formulation and motivation. We consider the following Dirichlet boundary problem, governed in a bounded domain Ω ⊂ Rn by the bilinear lumped control k = k(t) and the additive locally distributed control v(x, t)χω (x), supported in the given subdomain ω ⊂ Ω: (1)

∂u = ∆u + k(t)u − f (x, t, u, ∇u) + v(x, t)χω (x) in QT = Ω × (0, T ), ∂t u=0

in ΣT = ∂Ω × (0, T ), 139

u |t=0 = u0 ∈ L2 (Ω),

140

Khapalov k ∈ L∞ (0, T ), v ∈ L2 (QT ).

In the one space dimension we will also consider the case when both controls are lumped, that is, they are the functions of time only: k = k(t) and v = v(t). In this paper we are concerned with the issue of approximate controllability of system (1) in the (phase-) space L2 (Ω). Namely, given the initial state u0 , we want to know whether the range of the solution mapping (2)

L∞ (0, T ) × L2 (QT ) 3 (k, v) → u(·, T ) ∈ L2 (Ω)

is dense in L2 (Ω). (In fact, due to the possible nonuniquenes of solutions to (1) the situation here is more complex as we discuss it below in the subsection 1.2.) It is well known ([5], [9], [19], [8]) that a rather general semilinear parabolic equation, governed in a bounded domain by the classical either boundary or additive locally distributed controls only (i.e., no “changeable” bilinear control k(·) in (1)) is globally approximately controllable in L2 (Ω), provided that the nonlinearity is globally Lipschitz. The methods of these works make use of the fixed point argument and the fact that such semilinear equations can be viewed as “linear equations” with the coefficients uniformly bounded in some sense. Alternative approach employs the global inverse function theorem – we refer in this respect to the work [16] on the semilinear wave and plate equations. However, the situation is principally different if nonlinear terms admit polynomial superlinear growth at infinity. Such terms are in the focus of our attention in this paper. Given T > 0, we further assume that f (x, t, u, p) is Lebesgue’s measurable in x, t, u, p, and continuous in u, p for almost all (x, t) ∈ QT , and is such that (3a)

| f (x, t, u, p) |≤ β | u |r1 +β k p krR2n a.e. in QT for u ∈ R, p ∈ Rn ,

(3b) Z Z Z 2 f (x, t, φ, ∇φ) φdx ≥ (ν − 1) k ∇φ kRn dx − % (1 + φ2 )dx ∀φ ∈ H01 (Ω), Ω

Ω

Ω

where β, ν, % > 0, T % ≤ β, and (3c)

r1 ∈ (1, 1 +

4 ), n

r2 ∈ (1, 1 +

2 ). n+2

Here and below we use the standard notations for Sobolev spaces such as H01,0 (QT ) = {φ | φ, φxi ∈ L2 (QT ), i = 1, . . . , n, φ |ΣT = 0} and H01 (Ω) = {φ | φ, φxi ∈ L2 (Ω), i = 1, . . . , n, φ |∂Ω = 0}. (A simple example of a function f satisfying conditions (3a-c) is f (u) = u3 .) We refer, e.g., to [15] (p. 466), where it was shown that system (1), (3a-c) admits at least one generalized

Bilinear control for the semilinear equations

141

T T solution in C([0, T ]; L2 (Ω)) H01,0 (QT ) L2+4/n (QT ), while its uniqueness is not guaranteed. It turns out that in the superlinear case like (3a-c) the impact from the sole additive control v(t)χω (x) does not propagate “effectively” from its support to the rest of the space domain: regardless of how large the control applied on ω is, the corresponding solutions remain uniformly bounded on any closed subset of Ω\¯ ω . (In other words, given the initial state u0 , there exist target states, namely, “sufficiently large” on Ω\¯ ω , which are strictly separated from the range of the corresponding solution mapping.) This is true in any of the (phase- ) spaces Lp (Ω), 1 ≤ p < ∞ at any positive time, e.g., for the functions f = f (x, t, u) such that f (x, t, u)u > c1 | u |2+r −c2 for some constants c1 , c2 , r > 0 ([9], and the references therein). (On the other hand, for certain refinements of conditions (3a-c) a number of positive “superlinear” controllability results were obtained in [10], [12]-[14], see Remark 1.1 for details.) In this paper our goal is to show that the above-outlined principal difficulty with propagation of control impact can be overcome by using an additional bilinear lumped control k = k(t), entering the equation (1) as a coefficient and thus affecting the qualitative behavior of system (1),(3a-c) in the entire space domain. Remark 1.1: Some references. • We refer to the work [2] on controllability of the abstract infinite dimensional bilinear system as the only known to us on this subject in the framework of pde’s. In particular, in [2], under the additional assumption that all the modes in the initial data are active, the global approximate controllability of the rod equation utt + uxxxx + k(t)uxx = 0 with hinged ends and the wave equation utt − uxx + k(t)u = 0 with Dirichlet boundary conditions, where k is control, was shown making use of the Fourier series approach. (To the contrary, an extensive and thorough bibliography on controllability on bilinear ode’s is available, see, e.g., the survey [1].) We also refer to works [4], [17] (and the references therein) on the issue of optimal bilinear control for various pde’s. • Some examples of physical interpretation of bilinear controls can be found in the just-cited works. In particular, in the rod equation the control k(t) is the axial load [2]. In turn, the equation (1) is typically used to model the heat transfer and diffusion processes, with the semilinear term associated, e.g., with the porous media. In this context k(t) is respectively proportional to the heat or mass transfer coefficients. For example, in the heat transfer models involving fluids k(t) is sensitive to their velocity. In the diffusion processes, such as some biological models or the chain reaction, the values of k(t) can be both positive and negative. In general,

142

Khapalov the bilinear parameter k is linked to the physical properties of the process at hand and the use of it as a control has a potential to be associated with so-called “intelligent materials” that can change their physical properties under certain conditions. • In spite of the lack of global controllability of (1), (3a-c) discussed in the above, it was shown in [13] that this equation is actually globally approximately controllable at any time T > 0 solely by means of the additive locally distributed controls in the spaces that are weaker than any of Lp (Ω), 1 ≤ p < ∞. Moreover, under the additional assumption that the superlinear term is locally Lipschitz (which ensures the uniqueness of solutions in C([0, T ]; L2 (Ω))) the global finite dimensional exact controllability of (1), (3a-c) (i.e., not necessarily to the equilibrium) at any positive time T > 0 was also established in [13]. • The global approximate controllability of (1), (3a-c) with k = 0 was shown in [10] for the static controls v = v(x) supported in the entire Ω. • For the one dimensional version (5), (6a-c) (see below) of system (1) it was shown in [12] that, if in (6a) β = β(t) → 0 faster than any e−ν/t , ν > 0 as t → 0, then (5), (6a-c) with k = 0 is globally approximately controllable in L2 (Ω) at any time only by means of the lumped control v = v(t), provided that the endpoints of the interval (a, b) are the irrational numbers. This result was recently extended to the case of several dimensions and locally distributed controls in [14]. • The method of [10], [12]-[14] is based on the idea to “suppress” the effect of nonlinearity by applying the actual control action only during asymptotically short period of time. Similar idea is used in this article and in [11] in the context of bilinear ode’s. • Though in this article we discuss the global approximate controllability of the equation (1), we would also like to mention here some related works on the very close global exact null-controllability property (i.e., the exact steering to the origin) by means of the additive locally distruibuted controls only. In [7] the latter property was shown in L2 (Ω) (or appropriate Sobolev space) with the reaction term f = f (x, t, u) only, assuming that f can grow superlinearly at the logarithmic rate like lim|p|→∞ f (p)/(p log | p |) = 0. Assuming the dissipativity condition, this result was improved in [3] to the rate lim|p|→∞ f (p)/(p(log | p |)3/2 ) = 0. Also in [3] some interesting non-global exact null-controllability results were given.


143

1.2. Main results. The multidimensional case with additive locally distributed controls. Theorem 1.1. Let conditions (3a-c) hold. Then the range of the solution mapping (2) is dense in L2 (Ω). Note now that, since the boundary problem (1), (3a-c) admits multiple solutions, this result is qualitatively different from the classical understanding of the approximate controllability as steering (associated with applications in the first place), which is as follows: (1) is said to be globally approximately controllable in L2 (Ω) at time T if for any u0 , uT ∈ L2 (Ω), ξ > 0 there is a control pair (k, v) such that k u(·, T ) − uT kL2 (QT ) ≤ ξ. Clearly, this classical definition is ill-posed in our case of possible multiple solutions. Therefore, we will also use its adjustment which requires one to find a control pair which steers all the possible realizations of a solution to (1) in a uniform fashion. (This type of controllability was investigated in [10], [12]-[14].) Definition 1.1. We will say that the system (1), (3a-c), admitting multiple solutions, is globally approximately controllable in L2 (Ω) at time T if for every ξ > 0 and u0 , uT ∈ L2 (Ω) there is a control pair (k, v) ∈ L∞ (QT ) × L2 (QT ) such that for all (i.e., possibly multiple) solutions of (1), (3a-c), corresponding to it k u(·, T ) − uT kL2 (Ω) ≤ ξ.

(4)

Theorem 1.2. System (1), (3a-c) is globally approximately controllable in at any time T > 0 in the sense of Definition 1.1.

L2 (Ω)

The 1-D case with all lumped controls. Consider now the one dimensional version of problem (1), (3a-c) with all lumped controls: (5)

ut = uxx + k(t)u − f (x, t, u, ux ) + v(t)χ(a,b) (x) in QT = (0, 1) × (0, T ), u |x=0,1 = 0,

u |t=0 = u0 ∈ L2 (0, 1), k ∈ L∞ (0, T ), v ∈ L2 (0, T ).

Here both k = k(t) and v = v(t) are the functions of time only.

144

Khapalov

Distinguishing the 1-D case we pursue two goals. Firstly, the positive result for the case of lumped additive controls implies the same for the locally distributed ones (since the former controls are a degenerate subclass of the latter ones). Accordingly, our proof of Theorems 1.1/1.2 is given below as the immediate consequence of the 1-D-“lumped” case. Secondly, lumped controls are of special interest being closer to the engineering applications. Focusing on them, we can give somewhat more “explicit” feeling of our method and of the general conditions (3a-c), which for the equation (5) are as follows: (6a)

| f (x, t, u, p) | ≤ β | u |r1 + β | p |r2 Z1

Z1 f (x, t, φ, φx ) φdx ≥ (ν − 1)

(6b)

a.e. in QT for u, p ∈ R,

0

Z1 φ2x dx − %

0

φ2 dx ∀φ ∈ H01 (0, 1), 0

where β, ν, % > 0, %T ≤ β, and r1 ∈ (1, 5),

(6c)

r2 ∈ (1, 5/3).

Theorem 1.3. If a < b are any irrational numbers from (0,1), then system (5), (6a-c) is globally approximately controllable in L2 (Ω) at any time T > 0 in the sense of Definition 1.1. Clearly the assumption that the endpoints of the interval (a, b) are the irrational numbers makes the result of Theorem 1.1 unstable with respect to the choice of control support (a, b). We stress however that it is well known that this assumption is intrinsic for lumped controls even in the linear case. In this respect one may prefer its immediate “stable” corollary – Theorem 1.2. The paper is organized as follows. In the next two section we prove and recall some auxiliary results for the linear version of system (5). Theorem 1.3 is proven in Section 4. Theorems 1.1/1.2 are proven in Section 5. 2. Preliminaries. Consider the boundary problem (5), (6a-c) assuming that bilinear control k is constant, i.e., k(t) ≡ α on (0, T ): (7)

wt = wxx + αw − f (x, t, w, wx ) + v(t)χ(a,b) (x)

in QT ,

w |x=0,1 = 0,

w |t=0 = w0 ∈ L2 (0, 1), v ∈ L2 (0, T ). T Denote B(0, T ) = C([0, T ]; L2 (Ω)) H01,0 (QT ) and  k q kB(0,T ) =  max k q(·, t) k2L2 (0,1) + 2

ZT Z1

t∈[0,T ]

0

0

1/2 qx2 dxds

.


145

We have the following a priori estimate. Lemma 2.1. Given T > 0, α ≥ 0 and a positive number µ, any solution to (7), (6a-c) (if there are multiple solutions) satisfies the following two estimates: (8) k w kB(0,T ) , k w kL6 (QT ) ≤ C(µ)e(α+%+µ/2)T k w0 kL2 (0,1) + k v kL2 (0,T ) , where C(µ) does not depend on α. Here and below we routinely use symbols c and C to denote (different) generic positive constants or positive-valued functions. Proof. Indeed, recall ([15]) that f (·, ·, w, wx ) ∈ L6/5 (QT ) and that the following energy equality holds for (7) treated as a linear equation with the source term f (x, t, w, wx ) + v(t)χ(a,b) (x), e.g., [15] (p. 142): 1 k w k2L2 (0,1) |t0 + 2

Zt Z1 (wx2 − αw2 + f (x, s, w, wx )w 0

0

−v(t)χ(a,b) (x)w)dxds = 0

(9)

∀t ∈ [0, T ].

Here and everywhere below, if there exist several solutions to (5), we always deal separately with a selected one, while noticing that all the estimates hold uniformly. Combining (9) and (6b) yields: Zt Z1 k w(·, t)

k2L2 (0,1)

wx2 (x, s) dxds

+ 2ν 0

0

Zt Z1 ≤k

w0 k2L2 (0,1)

+ 2(α + %)

≤

w dxds + 2 0

k

Zt Z1 2

0

w0 k2L2 (0,1)

v(t)χ(a,b) (x)wdxds 0

0

b−a + k v k2L2 (0,T ) µ

Zt k w(·, τ ) k2L2 (0,1) dτ

+ 2(α + % + µ/2) 0

≤

k

w0 k2L2 (0,1)

b−a + k v k2L2 (0,T ) µ

146

Khapalov

(10)

Zt

+ 2(α+%+µ/2)

 k w(·, τ ) k2 2 L (0,1) +2ν

0

Zτ Z1 0

 wx2 (x, s) dxds dτ

∀t ∈ [0, T ].

0

In the above we have used Young’s inequality 1 (vχ(a,b) )2 + µw2 , µ

2wvχ(a,b) ≤

which holds for any positive µ. Applying Gronwall-Bellman inequality to (10) yields the first estimate in (8) with respect to the B(0, T )-norm. The second estimate (with properly arranged generic constant) follows by the continuity of the embedding B(0, T ) into L6 (QT ) (e.g., [15], pp. 467, 75). This ends the proof of Lemma 2.1. Remark 2.1. Note that if α < 0 in (7), then, as (10) implies, (8) holds with no α in it. We now intend to evaluate the difference between the solution w to (7) and that to its truncated version (11)

in QT , v ∈ L2 (0, T ),

yt = yxx + αy + χ(a,b) v(t) y |x=0,1 = 0,

y |t=0 = y0 ∈ L2 (0, 1),

assuming that w0 = y0 . Denote z = w − y, then zt = zxx + αz − f (x, t, w, wx ) z |x=0,1 = 0,

in QT ,

z |t=0 = 0.

Similar to (9) and (10) we have, Zt Z1 k z(·, t) k2L2 (0,1) + 2

zx2 (x, s) dxds 0

Zt Z1 ≤ 2α

0

Zt Z1 2

z dxds + 2 0

0

zf (x, s, w, wx )dxds 0

0

Zt Z1 ≤ 2α

z 2 dxds + 2 k z kL6 (Qt ) k f (·, ·, w, wx ) kL6/5 (Qt ) 0

0


147

Zt Z1 ≤ 2α

z 2 dxds + 2c k z kB(0,t) k f (·, ·, w, wx ) kL6/5 (QT ) 0

0

Zt ≤ 2α

k z k2B(0,s) ds + δ k z k2B(0,t) 0

(12)

+

c2 k f (·, ·, w, wx ) k2L6/5 (QT ) ∀t ∈ [0, T ], δ

where we have used H¨ older’s and Young’s inequalities and the continuity of the embedding B(0, T ) into L6 (QT ), due to which, k z kL6 (QT ) ≤ c k z kB (0, T ).

(13) From (12), we have

Zt max k z(·, τ )

τ ∈(0,t)

k2L2 (0,1)

≤ 2α

k z k2B(0,s) ds + δ k z k2B(0,t) 0

+

c2 k f (·, ·, w, wx ) k2L6/5 (QT ) ∀t ∈ [0, T ]. δ

Hence, again from (12), Zt kz

k2B(0,t)

≤ 4α

k z k2B(0,s) ds + 2δ k z k2B(0,t) 0

+

2c2 k f (·, ·, w, wx ) k2L6/5 (QT ) ∀t ∈ [0, T ] δ

and (1 − 2δ) k z k2B(0,t) Zt (14)

≤ 4α

k z k2B(0,s) ds +

2c2 k f (·, ·, w, wx ) k2L6/5 (QT ) . δ

0

Making use of Gronwall-Bellman inequality, we derive from (14) that √ 2α 2c T (15) k z kB(0,T ) ≤ e 1−δ √ k f (·, ·, w, wx ) kL6/5 (QT ) , δ

148

Khapalov

provided that 1 0 0, α ≥ 0, δ ∈ (0, 1/2), and w0 = y0 , we have the following two estimates for the difference z = w − y between any corresponding solution w to (7), (6a-c) (if there are multiple ones) and the unique corresponding solution to (11): r1 2αT 1 5 k z kB(0,T ) , k z kL6 (QT ) ≤ Ce 1−δ √ (T 6 (1− 5 ) k w krL16 (QT ) δ

(18)

5

+ T 6 (1−

3r2 5

)

k wx krL22 (Q ) ) T

∀δ ∈ (0, 1/2),

where C does not depend on α. 3. Controllability properties of the truncated linear system (11). Here we would like to remind the reader some controllability properties of the linear system (11). √ Denote by λk = −(πk)2 + α, ωk (x) = 2 sin πkx, k = 1, . . . the eigenvalues and orthonormalized in L2 (0, 1) eigenfunctions of the spectral problem: ωxx + αω = λω, ω ∈ H01 (0, 1). It is well known that the general solution to (11) admits the following representation: 1  Z ∞ X y(x, t) = eλk t  y0 (r)ωk (r)dr  ωk (x) + k=1

∞ Z X

t

(19)

k=1 0

 eλk (t−τ ) 

0

Z1 0

 v(τ )χ(a,b) (r) ωk (r) dr  dτ ωk (x),


149

where the series converge in L2 (0, 1) uniformly over t ≥ 0. λk τ }∞ Let {qT,k }∞ k=1 be a biorthogonal sequence to {e k=1 in the closed subspace cl span{eλk τ | k = 1, . . . } of L2 (0, T ) ([18], [6]):

ZT eλk τ qT,l (τ ) dτ =

1, if k = l, 0, if k 6= l,

0

where k qT,k kL2 (0,T ) =

(20)

I X

dk (α, T ) = inf{k eλk t +

1 , dk (α, T )

bi eλi t kL2 (0,T ) | bi ∈ R, I = 1, 2, . . . }.

i=1,i6=k

R b Assume that a ± b are the irrational numbers. We need this to ensure that a sin πkx dx 6= 0 for all k − 1, . . . . Denote √ Z vT,k (τ ) = qT,k (T − τ ) ( 2 sin πkx dx)−1 , τ ∈ (0, T ), b

(21)

a

so that ZT Z1 (22)

λk (T −τ )

e 0

vT,l (τ )χ(a,b) (r) ωk (r) dr dτ =

1, if k = l, 0, if k 6= l.

0

From (19) and (22) it follows that, given the positive integer L and the real numbers a1 , . . . , aL , if one applies control (23)

vˆT (t) =

L X

ak vT,k (t),

t ∈ (0, T )

k=1

in (11), then

(24)

y(x, T ) =

L X k=1

ai ωk (x) +

∞ X k=1

 (−(πkt)2 +α)T

e



Z1 0

 y0 (r)ωk (r)dr  ωk (x),

150

Khapalov

where, by (20) and (21),

(25)

k vˆT kL2 (0,T ) ≤ γ(α, T ) =

L X k=1

√ Z 1 | ak ( 2 sin πkx dx)−1 | . dk (α, T ) b

a

Remark 3.1. It follows from (20) that γ(α, T ) in (25) is nonincreasing in T > 0. Also, from (24), (26)

k y(·, T ) −

L X

ai ωk (·) kL2 (0,1) ≤ eαT k y0 kL2 (0,1) .

k=1

Now we are ready to prove Theorem 1.3. 4. Proof of Theorem 1.3. The scheme of the proof is as follows. (1) Given the initial and target state u0 and uT , we steer the system at hand “close” to the zero-state (equilibrium) employing the constant negative bilinear controls only. (2) Using a ( sort of) locally controllability technique with only additive controls active, we steer the system “close” to a state suT for some small parameter s > 0. (3) Again, employing only constant positive bilinear controls, we “stretch” the latter state to the desirable length uT . Step 1: Approximate null-controllability. Take any T ∗ > 0. Then it follows from the proof of Lemma 2.1 that if one applies control pair k(t) ≡ α < 0, v(t) = 0 on (0, T ∗ ), the corresponding solution(s) to (5), (6a-c) can be made arbitrarily small in L2 (0, 1) by selecting appropriately small negative α. Indeed, it follows from (10) that µ 2(−α − % − ) 2

ZT Z1 w2 dxds ≤ k w0 k2L2 (0,1) + 0

b−a k v k2L2 (0,T ) . µ

0

As α → −∞, this implies that we can make k u(·, t∗ ) kL2 (0,1) as small as we wish for some t∗ ∈ (0, T ) (in general, t∗ can be different for different multiple solutions). This “smallness” is preserved on [t∗ , T ] by Remark 2.1, applied with v = 0 and the same α on (t∗ , T ). In other words, we have the global approximate controllability to the origin in the sense of Definition 1.1, just by using constant bilinear controls.


151

Step 2. From Step 1 it follows that, without loss of generality, we may further assume that the initial state u0 in (1) is arbitrarily small in L2 (0, 1). (Otherwise, we need to apply the argument of Step 1 on some “small” timeinterval (0, T ∗ ), T ∗ < T .) To prove Theorem 1.1, it is sufficient to show that any function like uT (x) =

L X

ai ωk (x)

k=1

can be approached by u(·, T ) arbitrarily close in the sense of (4). Fix any positive integer L and the real numbers a1 , . . . ,L . Given T > 0, select a parameter s ∈ (0, 1) and also µ and δ in Lemmas 2.1 and 2.2. By Step 1, without loss of generality we may assume that k u0 kL2 (0,1) ≤ s2 . Consider any ε ∈ [0, T /2] and apply on the interval (0, T − ε) the control pair (see (21)-(23) for notations) k(t) = α = 0,

vs,T −ε (t) = sˆ vT −ε = s

L X

ak vk,T −ε (t),

t ∈ (0, T − ε).

k=1

Then, in notations of Section 2 with α = 0, u = w = y + z, and, see (24), (27)

u(·, T − ε) = suT + (y(·, T − ε) − suT ) + z(·, T − ε),

where in view of (26) and Lemmas 2.1 and 2.2, applied with α = 0 on (0, T − ε), (28)

k y(·, T − ε) − suT kL2 (0,1) ≤ k u0 kL2 (0,1) ≤ s2 , k z(·, T − ε) kL2 (0,1)

(29)

min{r1 ,r2 } ≤ C(T ) k u0 kL2 (0,1) + s k vˆT −ε kL2 (0,T −ε) ) ,

as s → 0, where C(T ) does not depend on ε. Since r1 , r2 > 1, (27)-(29) yields that (30)

u(·, T − ε) = suT + p(·, T − ε),

where, as it follows from Remark 3.1, uniformly over ε ∈ [0, T /2], (31)

k p(·, T − ε) kL2 (0,1) = o(s).

Step 3. On the interval (T − ε, T ) we apply controls k(t) = α > 0,

v(t) = 0,

t ∈ (T − ε, T ).

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Khapalov

Then, again in notations of Section 2, applied now on the interval (T − ε, T ), (32)

u(·, T ) = y(·, T ) + z(·, T ).

Here, according to (19) and (30), applied on (T − ε, T ), 1  Z ∞ X 2 y(x, T ) = e(−(πk) +α)ε  u(r, T − ε)ωk (r)dr  ωk (x) k=1

= eαε

∞ X

0

 2 e−(πk) ε 

k=1

(33)

Z1

 u(r, T − ε)ωk (r)dr  ωk (x)

0

= eαε (suT + p(·, T − ε) + h(ε)(suT + p(·, T − ε)) ,

where h(ε) → 0 as ε → 0 (by continuity of solutions to (7) in time). In other words, in view of (31), (34)

y(·, T ) = seαε uT + seαε g(·, s, ε),

where (35)

k g(·, s, ε) kL2 (0,1) → 0 as s, ε → 0.

On the other hand, by Lemma 2.2, applied on (T − ε, T ) with v = 0: 2αε

5

r1

k z(·, T ) kL2 (0,1) ≤ Ce 1−δ (ε 6 (1− 5 ) k w krL16 (Q

(36)

5

+ ε 6 (1−

3r2 ) 5

T −ε,T )

k wx krL22 (QT −ε,T ) ),

where QT −ε,T = (0, 1) × (T − ε, T ). In turn by Lemma 2.1, applied on (T − ε, T ) with v = 0: k w kL6 (QT −ε,T ) , k wx kL2 (QT −ε,T ) (37)

≤ Ce(α+%+µ/2)ε k u(·, T − ε) kL2 (0,1) ,

for some constant C > 0. Step 4. Summarizing the above estimates, we select parameters α, ε, and s so that (A) s → 0+; 1 (B) eαε = ; s


153

(C) ε → 0, so that 2αε

5

r1

5

e 1−δ εmin{ 6 (1− 5 ); 6 (1−

3r2 )} 5

5

r1

5

= εmin{ 6 (1− 5 ); 6 (1−

3r2 5

2 )} − 1−δ

s

→ 0.

Under these conditions we have, firstly, that, in view of (A), (B) and (30), (31), the right-hand side of (37) is bounded above by a constant and, secondly, that, by (C) and (36), k z(·, T ) kL2 (0,1) → 0. Then from (32), (34) and (35) this yields that (38)

k u(·, T ) − uT kL2 (0,1) → 0,

which completes the proof of Theorem 1.3. 5. Proof of Theorems 1.1/1.2. This proof, in fact, is identical to that of Theorem 1.3, with the following minor differences. • In the proof of Theorem 1.3, in Step 2, we can select control vˆT /2 first and then, as ε → 0, apply it only on the interval (T − ε, T − ε − 0.5), i.e., the same additive control (but shifted in time) for all ε ∈ [0, T /2]. In this way Remark 3.1 is not necessary to use in (31). Analogously, in the proof of Theorem 1.1/1.2, in place of vˆt in the above, we can select any function v = vˆ(x, t), t ∈ (0, T /2), x ∈ ω. Then the argument of Theorem 1.3 will lead us to the convergence as in (38) to uT = y(·, T /2), which is the state of the truncated multidimensional linear version of (2.5) with α = 0 generated by the selected vˆ(x, t). It remains to recall that the latter is approximately controllable in L2 (Ω) at time T /2 (or any other positive time, due to the dual unique continuation property from an open set ω × (0, T /2)), i.e., the set of such y(·, T /2) is dense in L2 (Ω). • In several space dimensions Lemmas 2.1 and 2.2 are principally no different from the one dimensional case.

6. Concluding remarks. • It seems quite possible that the results of this article can be extended at no extra cost to boundary controls in place of the additive ones. • In the proof of Theorem 1.3 we followed the Fourier series approach, which is due to the delicate nature of the lumped additive controls involving the Riesz’s basis properties of the sequence of exponentials (see Section 3). However, as we showed it in the (sketch of the) proof of Theorems 1.1/1.2,

154

Khapalov this approach can be avoided in part when we are dealing with the “stable” locally distributed controls. From this viewpoint it seems very plausible that these theorems can be extended to the case of more general parabolic equations with variable coefficients. References.

[1] A. Baciotti, Local Stabilizability of Nonlinear Control Systems, Series on Advances in Mathematics and Applied Sciences, vol. 8, World Scientific, Singapore, 1992. [2] J.M. Ball, J.E. Mardsen, and M. Slemrod, Controllability for distributed bilinear systems, SIAM J. Contr. Opt., 1982, pp. 575-597. [3] V. Barbu, Exact controllability of the superlinear heat equation, Appl. Math. Opt., 42 (2000), pp. 73-89. [4] M.E. Bradley, S. Lenhart, and J. Yong, Bilinear optimal control of the velocity term in a Kirchhoff plate equation, J. Math. Anal. Appl.,238 (1999), 451-467. [5] C. Fabre, J.-P. Puel and E. Zuazua, Approximate controllability for the semilinear heat equations, Proc. Royal Soc. Edinburg,125A (1995), pp. 31-61. [6] H.O. Fattorini and D.L. Russell, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations, Quarterly of Appl. Mathematics, April, 1974, pp. 45-69. [7] E. Fernandez-Cara, Null controllability for semilinear heat equation, ESAIM: Control, Optimization and Calculus of Variations, (1997), pp. 87-103. [8] L.A. Fernandez and E. Zuazua, Approximate controllability for the semilinear heat equation involving gradient terms, J. Optim. Theory Appl.,101 (1999), pp. 307-328. [9] A. Fursikov and O. Imanuvilov, Controllability of evolution equations, Lect. Note Series 34, Res. Inst. Math., GARC, Seoul National University, 1996. [10] A.Y. Khapalov, Some aspects of the asymptotic behavior of the solutions of the semilinear heat equation and approximate controllability, J. Math. Anal. Appl.,194 (1995), pp. 858-882. [11] A.Y. Khapalov and R.R. Mohler, Reachable sets and controllability of bilinear time-invariant systems: A qualitative approach, IEEE Trans. on Autom. Control, 41 (1996), pp. 1342-1346.


155

[12] A.Y. Khapalov, Approximate controllability and its well-posedness for the semilinear reaction-diffusion equation with internal lumped controls, ESAIM: COCV, 4 (1999), pp. 83-98. [13] A.Y. Khapalov, Global approximate controllability properties for the semilinear heat equation with superlinear term, Rev. Mat. Complutense, 12 (1999), pp. 511-535. [14] A.Y. Khapalov, A class of globally controllable semilinear heat equations with superlinear terms, J. Math. Anal. Appl., 242 (2000), pp. 271-283. [15] O. H. Ladyzhenskaja, V.A. Solonnikov and N.N. Ural’ceva, Linear and Quasi-linear Equations of Parabolic Type, AMS, Providence, Rhode Island, 1968. [16] I. Lasiecka and R. Triggiani, Exact controllability of semilinear abstract systems with application to waves and plates boundary control problems, Appl. Math. Optim.,23 (1991), pp. 109-154. [17] S. Lenhart, Optimal control of convective-diffusive fluid problem, Math. Models and Methods in Appl. Sci., 5 (1995), pp. 225-237. [18] W.A.J. Luxemburg, and J. Korevaar, Entire functions and M¨ untz-Sz´ asz type approximation, Trans. of the AMS,157 (1971), pp. 23-37. [19] D. Tataru, Carleman estimates, unique continuation and controllability for anizotropic PDE’s, Contemporary Mathematics, 209 (1997), pp. 267-279.

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A Nonoverlapping Domain Decomposition for Optimal Boundary Control of the Dynamic Maxwell System

J. E. Lagnese 1 , Georgetown University, Washington, DC 20057 USA

1

Introduction

Let Ω be a bounded, open, connected set in IR3 with piecewise smooth, Lipschitz boundary Γ, and let T > 0. We consider the Maxwell system ( εE 0 − rot H + σE = F µH 0 + rot E = 0 in Q := Ω × (0, T ) (1.1) ν ∧ E − δ ν ∧ (H ∧ ν) = J on Σ := Γ × (0, T ), δ > 0, E(0) = E0 ,

H(0) = H0 in Ω

Here 0 = ∂/∂t, ν denotes the exterior pointing unit normal vector to Γ and ε = (εjk (x)), µ = (µjk (x)) and σ = (σ jk (x)) are 3×3 Hermitian matrices with L∞ (Ω) entries with ε and µ uniformly positive definite on Ω and σ ≥ 0. When J = 0 the boundary condition on Σ is the so-called Silver-M¨ uller condition. The function F ∈ L1 (0, T ; L2 (Ω)) is given while J is a control input and is taken from the class U = L2τ (Σ) := {J| J ∈ L2 (0, T ; L2 (Γ)), ν · J(t) = 0 for a.a. x ∈ Γ and a.a. t ∈ (0, T )} Function spaces of C-valued functions are denoted by capital roman letters, while function spaces of C 3 -valued functions are denoted by capital script P letters. We use α · β to denote the natural scalar product in C 3 , i.e., α · β = 3j=1 αj β j , and write h·, ·i for the natural scalar product in various function spaces such as L2 (Ω) and L2 (Ω). A subscript may sometimes be added to avoid confusion. The spaces L2 (Ω) and L2 (Ω) denote the usual spaces of Lebesque square integrable C-valued functions and C 3 -valued functions, respectively. 1

Research supported by the National Science Foundation through grant DMS-9972034

157

158

J. E. Lagnese We set H = L2 (Ω) × L2 (Ω) with weight matrix M = diag(ε, µ). Thus k(φ, ψ)k2H = hεφ, φi + hµψ, ψi

It will be proved below that for J ∈ U and (E0 , H0 ) ∈ H, (1.1) has a unique solution (E, H) ∈ C([0, T ]; H). We shall consider the optimal control problem Z inf (1.2) |J|2 dΣ + kk(E(T ), H(T )) − (E1 , H1 )k2H , k > 0, J∈U

Σ

subject to (1.1), where (E1 , H1 ) ∈ H is given. Problem (1.1), (1.2) admits a unique optimal control Jopt which is given through the optimality system consisting of (1.1), ( εP 0 − rot Q − σP = 0 µQ0 + rot P = 0 in Q (1.3) ν ∧ P + δ ν ∧ (Q ∧ ν) = 0 on Σ (P (T ), Q(T )) = k((E(T ), H(T )) − (E1 , H1 )) in Ω, and (1.4)

Jopt = Qτ := ν ∧ (Q ∧ ν)|Σ = Q|Σ − (Q|Σ · ν)ν.

The purpose of this paper is to develop a domain decomposition procedure to approximate the solution of the above optimality system. Remark 1.1. One may work with controls J supported on Σ1 := Γ1 ×(0, T ), where Γ1 is a nonempty, relatively open subset of Γ. Outside of the support of J one may replace the Silver-M¨ uller boundary condition by (for example) ν ∧ E = 0. The term −δ ν ∧ (H ∧ ν) represents boundary damping and is included in order to improve the regularity of the solution of (1.1). Without it the domain decomposition procedure becomes considerably more complicated (see section 5 for a discussion of this case). Prior work on domain decomposition (DD) for the dynamic Maxwell systems seems to have been confined to the direct problem and to time harmonic solutions; see [1], [2], [10], [11]. There is also little previous work on DD in the context of optimal boundary control of dynamic partial differential equations with penalization of the final state. For one-dimensional problems, we can refer to the work of Leugering [14] - [18] dealing with networks of dynamic string or beam equations on 1-d graphs. For higher dimensional problems, there seems to be only the paper [13], which is concerned with transmission problems for wave equations. On the other hand, for DD in other types of optimal control problems let us mention the work by Benamou [3] - [8], and Benamou and Després [9], where elliptic, parabolic and hyperbolic problems with constant coefficients in Ω are considered together with a cost functional which involves

Domain Decomposition for the Maxwell System

159

the entire state over space and time (in addition to the control). In these papers the authors use an extension of P. L. Lions’ method [19], originally obtained for elliptic problems. This same principle was employed in [13] and will also be suitable adapted to the Maxwell system considered in the present paper. The remainder of this paper is organized as follows. Existence, uniqueness and regularity of solutions of (1.1) is examined in the next section. A domain decomposition procedure for the optimality system (1.1) - (1.4) is introduced in Section 3, and its convergence is studied in section 4. Remarks concerning the case δ = 0, and limit of the optimality system and its domain decomposition as δ → 0, are provided in section 5.

2

Existence and uniqueness of solution

We set V = {φ ∈ L2 (Ω) : rot φ ∈ L2 (Ω), ν ∧ φ ∈ L2τ (Γ)}, Z Z 2 2 2 kφkV = (|φ| + | rot φ| )dx + |ν ∧ φ|2 dΓ. Ω

Γ

Let us first consider the problem ( εφ0 − rot ψ + σφ = εf µψ 0 + rot φ = µg in Q (2.1) ν ∧ φ − δ ψτ = 0 on Σ φ(0) = φ0 ,

ψ(0) = ψ0 in Ω

Set A=M

−1

−σ rot , − rot 0

Dom(A) = {(φ, ψ) ∈ V × V : ν ∧ φ − δ ψτ = 0 on Γ}. The system (2.1) may be formally written φ f U = AU + F, where U = ,F = ψ g φ0 U (0) = U0 := ψ0 0

(2.2)

Lemma 2.1. A is a m-dissipative operator in H. Proof. A is densely defined and from the Green’s formula Z hφ, rot ψi = hrot φ, ψi − ν · (φ ∧ ψ)dΓ, (φ, ψ) ∈ V × V, (2.3) Γ

160

J. E. Lagnese

we obtain 2 hAU, U iH = −hσφ, φi − δ

Z

√ |ν ∧ φ|2 dΓ + 2 −1 Imhrot ψ, φi,

U ∈ D(A),

Γ

so RehAU, U iH ≤ 0. Let (f, g) ∈ H and let φ be the unique solution in V of the variational equation Z 1 −1 (2.4) h(ε + σ)φ, χi + hµ rot φ, rot χi + (ν ∧ φ) · (ν ∧ χ) dΓ δ Γ = hg, rot χi + hεf, χi, ∀χ ∈ V. Set ψ = g − µ−1 rot φ ∈ L2 (Ω). Then (2.4) reads Z 1 hψ, rot χi = h(ε + σ)φ, χi − hεf, χi + (ν ∧ φ) · (ν ∧ χ) dΓ, δ Γ

∀χ ∈ V.

It follows that rot ψ ∈ L2 (Ω) and that (ε + σ)φ − rot ψ = εf in Ω, δ ψτ = ν ∧ φ on Γ. φ φ f Therefore ∈ D(A) and (I − A) = . ψ ψ g Corollary 2.1. (1) If (φ0 , ψ0 ) ∈ H and (f, g) ∈ L1 (0, T ; H), then (2.1) has a unique mild solution (φ, ψ) ∈ C([0, T ]; H) and k(φ(t), ψ(t))kL∞ (0,T ;H) ≤ C k(φ0 , ψ0 )kH + k(f, g)kL1 (0,T ;H) . (2) If (φ0 , ψ0 ) ∈ D(A) and (f, g) ∈ C 1 ([0, T ]; H), then (φ, ψ) ∈ C([0, T ]; D(A)). Lemma 2.2. Let (φ0 , ψ0 ) ∈ H and (f, g) ∈ L1 (0, T ; H). Then the solution of (2.1) satisfies ν ∧ φ|Σ ∈ L2τ (Σ). Proof. First suppose that (φ0 , ψ0 ) ∈ D(A) and (f, g) ∈ C 1 ([0, T ]; H). We then have Z t Z Z 1 1 1 t 2 2 hσφ, φi dt + |ν ∧ φ|2 dΓdt k(φ(t), ψ(t))kH − k(φ0 , ψ0 )kH + 2 2 δ 0 Γ 0 Z t Z t √ − 2 −1 hrot ψ, φi dt = h(f, g), (φ, ψ)iH dt. 0

0

It follows easily that Z k(φ, ψ)k2L∞ (0,T ;H)

+ 0

T

1 hσφ, φi dt + δ

The result now follows by density.

Z

T

Z |ν ∧ φ|2 dΓdt

n o ≤ C k(φ0 , ψ0 )k2H + k(f, g)k2L1 (0,T ;H) . 0

Γ


161

By transposition, we have Theorem 2.1. If (E0 , H0 ) ∈ H, F ∈ L2 (0, T ; L2 (Ω)) and J ∈ L2τ (Σ), (1.1) has a unique solution (E, H) ∈ C([0, T ]; H). We remark that (E, H) satisfies Z (2.5) h(E(T ), H(T )), (φ0 , ψ0 )iH −

T

h(E(t), H(t)), (f (t), g(t))iH dxdt Z T Z 1 = h(E0 , H0 ), (φ(0), ψ(0))iH + hF (t), φ(t)idt + J · (ν ∧ φ) dΣ δ Σ 0 0

for all (φ0 , ψ0 ) ∈ H and (f, g) ∈ L1 (0, T ; H), where (φ, ψ) is the solution of ( εφ0 − rot ψ − σφ = εf µψ 0 + rot ψ = µg in Q (2.6) ν ∧ φ + δ ψτ = 0 on Σ φ(T ) = φ0 ,

ψ(T ) = ψ0 in Ω

Theorem 2.2. If (E0 , H0 ) ∈ H, F ∈ L2 (0, T ; L2 (Ω)) and J ∈ L2τ (Σ), the solution of (1.1) satisfies ν ∧ E|Σ ∈ L2τ (Σ). Proof. If J = 0 the result follows from Lemma 2.2. Suppose that F = 0 and assume that (E0 , H0 ) and J are such that (E, H) ∈ C([0, T ]; V × V ). ˆ Σ , where This will hold if, for example, (E0 , H0 ) ∈ V × V , J = ν ∧ J| 2 2 1 1 Jˆ ∈ C ([0, T ]; L (Ω)) ∩ C ([0, T ]; H (Ω)), and ν ∧ E0 − δ H0τ = J(0) on Γ. ˆ = E − J, ˆ H ˆ = H, one As may be seen by making the change of variables E ˆ ˆ has (E, H) ∈ C([0, T ]; D(A)) and therefore (E, H) ∈ C([0, T ]; V × V ). By calculating as in Lemma 2.2 we obtain Z t Z Z 1 1 t 2 hσE, Ei dt + |ν ∧ E|2 dΓdt k(E(t), H(t))kH + 2 δ 0 0 Γ Z tZ 1 1 2 = k(E0 , H0 )kH + Re J · (ν ∧ E) dΓdt. 2 δ 0 Γ From the boundary condition we have Re J · (ν ∧ E) =

1 |ν ∧ E|2 + |J|2 − δ2 |Hτ |2 , 2

from which one obtains Z (2.7) k(E(t), H(t))k2H + 2

t 0

The result now follows by density.

Z tZ

1 ( |ν ∧ E|2 + δ|Hτ |2 ) dΓdt 0 Γ δ Z Z 1 t 2 = k(E0 , H0 )kH + |J|2 dΓdt. δ 0 Γ

hσE, Ei dt +

162

3

J. E. Lagnese

Domain decomposition

In this section we describe an iterative domain decomposition for the optimality 3 system (1.1), (1.3), (1.4). Let {Ωi }m i=1 be bounded domains in IR with piecewise smooth, Lipschitz boudaries such that Ωi ∩ Ωj = ∅, i 6= j,

Ωi ⊂ Ω, i = 1, . . . , m,

Ω=

m [

Ωi .

i=1

We set Γij = ∂Ωi ∩ ∂Ωj = Γji , i 6= j,

Γi = ∂Ωi ∩ Γ,

γi =

[

Γij .

j:Γij 6=∅

Then ∂Ωi = γi ∪ Γi . It is assumed that each Γi and Γij is either empty or has a nonempty interior. We further set Qi = Ωi × (0, T ),

Σij = Γij × (0, T ),

Σi = Γi × (0, T ),

Si = γi × (0, T ).

Let (E, H), (P, Q) be the solution of the optimality system (1.1), (1.3), (1.4) with (E0 , H0 ) ∈ H. The global optimality system may be formally expressed as the coupled system ( εi Ei0 − rot Hi + σi Ei = Fi µi Hi0 + rot Ei = 0 in Qi (3.1) νi ∧ Ei − δHiτ = Qiτ on Σi Ei (0) = E0i ,

(3.2)

Hi (0) = H0i in Ωi ,

( εi Pi0 − rot Qi − σi Pi = 0 µi Q0i + rot Pi = 0 in Qi νi ∧ Pi + δQiτ = 0 on Σi (Pi (T ), Qi (T )) = k((Ei (T ), Hi (T )) − (E1i , H1i )) in Ωi ,

together with the interface conditions ( νi ∧ Ei = −νj ∧ Ej , νi ∧ Hi = −νj ∧ Hj , (3.3) νi ∧ Pi = −νj ∧ Pj , νi ∧ Qi = −νj ∧ Qj on Σij . The subscript i on E, H, P, Q indicates restriction to Qi ; for the coefficients and data the subscript indicates restriction to Ωi . The vector νi is the unit exterior pointing normal vector to ∂Ωi . The subscript τ indicates the tangential component, c.f. (1.4). The interface conditions (3.3) are realized in a weak sense


163

by the solution of the global optimality system and will hold in the sense of traces if rot E(t), rot H(t), rot P (t), rot Q(t) ∈ L2 (Ω), 0 < t < T. Remark 3.1. Although the tangential components of E, H, P, Q are continuous across an interface, in general there may be a discontinuity in their normal components. It is easy to see that (3.3) will hold if and only if Eiτ := νi ∧ (Ei ∧ νi ) = Ei − (Ei · νi )νi = Ejτ on Σij and similarly for the remaining three interface conditions. We further note that (3.3) is equivalent to (3.4)

νi ∧ Ei − αHiτ − βQiτ = −νj ∧ Ej − αHjτ − βQjτ νi ∧ Pi + αQiτ − βHiτ = −νj ∧ Pj + αQjτ − βHjτ ,

where α, β are nonzero constants. Indeed, by interchanging i and j in (3.4) and adding the results to (3.4) we find that E and P satisfy (3.3) and that αHiτ + βQiτ = αHjτ + βQjτ ,

αQiτ − βHiτ = αQjτ − βHjτ ,

hence Hiτ = Hjτ and Qiτ = Qjτ . Let us now consider the local iterations ( εi (Ein+1 )0 − rot Hin+1 + σi Ein+1 = Fi µi (Hin+1 )0 + rot Ein+1 = 0 in Qi (3.5) n+1 νi ∧ Ein+1 − δHiτ = Qn+1 on Σi iτ Ein+1 (0) = E0i , ( (3.6)

Hin+1 (0) = H0i in Ωi ,

εi (Pin+1 )0 − rot Qn+1 − σi Pi = 0 i n+1 0 n+1 µi (Qi ) + rot Pi =0 in Qi νi ∧ Pin+1 + δQn+1 = 0 on Σi iτ

(Pin+1 (T ), Qn+1 (T )) = k((Ein+1 (T ), Hin+1 (T )) − (E1i , H1i )) in Ωi , i ( n+1 n − βQn , νi ∧ Ein+1 − αHiτ − βQn+1 = −νj ∧ Ejn − αHjτ jτ iτ (3.7) n+1 n+1 n+1 n n n νi ∧ Pi + αQiτ − βHiτ = −νj ∧ Pj + αQjτ − βHjτ on Σij , Set n λnij = −νj ∧ Ejn − αHjτ − βQnjτ |Σij n ρnij = −νj ∧ Pjn + αQnjτ − βHjτ |Σij

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and let Hi = L2 (Ωi ) × L2 (Ωi ) with weight matrix Mi = diag(εi , µi ). We denote by h·, ·ii the natural scalar product in L2 (Ωi ). Theorem 3.1. Assume that α > 0, β > 0. If λnij , ρnij ∈ L2τ (Σij ), ∀j : Γij 6= ∅, problem (3.5) - (3.7) has a unique solution such that (Ein+1 , Hin+1 ) ∈ C([0, T ]; Hi ), (Pin+1 , Qn+1 ) ∈ C([0, T ]; Hi ), and all of the traces appearing in i the boundary conditions in (3.5) - (3.7) are in L2τ . Proof. This theorem is a consequence of the fact that (3.5) - (3.7) is the optimality system for the local optimal control problem Z

Z 1 X |Jij |2 + |βHiτ + ρnij |2 dΣ |Ji | dΣ + β Σi Σij 2

j:Γij 6=∅

+ kk(Ei (T ) − E1i , Hi (T ) − H1i )k2Hi → inf

Ji ,Jij

subject to (

(3.8)

εi Ei0 − rot Hi + σi Ei = Fi µi Hi0 + rot Ei = 0

in Qi

νi ∧ Ei − δHiτ = Ji on Σi νi ∧ Ei − αHiτ = Jij + λnij on Σij Ei (0) = E0i ,

Hi (0) = H0i in Ωi ,

where Ji ∈ L2τ (Σi ), Jij ∈ L2τ (Σij ), as may be directly verified. Since α > 0, problem (3.8) has the same structure as (1.1) and, from Theorems 2.1 and 2.2, its solution satisfies (Ei , Hi ) ∈ C([0, T ]; Hi ), νi ∧ Ei , Hiτ ∈ L2τ (Σi ) ∪ L2τ (Si ). Therefore (Pin+1 , Qn+1 ) satisfies a system of the form i ( εi Pi0 − rot Qi − σi Pi = 0 µi Q0i + rot Pi = 0 in Qi νi ∧ Pi + δQiτ = 0 on Σi νi ∧ Pi + αQiτ ∈ L2τ (Σij ) on Σij (Pi (T ), Qi (T )) ∈ Hi , and therefore (Pin+1 , Qn+1 ) ∈ C([0, T ]; Hi ), νi ∧ Pi , Qiτ ∈ L2τ (Σi ) ∪ L2τ (Si ). i As a consequence of Theorem 3.1, it follows that the iteration (3.5) - (3.7) is well defined if α > 0, β > 0, and λ0ij , ρ0ij ∈ L2τ (Σij ), ∀j : Γij 6= ∅.

4

Convergence

In this section we prove that the solutions {(Ein+1 , Hin+1 )}m i=1 , m {(Pin+1 , Qn+1 )} of the local optimality systems (3.5) (3.7) converge i=1 i


165

m to the solution {(Ei , Hi )}m i=1 , {(Pi , Qi )}i=1 of the global optimality system (3.1) - (3.3). Set

en , H e n ) = (E n − Ei , H n − Hi ), (E i i i i

en ) = (P n − Pi , Qn − Qi ), (Pein , Q i i i

n ≥ 1.

en , H e n ), (Pen , Q en ) satisfy Then (E i i i i ( n+1 e n+1 = 0 e e n+1 + σi E εi (E )0 − rot H i i i e n+1 )0 + rot E en+1 = 0 µi (H in Qi i i (4.1) e n+1 − δH e n+1 = Q e n+1 on Σi νi ∧ E i iτ iτ n+1 n+1 e e E (0) = H (0) = 0 in Ωi , i

i

( e n+1 − σi Pei = 0 εi (Pein+1 )0 − rot Q i en+1 )0 + rot Pen+1 = 0 µi (Q in Qi i i

(4.2)

en+1 = 0 on Σi νi ∧ Pein+1 + δQ iτ n+1 n+1 e e e e n+1 (T )) in Ωi , (Pi (T ), Qi (T )) = k(Ein+1 (T ), H i (

(4.3)

ñ , e n+1 − αH e n+1 − β Q en+1 = λ νi ∧ E ij i iτ iτ n+1 n+1 n+1 n e e e νi ∧ Pi + αQiτ − β Hiτ = ρ˜ij on Σij ,

where ˜ n = −νj ∧ E en − αH e n − βQ en , λ ij j jτ jτ n n n n e e e ρ˜ij = −νj ∧ Pj + αQjτ − β Hjτ . ˜ 0 , ρ˜0 ∈ L2 (Σij ), Lemma 4.1. Assume that α > 0, β > 0, and that λ τ ij ij ∀j : Γij 6= ∅. i = 1, . . . , m. Then E n+1 = E n − (F n+1 + F n ),

n = 1, 2, . . . ,

where 1X 2 m

E n+1 =

i=1

Z

α2 + β 2 en+1 |2 ) e n+1 |2 + |Q (|H iτ iτ β Si +

F

n+1

=

m X

1 e n+1 |2 + |νi ∧ Pen+1 |2 ) dΣ, (|νi ∧ E i i β

α en+1 (T ), H e n+1 (T ))k2 + α k(Pen+1 (0), Q en+1 (0))k2 (1−k2 ) k(E Hi Hi i i i i 2β 2β i=1 Z α T en+1 , E en+1 ii + hσi Pen+1 , Pen+1 ii ) dt + (hσi E i i i i β 0 Z α n+1 2 α e n+1 2 α e n+1 ) dΣ . e e n+1 · Q + δ |H | + (1 + δ | + )| Q Re ( H iτ iτ iτ β iτ β β Σi k+

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Proof. From Green’s formula (2.3) we have Z

T

(4.4) 0 = 0

e n+1 , Pen+1 ii en+1 )0 − rot H e n+1 + σi E {hεi (E i i i i

e n+1 )0 + rot E en+1 , Q en+1 ii } dt = kk(E en+1 (T ), H e n+1 (T ))k2 + hµi (H Hi i i i i i Z Z e n+1 | dΣ + Re e n+1 · (νi ∧ Pen+1 ) + (νi ∧ E e n+1 ) · Q en+1 } dΣ. + |Q {H Σi

iτ

Si

iτ

i

i

iτ

Use of (4.3) in the last integral in (4.4) gives Z n+1 n+1 2 e e en+1 | dΣ (4.5) 0 = kk(Ei (T ), Hi (T ))kHi + |Q iτ Σi Z ˜ n )} dΣ e n+1 |2 + |Q en+1 |2 ) + Re (H e n+1 · ρñij ) + Re (Q e n+1 · λ + {β(|H ij iτ iτ iτ iτ Si

We have 2 e n+1 · ρñ ) = 1 |νi ∧ Pe n+1 |2 + α |Q en+1 |2 − β |H e n+1 |2 Re (H ij iτ i 2β 2β iτ 2 iτ 1 n 2 α en+1 − |˜ ρ | + Re (νi ∧ Pein+1 ) · Q iτ 2β ij β α2 e n+1 2 β e n+1 2 n+1 2 ˜ n ) = 1 |νi ∧ E en+1 · λ e Re (Q | + | − |Qiτ | |H ij iτ i 2β 2β iτ 2 1 ñ 2 α e n+1 ) · H e n+1 . − |λij | − Re (νi ∧ E i iτ 2β β

Substitution into (4.5) yields

en+1 (T ), H e n+1 (T ))k2 + (4.6) 0 = kk(E Hi i i Z

Z Σi

en+1 | dΣ |Q iτ

α2 + β 2 e n+1 |2 ) + 1 (|νi ∧ E en+1 |2 + |νi ∧ Pen+1 |2 ) e n+1 |2 + |Q + (|H iτ iτ i i 2β 2β Si α e n+1 − (νi ∧ E en+1 ) · H e n+1 ) dΣ + Re((νi ∧ Pein+1 ) · Q iτ i iτ β Z 1 X ˜ n |2 + |˜ − (|λ ρnij |2 )dΣ. ij 2β Σij j:Γij 6=∅


167

From (4.1) and Green’s formula we calculate Z

T

0= 0

en+1 , E en+1 )0 − rot H e n+1 + σi E en+1 ii {hεi (E i i i i

e n+1 )0 + rot E e n+1 , H e n+1 ii } dt = 1 k(E e n+1 (T ))k2 en+1 (T ), H + hµi (H Hi i i i i i 2 Z T √ en+1 , E en+1 ii − 2 −1 ImhH e n+1 , rot E e n+1 i}dt + {hσi E i i i i 0 Z TZ e n+1 · (νi ∧ E e n+1 ) dΣ + H iτ i 0

∂Ωi

and therefore Z

(4.7)

e n+1 · (νi ∧ E e n+1 ) dΣ = 1 k(E e n+1 (T ))k2H en+1 (T ), H H iτ i i i i 2 Si Z T Z en+1 , E en+1 ii dt + e n+1 |2 + Re (H e n+1 · Q e n+1 )} dΣ. + hσi E {δ|H i i iτ iτ iτ

− Re

0

Σi

Similarly, Z

e n+1 · (νi ∧ Pen+1 ) dΣ = 1 k(Pen+1 (0), Q en+1 (0))k2 Q Hi iτ i i i 2 Si Z Z T k2 en+1 n+1 n+1 e n+1 2 e e e n+1 |2 dΣ. − k(Ei (T ), Hi (T ))kHi + hσi Pi , Pi ii dt + δ |Q iτ 2 0 Σi

(4.8) Re

Upon substituting (4.7) and (4.8) into (4.6) we obtain α en+1 (T ), H e n+1 (T ))k2H (1 − k2 ) k(E i i i 2β Z T α en+1 e n+1 , E en+1 (0))k2H + α en+1 ii + hσi Pen+1 , Pen+1 ii ) dt + (hσi E k(Pi (0), Q i i i i i i 2β β 0 Z α n+1 2 α en+1 2 α e n+1 ) dΣ e e n+1 · Q + δ |H iτ | + (1 + δ )|Qiτ | + Re (Hiτ iτ β β β Σi Z 1 X ˜ n |2 + |˜ − (|λ ρnij |2 )dΣ, ij 2β Σij

(4.9) 0 = Ein+1 + k +

j:Γij 6=∅

where Ein+1 =

1 2

Z

α2 + β 2 e n+1 |2 ) + 1 (|νi ∧ E en+1 |2 + |νi ∧ Pen+1 |2 ) dΣ. e n+1 |2 + |Q (|H iτ iτ i i β β Si

168

J. E. Lagnese We next calculate 1 ñ 2 1 en |2 + |νi ∧ Pen |2 ) ρnij |2 ) = (|λ | + |˜ (|νi ∧ E j j 2β ij 2β α2 + β 2 e n 2 n enjτ |2 ) + α Re{(νj ∧ E ejn ) · H e jτ enjτ } + − (νj ∧ Pejn ) · Q (|Hjτ | + |Q 2β β n ejn ) · Q e njτ + (νj ∧ Pejn ) · H e jτ + Re{(νj ∧ E }.

(4.10)

It follows from (4.9) and (4.10), upon summing over i, that m X

α en+1 (T ), H e n+1 (T ))k2H (1 − k2 ) k(E i i i 2β i=1 Z T α en+1 en+1 , E en+1 (0))k2 + α e n+1 ii + hσi Pe n+1 , Pen+1 ii ) dt + (hσi E k(Pi (0), Q Hi i i i i i 2β β 0 Z α n+1 2 α en+1 2 α e n+1 ) dΣ e e n+1 · Q + δ |H iτ | + (1 + δ )|Qiτ | + Re (Hiτ iτ β β β Σi Z α n ein ) · H e iτ e niτ }dΣ − Re{(νi ∧ E − (νi ∧ Pein ) · Q β Si Z n n n n e e e e − Re{(νi ∧ Ei ) · Qiτ + (νi ∧ Pi ) · Hiτ }dΣ , 0 = E n+1 − E n +

k+

Si

that is, (4.11) 0 = E n+1 − E n + F n+1 m Z X α n ein ) · H e iτ eniτ }dΣ − Re{(νi ∧ E − (νi ∧ Pein ) · Q β Si i=1 Z n n n n e e e e + Re{(νi ∧ Ei ) · Qiτ + (νi ∧ Pi ) · Hiτ }dΣ . Si

From (4.5) we have

(4.12)

−

m Z X i=1

n ein ) · Q e niτ + (νi ∧ Pein ) · H e iτ Re{(νi ∧ E }dΣ Si

Z m X n n 2 e e = kk(Ei (T ), Hi (T ))kHi + i=1

n 2 e |Qiτ | dΣ , Σi


169

and from (4.7) and (4.8) we obtain (4.13)

− =

Z m X α β i=1 m X i=1

en) · H e n − (νi ∧ Pen ) · Q e n }dΣ Re{(νi ∧ E i iτ i iτ Si

α ein (T ), H e in (T ))k2H + α k(Pein (0), Q eni (0))k2H (1 − k2 )|(E i i 2β 2β Z α T en, E en ii + hσi Pen , Pen ii ) dt + (hσi E i i i i β 0 Z α n 2 n 2 n n e e e e + (δ|Hiτ | + δ|Qiτ | + Re(Hiτ · Qiτ ))dΣ . β Σi

Thus the sum of (4.12) and (4.13) equals F n so that (4.11) may be written E n+1 = E n − (F n+1 + F n ).

(4.14)

We now prove that under some additional hypotheses, (4.14) implies convergence of the iteration scheme. Theorem 4.1. In addition to the assumptions of Lemma 4.1, suppose that ! r 1 β2 β δ> + 1 , α/β < 2k/(k2 − 1) if k > 1, − + 2 α α2 that ε and µ are scalar valued functions with εi , µi ∈ C 2 (Ωi ). i = 1, . . . , m,

Then for

ein , H e in ) → 0, (Pein , Q eni ) → 0 weakly* in L∞ (0, T ; Hi ) (E e n (T ), H e n (T )) → 0, (Pen (0), Q en (0)) → 0 strongly in Hi (E i i i i n n n e e e e niτ |Σ → 0 strongly in L2τ (Σi ) νi ∧ Ei |Σi → 0, Hiτ |Σi → 0, νi ∧ Pi |Σi → 0, Q i n n n n e e e e νi ∧ Ei |Σ → 0, Hiτ |Σ → 0, νi ∧ Pi |Σ → 0, Qiτ |Σ → 0 weakly in L2τ (Σij ) ij

ij

ij

ij

Proof. It follows from (4.14) that E

n+1

= E −2 1

n+1 X0

F p,

p=1

where

n+1 X0

cp = (c1 + cp+1 )/2 +

n X

cp .

Under the stated condition on the

p=2

p=1

parameter δ, the quadratic form α e n+1 2 α en+1 2 α e n+1 ) e n+1 · Q δ |H iτ | + (1 + δ )|Qiτ | + Re (Hiτ iτ β β β

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J. E. Lagnese

is positive definite. (Note that any δ > 1/2 will satisfy the hypothesis.) If k > 1 and if we further restrict α/β so that α k+ (1 − k2 ) > 0. 2β then ∞ X

F p converges and {E n }∞ n=1 is a bounded sequence.

p=1

The convergence of

P

F p then implies that

en (T ), H e n (T )) → 0 and (Pen (0), Q en (0)) → 0 strongly in Hi , i = 1, . . . , m, (E i i i i n n e |Σ → 0 and Q e |Σ → 0 strongly in L2 (Σi ), i = 1, . . . , m, H iτ

iτ

i

τ

i

en+1 → 0, σi Pe n+1 → 0 weakly in L2 (Qi ). σi E i i From (4.1) and (4.2), e n+1 |Σ → 0, νi ∧ Pe n+1 |Σ → 0 strongly in L2 (Σi ) νi ∧ E τ i i i i n n e e (P (T ), Q (T )) → 0 strongly in Hi . i

i

˜ n , ρñ are bounded in L2 (Σij ). Therefore The boundedness of E n implies that λ τ ij ij en+1 , H e n+1 ), (Pen+1 , Q en+1 ) are bounded in L∞ (0, T ; Hi ), i = 1, . . . , m. (E i i i i It follows that, on a subsequence n = nk of the positive integers, en+1 , H e n+1 ) → (E ei , H e i ) weakly* in L∞ (0, T ; Hi ) (E i i en+1 ) → (Pei , Q ei ) weakly* in L∞ (0, T ; Hi ) (Pen+1 , Q i

(4.15)

i

en+1 → Ai , νi ∧ E i e νi ∧ P n+1 → Ci , i

e n+1 → Bi weakly in L2τ (Si ) H iτ n+1 e Qiτ → Di weakly in L2τ (Si )

ei , H e i ), (Pei , Q ei ) ∈ L∞ (0, T ; Hi ). for some Ai , Bi , Ci Di ∈ L2τ (Si ), (E ∞ Let (φ, ψ) ∈ C (Ω × [0, T ]). We have Z

T

en+1 , φi + hµi (H en+1 )0 − rot H e n+1 + σi E e n+1 )0 + rot E en+1 , ψi] dt [hεi (E i i i i i Z T en+1 (T ), H e n+1 (T )), (φ(T ), ψ(T ))iH − en+1 , εi φ0 − rot ψ − σi φi = h(E [hE i i i i 0 Z e n+1 , µi ψ 0 + rot φi] dt + e n+1 · (νi ∧ φ) + (νi ∧ E e n+1 ) · ψτ }dΣ + hH {H i i i Σi Z e n+1 · (νi ∧ φ) + (νi ∧ E e n+1 ) · ψτ }dΣ + {H i i

0=

0

Si


171

Upon passing to the limit through the subsequence n = nk we obtain Z

T

0

ei , εi φ0 − rot ψi + hH e i , µi ψ 0 + rot φi] dt [hE Z = {Bi · (νi ∧ φ) + Ai · ψτ }dΣ,

∀(φ, ψ) ∈ C ∞ (Ω × [0, T ]).

Si

Therefore ( e0 − rot H ei = 0 εi E i e 0 + rot E ei = 0 µi H i

in Qi

ei = H e iτ = 0 on Σi νi ∧ E ei = Ai , H e iτ = Bi on Si νi ∧ E

(4.16)

ei (0) = H e i (0) = E ei (T ) = H e i (T ) = 0 in Ωi . E Similarly, ( ei = 0 εi Pei0 − rot Q e 0 + rot Pei = 0 in Qi µi Q i e iτ = 0 on Σi νi ∧ Pei = Q e iτ = Di on Si νi ∧ Pei = Ci , Q

(4.17)

e i (0) = Pei (T ) = Q e i (T ) = 0 in Ωi . Pei (0) = Q ei = We may now use a unique continuation argument to conclude that E e e e Hi = 0 and Pi = Qi = 0 in Qi for i = 1, . . . , m, provided that (4.16) and (4.17) hold. Indeed, suppose that Ωi is a region adjacent to Γ, that is Γi 6= ∅. From (4.16) we have ei (t) = εi E

Z

t

0

e i (s) ds, rot H

e i (t) = − µi H

Z

t

ei (s) ds, rot E

0

hence (4.18)

ei ) = div(µi H e i ) = 0 in Qi , div(εi E ei ) = νi · (µi H e i ) = 0 on Σi , νi · (εi E

since νi · rot φ is a tangential differential operation on νi ∧ φ on Σi . Since eiτ = E ei − (E ei · νi )νi on Σi 0=E

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J. E. Lagnese

we have ei ) = (E ei · νi )νi · (εi νi ) on Σi , 0 = νi · (εi E ei = 0 on Σi since µi is positive definite. Therefore which implies that νi · E ei = νi · E ei = 0 on Σi νi ∧ E e i . Therefore (E ei , H e i ) is a solution of the dynamic Maxwell and, similarly, for H system and of (4.18) in Qi , and has zero Cauchy data on Σi . Moreover, since ei , H e i ) vanish at t = 0 and t = T , they may be continued by zero to Ω×{t < 0} (E and to Ω × {t > T } as solutions of the Maxwell system satisfying (4.18) having zero Cauchy data on Γi × (−∞, ∞). It then follows from a result of Eller [12, ei = H e i = 0 in Qi . Therefore, E ei = H e i = 0 in Qi for every Corollary 5.2] that E e i ). It follows that index i such that Γi 6= ∅. The argument is the same for (Pei , Q for such i we have Ai = Bi = Ci = Di = 0 and therefore the convergence in (4.15) is through the entire sequence of positive integers. Now suppose Ωj is a region adjacent to a boundary region, i.e., to a region Ωi such that such that Γi 6= ∅. Then Σij = ∂Ωi ∩ ∂Ωj 6= ∅ and we have (4.19)

en+2 − αH en+1 − αH e n+2 − β Q e n+2 = −νj E e n+1 − β Q e n+1 , νi E i iτ iτ j jτ jτ

(4.20)

e n+1 − αH en − αH e n+1 − β Q en+1 = −νi E e n − βQ en on Σij . νj E i iτ iτ j jτ jτ

Since for the index i, convergence is through the entire sequence of positive integers, if we pass to the weak L2τ (Σij ) limit in (4.19) and (4.20) through the subsequence n = nk we obtain −Aj − αBj − βDj = 0,

Aj − αBj − βDj = 0 on Σij .

−Cj + αDj − βBj = 0,

Cj + αDj − βBj = 0 on Σij ,

Similarly,

hence Aj = Bj = Cj = Dj = 0, that is to say (4.21)

ej = αH e jτ = νj ∧ Pej = β Q e jτ = 0 on Σij . νj ∧ E

ej = H e j = Pej = Q ej = 0 The same unique continuation argument as above gives E in Qj . One may now proceed step-by-step into the remaining interior regions ej = H e j = Pej = Q ej = 0 in Qj for j = 1, . . . , m. Ωj and conclude that E

5. FURTHER COMMENTS

5

173

Further comments

Consider the problem (1.1) without boundary damping: ( εE 0 − rot H + σE = F µH 0 + rot E = 0 in Q (5.22) ν ∧ E = J on Σ E(0) = E0 ,

H(0) = H0 in Ω

For (E0 , H0 ) ∈ H, F ∈ L1 (0, T ; L2 (Ω)) and J ∈ L2τ (Σ), (5.22) has a unique solution which is continuous on [0, T ] into X 0 , where X ,→ H is given by X = {(φ, ψ) ∈ V × V : ν × φ|Γ = 0, div(µψ) ∈ L2 (Ω), ν · (µψ)|Γ = 0} and X 0 denotes the dual space of X with respect to H. The appropriate cost functional for (5.22) analogous to (1.2) is therefore Z J (J) = (5.23) |J|2 dΣ + kk(E(T ), H(T )) − (E1 , H1 )k2X 0 Σ

where (E1 , H1 ) ∈ H, and the optimality system for the problem inf J∈U J (J) subject to (5.22) consists of (5.22), ( εP 0 − rot Q − σP = 0 µQ0 + rot P = 0 in Q (5.24) ν ∧ P = 0 on Σ (P (T ), Q(T )) = kA−1 ((E(T ), H(T )) − (E1 , H1 )) in Ω, Jopt = Qτ |Σ ,

(5.25)

where A : X 7→ X 0 is the canonical isomorphism. Note that (P (T ), Q(T )) is determined by solving a stationary Maxwell-type system of 6 first order partial differential equations. Therefore any DD for the optimality system (5.22), (5.24) and (5.25) must include a DD procedure for the approximation of (P (T ), Q(T )). This makes the overall approximation scheme considerably more complicated than that given above for the regularized optimality system (see [13], where a DD is given for Neumann boundary optimal control of the wave equation with penalization of the final state, and which leads to similar considerations). Now consider the regularized problem ( ε(E δ )0 − rot H δ + σE δ = F µ(H δ )0 + rot E δ = 0 in Q (5.26) ν ∧ E δ − δHτδ = J on Σ E δ (0) = E0 ,

H δ (0) = H0 in Ω.

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J. E. Lagnese

The solution (E δ , H δ ) is continuous on [0, T ] into H. Suppose one considers the optimal control problem inf J∈U J (J) subject to (5.26), where J (J) is given by (5.23). The optimality system is then given by (5.26), ( ε(P δ )0 − rot Qδ − σP δ = 0 µ(Qδ )0 + rot P δ = 0 in Q (5.27) ν ∧ P δ + δQδτ = 0 on Σ (P δ (T ), Qδ (T )) = kA−1 ((E δ (T ), H δ (T )) − (E1 , H1 )) in Ω, (5.28)

δ Jopt = Qδτ |Σ .

It is possible to prove by compactness arguments that the solution of (5.26) (5.28) converges in a certain sense as δ → 0 to the solution of the optimality system (5.22), (5.24), (5.25). However, by penalizing in the X 0 norm rather than the H norm, a significant complication is introduced into the DD as noted above, and the simplifications gained by regularizing the problem are lost. From this point of view, it makes more sense to work with the cost functional (1.2) when optimally controlling the damped system (5.26). But it cannot be expected that the solution of the corresponding optimality system converges in any sense as δ → 0 to the solution of (5.22), (5.24), (5.25) and, indeed, it does not. Finally, let us remark that we have also considered the case k = ∞, that is, the problem of minimum L2τ (Σ) exact controllability of the solution of (5.26) to (E1 , H1 ) at time T . We have proved that, in this case, the solution of the optimality system does converge as δ → 0 to the solution of the optimality system for the limit problem, that is, the optimality system for the problem of minimum L2τ (Σ) exact controllability of the solution of (5.22) to (E1 , H1 ) at time T . Details will be published in a forthcoming paper.

Acknowledgment The author thanks Matthias Eller for very helpful discussions regarding the unique continuation argument used in section 4.

References [1] A. Alonso and A. Valli, “A domain decomposition approach for heterogeneous time-harmonic Maxwell equations,” Comput. Methods Appl. Mech. Engrg., 143 (1997), 97 - 112. [2] A. Alonso and A. Valli, “An optimal domain decomposition preconditioner for low frequency time-harmonic Maxwell equations,” Math. Comp., 68 (1999), 607 631.

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[3] J.-D. Benamou, “Décomposition de domaine pour le contrˆ ole de systèmes gouvernés par des équations d’evolution,” CRAS Paris, Série 1, 324 (1997), 10651070. [4] J.-D. Benamou, “A domain decomposition method for the optimal control of systems governed the Helmholtz equation,” Mathematical and Numerical aspects of Wave Propagation, (G. Cohen, Ed.), SIAM 1995, 653–662. [5] J.-D. Benamou, “A domain decomposition method with coupled transmission conditions for the optimal control of systems governed by elliptic partial differential equations,” SIAM J. Num. Anal., 36 (1995), 2401-2416. [6] J.-D. Benamou, “A domain decomposition method for control problems,” in DD9 Proceedings 1996, Bergen, (P. Bjørstad et al., Ed.), DDM.org, 1998, 266–273. [7] J.-D. Benamou, ”Résolution d’un cas test de contrˆ ole optimal pour un système gouvernè par l’équation des ondes a` l’aide d’une méthode de décompostion de domaine,” INRIA, Rapport de Rechereche No. 3095, 1997. [8] J.-D. Benamou, ”Domain decomposition, optimal control of systems governed by partial differential equations and synthesis of feedback laws,” J. Opt. Theory Appl., 102 (1999), 15-36. [9] J.-D. Benamou and B. Desprès, “A domain decomposition method for the Helmholtz equation and related optimal control problems,” J. Comp. Physics, 136 (1997), 68-82. [10] P. Collino, G. Delbue, P. Joly and A. Piancenti, “A new interface condition in the non-overlapping domain decomposition method for the Maxwell equations,” Comput. Methods Appl. Mech. Engrg., 148 (1997), 195 - 207. [11] B. Després, P. Joly and E. Roberts, “A domain decomposition for the harmonic Maxwell equation,” in Iterative Methods in Linear Algebra, Elsevier, Amsterdam, 1992, 475 - 484. [12] M. Eller, “Uniqueness of continuation theorems,” in Direct and Inverse Problems of Mathematical Physics, R. P. Gilbert, J. Kajiwara and Y. S. Xu, Eds., Kluver, 1999. [13] J. Lagnese and G. Leugering, “Dynamic domain decomposition in approximate and exact boundary control in problems of transmission for wave equations,” SIAM J. Control Opt., 38 (2000), 503 - 537. [14] G. Leugering, “On domain decomposition of controlled networks of elastic strings with joint masses,” in Control and Estimation of Distributed Parameter Systems (Vorau 1996), (F. Kappel, Ed.), ISNM, 126, Birkh¨ auser Verlag, Basel, 1998, 191205. [15] G. Leugering, “On domain decomposition of optimal control problems for dynamic networks of elastic strings,” Computational Optimization and Applications, to appear. [16] G. Leugering, “Dynamic domain decomposition of optimal control problems for networks of strings and beams,” SIAM J. Control Opt., 37 (1999), 1649 - 1675. [17] G. Leugering, “Domain decomposition of optimal control problems for multi-link structures,” in ENUMATH 97 (Heidelberg), World. Sci. Publishing, River Edge, NJ, 1998, 38 - 53. [18] G. Leugering, “Dynamic domain decomposition of controlled networks of elastic strings and joint masses,” in Control and partial differential equations (MarseilleLuminy, 1997), ESIAM Proc., 4 (1998), 223 - 233. [19] P.-L. Lions, “On the Schwarz alternating method 3”, The Third International

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Symposium on Domain Decomposition Methods for Partial Differential Equations, T. Chan and R. Glowinski, Eds.), Society for Industrial and Applied Mathematics, Philadelphia, PA, 1990, 202-223. [20] Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bull. A.M.S., 73 (1967), 591-597.

Boundary Stabilizibility of a Nonlinear Structural Acoustic Model Including Thermoelastic Effects

Catherine Lebiedzik, University of Virginia, Charlottesville, Virginia Abstract We are interested in a three-dimensional coupled PDE system arising in problems dealing with the active control of structural acoustic systems. A wave equation defined on a three-dimensional domain is coupled with a nonlinear thermoelastic plate equation on a portion of the boundary. The major issue studied here is the uniform stabilizibility of the entire interactive model. Our principal result states that boundary nonlinear dissipation placed on a suitable portion of the boundary suffices to stabilize the system.

1

Introduction

Structural acoustic systems are typically modeled by a a three-dimensional interactive system of partial differential equations(PDE) that consists of a wave equation coupled at an interface with some sort of plate equation. An undamped wave equation is defined on a three-dimensional bounded domain Ω with boundary Γ. On a portion of the boundary (the interface labelled Γ0 ), the wave equation in the chamber is coupled with a plate equation. Though the issue of stability of wave and plate equations has attracted much attention, there is less known about the behavior of these coupled (hybrid) PDE structures. Only recently has significant progress been made in understanding the nature of these interactive models [3, 1, 7, 23, 22, 16]. It becomes quickly clear that understanding the interactions between different types of dynamics is the key to stabilizing these systems. The works cited above deal exclusively with linear constitutive laws corresponding to the PDE’s. This paper, on the other hand, pertains to a nonlinear , large displacement theory in the context of stabilization of hybrid structures. We consider the case where the interactive portion of the boundary, Γ0 , is represented by a nonlinear thermoelastic plate equation. Our main aim is to study the question of stability/stabilizibility of the entire coupled model. A five-page announcement of the results of this paper will appear in [15]. 177

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Fig. 1. Cross section of the domain Ω. The thick line denotes the area subject to frictional damping g(zt ).

1.1

Statement of the Problem

Let Ω ∈ R3 be an open bounded domain with two dimensional boundary Γ which is assumed sufficiently smooth. The boundary Γ consists of three ¯1 ∪ Γ ¯2 ∪ Γ ¯ 0 , with Γ2 possibly empty. The pressure in connected regions Γ = Γ the acoustic medium is defined on Ω, whereas the displacement of the flexible wall is defined on Γ0 . The other portions of the boundary, Γ1 and Γ2 , represent ‘hard’ walls, with Γ1 being the section subject to frictional forces. The PDE model considered consists of the wave equation in the variable z (where the quantity ρzt is the acoustic pressure, and ρ is the density of the fluid)

(1.1)

ztt = c2 ∆z in Ω × (0, ∞) ∂ z = 0 on Γ2 × (0, ∞) ∂ν ∂ z = −g(zt ) − d z on Γ1 × (0, ∞) ∂ν ∂ z = wt on Γ0 × (0, ∞) ∂ν

and the elastic equation representing the displacement of the wall w subject to

Stability of a nonlinear structural acoustic model

179

thermal effects [See, e.g., [11]]: (1.2)

 wtt − γ∆wtt + ∆2 w = −∆θ − ρzt + [F(w), w]  ∆2 F(w) = −[w, w]  θt − ∆θ = ∆wt w = ∂∂ν˜ w = 0; θ = 0 F(w) = ∂∂ν˜ F(w) = 0

on

Γ0 × (0, ∞)

on

∂Γ0 × (0, ∞)

Here θ is the temperature, c2 is the speed of sound as usual, and the constant γ ≥ 0 accounts for rotational forces. The vector ν (respectively, ν˜ denotes the outward unit normal vector to the boundary Γ, (respectively, ∂Γ0 ), and [u, v] denotes the usual von K´ arm´ an bracket, i.e. [u, v] = uxx vyy + uyy vxx − 2uxy vxy . The function g, a nonlinear boundary feedback control, represents frictional damping and here is assumed continuous, monotone increasing, and zero at the origin. The boundary conditions given are those for a clamped plate, though hinged boundary conditions can be considered as well with no increase in complexity. For convenience, and without loss of generality, in what follows we will choose c = ρ = d = 1. Our goal is to show the uniform stability of the coupled PDE system (1.1)-(1.2). To accomplish this goal we shall use differential multipliers developed in the context of stability analysis for the wave equation [12] together with the operator multiplier method introduced in [2].

1.2

Previous literature and the contribution of this paper.

Models of structural acoustic interactions as a coupling of wave and plate equations go back to [24] and earlier. More recently, these models have become a source of interest as engineers use them to try to control the noise in an acoustic chamber [4, 5, 8]. The coupling of the wave equation in the chamber with the equation of the wall or plate provides the essential mechanism for control of the system. The first system thus studied was the case where the wall was modelled by a structurally damped plate[4, 3, 1]. Since structural damping is such a strong effect, the mathematical properties of such a system are much richer. However, modelling of structural damping is poorly understood and structural dampers added onto the active wall may produce a local ‘over-damping’ effect. Thus, there is a great deal of appeal in models which do not depend on structural damping to provide the necessary stabilizing effect. The existing literature on interactive structural acoustic models (see [3, 1, 7, 23, 22, 14, 16] and references therein) deals with stabilization in the context of linear plate models only. In contrast, this paper presents a model which allows for large displacements of the active wall, thus addressing nonlinear dynamics in the context of interactive structures. In addition, we do not assume any source of structural damping or additional mechanical damping on the active

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wall Γ0 . Not only is this physically appealing, but in fact it leads to interesting mathematical difficulties. In the case where structural damping is added to the wall, the corresponding dynamics are analytic, and the mathematical analysis is much more straightforward [1]. In the case of mechanical damping on the interface, we have certain regularizing effects on the traces of the wave equation. In our situation, however, we have neither of these effects, and thus we need a much more subtle mathematical analysis (particularly at the level of “sharp” trace theory for waves and plates). Hence, the main novelty of our contribution is the consideration of nonlinear dynamics and thermoelastic effects in the context of hybrid PDE structures. We wish to show that the addition of thermal effects on the flexible wall Γ0 , as well as boundary dissipation affecting a part of the hard wall of the acoustic medium, suffices to stabilize the system. From a mathematical standpoint, the fact that we do not assume and damping affecting Γ0 is critical. The lack of damping on Γ0 has important implications regarding the regularity of solutions. To appreciate this point, it is enough to notice that the presence of the damping on the wall Γ0 provides a priori L2 regularity on the trace of the pressure zt |Γ0 , which is the coupling term between the wall and acoustic medium. If there is no damping on Γ0 , the term zt |Γ0 is not even defined ( recall zt ∈ L2 (Ω)). Thus, one of the fundamental task is to provide appropriate estimates (in appropriate negative norms) for this term, as well as for the tangential derivatives of z on Γ0 .

1.3

Statement of Main Results

We begin with a preliminary result that the system is well-posed. Theorem 1.1. (well-posedness) Let Ω be a bounded open domain in R3 with boundary Γ as previously described. For all initial data y0 = [z0 , z1 , w0 , w1 , θ0 ] ∈ Y where 1 Y ≡ HΓ11 (Ω) × L2 (Ω) × H02 (Γ0 ) × H0,γ (Γ0 ) × L2 (Γ0 )

the solution y(t) = [z, zt , w, wt , θ] of the model (1.1)-(1.2) exists in C([0, ∞); T ) and is unique. Here the function spaces used are defined as HΓ11 (Ω) = {f ∈ 1 (Γ ) = H 1 (Γ ) f or γ > 0; L2 (Γ ) f or γ = 0 H 1 (Ω) with f |Γ1 = 0} and H0,γ 0 0 0 0 with inner product (1.3) (ω1 , ω2 )H0,γ 1 (Γ ) ≡ (ω1 , ω2 )L2 (Γ ) + γ(∇ω1 , ∇ω2 )L2 (Γ ) 0 0 0

1 ∀ ω1 , ω2 ∈ H0,γ (Γ0 )

Proof. Since the problem is a locally Lipschitz perturbation of a m-monotone system with a priori bounds, the result follows from general theory of mdissipative operators(see, e.g. [6]). The full details of this argument are given in [13].


181

In order to formulate our main result on stability, we present some notation. We define the energy functional associated with the model as Z 2 |zt | + |∇z|2 dΩ + z 2 dΓ1 Ω Γ1 Z 1 2 2 2 2 2 |wt | + γ|∇wt | + |∆w| + |∆F(w)| + |θ| dΓ0 + 2 Γ0

Z (1.4) Eγ (t) =

Next, we introduce the function h(s) which is assumed concave, strictly increasing, zero at the origin, and such that the following inequality is satisfied for all |s| ≤ 1: (1.5)

h(s g(s)) ≥ s2 + |g(s)|2

Such a function can easily be constructed in view of the monotonicity assumption imposed on g, [17]. Additionally, we will impose a geometric condition on the ‘clamped’ portion of the boundary Γ2 . Γ2 is assumed convex (that is, the level set function representing Γ2 has a nonnegative Hessian in the neighborhood of Γ2 on the side of Ω) and there exists a point x0 ∈ R3 such that (1.6)

(x − x0 ) · ν ≤ 0,

x ∈ Γ0 ∪ Γ2

Note that this condition is automatically satisfied if Γ2 is empty. In fact, one can choose x0 to be any point in Γ0 . If Γ2 is non-empty, then x0 is a suitably selected point in the hyperplane containing Γ0 (see Figure 1). Our main result is the following. Theorem 1.2. (uniform stability) Let Ω be a bounded open domain in R3 with boundary Γ as previously described. Assume that the nonlinear function g satisfies (1.7)

m s2 ≤ g(s)s ≤ M s2 ;

|s| ≥ 1

Then, with the constant γ ≥ 0, every weak (finite energy) solution of (1.1)-(1.2) decays uniformly to zero, i.e.: (1.8)

Eγ (t) ≤ C sγ (t/T0 − 1));

t ≥ T0

where the real variable function sγ (t), which may depend on Eγ (0) and γ converges to zero as t → ∞ and satisfies the following ordinary differential equation (ODE): (1.9)

d sγ (t) + q(sγ (t)) = 0, dt

sγ (0) = Eγ (0)

The nonlinear monotone increasing function q(s) is determined entirely from the behavior at the origin of the nonlinear function g and is given by the

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following algorithm: (1.10) (1.11) (1.12)

q ≡ I − (I + p)−1 · p ≡ (I + h0 )−1 K h0 (x) ≡ h (x/mes(0, T ) × Γ1 )

where h is given by (1.5) and the constant K > 0 may depend on Eγ (0) and γ. The main mechanisms which allow us to stabilize the system are the thermal effects in the plate Γ0 and the nonlinear dissipation on Γ1 . Thus, the decay rates will be determined by the strength of the nonlinear function g(zt ). In fact, once the behavior of g(s) at the origin is specified, the decay rates can be explicitly solved for using the nonlinear ODE (1.9). If g is bounded from below by a linear function, then it can be shown that the decay rates predicted are exponential. If, instead, g has polynomial growth (or is exponentially decaying) at the origin, then the decay rates are algebraic (or logarithmic). This can be demonstrated by solving (1.9) (see [17]). It is important also to note that though the decay rates in general may depend on Eγ (0), and thus on the norm of the initial conditions, they are independent of the profile of the initial conditions. In addition, we have not assumed any source of boundary damping on the active wall Γ0 . This is a source of additional mathematical difficulty, as the existence of boundary damping on Γ0 gives rise to a priori L2 regularity on zt |Γ0 . This not only has an additional stabilizing effect but also substantially improves the regularity of the hyperbolic traces. In the absence of this property, a more sophisticated method is needed to reconstruct the appropriate energy estimates, mainly at the level of treating the traces in the case γ = 0. It is necessary to consider the two cases γ = 0 and γ > 0 separately, and thus our estimates are not uniform as γ → 0. We have not assumed any geometric conditions imposed on Γ1 , the portion of the boundary subject to dissipation. This makes physical sense, since the geometry of the boundary should only be an issue where the boundary is ‘uncontrolled’. Moreover, we have not imposed any conditions on the growth of the nonlinearity g at the origin. This is in contrast with most of the literature on boundary stabilization of wave and plate equations alone. (see [10] and references therein). Finally, we have assumed Neumann, rather than Dirichlet, boundary data on the ‘uncontrolled’ portion of the boundary Γ2 . This is a source of technical difficulty, due to the fact that the Lopatinski condition is not satisfied. On the other hand, in the context of the structural acoustic problem, it is desirable to have Neumann data on Γ2 . Indeed, if we assumed Dirichlet conditions on Γ2 , regularity for the corresponding elliptic problem would force the assumption that Ω was not simply connected! Clearly this is not what we want. Our techniques provide the result under the additional geometric assumption that Γ2 is convex. It is not known if the same result can be shown without this assumption (i.e. assuming only a “star-shaped” condition).

Stability of a nonlinear structural acoustic model We shall adopt the following notation:

183

Z

|w|s,Ω ≡ |w|H s (Ω) ; (u, v)Ω ≡

uvdΩ Ω

The same notation will be used with Ω replaced by Γ etc. The negative Sobolev spaces H −s (Ω) are defined as dual spaces to H0s (Ω). In addition, we will make use of the following properties of Airy’s stress function. (1.13) |[F(w), w]|−θ,Γ0 ≤ C|F(w)|3−β,Γ0 |w|2−θ+β,Γ0 , (1.14)

|F(w)|3−ε,Γ0 ≤ C|w|22,Γ0 ,

0 < β < 1,

0≤θ≤1

for any ε > 0

Moreover, straightforward application of (1.13) gives that (1.15)

2

|[F(w), w]|−1,Γ0 ≤ C|w|22,Γ0 |w|2−ε,Γ0

ε>0

Uniform Stabilization – proof of Theorem 1.2

Our goal is to show the uniform stability of the coupled PDE system (1.1)-(1.2). We begin with a preliminary energy identity which illustrates the fact that the system is dissipative. Proposition 2.1. With respect to the system of equations (1.1)-(1.2) , the following energy equality holds for all T > 0, s < T : Z T Z T Eγ (s) = Eγ (T ) + 2 (2.1) (g(zt ), zt )Γ1 dt + 2 |∇θ|20,Γ0 dt s

s

where the ‘energy’ Eγ (t) is defined by (1.4). Proof. By running the multipliers zt on the wave equation, wt on the elastic equation, θ on the thermal equation, and then integrating by parts, we obtain the above equality for smooth solutions. A density argument allows us to extend this inequality to all solutions of finite energy. In order to prove Theorem 1.2, our strategy is to study the thermoelastic equations on Γ0 and the wave equation on Ω separately and combine the results. In the case of the thermoelastic plate, we will run the multiplier A−1 D θ (introduced in [2]) on the elastic equation to yield an estimate of the plate energy. For the wave equation, we run the multipliers z , z∇ · h , and h · ∇z . This leads to an estimate of energy plus lower-order terms, which are then absorbed via a standard uniqueness/compactness argument. Though the multiplier used are “standard” by now in the context of the wave and thermoelastic plate equations, the interactions between the two media brings forward several technical difficulties due to the appearance of boundary traces which are not a priori bounded by the energy. Handling of these terms constituties the bulk of the the techincal content of this paper.

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2.1

Catherine Lebiedzik

Thermoelastic Equations

We define the plate energy Ew,γ (t) as 1 Ew,γ (t) = |wt |20,Γ0 + γ|∇wt |20,Γ0 + |∆w|20,Γ0 + |F(w)|20,Γ0 + |θ|20,Γ0 2

(2.2)

Next, we state several Lemmas which will be necessary for our estimates. Proof of these Lemmas is in the Appendix. First, we consider the case where γ = 0 and the dynamics of the plate are analytic: wtt + ∆2 w = ht − ∆θ + k1 on ∂ w= w = 0 on ∂Γ0 × (0, T ) ∂ν

(2.3)

Γ0 × (0, T )

Lemma 2.1. With reference to (2.3), where θ is the solution to the heat equation in (1.2), we obtain the following regularity with an index α ≤ 1/2: (2.4) Z T Z 2 2 [|w(t)|2+2α,Γ0 + |wt (t)|2α,Γ0 ]dt ≤ CT Ew (0) + CT 0

0

T

[|ht |22α−2,Γ0 + |k1 |22α−2,Γ0 ]dt

where Ew (t) ≡ Ew,0 (t). Lemma 2.2. Let γ = 0. We consider the original system given by (1.1), (1.2). Z (2.5)

T

[|w(t)|23,Γ0 + |wt (t)|21,Γ0 ]dt ≤ CT [Ew (0) + Ez (0)] 0 Z T Z T 2 2 2 + CT [|wt |−1,Γ0 + |z|0,Γ1 + |g(zt )|0,Γ1 ]dt + C |w|42,Γ0 |w|22−ε,Γ0 dt 0

0

(2.6) Z T Z T 2 2 [|w(t)|3−δ,Γ0 +|wt (t)|1−δ,Γ0 ]dt ≤ ε[Ew (0)+Ez (0)]+C |w|42,Γ0 |w|22−ε,Γ0 dt 0 0 Z T Z T + +CT,ε,δ [|wt |2−1,Γ0 + |w|21,Γ0 ]dt + CT [|g(zt )|20,Γ1 + |z|20,Γ1 ]dt 0

0

Next, we give a result for the clamped plate (for γ ≥ 0) which does not follow from standard Sobolev trace theory. We note that the analogous result was shown for the linear thermoelastic plate only in [2]. We have extended this to account for the nonlinear von K´ arm´ an term and for the structural acoustic interaction. Lemma 2.3. With respect to the system of equations (1.1)-(1.2), the component

185


w of the solution [z, zt , w, wt , θ] satisfies ∆w|∂Γ0 ∈ L2 (0, T ; L2 (∂Γ0 )) with the estimate (2.7) Z T

Z T |∆w|20,Γ0 + |wt |20,Γ0 + γ|∇wt |2Γ0 + |∇θ|20,Γ0 + |z|20,Γ1 |∆w|20,∂Γ0 dt ≤ CT 0 0 +|g(zt )|20,Γ1 dt + C(Ew,γ (T ) + Ew,γ (0)) + CT [Ez (0) + Eγ (0)lotγ (w, θ)]

where C, CT do not depend on the parameter γ, and lotγ (w, θ) is given by (2.8) lotγ (w, θ) ≤ Cδ sup t∈[0,T ]

h

i |w|22−δ,Γ0 + |wt |2−δ,Γ0 + γ|wt |21−δ,Γ0 + |θ|2−1/4,Γ0 , δ > 0

The major result of this section is the following estimate. Lemma 2.4. With respect to the thermoelastic component of the model (1.2), for all ε > 0, there exist CT , CEγ (0),T,ε such that the following inequality holds: (2.9) Z T 0

Z

T |z|20,Γ1 + |g(zt )|20,Γ1 dt Ew,γ (t) dt ≤ εCT [Ew,γ (0) + Ew,γ (T ) + Ez (0)]+εCT 0 Z T + CEγ (0),T,ε |θ|21,Γ0 dt + lotγ (w, θ) + lot(z) 0

where lotγ (w, θ) is as given in (2.8), lot(z) ≤ Cδ sup |z(t)|21−δ + |zt (t)|2−δ , δ > 0 (2.10) t∈[0,T ]

and the constants CT and CEγ (0),T,ε are uniformly bounded in γ ≥ 0. Proof. We multiply the first equation in (1.2) by A−1 D θ and integrate from 0 to T to obtain Z T (wtt − γ∆wtt + ∆2 w + ∆θ + zt − [F(w), w], A−1 D θ)Γ0 dt = 0. 0

We deal with each part separately: (1) Using integration by parts, substitution of boundary conditions, the second equation of (1.2), and the fact that A−1 D is smoothing give(detailed calculations are in [2]) (2.11) Z Z Th T i |wt |20,Γ0 + γ|∇wt |20,Γ0 dt wtt − γ∆wtt , A−1 D θ L2 (Γ0 ) dt − 0 0 Z Z Th i 2 2 ≤ εC Ew,γ (0) + Ew,γ (T ) + ε |wt |0,Γ0 + γ|∇wt |0,Γ0 dt + Cε 0

0

T

|θ|21,Γ0

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Note that neither of the constants C and Cε depend on T or γ. (2) Another integration by parts and application of boundary conditions gives (2.12) Z T Z 2 −1 ∆ w, AD θ 0,Γ0 dt = − 0

T

∆w, ∇A−1 D θ 0,∂Γ0

0

Z

T

∆w, ∆A−1 D θ

dt + 0

0,Γ0

dt

In order to estimate this term we need to use the trace regularity result from Lemma 2.3. (2.13) Z Z T T −1 2 ∆ w, AD θ Γ0 dt ≤ Cε |θ|21,Γ0 dt+εCT [Ew,γ (0) + Ew,γ (T ) + Ez (0) 0 0 Z T Z T 2 2 + |z|0,Γ1 + |g(zt )|0,Γ1 dt + Eγ (0)lotγ (w, θ) Ew,γ (t) dt + 0

0

(3) For the next term we just use integration by parts: Z T Z T −1 ∆θ, AD θ L2 (Γ0 ) dt ≤ C (2.14) |θ|20,Γ0 0

0

(4) For the term with zt |Γ0 , we will need to first integrate by parts, use trace theory, and then substitute in the heat equation of (1.2): Z T −1 zt , AD θ Γ0 dt 0 Z T Z T −1 ≤ sup |z(t)|1/2+δ,Ω |AD θ|0,Γ0 + (z, θ)Γ0 dt + (z, wt )Γ0 dt t∈[0,T ]

0

Z

T

(2.15) ≤ CT,ε lot(z) + lotγ (w, θ) + 0

(5) Finally, using (1.15) gives: Z T Z −1 ([F(w), w], AD θ)Γ0 dt ≤ C 0

0

T

|θ|21,Γ0 dt + ε

0

Z

T 0

|wt |20,Γ0 dt

|[F(w), w]|−1,Γ0 |A−1 D θ|1,Γ0

≤ C|w|22,Γ0 |w|2−ε |θ|0,Γ0 ≤ C Eγ (0)lotγ (w, θ)

(2.16)

Combining equations (2.11) - (2.16) results in the fact that for ε small enough there exists a constant CT > 0 so that (2.17)

Z

i T h (1 − 2ε) |wt |20,Γ0 + γ|∇wt |20,Γ0 dt ≤ εCT [Ew,γ (0) + Ew,γ (T ) + Ez (0)] 0 Z T Z T 2 +CT,ε E(t) + |z|20,Γ1 + |g(zt )|20,Γ1 dt |θ|1,Γ0 dt+CT,Eγ (0) lotγ (w, θ)+εCT 0

0


187

where the non-crucial dependence of CT on ε has not been noted. Next, we multiply the same equation of (1.2) by w and integrate from 0 to T to obtain T T Z T h i wt , w 0,Γ0 + γ ∇wt , ∇w 0,Γ0 − |wt |20,Γ0 + γ|∇wt |20,Γ0 dt = Z

−

0

T

0

|∆w|20,Γ0

0

0

Z

T

dt+ 0

([F(w), w], w)0,Γ0 dt+

Z

0

T

Z

(∇θ, ∇w)0,Γ0 dt−

0

T

(zt , w)0,Γ0 dt

Taking norms and using the trace theorem gives T T wt , w 0,Γ0 + γ ∇wt , ∇w 0,Γ0 ≤ ε[Ew,γ (0) + Ew,γ (T )] + Cε lotγ (w, θ) 0 0 and

Z

T 0

Z (zt , w)0,Γ0 dt ≤ CT (lot(z) + lotγ (w, θ)) + C

0

T

|wt |20,Γ0 dt

Combining these and using the divergence theorem as well as symmetricity of the bracket on the term involving F(w) gives Z T |∆w|20,Γ0 + |∆F|20,Γ1 dt 0 Z Th h i i 2 2 ≤ ε Ew,γ (0) + Ew,γ (T ) + C |wt |0,Γ0 + γ|∇wt |0,Γ0 dt 0 Z T Z T +ε |∆w|20,Γ0 dt + Cε |θ|21,Γ0 dt + CT (lot(z) + lotγ (w, θ)) 0

0

Thus, we have that there exist constants C, CT , Cε > 0 such that for ε > 0 small enough, Z

T (2.18) (1 − ε) |∆w|20,Γ0 + |∆F|20,Γ1 dt ≤ ε Ew,γ (0) + Ew,γ (T ) 0 Z Th Z T i 2 2 +C |wt |0,Γ0 + γ|∇wt |0,Γ0 dt + Cε |θ|21,Γ0 dt + CT (lot(z) + lotγ (w, θ)) 0

0

If the ε of equations (2.17) and (2.18) is small enough, they can be combined to produce the inequality (2.9), which is the desired result.

2.2

Wave Equation

Let Ez (t) be the energy defined by (2.19)

Ez (t) = |zt (t)|20,Ω + |∇z(t)|20,Ω + |z(t)|20,Γ1

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Catherine Lebiedzik

We quote here a sharp trace result for the wave equation that will be necessary to our proof. Lemma 2.5. Let z be a solution to ztt = ∆z in Ω×(0, T ) with interior regularity z ∈ C(0, T ; H 1 (Ω)) ∩ C 1 (0, T ; L2 (Ω)); and the following boundary regularity ∂ z ∈ L2 ((0, T ) × Γ); zt |Γ1 ∈ L2 ((0, T ) × Γ1 ); ∂ν Let T > 0 be arbitrary and let α be an arbitrary small constant such that α < Then, we have: Z

T −α

(2.20) α

Z (2.21) 0

T

T 2.

Z T ∂ 2 ∂ | z|0,Γ1 dt ≤ CT,α [ [| z|20,Γ + |zt |20,Γ1 ]dt + lot(z)] ∂τ ∂ν 0 Z |zt |2−4/5,Γ dt

T

≤ CT [Ez (0) + 0

|

∂ 2 z| dt] ∂ν 0,Γ

where lot(z) is defined in (2.10). Proof. Inequality (2.20) follows similarly as Lemma 7.1 in [18]. Though here we are evaluating the tangential derivative and zt on Γ1 , rather than the whole of the boundary Γ, the argument given in [18] can still be used. It is necessary to note only that the measurements of zt |Γ1 are needed only in the nonelliptic sector (after microlocalization), where the argument is purely local. Inequality (2.21) is given in Theorems A and C of [20] (see also [19]). The main result of this section is the following ”recovery” estimate for the wave equation. Lemma 2.6. Assume that the geometric condition (1.6) is in force. Consider the wave equation (1.1) with finite energy solutions, where w is a finite energy solution to (1.1) and T > 0 is arbitrary. Then, for any α < T /2 there exist positive constants C, possibly depending on α, such that if γ > 0 Z

T −α

(2.22) α

Ez (t) dt ≤ C [Ez (α) + Ez (T − α) + Ez (0)] Z T + CT [|zt |20,Γ1 + |g(zt )|20,Γ1 ] dt + CT lot(z) + CT,γ lotγ (w, θ)] 0

and if γ = 0, Z

T −α

(2.23) α

+ CT,Eγ (0)

Z

T 0

Ez (t) dt ≤ C [Ez (α) + Ez (T − α) + Ez (0) + Ew (0)] [|zt |20,Γ1 + |g(zt )|20,Γ1 + |wt |20,Γ0 ] dt + CT,Eγ (0) [lot(z) + lotγ (w, θ)]


189

Proof. The first step of the proof involves the use of a multiplier method. As usual, in order to apply this method it is necessary to have solutions which are smooth enough that we can use standard differential calculus. In the nonlinear case, our solutions may not have enough regularity, even if the initial data are taken sufficiently smooth. We can bypass this difficulty by using a regularization argument proposed in [17]. We will perform the necessary PDE calculus on solutions of the ’regularized’ wave equation and obtain estimates (2.22) and (2.23). Then we will pass through the limit, using an appropriate regularization parameter, and in this way we will reconstruct these inequalities for the original problem. The Lemma below states the result necessary for this limit passage. Lemma 2.7. Given any solution of the wave equation ztt = ∆z; in Ω × (0, T ) ∂ z = 0; on Γ2 × (0, T ) ∂ν

(2.24)

with the following regularity properties 1. z ∈ C[0, T ; H 1 (Ω)] ∩ C 1 [0, T ; L2 (Ω)] 2.

∂ ∂ν z|Γ1 , zt |Γ1

3.

∂ ∂ν z|Γ0

∈ L2 (Σ1 )

∈ L2 (0, T ; H 1/2 (Γ0 ))

there exists a sequence z n ∈ C[0, T ; H 2 (Ω)] ∩ C 1 [0, T ; H 1 (Ω)] of solutions to the wave equation n ztt = ∆z n ; in Ω × (0, T ) ∂ n z = 0; on Γ2 × (0, T ) ∂ν

(2.25)

such that the following convergence holds • z n → z in C[0, T ; H 1 (Ω)] ∩ C 1 [0, T ; L2 (Ω)] •

∂ n ∂ν z |Γ1

→

∂ ∂ν z|Γ1

in L2 (Σ1 )

•

∂ n ∂ν z |Γ0

→

∂ ∂ν z|Γ0

in L2 (0, T ; H 1/2 (Γ0 ))

• ztn |Γ1 → zt |Γ1 in L2 (Σ1 ) Proof. The proof of this Lemma follows from the method in [17] and is detailed in [16]. Lemma 2.7 applies to any finite energy solutions of (1.1). From the energy relation (2.1) and the properties of g, it can be immediately seen that zt |Γ1 ∈ L2 (Σ1 ). Applying the boundary conditions on Γ1 and the Trace ∂ ∂ Theorem gives that ∂ν z|Γ1 ∈ L2 (Σ1 ). Finally, ∂ν z|Γ0 ∈ L2 (0, T ; H 1/2 (Γ0 )) is 1 given by the fact that w ∈ L2 (0, T ; H (Γ0 )).

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Catherine Lebiedzik

In order to apply the result of Lemma 2.7 we will consider a sequence of smooth solutions z n to (2.25), to which we apply the multipliers h · ∇z n and div h z n . The vector field h here is is such that h · ν = 0 on Γ0 ∪ Γ2 ,

J(h) > c0 > 0 on Ω

where J(h) denotes the Jacobian of h. Such a vector field exists (see [14, 21]) as long as the geometrical condition (1.6) is satisfied and Γ0 ∪ Γ2 is convex. Applying these multipliers and performing familiar computations (see, e.g., [17]) gives (2.26) Z T s

Z T ∂ ∂ Ez n (t)dt ≤ C[Ez n (s)+Ez n (T )]+C [|ztn |20,Γ1 +| z n |20,Γ1 +| z n |20,Γ1 ]dt ∂ν ∂τ s 2 Z T Z Z T ∂ n ∂ n ∂ n z +C dt + C z z h · τ dΓ0 dt + CT lot(z n ) ∂ν ∂τ s s Γ0 ∂ν 0,Γ0

The main issue and difficulty here is to provide the estimates for the tangential derivatives of z n on Γ1 and Γ0 . Indeed, these terms are not bounded by the energy and the sharp trace regularity theory of hyperbolic solutions, recalled in Section 2, is necessary. Tangential derivatives on Γ1 are estimated with the help of Lemma 2.5, inequality (2.20). By applying the Trace Theorem and Young’s Inequality we obtain estimates for the tangential derivatives on Γ0 . Z

T s

Z T ∂ n2 ∂ | z |−1/2,Γ0 dt + Cε | z n |21/2,Γ0 dt ∂τ ∂ν s s Z T Z T ∂ ≤ ε |z n |21,Ω dt + Cε | z n |21/2,Γ0 dt ∂ν s s Z T Z T ∂ ≤ εCT [Ez n (0) + Ez n (t)dt + Cε | z n |21/2,Γ0 ∂ν s 0

∂ ∂ n ( zn , z )Γ0 dt ≤ ε ∂ν ∂τ

(2.27)

Z

T

Combining (2.26),(2.20), (2.27), applied with s = α, T = T − α and taking ε small enough so that εCT < C yields Z (2.28)

T −α

Ez n (t)dt ≤ C[Ez n (α) + Ez n (T − α) + Ez n (0)] Z T Z T ∂ n2 ∂ n 2 + CT [|zt |0,Γ1 + | z |0,Γ ]dt + CT | z n |21/2,Γ0 dt + CT lot(z n ) ∂ν ∂ν 0 0 α

Since, as previously explained, the function z in (1.1) satisfies all the requirements in Lemma 2.7, we may apply this Lemma and pass with the limit on all


191

terms in (2.28). In addition, we apply the boundary conditions in (1.1). This gives the estimate Z

T −α

(2.29) α

Ez (t)dt ≤ C[Ez (α) + Ez (T − α) + Ez (0)] Z T + CT [ [|zt |20,Γ1 + |g(zt )|20,Γ1 + |wt |21/2,Γ0 ]dt + lot(z)] 0

In the case γ > 0, this yields Z

T −α

(2.30) α

Ez (t)dt ≤ C[Ez (α) + Ez (T − α) + Ez (0)] Z T + CT [|zt |20,Γ1 + |g(zt )|20,Γ1 dt + Cγ,T lotγ (w, θ) + CT lot(z) 0

For γ = 0 , it is necessary to estimate the last integral on the RHS of (2.29), since it no longer contributes lower order terms. At this point, the analyticity of the w component plays a critical role. Indeed, by the second statement in Lemma 2.2 applied with δ = 1/2 we obtain: Z

T

(2.31) CT 0

|wt |21/2,Γ0 dt ≤ ε0 CT [Ew (0) + Ez (0)] + Cε0 ,Eγ (0),T [lotγ (w, θ) + lot(z) +

Z

T

0

|g(zt )|20,Γ1 dt]

Making a suitable choice of ε0 = ε0 (T ) and collecting (2.30) (for γ > 0) and (2.29) (for γ = 0) combined with (2.31) yields desired conclusion in Lemma 2.6.

2.3

Uniform stability analysis for the coupled system

In the final analysis, we will combine the energy estimates on plate and wave equations, and then absorb the lower order terms by means of a standard compactness/uniqueness argument. Proposition 2.2. With respect to the coupled PDE system (1.1)-(1.2), the following estimate holds: Z (2.32) Eγ (T ) ≤ CT,Eγ (0),γ

0

T

|θ|21,Γ0 + |zt |20,Γ1 + |g(zt )|20,Γ1 dt + CT,Eγ (0),γ [lotγ (w, θ) + lot(z)]

Here the energy Eγ (t) is defined as in (1.4). Proof. Here the argument is the same for γ > 0 and γ = 0, so we give only the argument for γ = 0. The inequality (2.32) follows from equations (2.9)

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Catherine Lebiedzik

and (2.22). First, we have added (2.9) and (2.22) after multiplying (2.9) by a suitable constant AT in order to consolidate the |wt |20,Γ0 and |z|20,Γ1 terms. This gives: (2.33) Z AT

Z

T

T −α

Ez (t) dt ≤ εAT CT [Ew,γ (0) + Ew,γ (T ) + Ez (0)] Z T +C [Ez (α) + Ez (T − α) + Ez (0) + Ew (0)]+(CT,Eγ (0),γ +εAT CT ) |g(zt )|20,Γ1 dt 0 Z T Z T Z T +εAT CT |z|20,Γ1 dt+AT CT,Eγ (0) |θ|21,Γ0 dt+CT,Eγ (0),γ (|wt |20,Γ0 +|zt |20,Γ1 ) dt Ew,γ (t) dt +

0

α

0

0

0

+ (CT,Eγ (0),γ + AT CT,Eγ (0) ) (lotγ (w, θ) + lot(z)) Choosing AT > 2CT and ε, ε small enough (so that εAT CT ≤ CT ) and recalling the definition of Eγ (t) given in (1.4) gives Z T −α (2.34) Eγ (t) dt ≤ CT [Eγ (0) + Eγ (T ) + Eγ (α) + Eγ (T − α)] α Z T +CT,Eγ (0),γ |θ|21,Γ0 + |zt |20,Γ1 + |g(zt )|20,Γ1 dt+CT,Eγ (0),γ (lotγ (w, θ) + lot(z)) 0

Next, we use the dissipativity property to eliminate the parameter α. Using the identity (2.1) and the simple inequality Z α Z T ! Eγ (t) dt ≤ 2αEγ (0) + 0

gives

Z

(T −α)

T

Eγ (t) dt 0

Z

≤ CT [Eγ (0) + Eγ (T )] + CT,Eγ (0),γ (2.35)

0

T

|θ|21,Γ0 + |zt |20,Γ1 + |g(zt )|20,Γ1 dt

+CT,Eγ (0),γ (lotγ (w, θ) + lot(z))

Again, dissipativity gives that for t < T ,Eγ (T ) ≤ Eγ (t), so that T Eγ (T ) ≤ RT 0 Eγ (t) dt. Substituting this fact and (2.1) into (2.35) and taking T > 2CT leads to the desired conclusion in Proposition 2.2. Our next step is to eliminate the lower order terms from equation (2.32). Proposition 2.3. With respect to the coupled PDE system (1.1)-(1.2), there exists a constant CT,Eγ (0) > 0 such that Z T (2.36) lotγ (w, θ) + lot(z) ≤ CT,Eγ (0),γ |zt |20,Γ1 + |g(zt )|20,Γ1 + |θ|21,Γ0 dt 0


193

Proof. The conclusion follows by contradiction via the usual compactness and uniqueness argument. Since this argument is standard, we shall only point out the main steps. The compactness of lotγ (w, θ) + lot(z), with respect to the topology induced by the energy Eγ , γ ≥ 0, follows from the compact imbeddings H 2−ε (Γ0 )×H 1−ε (Γ0 )×H 1−ε (Ω)×H −ε (Ω) ⊂ H 2 (Γ0 )×H 1 (Γ0 )×H 1 (Ω)×L2 (Ω); for ε > 0. As for the uniqueness part, we deal with the following overdetermined system (here we consider only the more difficult case γ > 0):

(2.37)

z˜tt = ∆˜ z ∂ z˜t = 0; ∂ν z˜ + z˜ = 0 ∂ ˜=w ˜t ; θ˜ = 0 ∂ν z ∂ z˜ ∂ν = 0 ∆w ˜t = 0

on on on on on

[0, T ] × Ω [0, T ] × Γ1 [0, T ] × Γ0 [0, T ] × Γ2 [0, T ] × Γ0

∂ Since z˜t = 0; ∂ν z˜t + z˜t = 0 on Γ1 × (0, T ), a version of Holmgren’s Uniqueness Theorem applies (see Thm 3.5 in [9]) to conclude z˜t ≡ 0. Feeding back this information into the plate equation yields the following overdetermined system for the variable w. ˜

w ˜tt + ∆2 w ˜ = [F(w), ˜ w] ˜ on [0, T ] × Γ0 θ˜ ≡ 0 on [0, T ] × Γ0 ∆w ˜t = 0 on [0, T ] × Γ0 ∂ w ˜ = ∂ν w ˜=0 on [0, T ] × ∂Γ0 ˜ {˜ z (0), z˜t (0), w(0), ˜ w˜t (0), θ(0)} = {˜ z0 , z˜1 , w ˜0 , w ˜1 , θ˜0 } ∈ Y ∂ Since θ˜ ≡ 0, ∆w ˜t = 0. However, we have that w ˜t = ∂ν w ˜t = 0 on ∂Γ0 . Thus, by elliptic theory w ˜t ≡ 0. Substituting this into (2.37) gives the following system:

(2.38)

∆˜ z=0 ∂ ˜=0 ∂ν z ∂ z ˜=0 ∂ν ˜ + z ∂ z˜ ∂ν = 0

on on on on

[0, T ] × Ω [0, T ] × Γ0 [0, T ] × Γ1 [0, T ] × Γ2

∆2 w ˜ = [F(w), ˜ w] ˜ on [0, T ] × Γ0 ˜ θ≡0 on [0, T ] × Γ0 ∂ w ˜ = ∂ν w ˜=0 on [0, T ] × ∂Γ0 ˜ ≡ 0 , for all By the uniqueness of elliptic solutions, we conclude that {˜ z , w, ˜ θ} t > t0 > 0. This allows us to assert (2.36) as desired. To finish off the proof of Theorem 1.2, we use the inequality (2.36) to combine terms on the right hand side of (2.32). Z T Eγ (T ) ≤ CT,Eγ (0),γ (2.39) |zt |20,Γ1 + |g(zt )|20,Γ1 + |θ|21,Γ0 dt 0

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By using the assumptions imposed on the nonlinear function g and splitting the region of integration into two: zt ≤ 1 and zt > 1 we also obtain (see [17]): Z

T

(2.40) 0

[|zt |20,Γ1 + |g(zt )|20,Γ1 + |θ|21,Γ0 ]dt Z

T

≤ CT,m,M [I + h0 ]

Z [

0

Γ1

g(zt )zt dx + |∇θ|20,Γ0 ]dt

where we have used Jensen’s inequality. Combining (2.39) and (2.40) and recalling monotonicity of h0 we obtain: Z T Z Eγ (T ) ≤ CT,γ,m,M [I + h0 ] g(zt )zt dΓ1 + |∇θ|20,Γ0 dt 0

Γ1

= CT,γ,m,M,Eγ (0) [I + h0 ][Eγ (0) − Eγ (T )]

(2.41)

where in the last step we have used the energy relation. Since [I + h0 ] is invertible, this gives −1 [I + h0 ]−1 (Cγ,T,m,M,E Eγ (T )) ≤ Eγ (0) − Eγ (T ) γ (0)

(2.42) this gives

p(Eγ (T )) + Eγ (T ) ≤ Eγ (0) with p defined by the Theorem 1.2. The final conclusion of Theorem 1.2 follows now from application of Lemma 3.1 in [17]. The argument for γ > 0 is identical.

3

Appendix

Here we cite the necessary proofs of the Lemmas introduced in Section 2.1. Proof. (Lemma 2.1) The proof of this result is identical to that given in [16], though [16] deals with the case of free boundary conditions. Proof. (Lemma 2.2) We apply Lemma 2.1 with h ≡ z, α = 1/2, k1 ≡ [F(w), w], and use (1.15). (3.1) Z T Z 2 2 [|w(t)|3,Γ0 + |wt (t)|1,Γ0 ]dt ≤ CT Ew (0) + CT 0

0

T

[|zt |2−1,Γ0 + |w|42,Γ0 |w|22−ε,Γ0 ]dt

But from (2.21) in Lemma 2.5 and the boundary conditions imposed on the wave equation we have Z T (3.2) |zt |2−1,Γ0 dt 0 Z T Z T 2 ≤ |zt |−4/5,Γ0 dt ≤ CT [Ez (0) + [|wt |20,Γ0 + |g(zt )|20,Γ1 + |z|20,Γ1 ]dt] 0

0


195

Combining the two inequalities gives: (3.3) Z T 0

Z [|w(t)|23,Γ0

|wt (t)|21,Γ0 ]dt

+ Z +ε

T 0

T

≤ CT [Ew (0) + Ez (0) + |w|42,Γ0 |w|22−ε,Γ0 ]dt] 0 Z T |wt |21,Γ0 dt + CT,ε [ |wt |2−1,Γ0 dt + |g(zt )|20,Γ1 + |z|2Γ1 ]dt] 0

where in the last step we have used interpolation inequalities. Taking in (3.3) ε sufficiently small gives the first inequality in Lemma 2.2. As for the second inequality, this follows from the first after an additional use of interpolation inequalities. Proof. (Lemma 2.3) We will multiply the first equation of (1.2) by the ¯ 0 )]2 vector field quantity h · ∇w, where h(x, y) ≡ [h1 (x, y), h2 (x, y)] is a [C 2 (Γ such that h|∂Γ0 = [ν1 , ν2 ], and then integrating from 0 to T , i.e. Z

T

wtt − γ∆wtt + ∆2 w + ∆θ + zt − [F(w), w], h · ∇w

(3.4) 0

Γ0

dt = 0

By integration by parts, application of the divergence theorem and the estimates in [2], we have that (3.5) Z T 0

Z Th |w|22,Γ0 +|wt |20,Γ0 +γ|∇wt |20,Γ0 |∆w|20,∂Γ0 dt ≤ C[Ew,γ (0)+Ew,γ (T )]+C 0 Z T Z T i 2 + |θ|1,Γ0 dt + 2 |(zt , h · ∇w)Γ0 | dt + 2 |([F(w), w], h · ∇w)Γ0 | dt 0

0

In order to estimate the last two terms we need to use (2.21) and (1.13), respectively. First, by means of (2.21) and the boundary conditions on z, we have that Z

T

(3.6) 0

Z

|(zt , h · ∇w)Γ0 | dt

Z T |zt |2−4/5,Γ0 dt + C |h · ∇w|24/5,Γ0 dt 0 0 Z Z T 2 2 2 ≤ CT Ez (0) + |wt |0,Γ0 + |g(zt )|0,Γ1 + |z|0,Γ1 dt + C

≤ C

T

0

T 0

|w|22,Γ0

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Catherine Lebiedzik

Next, (1.13) and (1.14) give Z T Z T |([F(w), w], h · ∇w)Γ0 | , dt ≤ C |[F(w), w]|−1/2,Γ0 |h · ∇w|1/2,Γ0 0 0 Z T (3.7) ≤ C |F(w)|3−ε,Γ0 |w|3/2+ε,Γ0 |w|3/2,Γ0 0 Z T ≤ C |w|22,Γ0 |w|23/2+ε,Γ0 0

≤ CT sup |w|22,Γ0 |w|23/2+ε,Γ0 t∈[0,T ]

≤ CT E(0)lot(w, θ, z) Combining estimates (3.5),(3.6), and (3.7) gives the desired result (2.7).

References [1] G. Avalos, The exponential stability of a coupled hyperbolic/parabolic system arising in structural acoustics, Abstr. Appl. Anal., 1 (1996), pp. 203–217. [2] G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation. Dedicated to the memory of Pierre Grisvard, Rend. Istit. Mat. Univ. Trieste, 28 (1997), pp. 1–27. [3] , Uniform decay rates of solutions to a structural acoustic model with nonlinear dissipation, Appl. Math. Comput. Sci., 8 (1998), pp. 287–312. [4] H. T. Banks, R. J. Silcox, and R. C. Smith, The modeling and control of acoustic/structure interaction problems via piezoceramic actuators: 2-d numerical examples, ASME Journal of Vibration and Acoustics, 116 (1994), pp. 386–396. [5] H. T. Banks, R. C. Smith, and Y. Y. Wang, The modeling of piezoceramic patch interactions with shells, plates, and beams, Quarterly of Applied Mathematics, 53 (1995), pp. 353–381. [6] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976. [7] M. Camurdan, Uniform stability of a coupled structural acoustic system by boundary dissipation, Abstract and Applied Analysis, (1999). [8] C. R. Fuller, G. P. Gibbs, and R. J. Silcox, Simultaneous active control of flexural and extensional power flow in beams, Journal of Intelligent Materials, Systems and Structures, 1 (1990). [9] V. Isakov, Inverse Problems for Partial Differential Equations, Springer Verlag, New York, 1998. [10] V. Komornik, Exact controllability and Stabilization – the Multipliers Method, Masson, Paris, 1994. [11] J. Lagnese, Boundary Stabilization of Thin Plates, SIAM, Philadelphia, PA, 1989. [12] I. Lasiecka, Boundary stabilization of a 3-dimensional structural acoustic model, Journal de Mathematiques Pure et Applique, 78 (1999), pp. 203–232. , Mathematical Control Theory of Coupled PDE Systems: NSF-CMBS [13] Lecture Notes, SIAM, 2000.


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[14] I. Lasiecka and C. Lebiedzik, Uniform stability in structural acoustic systems with thermal effects and nonlinear boundary damping, Control and Cybernetics, 28 (1999), pp. 557–581. [15] , Boundary stabilizibility of nonlinear acoustic models with thermal effects on the interface, C.R Acad. Sci. Paris, 328 (2000), pp. 187–192. [16] , Decay rates in interactive hyperbolic-parabolic pde models with thermal effects on the interface, Appl. Math. Optim., to appear (2000). [17] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Diff. Int. Eq., 6 (1993), pp. 507–533. [18] I. Lasiecka and R. Triggiani, Regularity theory of hyperbolic equations under Dirichlet boundary terms, Appl. Math. and Optim., 10 (1983), pp. 275–286. [19] , Sharp regularity theory for second order hyperbolic equations of Neumann type. part i. nonhomogenous data, Ann.Mat. Pura Applicata IV, CLVII (1990), pp. 285–367. [20] , Sharp regularity theory for second order hyperbolic equations of Neumann type. part ii: general boundary data, J. Diff. Eq, 94 (1991), pp. 112–164. [21] I. Lasiecka, R. Triggiani, and X. Zhang, Exact controllability and unique continuation for wave equations with Neumann uncontrolled boundary conditions, Proceedings of the AMS, (to appear). [22] C. Lebiedzik, Uniform stability of a coupled structural acoustic system with thermoelastic effects, Dynamics of Continuous, Discrete, and Impulsive Systems, (to appear). [23] W. Littman and B. Liu, On the spectral properties and stabilization of acoustic flow, SIAM J. Appl. Math., 59 (1999), pp. 17–34. [24] P. Morse and K. Ingard, Theoretical Acoustics, McGraw-Hill, 1968.

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On Modelling, Analysis and Simulation of Optimal Control Problems for Dynamic Networks of Euler-Bernoulli-and Rayleigh-beams

G¨ unter Leugering, TU Darmstadt, Darmstadt, Germany Wigand Rathmann, inuTech GmbH, Seukendorf, Germany Abstract We consider a network of Euler-Bernoulli- and Rayleigh-beams. For the sake of simplicity, we concentrate on scalar displacements coupled to torsion. We show that the model is well-posed in appropriate ramification spaces. We then describe a dynamic nonoverlapping domain decomposition procedure of the network into its individual edges and provide a proof of convergence. Further, we formulate typical optimal control problems, related to exact controllability. The optimality system is solved using conjugate gradients. Various numerical examples illustrate the method.

AMS-Classification: 49M27, 73K12, 73K50, 93C20 Key-word: Euler-Bernoulli- and Rayleigh beams, network models, nonoverlapping domain decompositions, optimal controls.

1

Introduction

Beginning with the work of Chen et. al. [5], the question of controllability and stabilizability of connected beams and more general flexible structures has become a major field of research in the last ten years. Whereas such structures have been investigated in the engineering literature mainly in terms of the Finite-Element-Method (FEM), the mathematical literature is dominated by the original continuum mechanical formulation in terms of partial differential equations (PDE’s). It has become more and more apparent that controllability properties as well as their counterparts in terms of stabilizability are very much dependent on the underlying PDE-framework and can not be predicted or analyzed on the FEM-level. Therefore, rather than to first discretize and then apply standard optimal-control- or simulation-software (of which an abondance is currently available), we insist on simulation- and optimal-control procedures that are developed on the continuous level and are then discretized. In this 199

200

Leugering and Rathmann

paper we describe the continuous modelling and algorithms applied to the continuous systems. The discretization is then based on classical FEM-tools. Networks of strings, Timoshenko beams and combination there of have been investigated by Lagnese, Leugering and Schmidt [13]. Networks of EulerBernoulli-beams have been introduced in Leugering and Schmidt [20] and preliminary controllability results have been shown there. About ten years later Dekoninck and Nicaise [6] considered a scalar-displacement Euler-Bernoullibeam model with umbrella-type node conditions and showed controllability in some cases where Ingham’s inequality and a certain uniqueness result apply. Almost simultaneously Briffaut [4], in his thesis, considered in-planebeams for some particular configurations and provided numerical evidence for controllability and stabilizability. His analysis and also the numerical work heavily depends on the nodes of the system and a modal approximation to the HUM-operator. In summary, it is fair to say that up to know no satisfactory theory of controllability/stabilizability of networks of Euler-Bernoulli-beams, let alone Rayleigh-beams, exists. Furthermore, very few papers deal with numerical simulations of such networks. Of course, this has to do with the enormuous complexity of PDE-systems describing the motion of the such networks. Our philosophy here is to decompose the networks under consideration into smaller pieces, in fact, into the edges of the underlying graph. This is done by a nonoverlapping domain decomposition procedure, inspired by the work of P. L. Lions [21] and Benamou [1],[2]. The method is not a substructering method in the sense of Shur-complement-preconditioning. Rather it is an iterative procedure which, in the static case, can be derived from an augmented Lagrangian-saddle point algorithms of modified Uzawa-type, see [11] for the Laplace-operator. For in-plane-model the appropriate algorithm has been for Euler-Bernoulli-networks given in Leugering [16]. See also [3], [14], [15], [19], [17], [18] for string- and Timoshenko-beam networks. In this paper we also introduce torsional motion, while, for simplicity, we restrict ourselves to scalar displacements. The more general case has been considered in [23]. In contrast to [3], [14], [15],[16], [19], [17] [18], we do not decompose the optimality system, but rather we use the domain decomposition method (DDM) as a network solver. The optimal control problems which we consider are related to exact controllability, but other cost functionals can easily be considered. The method of choice here is a penalization of the final states which acts as a Tychonov regularization of the (illposed) controllability problem. We use a conjugate gradient (CG) approach in the spirit of R. Glowinski and J. L. Lions [9], [10]. The numerical results are very encouraging as far as the CG-iteration is concerned. As for the network solver based on DDM we have to say that the convergence, which is linear by the method of choice, is strongly dependent of the proper choice of the various penalty and relaxation parameters. This,

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201

of course, comes with no surprise. We currently perform research in the direction of optimal interface conditions in the sense of de Sturler, Nataf and Rogier [22]. This amounts to analyze and approximate Steklov-Poincaréoperators of Petrovski-systems. This will be the subject of a forthcoming publication. Analysis in this direction appears to be very important for realtime-applicability of the algorithms. The paper is organized as follows. We first introduce some notation (section 2). In section 3 we show that scalar non collinear networks of Euler-Bernoullibeams with rigrid joints but without torsion are inconsistent with mechanics. In section 4 we develop our main model including torsion. There we also give the existence and uniqueness result, the proof of which is shifted to section 7. In section 5 we describe our domain decomposition procedure and state the convergence result. The proof is shifted to section 8. Section 6 is devoted to the numerical treatment of optimal control problems.

2

Notations

We consider a network of beams. A planar graph G = (V, E) with nodes V and edges E is taken to describe the configuration at rest. The J-th node and the centerline of the i-th beam are identified with the vertex vJ ∈ V and edge ei ∈ E, respectively. The nodes (edges) are labeled 1, . . . , nv = ]V (1, . . . , ne = ]E).

◦

V denotes the set of inner nodes and ∂V the set of ◦ boundary nodes. The sets V and ∂V are defined by the degree of a node d(vJ ) := {number of the beams, which are incident at vJ }, as (2.1a)

◦

V := {vJ ∈ V :

d(vJ ) > 1}

and (2.1b)

◦

∂V := V \ V .

Clamped or free ends of single beams correspond to nodes vJ with d(vJ ) = 1 and, thus, to elements of ∂V . The position of a single beam (i = 1, . . . , ne ) is given in the reference configuration in R3 by (2.2)

Ri (xi , t) = Ri0 + xi1 ei1 + xi2 ei2 + xi3 ei3 ,

with the local coordinate system ei . ei1 is a unit vector directed along the centerline of the i-th beam. We assume, that the beams in the reference configuration are straight and untwisted. The cross section area at xi1 is defined by Ai (xi1 ) = {(xi2 , xi3 )| − b/2 6 xi2 6 b/2, −h/2 6 xi3 6 h/2}.

202


Ri0 is the offset of the point (0, 0, 0) of i-th beam in the rest configuration with respect to the global fixed coordinate system e0 .We assume e03 kei3 ,

∀i = 1, . . . , ne ,

such that the representing planar graph lies in the e01 − e02 plane. The out-ofthe-plane displacement is taken in the ei3 direction. We denote the deformed centerline in the following way ˆ i (xi , t) = Ri0 + xi ei + wi (xi , t)ei . R 1 1 1 3

(2.3)

We do not consider lateral displacement in the direction of ei2 , i.e. no in-plane motion is considered here. The more general case will be treated in [23]. E(vJ ) denotes the set of all beams being incident at vJ ∈ V , i.e. |E(vJ )| = d(vJ ). We distinguish the boundary nodes between Dirichlet- und Neumanntype nodes denoted by VD and VN , respectively. They are defined in the following way: (2.4)

VD := {vJ ∈ ∂V : wi (vJ ) = 0,

wi 0 (vJ ) = 0, i ∈ E(vJ )}

(2.5)

VN := {vJ ∈ ∂V : wi 00 (vJ ) = 0,

wi 000 (vJ ) = 0, i ∈ E(vJ )}.

Beyond the lateral displacements we introduce the angle φi (x, t) of Ai (x, t) about ei2 with respect to the rest configuration. We assume that φ(x) = w0 (x),

(2.6)

according to the Euler-Bernuolli-hypotheses. We use the following material constants on the i’th beam: Ei Gi ρi Ai Ii1 Ii2 Ei Ii2 Gi Ii1

Youngs modulus, shear modulus (not to mixed with transversal shear), density, cross section area, polar inertia of cross section area, inertia of cross section area, bending stiffness, torsional stiffness.

We consider the beams as onedimesional continua, denote the outer normal by ±1 and introduce ( −1 if R(vJ ) = Ri (0), εiJ = (2.7) +1 if R(vJ ) = Ri (li ).

203

Networks of Beams

3

Scalar beam networks without torsion are inconsistent

In this section we consider a model for the motion of the network based on conversation of the energy, as follows. The kinetic energy Ki of the i’th beam is given by 1 Ki = 2

Z

li

0

1 ρi Ai w˙ i dx + 2

Z

2

li

ρi Ii2 (w˙ i 0 )2 dx

0

and the potential energy Ui by 1 Ui = 2

Z

li

Ei Ii2 wi00 dx. 2

0

Therefore we obtain for the total energy Ei of i-th beam (3.1) 1 Ei = Ki + Ui = 2

Z 0

li

1 ρi Ai w˙ i dx + 2

Z

2

li

0

1 ρi Ii2 (w˙ i ) dx + 2 0 2

Z

li

Ei Ii2 wi00 dx 2

0

and, hence, for the total energy E of the entire system: nE X

E=

Ei .

i=1

The shear force F J and moment M J acting on vJ are defined as X FJ = (3.2a) εiJ Ei Ii2 wï 0 − Ei Ii2 wi 000 (vJ ) i∈E(vJ ) J

M =

(3.2b)

X

εiJ Ei Ii2 wi 00 (vJ )ei2 .

i∈E(vJ )

P J w(v The time deriviattive of the total energy is given by E˙ = ˙ J) + F vJ ∈V P J w ˙ 0 (vJ ). In order to obtain the equations of motion and proper vJ ∈V M compatibility conditions at the nodes, we assume that E˙ = 0 (F J = 0, M J = ◦

0 ∀vJ ∈V ) E˙ =

nE n Z X ∂i

i=1

Ei Ii2 wi00 w˙ i0

Z

Z dΩi −

ρi Ii2 w ï0 w˙ i dΩ −

+ ∂i

= 0.

Z

000

li

Ei Ii2 wi w˙ i dΩi + ∂i

Z

0

li

ρi Ii2 w ï00 w˙ i dx

Z

Ei Ii2 wi 0000 w˙ i dx

0 li

+ 0

o ˙ ρi Ai w ï widx

204

Leugering and Rathmann Rewriting the boundary terms as sums over the nodes we obtain (Z ) nE li X 00 0000 ¨ + ρi Ai wï dx E˙ = Ei Ii2 wi − ρi Ii2 wi i=1

+

X

0

εiJ Ei Ii2 wï 0 − Ei Ii2 wi 000 w˙ i (vJ )

X

vJ ∈V i∈E(vJ )

+

X

X

εiJ Ei Ii2 wi 00 w˙ i 0 (vJ ).

vJ ∈V i∈E(vJ )

Assuming continuity of the network we derive as another compatibility condition w˙ i 0 ei2 = w˙j 0 ej2

(3.3)

∀i, j ∈ E(vJ ), vJ ∈ V .

Finally, we arrive at the system (3.4a)

¨ 00 + ρi Ai wï = 0 Ei Ii2 wi 0000 − ρi Ii2 wi

i = 1, . . . , ne ,

0

vJ ∈ VD , i ∈ E(vJ ),

wi (vJ ) = wi (vJ ) = 0

(3.4b) (3.4c)

w˙ i = w˙j

i, j ∈ E(vJ ),

(3.4d)

w˙ i 0 ei2 = w˙j 0 ej2

i, j ∈ E(vJ ),

J

(3.4e)

F = 0,

(3.4f)

M J = 0,

J = 1, . . . , nJ , J = 1, . . . , nJ ,

(3.4g)

wi (0, x ) = wi (x )

xi ∈ (0, li ),

(3.4h)

w˙ i (0, xi ) = wi 1 (xi )

xi ∈ (0, li ).

0

i

i

We find a nontrivial solution for (3.3) (w˙ i 0 , w˙j 0 ) only if ei2 and ej2 are linear independent, that means only for beams in a collinear configuration. The problem is that this model does not account for torsion.

4

Second Model for out-of-plane dynamics

Here, we consider the displacements again in the ei3 -direction. However, now we take torsion (denoted by θi ) into account. The kinetic and potential energy is then given by Ki =

1 2

Z

li

ρi Ai w˙ i 2 dx +

0

1 2

Z

li

ρi Ii2 w˙ i 02 dx +

0

1 2

Z

li

0

and 1 Ui = 2

Z

li 0

2 Ei Ii2 wi00 dx

1 + 2

Z

li 0

Gi Ii1 θi 0 dx. 2

0

ρi Ii1 θ˙i 2 dx

Networks of Beams

205

The total energy of the system is nE nE X X E= (Ki + Ui ) = Ei . i=1

i=1

The shear force F J and the bending moment M J are now defined as X FJ = (4.1a) εiJ ρi Ii2 wï 0 − Ei Ii2 wi 000 , i∈E(vJ )

X

J

M =

(4.1b)

εiJ Gi Ii1 θi 0 ei1

+ Ei Ii2 wi00 ei2

.

i∈E(vJ )

In the next step we derive the equation of motion and the nodal conditions. We start with the condition E˙ = 0: Z Z li nE n Z X E˙ = Ei Ii2 wi00 w˙ i0 dΩi − Ei Ii2 wi 000 w˙ i dΩi + Ei Ii2 wi 0000 w˙ i dx ∂i

i=1

Z + −

ρi Ii2 w ï0 w˙ i dΩi −

∂i Z li

Gi Ii1 θ˙i00 θ˙i dx +

0

Z Z

∂i

0

li

ρi Ii2 w ï00 w˙ i dx

0 li

Z

li

+

ρi Ai w ï w˙ i dx 0

o

ρi Ii1 θï θ˙i dx

0

= 0. This gives ρi Ii1 θï − Gi Ii1 θi 00 = 0

(4.2a) (4.2b)

ρi Ai wï −

ρi Ii2 w ï00

(torsion)

00

+ Ei Ii2 wi = 0

(lateral displacement)

and, at the nodes, X X X 0= εiJ ρi Ii2 wï 0 − Ei Ii2 wi 000 w˙ i + vJ ∈V i∈E(vJ )

X

εiJ Ei Ii2 wi00 w˙ i 0 + Gi Ii1 θi 0 θ˙i .

vJ ∈V i∈E(vJ )

The transmission of moments reads like X X 0 ˙ 00 0 εiJ Ei Ii2 wi w˙ i + Gi Ii1 θi θi vJ ∈V i∈E(vJ )

=

X

X

εiJ Ei Ii2 wi00 ei2 + Gi Ii1 θi 0 ei1 θ˙i ei1 + w˙ i 0 ei2 .

vJ ∈V i∈E(vJ )

Since the resulting moments M J and forces F J are zero at the node we obtain (4.3a)

θ˙i ei1 + w˙ i 0 ei2 = θ˙j ej1 + w˙ j0 ej2 ,

206

Leugering and Rathmann ∀i, j ∈ E(vJ ).

w˙ i = w˙ j ,

(4.3b)

The description becomes complete with the initial data (4.4a)

θi (0, xi ) = θi 0 (xi ),

θ˙i (0, xi ), = θi 1 (xi )

xi ∈ (0, li ),

(4.4b)

wi (0, xi ) = wi 0 (xi ),

w˙ i (0, xi ) = wi 1 (xi ),

xi ∈ (0, li ).

The difference of the models presented in this and the previous section becomes obvious in the case of a carpenter square. Theorem 4.1. A network consisting of ne Rayleigh-beams, described by the system, ρi Ii1 θï − Gi Ii1 θi 00 = 0,

(4.5a) (4.5b)

ρi Ai wï − ρi Ii2 w ï00

+ Ei Ii2 wi

0000

i = 1, . . . , ne ,

= 0,

i = 1, . . . , ne ,

(4.5c)

θi ei1 + wi 0 ei2 = θj ej1 + wj0 ej2 ,

∀i, j ∈ E(vJ ),

(4.5d)

wi = wj ,

∀i, j ∈ E(vJ ),

(4.5e)

0

i ∈ E(VD ),

θi (vJ ) = wi (vJ ) = wi (vJ ),

with

X

(4.5f)

εiJ ρi Ii2 wï 0 − Ei Ii2 wi 000 = 0,

i∈E(vJ )

X

(4.5g)

εiJ Gi Ii1 θi 0 ei1 + Ei Ii2 wi00 ei2 = 0,

i∈E(vJ )

and initial data xi ∈ (0, li ),

(4.6b)

θi (0, xi ) = θi 0 (xi ) ∈ H 2 (0, li ), θ˙i (0, xi ) = θi 1 (xi ) ∈ H 1 (0, li ),

(4.6c)

wi (0, xi ) = wi 0 (xi ) ∈ H 3 (0, li ),

xi ∈ (0, li ),

(4.6d)

w˙ i (0, xi ) = wi 1 (xi ) ∈ H 2 (0, li ),

xi ∈ (0, li ),

(4.6a)

xi ∈ (0, li ),

admits a unique strong solution (w, θ) with θi ∈ C(R, H 1 (0, li )),

θ˙i ∈ C(R, H 1 (0, li )),

wi ∈ C(R, H 2 (0, li )),

w˙ i ∈ C(R, H 2 (0, li )).

satisfying (4.5). Proof. A proof of an analogous result for 3-d beams is presented in [23]. In the statement of this theorem we do not insist on optimal regularity setups with respect to the spaces of initial conditions. the proof is based on a generation result a C0 -(semi)group. For the sake of simplicity we do not describe the domain of the generator in full detail, and state sufficient conditions, only.

Networks of Beams

5

207

Numerical Investigations

In section 4 we deduced a model for the out-of-the-plane motion of networks of beams. In this section we first present an algorithm for dynamic domain decomposition for networks of such beams which is implemented in a software package, and then we present some numerical results.

5.1

Dynamic Domain Decompositon Method

The idea is, not to compute the response of the entire network at once, that is on the global network level, but rather perform the computation on each single edge (beam) and match the the edges in a iterative procedure. Therefore we introduce the augmented Langrangian Z T "X nE n Z li 2 1 2 L= −ρi Ii1 θ˙i + ρi Ai w˙ i 2 − ρi Ii2 w˙ i0 2 0 0 i=1 o + Gi Ii1 (θi 0 )2 + Ei Ii2 (wi00 )2 dx dt X σJ X + |wi (vJ ) − ζJ |2 2 J i∈E(vJ ) X νJ X + |θi (vJ )ei1 + wi 0 (vJ )ei2 − ηJ |2 2 J i∈E(vJ ) X X + µiJ (wi (vJ ) − ζJ ) J i∈E(vJ )

+

X X

# ρiJ (θi (vJ )ei1

+

0

wi (vJ )ei2

− ηJ ) dt

J i∈E(vJ )

which is of the form L = −K +U +penalty+Lagrangian multipliers for the nodal conditions. To this Lagrangian we apply the ALG3 of Glowinski/Le Tallec [11] to determine a saddle point: 1) ∂1 L((θ, w)n , (ζ, η)n−1 , (µ, ρ)n ) = 0, n+ 12

= µniJ + σJ ∂31 L((θ, w)n , (ζ, η)n−1 , µniJ ),

n+ 12

= ρniJ + νJ ∂32 L((θ, w)n , (ζ, η)n−1 , ρniJ ),

2) µiJ ρiJ

1

3) ∂2 L((θ, w)n , (ζ, η)n , (µ, ρ)n+ 2 ) = 0, n+ 12

4) µn+1 iJ = µiJ

n+ 21

ρn+1 = ρiJ iJ

+ σJ ∂31 L((θ, w)n , (ζ, η)n , µniJ ), + νJ ∂32 L((θ, w)n , (ζ, η)n , ρniJ ).

Eliminating the Lagrange multipliers we deduce the following iteration procedure in n for i = 1, . . . , ne (5.1a)

n

ρi Ii1 θï − Gi Ii1 θi n00 = 0,

208

Leugering and Rathmann ρi Ai wï n − ρi Ii2 w ïn 00 + Ei Ii2 wi n0000 = 0,

(5.1b)

n εiJ (ρi Ii2 wï n00 − Ei Ii2 wi n000 ) + σJ wi n = giJ ,

(5.1c)

n 0 εiJ (Gi Ii1 θi n0 ei1 + Ei Ii2 wi n00 ei2 ) + νJ (θi n ei1 + wi n0 ei2 ) = giJ ,

(5.1d) with (5.1e)

(5.1f)

n−1 giJ − 2σJ wi n−1 (vJ ) −

−

2 dJ

X

0 n−1 giJ 0 (gjJ

n−1

2 dJ

X

n−1 n (gjJ − 2σJ wjn−1 (vJ )) = giJ ,

j∈E(vJ )

− 2ρJ (θi n−1 (vJ )ei1 + wi0

− 2ρJ (θi n−1 (vJ )ei1 + wj0

n−1

n−1

(vJ )ei2 )

0 (vJ )ei2 )) = giJ . n

j∈E(vJ )

This iteration is analoguous to the one obtained in [16] with the updates written in the format of [7]. In each single iteration we have to solve a system of 1-d hyperbolic and quasi-hyperbolic or even Petrovski-type PDE’s on each single n and g 0 n . We implement this algorithm in a edge followed by an update of giJ iJ slightly different form: we formulate the Robin-boundary data and the update for giJ on the velocity level n

(5.2a)

ρi Ii1 θï − Gi Ii1 θi n00 = 0,

(5.2b)

ρi Ai wï n − ρi Ii2 w ¨n00 + Ei Ii2 wi n0000 = 0, n εiJ (ρi Ii2 wï n00 − Ei Ii2 wi n000 ) + σJ w˙ i n = giJ , n i n n0 i n00 i n0 i 0 εiJ (Gi Ii1 θi e1 + Ei Ii2 wi e2 ) + νJ (θ˙i e1 + w˙ i e2 ) = giJ ,

(5.2c) (5.2d) with (5.2e)

(5.2f)

n−1 giJ − 2σJ w˙ i n−1 (vJ ) −

−

2 dJ

X

2 dJ

X

n−1 n (gjJ − 2σJ w˙ j n−1 (vJ )) = giJ ,

j∈E(vJ ) n−1

(vJ )ei1 + w˙ i 0n−1 (vJ )ei2 )

0 giJ

n−1

− 2ρJ (θ˙i

0 (gjJ

n−1

n−1 0 n − 2ρJ (θ˙i (vJ )ei1 + w˙ i 0n−1 (vJ )ei2 ) = giJ .

j∈E(vJ )

The incorporation of velocities rather then displacements is motivated by the work of [8] and [22]. Theorem 5.1. The algorithm (5.2) converges to a solution (θ, w) in an H 1 × H 2 -sense. Proof. The proof is presented in [23]. Some numerical results are presented in the next part of this section 5.2.

5.2

Numerical results

The dynamic domain decomposition algorithm (5.2) is been implemented using the finite element method. We solve the equations (5.2a),(5.2b) by FEM with

Networks of Beams

209

linear elements for the torsion and hermit-cubic elements for the wi ’s and use a Newmark-algorithm to solve the equations in time. The semi-discrete form of the algorithm after space discretization reads as follows: n + G I K =f , ρi Ii1 Mθi θ¨hi i i1 θi θi

(5.1a)

(5.1b) n +E I K ρi Ai Mwi + ωR ρi Ii2 IMwi w¨hi i i2 wi =fwi ,

t ∈ [0, T ], i = 1, . . . nE ,

The right hand sides fθi , fwi include the right hand sides of the PDE, n and g n0 . The updates for g n and g n0 are applied moments and forces and giJ iJ iJ iJ implemented in a relaxed form n−1 λJ giJ − 2σJ w˙ i n−1 (vJ ) (5.1c) −

X

2 dJ

n−1 n−1 n (gjJ − 2σJ w˙ j n−1 (vJ )) + (1 − λJ )giJ = giJ ,

j∈E(vJ ) 0 µJ giJ

2 − dJ

λJ ∈ (0, 1),

X

n−1

0 n−1 gjJ

n−1 − 2ρJ θ˙i (vJ )ei1 + w˙ i 0n−1 (vJ )ei2

− 2ρJ (θ˙i

n−1

(vJ )ei1

+ w˙ i

0n−1

!

(vJ )ei2 )

j∈E(vJ ) 0 +(1 − µJ )giJ

(5.1d)

n−1

0 = giJ , µJ ∈ (0, 1). n

For the updates we only need the velocities at the nodes, which we get from the Newmark-algorithm. It is not necessary to compute deriviatives of θi or wi . We mention, that the space discretization is allowed to differ on each edge. Hence, the dimension of the matrices might be different on each edge. For time discretization we choose the Newmark-algorithm in a predictor corrector form. The time step size is denoted by ∆t = nTt , ti = i∆t,

i = 0, . . . , nt .

We use the property of conservation of energy guaranteed by the Newmark n and g n0 act as artifical forces and moments. scheme. In the first iterates the giJ iJ One observes max Eh (t) − min Eh (t) → 0 ti ∈[0,T ]

t∈[0,T ]

over the iterations for a closed system, but we do not use this as a stop-criteria. In fact, we use the maximal error at the node max max |wi − wj | 6 tol, ◦ ti ∈[0,T ] i,jE(vJ )

vJ ∈V

210


and max kXn − Xn−1 kE 6 tolE ,

t∈0:∆t:T

Xn =

n θhi n whi

. We choose as norm the discrete energy norm 1:nE

(5.2) kXn (ti )kE ( nE X T n (t )T (ρ A M ˙n ˙n ˙n w˙hi = i i i wi + ωR ρi Ii2 IMwi )whi (ti ) + θhi (ti ) ρi Ii1 Mθi θhi (ti ) i=1

) +

n n whi (ti )T Ei Ii2 Kwi whi (ti )

+

n n θhi (ti )T Gi Ii1 Kθi θhi (ti )

.

The choice of the parameter σJ , νn , λJ , µJ is not obvious at all, an analytical / numerical treatment of optimal transmission conditions in the spirit of [22] is under way. On the other side the dependence of material constants is more transparent and will be examined in below. Our experiences suggest a relaxation parameter between 0.8 and 0.9. The implementation of a domain decomposition algorithm (DDMalgorithm) to solve the elliptic (static) case is similar to the DDDM-algorithm above. The only difference is with the update of traces rather of velocity-traces. The choice of the numerical parameters can be taken from DDDM-algorithm (and vice versa). We now discuss some examples. Loaded clamped free beam. We consider the configuration showed in fig. 1. The beam is clamped at 0 and free at l = 0.8m. The material constants are given in tab. 1. F=-17.5N q=-9.37N/m

500

300

Fig. 1. Clamped-free beam with loads.

Networks of Beams

211

Table 1 Material constants.

density Young’s modulus cross-section area 25mm inertia of the cross-section

ρ E A I2

= = = =

7584 kg/m3 2.1 · 1011 N/m2 0.012 m2 4.36 · 10−9 m4

The explicit solution for the given loading is (5.3)

w(x) =

          

q 24EI ((0.8

− x)4 + 4 · 0.83 x − 0.84 ) +

− x)3 + 3 · 0.52 x − 0.53 ) x ∈ [0, 0.5], q F 4 3 4 2 3 24EI ((0.8 − x) + 4 · 0.8 x − 0.8 ) + 6EI (3 · 0.5 x − 0.5 ) x ∈ (0.5, 0.8]. F 6EI ((0.5

The solution is plotted in fig. 2(a). This problem was solved with a domain decomposition method, where we considered this configuration as a serial network of two beams. The relative error is about 10−4 (see fig. 2(b)). The graph is given in fig. 3. The material coefficients are equal for both beams. We choose for the numerical parameter σJ = νJ = 1000, λJ = µJ = 0.9, tol = 10−30 , tolE = 10−30 , ωR = 0. ωR = 0 means, that we consider the Euler-Bernoulli-beam model. 266 Iterations where needed for convergence . In fig. 4(a) the total error on node 2 is plotted over the number of iterations. The relative error of about 10−4 was reached once the error at the inner node settled at about 10−10 . Fig. 4 shows, that we obtain linear convergence. Vibrating beam. The lowest eigenfrequency of the beam given above is approximately ω = 7.77719s−1 and the related lowest period is T1 = 0.128581s. We choose as initial value the solution of the static problem 5.2 in fig. 2(a), and w˙ 1 = w˙ 2 = 0. With the numerical parameters T = 0.26, ∆t = 0.01, σJ = νJ = 200,

212


relative error exact and numerical solution

−4

solution 0

1.115

−0.005

1.105

x 10

1.11

1.1 −0.01

1.095 1.09

−0.015

1.085 1.08

−0.02

1.075 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1.07 0

(a) Solution for clamped free beam.

0.1

0.2

0.3

0.4

0.5

(b) Relativ error.

Fig. 2. Solution of the problem.

1 0.8 0.6 0.4

y

0.2 0 1

1

2

2

3

−0.2 −0.4 −0.6 −0.8 −1

0

0.2

0.4 x

0.6

0.8

Fig. 3. Graph and material constants of the network.

λJ = µJ = 0.9, ωR = 0,

0.6

0.7

0.8

213

Networks of Beams i=266, |Xi(vJ)−Xj(vJ)|=0 0

0

10

−5

0

10 ||Xi−Xi−1||E

max|Xi(vJ)−Xj(vJ)|

10

−10

10

−10

10

−15

−20

10

10

−20

10

i=266, ||Xi−Xi−1||E=1.2835e−28

10

10

−30

0

50

100

150 Iterationen i

200

250

300

10

0

50

(a) Error on node 2. − maxt |w1n (vJ ) − 0n

150 Iterationen i

200

250

300

1000

1200

(b) kXn − Xn−1 kE .

0n

100

w2n (vJ )|, - - maxt |w1 (vJ ) − w2 (vJ )|

Fig. 4. Convergence of static domain decomposition.

−2

maxt ||Xi(t)−Xi−1(t)||E

2

10

10

0

10 −4

10

−2

10 −6

10

−4

10

−6

10

−8

10

−8

10 −10

10

−10

10 −12

10

0

−12

200

400

600 iterations n

800

1000

1200

(a) Error on node 2. − maxt |w1n (vJ ) − w2n (vJ )|,

-.

0

maxt |w1n (vJ )

−

0

10

0

200

400

600 iterations n

800


w2n (vJ )|

Fig. 5. Convergence of dynamic domain decomposition.

the algorithm stops after 1182 iterations with the following errors: max kw1 (v2 ) − w2 (vJ )k = 9.99033 · 10−11 , t

max kw1 (v2 ) − w2 (vJ )k = 4.47642 · 10−11 , t

max kX1182 − X1181 kE = 3.93958 · 10−12 , t

max E(t) − min E(t) = 3.19250 · 10−6 . t

t

214


The meshplot in fig. 6 shows that the lowest eigenfrequence with period T1 ≈ 0.13s is recovered numerically. 0

10

−2

10

0.03 0.02 0.01

−4

0

10

−0.01 −0.02 −0.03 0

−6

10

0.5 −8

10

0

200

400

600 800 iterations n

1000

1200

(a) maxt En (t) − mint En (t)

x

1

0

0.05

0.1

0.15

0.2

0.25

time t

(b) Dynamic responce of the beam.

Fig. 6. Variation of energy in time and dynamical response.

Carpenter square with loads. In the next example we consider the carpenter-square with loads illustrated in fig. 7, with material parameters in table 2. The initial value is the static solution of a domain decomposition for w1 , w2 and θ1 , θ2 . The velocities are all equal to zero. We now observe torsional effects in the beam, since the lateral elongation couples to the torsion angle at the common node. With the numerical parameters T = 1.23, ∆t = 0.025, σJ = νJ = 160, λJ = µJ = 0.9, ωR = 1, the algorithm stops after 296 iterations with the following errors: max kw1 (v2 ) − w2 (vJ )k = 5.12849e · 10−9 , t

max kw10 (v2 ) − w20 (vJ )k = 2.63987 · 10−9 , t

max kX296 − X295 kE = 8.01305 · 10−11 , t

max E(t) − min E(t) = 3.74158 · 10−5 . t

t

Networks of Beams

215

We choose the Rayleigh-beam model for the example. Fig. 8 shows the convergence of the two stop-criteria. The behaviour of the algorithm is linear. It is easy to see, that the error at the common node and the difference of two successive iterates Xn is correlated. If we look at the loss of energy in fig. 9(a) we observe the same. In fig. 9(b) the response in time is displayed. The state for t = 0 is displayed at the buttom corner of the right side, while the final time T = 1.23s upper left hand side. F=150 N

500 N/m

2500

Fig. 7. Carpenter square with loads.

density Young’s modulus shear modulus cross-section area ineratia of the cross-section polar inertia of the cross-section

ρ E G A I2 I1

= = = = = =

2690 kg/m3 7.3 · 1010 N/m2 2.54 · 1010 N/m2 0.000625 m2 3.26 · 108 m4 6.51 · 10−8 m4

Table 2 Material constants for carpenter square.

6

Control of Networks

We now proceed to apply the domain decomposition procedure above to optimal control problem. Since the model is time reversible it is sufficient to look at the reachability problem. We give a CG-Algorithm to compute the control from the adjoint system. We tested the method for serially connected beams and the carpenter square. For the latter no exact controllabilty results are known.

6.1

The control problem

We consider a given network of Rayleigh/ Euler–Bernoulli–beams as above with at least one boundary node clamped. We want to control at the inner (multiple)

216


maxt ||Xn(t)−Xn−1(t)||E

10

carpenter3; Penalty: [2e+02 2e+02]; Relaxation: [0.9 0 10

0.9]; Iterationen: 296

10

5

10 −2

10

0

10 −4

10

−5

10 −6

10

−10

10

−8

10

−15

10

−10

10

0

50

100

150

200

250

0

(a) Error on node 2. − maxt |w1n (vJ ) − w2n (vJ )|,

-.

0

maxt |w1n (vJ )

−

0

50

100

150 iterations n

300

200

250

300


w2n (vJ )|

Fig. 8. Stop criteria for DDDM.

or free nodes. The forward (in time) running system is − Gi Ii1 θi 00 = 0,

(6.1a)

ρi Ii1 θï

(6.1b)

ρi Ai wï − ρi Ii2 wï 00 + Ei Ii2 wi 0000 = 0, ◦

θi ei1 + wi 0 ei2 = θj ej1 + wj0 ej2 , ∀i, j ∈ E(vJ ), vJ ∈V ,

(6.1c)

wi = wj ,

(6.1d) (6.1e)

X

(6.1f)

θi (vJ ) = wi (vJ ) = wi 0 (vJ ) = 0, εiJ (ρi Ii2 wï 0 − Ei Ii2 wi 000 ) = F J ,

◦

∀i, j ∈ E(vJ ), vJ ∈V , i ∈ E(VD ), ∀vJ ∈ VC ,

i∈E(vJ )

(6.1g) X

εiJ (Gi Ii1 θi 0 ei1 + Ei Ii2 wi 00 ei2 ) = M J ,

∀vJ ∈ VC

i∈E(vJ )

with initial conditions

(6.1h)

θi (x, 0) = θi 0 ,

θ˙i (x, 0) = θi 1 ,

(6.1i)

wi (x, 0) = wi 0 ,

w˙ i (x, 0) = wi 1 .

217

Networks of Beams ∆ E(n)

4

10

2

10

0

10

7

−2

10

6 5

1 −4

w

10

0 −1 −5

−6

10

0

50

100

150 200 iterations n

250

(a) maxt En (t) − mint En (t)

4 3

t 2 −4

−3

300

−2

−1

1 0

1

2

0

(b) Dynamic responce of the carpenter square.

Fig. 9. Results of DDDM for the carpenter square.

The proper function spaces are given by ( θi θi 1 H := ∈ Hi0 ⊕ Hi[3] : wi i=1,...,n wi E

wi (vJ ) = 0 ∀i ∈ E(vJ ), vJ ∈ VD

)

wi (vJ ) = wj (vJ ) ∀i, j ∈ E(vJ ), vJ ∈ V \VD , and ( H1 :=

θi wi

i=1,...,nE

θi 2 ∈ Hi1 ⊕ Hi[3] , wi θi (vJ ) = wi (vJ ) = 0 ∀ i ∈ E(vJ ), vJ ∈ VD

θi (vJ )ei1

0

+ wi (vJ )ei2 = θj (vJ )ej1 + wj0 (vJ )ej3 , wi (vJ ) = wj (vJ ),

)

∀i, j ∈ E(vJ ), vJ ∈ V \VD .

2 (i = 1, . . . , n ) H0 und H10 denote the corresponding dual spaces. Hi1 , Hi[3] E are defined by

Hi1 := {v | v, v 0 ∈ L2 ([0, li ]; [ei1 ])},

218

Leugering and Rathmann 2 Hi[3] := {v | v, v 0 ∈ L2 ([0, li ]; [ei3 ]), v 00 = L2 ([0, li ]; [ei3 ])}.

Further we introduce the Riesz-isomorphisms *

+ θ φ AH , w ψ

(6.2a) =

nE X

(ρi Ii1 θi , φi ) +

0

(ρi Ai wi , ψi ) + (ρi Ii2 wi , ψi0 )

i=1

*

+ θ φ AH1 , w ψ

(6.2b) =

nE X

0

(Gi Ii1 θi , φ0i )

00

(Ei Ii2 wi , ψi00 )

+

θ φ , ∀ , ∈ H, w ψ

θ φ ∀ , ∈ H1 . w ψ

,

i=1

The Riesz-isomorphism between the product space A : H1 × H → H10 × H0 is given by

A=

AH1 . AH

The right hand sides of (6.2) are the scalar products of the corresponding spaces. We consider the solutions of equation (6.1) with initial data in H0 × H10 . Therefore we use transposition, see [12]. Consider the solution of

(6.3a) (6.3b)

− Gi Ii1 θ¯i00

ρi Ii1 θ¯ï 00

ρi Ai w ¯ï − ρi Ii2 w ¯ï + Ei Ii2 wi

X

(6.3c)

= 0, 0000

= 0,

0 ¯ï − Ei Ii2 w εiJ (ρi Ii2 w ¯i000 ) = F J ,

i∈E(vJ )

(6.3d)

X

εiJ (Gi Ii1 θ¯i0 ei1 + Ei Ii2 w ¯i00 ei2 ) = M J ,

i∈E(vJ )

for fixed

1 θ¯0 θ¯ ∈ H ∈ H. , 1 w ¯0 w ¯1

Networks of Beams

219

Thus we have

0=

nE X

(Z

T 0

=

i=1 (Z nE li X

Z

li

) 00

¯ï − ρi Ii2 w ¯ï + Ei Ii2 w (ρi Ii1 θ¯ï − Gi Ii1 θ¯i00 )θi + (ρi Ai w ¯i0000 )wi dx dt

0

ρi Ii1 θ¯˙i (T )θi (T ) − ρi Ii1 θ¯i (T )θ˙i (T )

0

i=1

0 + ρi Ai w ¯˙ i (T )wi (T ) + ρi Ii2 w ¯˙ i (T )wi 0 (T ) − ρi Ai w ¯i (T )w˙ i (T ) − ρi Ii2 w ¯i0 (T )w˙ i 0 (T ) dx Z li − ρi Ii1 θ¯i1 θi 0 − ρi Ii1 θ¯i0 θi 1 + ρi Ai w ¯i1 wi 0 + ρi Ii2 w ¯i1 0 wi 00 0 ) X Z T 0 1 0 0 10 − ρi Ai w ¯i wi − ρi Ii2 w ¯i wi dx + FJw ¯i + M J (θ¯i ei1 + w ¯i0 ei2 ) dt. vJ ∈VC

0

Upon defining LS (6.4)

" ˙ #! X Z S θ¯0 θ¯1 := , ˙1 FJw ¯i + M J (θ¯i ei1 + w ¯i0 ei2 ) dt w ¯0 w ¯ v∈VC 0  1   ¯0  θ + * θ 0  w 1  w  ¯  +   −θ 0  ,  θ¯1  H01 ×H0 ,H1 ×H −w0 w ¯1

we rewrite this identity as * LT ((θ¯0 , w ¯0 ), (θ¯1 , w ¯1 )) =



 ¯  ˙ ) θ(T ) + θ(T  w(T   ¯ )  ˙ )  , w(T  .  −θ(T )   θ(T ¯˙ )  H01 ×H0 ,H1 ×H −w(T ) w(T ¯˙ )

˙ w)] Definition 6.1. [(θ, w), (θ, ˙ ∈ H0 × H10 is called a solution of (6.1) if   θ w  0 0  i)   θ˙  ∈ C(R, H × H1 ), w˙

ii) (6.4) is fulfilled for all S ∈ R+

 ¯0  θ w ¯0   and all   θ¯1  ∈ H1 × H, w ¯1

220


(6.5)   ¯  ˙ θ(S) + * θ(S)  w(S)    w(S)  ˙  ¯   −θ(S)  ,  θ(S)  ¯˙ H01 ×H0 ,H1 ×H −w(S) w(S) ¯˙ X Z

=

v∈VC

S 0

* FJw ¯i + M J (θ¯i ei1 + w ¯i0 ei2 ) dt +

  ¯0  θ + θ1 0  w 1  w , ¯   .  −θ 0   θ¯1  0 0 H1 ×H ,H1 ×H −w0 w ¯1 

We choose as cost functional (6.6)

Z 1 X T J= kF J k2 + kM J k2 dt + 2 0 J∈VC

2

2

˙ ) k k

θ(T )

θ(T 0 1 −z + −z .

0 2 w(T

0 ˙ ) 2 w(T ) H

H1

z 0 denotes the desired state at time T and z 1 the velocity at the final time. The necessary first order optimality condition is (6.7) 0=

X Z

J∈VC

T 0

D

+k

ˆ J dt F J Fˆ J + M J M

" # θ(T ) E D ˆ˙ ˙ E ˆ θ(T ) θ(T ) θ(T ) −1 −1 0 , AH −z , +k − z1 . , AH1 w(T ) w(T ˙ ) w(T ˆ ) w(T ˆ˙ )

J J ˆ w] ˆJ ˆJ [θ, ˆ solves (6.1) with F = F and M = M and homogenous initial φ data. solves the equations ψ

(6.8a) (6.8b)

ρi Ii1 φï − Gi Ii1 φ00i = 0, ρi Ai ψï − ρi Ii2 ψï00 + Ei Ii2 ψi0000 = 0, X

(6.8c)

εiJ (ρi Ii2 ψï0 − Ei Ii2 ψi000 ) = 0,

i∈E(vJ )

(6.8d)

X

i∈E(vJ )

εiJ (Gi Ii1 φ0i ei1 + Ei Ii2 ψi00 ei2 ) = 0,

Networks of Beams

221

with the final data at T ! ˙ ) φ(T ) θ(T −1 1 = kAH1 (6.9a) −z , ψ(T ) w˙ i (T ) ! ˙ ) φ(T θ(T ) −1 0 (6.9b) −z . ˙ ) = −kAH wi (T ) ψ(T From this we calculate F J and M J in equation (6.1) as (6.10a)

F J = −ψ¯i ,

(6.10b)

M J = −φ¯i ei1 − ψ¯i0 ei3 . ¯

¯

The dual problem of (6.7) is (Rockefellar-Fenchel duality) 1 1 −z ˆ ˆ (6.11) ,f , hAf, f i + hΛf, f i = z0 k with



 φ(T )  ψ(T )  f   f = 0 =  , f1  φ(T ˙ )  ˙ ) ψ(T

and Λ the so called HUM-operator associated with the problem. Λ maps f to the final state of the forward running System:

(6.12)

Λ : H1 × H → H10 × H0 ,   ˙ θ(T ) − w(T ˙ )    f 7→ Λf =  .  θ(T )  w(T )

The duality h·, ·i operates between H10 × H0 and H1 × H. Equation (6.9) can be rewritten as (6.13)

a(f, fˆ) = l(fˆ),

where a(·, ·) is a bilinear form on (H1 × H)2 and l(·) a linear form on H1 × H given as follows (6.14a) (6.14b)

1 1 a(f, fˆ) = hAf, fî + hΛf, fî = (f, fˆ) + (A−1 Λf, fˆ), k k 1 1 ! −z −z l(fˆ) = , fˆ = A−1 , fˆ . z0 z0

222

6.2


CG-algorithm for the control problem

We want to solve (6.12) with a CG-algorithm. Glowinski and Lions suggested in [9] and [10] a CG-Algorithmus for parabolic and hyperbolic equations for diffenrent controls (Dirchlet or Neumann type controls). Their cost functional is similiar to (6.6). The difference of both the final state and the desired state, however, was taken in the L2 -norm. The CG-algorithm for the situation considered here is given in a abstract setting. (0) Initialization. Choose a value f0 ∈ H1 × H, calculate (g0 , fˆ) = a(f 0 , fˆ) − l(fˆ), or in dualities 1 hAg0 , fî = hAf 0 , fî + hΛf 0 , fî − l(fˆ) k * 1 0 ˆ −z 1 ˆ 0 ˆ = ,f . Af , f + hΛf , f i − z0 k For the evaluation of Λf0 we have to solve ρi Ii1 φï − Gi Ii1 φ00i = 0, ρi Ai ψï − ρi Ii2 ψï00 + Ei Ii2 ψi0000 = 0, with homogenous nodal conditions X εiJ (ρi Ii2 ψï0 − Ei Ii2 ψi000 ) = 0, i∈E(vJ )

X

εiJ (Gi Ii1 φ0i ei1 + Ei Ii2 ψi00 ei2 ) = 0,

i∈E(vJ )

and the final data at T (6.15a) (6.15b)

φ(T ) = f00 , ψ(T ) ˙ ) φ(T 1 ˙ ) = f0 . ψ(T

With the control given by (6.16a)

F J = −ψ¯i (vJ ),

(6.16b)

M J = −φ¯i (vJ )ei1 − ψ¯i0 (vJ )ei3 , ¯

¯

∀vJ ∈ VC ,

Networks of Beams

223

we solve − Gi Ii1 θi 00 = 0,

ρi Ii1 θï

ρi Ai wï − ρi Ii2 wï 00 + Ei Ii2 wi 0000 = 0,

X

εiJ (ρi Ii2 wï 0 − Ei Ii2 wi 000 ) = F J ,

i∈E(vJ )

X

εiJ (Gi Ii1 θi 0 ei1 + Ei Ii2 wi 00 ei2 ) = M J

i∈E(vJ )

with homogenous initial conditions θi (x, 0) = 0,

θ˙i (x, 0) = 0,

wi (x, 0) = 0,

w˙ i (x, 0) = 0.

In order to determine g0 = (g00 , g01 ) ∈ H1 × H, the equations ˙ ) θ(T −1 ˙ −1 1 1 g00 = −A−1 + A−1 (6.17) H1 w(T H1 z = −AH1 X(T ) + AH1 z , ˙ ) 1 −1 θ(T ) −1 0 g0 = AH − AH−1 z 0 = A−1 (6.18) H X(T ) − AH z , w(t) are to be solved. The Riesz-isomorphism from H to H0 is the identity. To obtain g01 we solve ˙ ) + z1 , −Gi Ii1 gθ0i 0 00 = −θ(T θi

(6.19a)

0 0000 1 Ei Ii2 gw = −w(T ˙ ) + zw , i0 i

(6.19b)

with homogenous node conditions X

(6.19c) (6.19d)

X

0 000 εiJ Ei Ii2 gw0 = 0,

i∈E(vJ ) 0 0 i 0 00 i εiJ (Gi Ii1 gθ0 e1 + Ei Ii2 gw0 e2 ) = 0.

i∈E(vJ )

Hence, we obtain a solution of the equation (g0 , fˆ) = a(f0 , fˆ) − l(fˆ). Set p0 = g0 .

224


(1) Descent step (n ≥ 0). Compute hA¯ gn , vi =

1 hApn , vi + hΛpn , vi, k

respectively "

1 (¯ gn , v) = (pn , v) + k

A

−1

# ! ¯˙ ) −X(T ¯ ) ,v . X(T

*"

Evaluate

# + ¯˙ ) −X(T ¯ ) ,v X(T

hΛpn , vi =

and solve the backward running system − Gi Ii1 φ¯00i = 0,

(6.20a)

ρi Ii1 φ¯ï

(6.20b)

ρi Ai ψ¯ï − ρi Ii2 ψ¯ï + Ei Ii2 ψ¯i0000 = 0,

00

X

(6.20c)

0

εiJ (ρi Ii2 ψ¯ï − Ei Ii2 ψ¯i000 ) = 0,

i∈E(vJ )

(6.20d)

X

εiJ (Gi Ii1 φ¯0i ei1 + Ei Ii2 ψ¯i00 ei2 ) = 0,

i∈E(vJ )

with final data at T

φ¯i (T ) = p0n , ψ¯i (T ) " # φ¯˙i (T ) = p1n . ψ¯˙ (T )

(6.20e) (6.20f)

i

The forward system is (6.21a) (6.21b)

− Gi Ii1 θ¯i00

ρi Ii1 θ¯ï

= 0,

00

ρi Ai w ¯ï − ρi Ii2 w ¯ï + Ei Ii2 wi 0000 = 0,

X

(6.21c)

0 ¯ï − Ei Ii2 w εiJ (ρi Ii2 w ¯i000 ) = −ψ¯¯i ,

i∈E(vJ )

(6.21d)

X

i∈E(vJ )

¯ ¯ εiJ (Gi Ii1 θ¯i0 ei1 + Ei Ii2 w ¯i00 ei2 ) = −ψ¯¯i0 ei1 − φ¯¯0i ei3 ,

Networks of Beams with initial data

θ¯0 = 0, w ¯0 1 θ¯ = 0. w ¯1

(6.21e) (6.21f) Now solve (¯ gn , v) =

(6.22)

225

" # ˙ ¯ 1 −X(T ) ,v , pn + A−1 ¯ k X(T )

or (6.23)

(¯ gn0 , v 0 )H1 + (¯ gn1 , v 1 )H =

1 0 0 ¯˙ p − A−1 H1 X(T ), v k n

+ H1

1 1 1 ¯ p + A−1 H X(T ), v k n

Therefore, compute 0 00 ¯˙ ), −Gi Ii1 gˆθn = −θ(T 0 0000 Ei Ii2 gˆwn = −w(T ¯˙ ),

(6.24a) (6.24b) and set

g¯n =

(6.25)

1 pn + gˆn . k

The new stepsize ρn is computed by ρn =

kgn k2H1 ×H . (¯ gn0 , p0n )H1 + (¯ gn1 , p1n )H

The inner products are defined as (6.26a) (¯ gn0 , p0n )H1

( ) nE X 0 0 0 0 0 00 0 00 Gi Ii1 g¯θi n , pθi n + (Ei Ii2 g¯wi n , pwi n , = i=1

(6.26b) (¯ gn0 , p0n )H

=

nE X

( ρi Ii1 g¯θ1i n , p1θi n

+

1 ρi Ai g¯w , p1wi n in

+

1 0 1 0 ρi Ii2 g¯w , pw i n in

) ,.

i=1

Finally, we obtain the updates for the final state of the adjoint system fn+1 and the residuum gn+1 by letting fn+1 = fn − ρn pn , gn+1 = gn − ρn g¯n .

. H

226


(2) Convergence. If

kgn+1 kH1 ×H 6 ε −→ STOP, kg0 kH1 ×H

else compute γn =

kgn+1 k2H1 ×H kgn k2H1 ×H

and pn+1 = gn+1 + γn pn , update n = n + 1 and go to 1). As the error at the nodes (after domain decomposition) is not zero, we use the averages for comuptation of the control from the backward system 1 dJ

(6.27a)

FJ = −

(6.27b)

1 MJ = − dJ

6.3

X

ψi (vJ ),

i∈E(vJ )

X

(φ(vJ )ei1 + ψi0 (vJ )ei2 ).

i∈E(vJ )

Numerical results

The algorithm described above was applied to two serially connetcted beams like in 5.2 and the carpenter squar from 5.2. We will present the results below. Two serially connetcted beams. The initial state is given as an equlibrium due to the load as shown in fig. 10. The matrial constants are displayed in table 3. The aim is to control the system to rest. On the free end a force in vertical direction and a moment is applied. In order to solve the backward and forward running systems we use the same simulation routine as we described in sec. 5.

Networks of Beams

227

F=15oN

q=5ooN/m 25

1 0 0 1 0 1 0 1 0 1 0 1 0 1

125o 25oo 5ooo

Fig. 10. Clamped-free beam with loads.

density Young’s modulus cross-section area inertia of the cross-section

ρ E A I2

= = = =

2960 kg/m3 7.3 · 1010 N/m2 4.91 · 10−4 m2 1.917 · 10−8 m4

Table 3 Material constants of controlled single beam.

We started the CG-algorithm with the following parameters: 53 , 64 σJ = νJ = 160, T =

tol = 10−10 , ωR = 1,

1 , 128 λJ = µJ = 0.9, ∆t =

tolE = 10−10 , k = σJ /∆T = 20480.

The figures 11 and 12 show the results for the single beam. The same computation was done with two serially connected beams of length 0.4m. The results of the CG-algorithm are shown in figures 13 and 14. In this serial case we took 10 CG-iterations. Often already 5 iterations where sufficent. The choice for k = σJ /(∆t) appeared appropreate.

228


max. error ||w|| =2.28528e−01 ∞

final state of the adjoint system 40

max. error ||w’||∞=7.99717e−02

φ(T) φ’(T)

20

0.25 0.2

0

0.15

−20 0

1

2

3

4

0.1

5

x 20

0.05

10

0 0

1

0

2

3

4

5

4

5

max. error ||d/dt w|| = 5.69285e−01 ∞

max. error ||d/dt w’||∞= 4.51186e+00

d/dt φ(T) d/dt φ’(T)

−10 −20 0

1

0.2

2

control

3

4

5 0

40 20

−0.2

0 −0.4

−20

moment F moment M

−40 0

0.2

0.4

0.6

0.8

−0.6 0

1

1

(a) Results of adjoint system.

2

3

(b) Results of adjoint system.

Fig. 11. Results of CG-Algorithm for the single beam. 2

E(T)/E(0)= 0.0174 250

2

||gn|| /||g0||

0

10

energy uncontrolled energy controlled 200

−1

10

150 −2

10

100 −3

10

50

−4

0 0

10

0.2

0.4

0.6

0.8

(a) Dissipation of energy.

1

1

1.5

2

2.5

3 3.5 iterations n

4

(b) Dissipation of energy.

Fig. 12. Convergence of CG-Algorithm.

4.5

5

Networks of Beams

final state adjoint system

max. error ||w||∞=1.05017e−02

40

max. error ||w’||∞=2.52588e−02

−3

φ(T) φ’(T)

30

229

12

20

x 10

10

10

8

0

6

−10

4 −20 0

1

2

3

4

5

2 0

40

−2 20

−4 0

1

2

3

4

5

4

5

0

max. error ||d/dt w|| = 3.60220e−01 ∞

−20

max. error ||d/dt w’||∞= 4.55007e+00


−40 0

1

0.4 2

3

4

5

0.3

x controls

0.2

40

0.1

force F2 moment M2

20

0

0

−0.1 −0.2

−20

−0.3 −40 0

0.2

0.4

0.6

0.8

1

−0.4 0

t

(a) Results of adjoint system.

1

2

3

(b) Reached final state of the system.

Fig. 13. Results of CG-Algorithm for the serial case. 2

E(T)/E(0)=0.0075 250

2

||gn|| /||g0||

0

10

energy uncontrolled energy controlled 200

−1

10

150 −2

10

100 −3

10

50

−4

0 0

10

0.2

0.4

0.6

0.8

(a) Dissipation of energy.

1

0

2

4

6

8

iterations n

(b) Convergence of CG-Algorithm.

Fig. 14. Convergence of CG-Algorithm for the serial case.

10

230


Carpenter square. In 5.2 we described a carpenter square with loads. This system is controlled to rest by a force and a moment on the free end. The results are presented in figures 15 and 16. We took five CG-Iteration steps.

References [1] J.-D. Benamou. A domain decomposition method for control problems. In P. Bjørstad, editor, DD9 Proceedings, Bergen. John Wiley & Sons, 1996. [2] J.-D. Benamou. A domain decomposition method with coupled transmission conditions for the optimal control of systems governed by elliptic partial differential equations. SIAM Journal Numerical Analysis, 33(6):2401–2416, 1996. [3] U. Brauer and G. Leugering. Semi-discretization of control and observation problems for a networks of elastic strings. Control and Cybernatics, 28(3):421–447, 1999. [4] J.-S. Briffaut. Méthodes numérique pour le contrˆ ole et la stabilisation rapide des ´ grandes structures flexibles. PhD thesis, Ecole Nationale des Ponts et Chaussées, 1999. [5] G. Chen, M. C. Delfour, A. M. Krall, and G. Payre. Modeling, Stabilization and Control of Serially Connected Beams. SIAM J. Control and Optimization, 25(3):526–546, 1987. [6] B. Dekoninck and S. Nicaise. Control of Networks of Euler-Bernoulli-Beams. ESAIM-COCV, 4:57–81, 1999. [7] Q. Deng. An analysis for a nonoverlapping domain decomposition iterative procedure. SIAM J. Sci. Comput., 18(5):1517–1525, 1997. [8] B. Despres. Methodes de decomposition de domaine pour les problemes de propagation d’ondes en regime harmonique. Le theoreme de Borg pour l’equation de Hill vectorielle. (Domain decomposition methods for harmonic wave-propagtion problems. The Borg theorem for vectorial Hill equation). PhD thesis, Univ. de Paris IX, 1991. [ISBN 2-7261-0706-0]. [9] R. Glowinski and J.L. Lions. Exact and approximate controllability for distributed parameter systems (I). Acta Numerica, pages 269–378, 1994. [10] R. Glowinski and J.L. Lions. Exact and approximate controllability for distributed parameter systems (II). Acta Numerica, pages 159–333, 1995. [11] R. Glowinski and P. Le Tallec. Augmented Lagrangian Interpretation of the Nonoverlapping Schwarz Alternating Method. In T.F. Chan, R. Glowinski, J. Reriaux, and O. B. Widlund, editors, The Third International Symposium on Domain Decompostion Methods for Partial Differential Equations, pages 224–231. SIAM, Philadelphia, 1990. [12] V. Komornik. Exact Controllability and Stabilization, volume 36 of RAM. John Wiley & Sons, Chichester, 1994. [13] J. Lagnese, G. Leugering, and E.J.P.G. Schmidt. Modelling, Analysis and Control of Multi-Link Flexible Structures. Systems & Control: Foundations & Applications. Birkhäuser Basel, 1994. [14] J.E. Lagnese. Domain decomposition in exact controllability of second order hyperbolic systems on 1-d-networks. Control and Cybernetics, 28(3):531–556, 1999.

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[15] J.E. Lagnese and G. Leugering. Dynamic domain decomposition in approximate and exact boundary control problems of transmission for wave equations. SIAM Journal Control and Optimization, 38(2):503–537, 2000. [16] G. Leugering. Domain Decomposition of Optimal Control Problems for Networks of Euler-Bernoulli-Beams. In J.P. Puel and M. Tucsnak, editors, Control and Partial Differential Equations, ESAIM Proc. 4, pages 223–233. ESAIM Paris, 1998. [17] G. Leugering. Domain decomposition of optimal control problems of networks of strings and timoshenko-beams. SIAM Journal Control and Optimization, 37(6):1649–1675, 1999. [18] G. Leugering. On the semi-discretization of optimal control problems for networks of elastic strings: global optimality systems and domain decomposition. Journal of Computational and Applied Mathematics, 1999. to appear. [19] G. Leugering. A domain decomposition of optimal control problems for dynamic networks of elastic strings. Computational Optimization and Applications, 16:5– 27, 2000. [20] G. Leugering and E.J.P.G. Schmidt. On the control of networks of vibrating strings and beams. In Proceedings of the 28th IEEE Conference on Decision and Control, volume 3, pages 2287–2290. IEEE, 1989. [21] P. L. Lions. On the Schwarz Alternating Method III: A Variant for Nonoverlapping Subdomains. In Chan, T.F., Glowinski, R., Reriaux, J. Widlund, O. B., editor, The Third International Symposium on Domain Decompostion Methods for Partial Differential Equations, pages 202–223. SIAM, Philadelphia, 1990. [22] F. Nataf, F. Rogier, and E. de Sturler. Optimal interface conditions for domain decomposition methods. Technical report, I.R. CMAP (Ecole Polytechnique), 1994. [23] W. Rathmann. Modellierung, Simulation und Steuerung von Netzwerken aus schwingenden Balken mittels Bereichszerlegung. PhD thesis, Universit¨ at Bayreuth, 2000.

232

Leugering and Rathmann E(T)/E(0)= 0.0221 450 energy uncontrolled energy controlled

400 350 300 250 200

final state of adjoint system 0.4

150

0.2

100

φ(T) φ’(T)

0

50 0 0

−0.2 0 0

0.5

1

1.5

2

2.5

1 controls 1.5

2

2.5

0.2

0.4

0.6

0.8

1

1.2

1.4

t

x

−200 −400


−600 0 50

0.5

8

6 2

force F moment M −50 0

4

1

0

0.2

0.4

0.6

0.8

1

1.2

0

t

−1 −5

1.4

−4

t

−3

−2

2

−1

0

1

2

3

0

Fig. 15. Final adjoint state and control. max. error ||w||∞=9.87220e−02

max. error ||w||∞=9.87220e−02

max. error ||w’||∞=3.51766e−02

max. error ||w’||∞=5.35614e−02

−0.03

0

−0.04 −0.02

−0.05 −0.04

−0.06 −0.07

−0.06

−0.08 −0.08

−0.1 0

−0.09

0.5

1

1.5

2

2.5

−0.1 0

0.5

0.7 0.6

0.8

0.5

0.6

0.4

0.4

0.3

0.2

0.2

0

0.1

−0.2 1.5

2

(a) Error on edge 1

2.5

3

2.5

3

max. error ||d/dt ’||∞=3.36724e+00 1

1

2 ∞

∞

max. error ||d/dt w’||∞= 7.71978e−01

0.5

1.5

max. error ||d/dt w|| =8.63234e−01

max. error ||d/dt w|| = 5.61083e−01

0 0

1

2.5

−0.4 0

0.5

1

1.5

2

(b) Error on edge 2

Fig. 16. Error at the edges.

Local Characterizations of Saddle Points and Their Morse Indices

Yongxin Li , IBM Watson Research Center, Yorktown Hts, NY Jianxin Zhou1 , Texas A&M University, College Station, Texas. Abstract In this paper, numerically computable bound estimates of the Morse indices of saddle points are established through their new local minimax type characterizations. The results provide methods for measuring instability of unstable solutions in system design and control theory.

1

Introduction

Let H be a Hilbert space and J : H → R be a Frechet differentiable functional. Denote by J 0 its Frechet derivative and J 00 its second Frechet derivative if it exists. A point u ˆ ∈ H is a critical point of J if J 0 (ˆ u) = 0 as an operator J 0 : H → H. A number c ∈ R is called a critical value of J if J(ˆ u) = c for some critical point u ˆ. For a critical value c, the set J −1 (c) is called a critical level. When the second Frechet derivative J 00 exists at a critical point u ˆ, u ˆ is said to be non-degenerate if J 00 (ˆ u) is invertible as a linear 00 operator J (ˆ u) : H → H. Otherwise u ˆ is said to be degenerate. The first candidates for a critical point are the local maxima and minima to which the classical critical point theory was devoted in calculus of variation. Traditional numerical methods focus on finding such stable solutions. Critical points that are not local extrema are unstable and called saddle points, that is, critical points u∗ of J, for which any neighborhood of u∗ in H contains points v, w s.t. J(v) < J(u∗ ) < J(w). In physical systems, saddle points appear as unstable equilibria or transient excited states.

1

Supported in part by NSF Grant DMS 96-10076. E-mail: [email protected]

233

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Li and Zhou

Assume that at a critical point u∗ , J 00 (u∗ ) is a self-adjoint, Fredholm operator from H → H, according to the spectral theory H has an orthogonal spectral decomposition H = H− ⊕ H0 ⊕ H+ where H − , H 0 and H + are respectively the maximum negative definite, the null and the maximum positive definite subspaces of J 00 (u∗ ) in H with dim(H 0 ) < ∞ and, H − , H 0 and H + are closed invariant subspaces under J 00 (u∗ ). Following the Morse theory, the Morse index of the critical point u∗ of J is MI(u∗ ) = dim(H − ). So for a non-degenerate critical point, if its MI = 0, then it is a local minimizer and a stable solution, and if its M I > 0, then it is a min-max type and unstable solution. Throughout this paper, when the Morse index is involved, we always assume that at a critical point u∗ , J 00 (u∗ ) is a self-adjoint, Fredholm operator from H → H. So the orthogonal spectral decomposition of H will always be available. When saddle points are introduced, one may mention another saddle point approach in optimization and game theory. Note that in optimization and game theory, saddle points are defined differently. For given vector spaces X and Y , let f : X × Y → R be a function. A saddle point of f is a point (x∗ , y ∗ ) ∈ X × Y such that there exist neighborhoods N (x∗ ) of x∗ in X and N (y ∗ ) of y ∗ in Y satisfying f (x∗ , y) < f (x∗ , y ∗ ) < f (x, y ∗ ) ∀ x ∈ N (x∗ ), ∀ y ∈ N (y ∗ ). Such a saddle point is usually also a critical point of f in X ×Y . The significant difference here is that f has a splitting structure on X × Y and such a splitting structure is known beforehand and fixed. Interactions between X and Y are therefore very limited. While saddle points defined in critical point theory are much more general. Such a splitting structure is in general not available. If u∗ ∈ H is a nondegenerate critical point of a generic energy function J with MI(u∗ )/geq1, by the spectral theory, we have H = H − ⊕ H + . Then we have the decomposition u∗ = u− + u+ for u− ∈ H − , u+ ∈ H + and there are neighborhoods N (u− ) of u− in H − and N (u+ ) of u+ in H + such that J(v + u+ ) < J(u− + u+ ) < J(u− + w) ∀v ∈ N (u− ), w ∈ N (u+ ). A formation close to a saddle point in optimization and game theory. However, here the splitting structure H = H − ⊕ H + is only partial, interactions between H − and H + are allowed. The most important difference here is that the subspaces H − and H + cannot be available before we physically find the saddle point u∗ . Therefore such an approach plays little role in finding or searching saddle points in the critical point theory. Multiple solutions with different performances and instabilities exist in many nonlinear problems in natural and social sciences [22], [19], [14], [24],

Saddle Points and Their Morse Indices

235

[13]. Stability is one of the main concerns in system design and control theory. For instance, traveling waves have been observed to exist in suspended bridges-a nonlinear beam equation [6] and showed as saddle points, therefore unstable solutions, to their corresponding variational problem ((10) and (11) in [6]). Those unstable solutions have been observed to have different instability properties. How to mathematically measure their instability properties becomes an interesting problem. As matter of fact, travelling waves make up an important class of solutions to both reaction-diffusion equation and nonlinear hyperbolic equations with “viscosity”. They are solutions of the form u = u(x−ct) where c is a constant, the speed of the wave. Many phenomena arising in various physical, or biological context can be modelled by travelling waves; such as shock waves, nerve impulses, and various oscillatory chemical reactions. The nice mathematical feature associated with such solutions is that the problem often reduces to a nonlinear ordinary differential equations and the solutions correspond to saddle points of their generic energy functions (ref. Chapter 24 in [21]). Stability (instability) analysis of such solutions is interesting to many engineers and researchers and has been carried out in the literature. Our objective is to develop some mathematical tool to measure instability properties for saddle points. Usually stability describes certain property possessed by a solution to a dynamic system. When one says that a solution u∗ is a stable solution to a dynamic system, if u0 is near u∗ , the solution to the dynamic system through u0 tends to u∗ as t → +∞. One may show that a solution u∗ to a dynamic system is stable if the spectrum of the linearized operator of the dynamic system at u∗ lies in the left-hand complex plane. When the system is variational, an associated energy function is available. One may also use the energy function to define stability. In this paper, we say that a solution (critical point) u∗ is stable if it is a local minimizer of the associated generic energy function. In this case MI(u∗ ) = 0. Thus any local perturbation of a stable solution in an associated feasible function space will increase the energy level. For an unstable solution (saddle point) u∗ , we may also use the maximum dimension of a subspace in which a local perturbation of the unstable solution u∗ in an associated feasible function space will always decrease the energy level to define its instability index. Since by the definition, any local perturbation of u∗ in H − will always decrease its energy level, for such variational problems, the Morse index (= dim(H − )) of a solution can be used to measure its instability (ref. [4],[21]). It is clear that the Morse index serves as a lower bound of the instability index. On the other hand, in many applications, performance or maneuverability is more desirable, in particular, in system design or control of emergency or combat machineries. Usually unstable solutions have much higher maneuverability or performance indices. For providing choice or balance between stability and maneuverability or performance, it is important to solve for multiple solutions and their Morse indices. When a saddle point u0 is

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degenerate, the nullspace H 0 of J 00 (u0 ) is nonempty. Since many different situations may happy in H 0 , it is extremely difficult to determine the Morse index in this case. On the other hand, whether or not a solution is degenerate also depends on the domain of the solution. So we can not simply exclude such situation. When cases are variational, we will be dealing with multiple critical point problems. So it is important for both theory and applications to numerically solve for multiple critical points and their Morse indices in a stable way. So far, little is known in the literature to devise such a feasible numerical algorithm. Minimax principle is one of the most popular approaches in critical point theory. However, most minimax theorems in the literature (See [1], [16], [17], [19], [24]), such as the mountain pass, various linking and saddle point theorems, require one to solve a two-level global optimization problem and therefore not for algorithm implementation. In [11], motivated by the numerical works of Choi-McKenna [7] and DingCosta-Chen 9, the Morse theory and the idea to define a solution submanifold, new local minimax theorems which characterize critical points as solutions to a two-level local optimization problem are established. Based on the local characterization, a new numerical minimax method for finding multiple critical points is devised. The numerical method is implemented successfully to solve a class of semilinear elliptic PDE on various domains for multiple solutions [11]. Although Morse index has been printed for each numerical solution obtained in [11], their mathematical verifications have not been established. In [2], by using a global minimax principle, A. Bahri and P.L. Lions established some lower bound estimates for the Morse indices of solutions to a class of semilinear elliptic PDE. There are also some efforts in the literature to numerically compute the Morse index of a solution to a class of semilinear elliptic PDE. In addition to finding a saddle point, one has to solve a corresponding linearized elliptic PDE at the solution for its first a few eigen-values. It is very expensive and not much success has been documented. Since Morse index reflects local structure of a critical point, in this paper we show that our local minimax characterization enables us to establish more precise estimates for the Morse index of a saddle point in a more general setting. By our results the Morse index of a saddle point based on the local minimax method can be estimated even before we numerically compute the saddle point. So no extra work is required in addition to computation for the saddle point. In the last section, new local characterization of saddle points which are more general than minimax solutions and bound estimates for their Morse indices will be developed. In the rest of this section, we introduce some notations and theorems from [11] for future use. For any subspace H 0 ⊂ H, let SH 0 = {v|v ∈ H 0 , kvk = 1} be the unit sphere in H 0 . Let L be a closed subspace in H, called a base space, and


237

L H = L L⊥ be the orthogonal decomposition where L⊥ is the orthogonal complement of L in H. For each v ∈ SL⊥ , we define a closed half space [L, v] = {tv + sw|w ∈ L, t > 0, s ∈ R}. H Definition 1.1. A set-valued L ⊥ mapping P : SL⊥ → 2 is called the peak mapping of J w.r.t. H = L L if for any v ∈ SL⊥ , P (v) is the set of all local maximum points of J on [L, v]. A single-valued mapping p : SL⊥ → H is a peak selection of J w.r.t. L if p(v) ∈ P (v)

∀v ∈ SL⊥ .

For a point v ∈ SL⊥ , we say that J has a local peak selection w.r.t. L at v, if there is a neighborhood N (v) of v and a mapping p : N (v) ∩ SL⊥ → H such that p(u) ∈ P (u)

∀u ∈ N (v) ∩ SL⊥ .

Most minimax theorems in critical point theory require one to solve a twolevel global minimax problem and not for algorithm implementation. Our local minimax algorithm requires one to solve only unconstrained local maximizations at the first level. As pointed in [11], numerically it is great. However, theoretically, it causes three major problems: (a) for some v ∈ SL⊥ , P (v) may contain multiple local maxima in [L, v]. In particular, P may contain multiple branches, even U-turn or bifurcation points; (b) p may not be defined at some points in SL⊥ ; (c) the limit of a sequence of local maximum points may not be a local maximum point. So the analysis involved becomes much more complicated. We have been devoting great efforts to solve these three problems. We solve (a) and (b) by using a local peak selection. Numerically it is done by following certain negative gradient flow and developing some consistent strategies to avoid jumps between different branches of P . As for Problem (c), numerically we showed in [12] that as long as a sequence generated by the algorithm converges, the limit yields a saddle point. New local characterization of saddle points in this paper will further help us to solve those problems. The following two local characterizations of saddle points are established in [11] and played important role in our local theory. We then provide some bound estimates of Morse indices of solutions based upon these two local characterizations. Lemma 1.1. Let vδ ∈ SL⊥ be a point. If J has a local peak selection p w.r.t. L at vδ such that p is continuous at vδ and dis(p(vδ ), L) > α > 0 for some α > 0, then either J 0 (p(vδ )) = 0 or for any δ > 0 with kJ 0 (p(vδ ))k > δ, there exists s0 > 0, such that J(p(v(s))) − J(p(vδ )) < −αδkv(s) − vδ k for any 0 < s < s0 and v(s) =

vδ + sd , kvδ + sdk

d = −J 0 (p(vδ )).

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Li and Zhou

The above result indicates that v(s) defined in the lemma represents a direction for certain negative gradient flow of J(p(·)) from v. So it is clear that if p(v0 ) is a local minimum point of J on any subset containing the path p(v0 (s)) for some small s > 0 then J 0 (p(v0 )) = 0. In particular, if we define a solution manifold n o M = p(v) : v ∈ SL⊥ , we have p(v(s)) ⊂ M. A solution submanifold was first introduced by Nehari in study of a dynamic system [15] and then applied by Ding-Ni in study of semilinear elliptic PDE [17]. Although their definitions of a solution submanifold are closely related to the problems that they were dealing with and our definition of a solution submanifold is given in a quite general setting, it is easy to check that their solution submanifold coincides with ours with L = {0}. While for our solution submanifold, L can be any closed subspace. So our definition of a solution submanifold generalizes their notions. Furthermore, they prove that a global minimum point of the generic energy function J on the solution submanifold M w.r.t. L = {0} yields a saddle point basically with MI= 1. While our next theorem proved in [11] shows that a local minimum point of the generic energy function J on our solution submanifold M also yields a saddle point and the Morse index of such a saddle point can be greater than one. Since for such an unstable saddle point u∗ , its generic energy is minimized on the corresponding solution submanifold, the solution submanifold becomes a stable submanifold of the unstable saddle point u∗ , in the sense that any small perturbation of u∗ will increase the value of the generic functional J as long as the perturbation stays on the submanifold M. Since u∗ is a saddle point, a perturbation of u∗ off the submanifold may decrease the value of the generic energy functional J. So the solution submanifold is also called a stable submanifold. Theorem 1.1. [11] Let v0 ∈ SL⊥ be a point. If J has a local peak selection p w.r.t. L at v0 s.t. (i) p is continuous at v0 , (ii) dis(p(v0 ), L) > 0 and (iii) v0 is a local minimum point of J(p(v)) on SL⊥ . Then p(v0 ) is a critical point of J. The following PS condition will be used to replace the usual compact condition. Definition 1.2. A function J ∈ C 1 (H) is said to satisfy the Palais-Smale (PS) condition, if any sequence {un } ∈ H with J(un ) bounded and J 0 (un ) → 0 has a convergent subsequence.


2

239

Bound Estimates for Morse Index

The Morse index provides understanding of the local structure of a saddle point and is used as an instability index for an unstable solution [4],[21]. It is an important notion in stability analysis [4]. Although we have printed the Morse index for each numerical solution computed by our minimax method in [11], their mathematical justifications have not been verified. In this section, we establish several bound estimates for the Morse index of a critical point based on our minimax method. Lemma 2.1. Let v0 ∈ SL⊥ be a point. If there exist a neighborhood N (v0 ) of v0 and a locally defined mapping p : N (v0 ) ∩ SL⊥ → H such that p(v) ∈ {L, v} ∀v ∈ N (v0 ) ∩ SL⊥ . If p is differentiable at v0 and u0 = p(v0 ) ∈ / L, then p0 (v0 )({L, v0 }⊥ ) + {L, v0 } = H. v0 + sw . Then there kv0 + swk exists s0 > 0 such that when |s| < s0 , we have vs ∈ N (v0 ) ∩ SL⊥ . Consider the one dimensional vector function α(s) = PL⊥ (p(vs )), where PL⊥ is the projection onto L⊥ . Since p is differentiable at v0 and vs smoothly depends on s, α is differentiable at 0 and

Proof. For any w ∈ {L, v0 }⊥ , kwk = 1, denote vs =

α0 (0) = PL⊥ (p0 (v0 )(

∂vs )) = PL⊥ (p0 (v0 )(w)). ∂s

On the other hand, p(vs ) ∈ {L, vs }, we have α(s) = ts vs , where ts = hα(s), vs i is differentiable. So α0 (0) = t00 v0 + t0 w, where due to our assumption that u0 = p(v0 ) ∈ / L, we have t0 6= 0. The two different expressions of α0 (0) imply PL⊥ (p0 (v0 )(w)) = t0s (0)v0 + t0 w. Then it leads to w ∈ {p0 (v0 )(w), L, v0 }. Since w is an arbitrary vector in {L, v0 }⊥ , it follows that {L, v0 }⊥ ⊂ {p0 (v0 )({L, v0 }⊥ ), L, v0 } or (2.1)

p0 (v0 )({L, v0 }⊥ ) + {L, v0 } = H.

Lemma 2.2. Let v0 ∈ SL⊥ be a point. If there exist a neighborhood N (v0 ) of v0 and a locally defined mapping p : N (v0 ) ∩ SL⊥ → H such that p(v) ∈ {L, v} ∀v ∈ N (v0 ) ∩ SL⊥ . Assume that p is differentiable at v0 and u0 = p(v0 ) ∈ / L. If u0 is a critical point of J with M I(u0 ) > dim L + 1, then p0 (v0 )({L, v0 }⊥ ) ∩ H − 6= {0}.

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Proof. Denote H − the negative subspace of J 00 (u0 ) and k = dim L + 1. Then dim H − > k. By applying Lemma 2.1, there exit linearly independent vectors e0 , e1 , . . . , ek ∈ H − which can be represented as ei = gi + fi with gi ∈ p0 (v0 )({L, v0 }⊥ ) and fi ∈ {L, v0 }. f0 , f1 , . . . , fk have to be linearly dependentP because k = dimP L + 1. So we can find real numbers a0 , a1 , . . . , ak k 2 such that i=0 ai 6= 0 and ki=0 ai fi = 0. Therefore k X

ai ei =

i=0

k X

ai gi ∈ p0 (v0 )({L, v0 }⊥ ) ∩ H − .

i=0

Pk Because, e0 , e1 , . . . , ek are linearly independent, i=0 ai ei 6= 0. Thus, the conclusion of the lemma is verified. Theorem 2.1. Let v0 ∈ SL⊥ be a point. If J has a local peak selection p w.r.t. L at v0 such that p is differentiable at v0 and u0 = p(v0 ) ∈ / L. If v0 is a local minimum point of J ◦ p on SL⊥ , then u0 is a critical point of J with M I(u0 ) 6 dim L + 1. Proof. Since p is a local peak selection of J w.r.t. L at v0 , there exists a neighborhood N (v0 ) of v0 such that p(v) ∈ {L, v}, ∀v ∈ N (v0 ) ∩ SL⊥ . By applying Lemma 2.1, we have p0 (v0 )({L, v0 }⊥ ) + {L, v0 } = H or

codim(p0 (v0 )({L, v0 }⊥ )) 6 dim L + 1.

Now suppose that M I(u0 ) > dim L + 1. Denote H − the negative subspace of J 00 (u0 ). By Lemma 2.2, we have p0 (v0 )({L, v0 }⊥ ) ∩ H − 6= {0}.

(2.2)

Choose any w ∈ {L, v0 }⊥ , kwk = 1, such that p0 (v0 )(w) ∈ H − . Around u0 = p(v0 ), we have the second order Taylor expansion (2.3)

1 J(u) = J(u0 ) + hJ 00 (u0 )(u − u0 ), u − u0 i + o(ku − u0 k2 ) 2

Denote vs =

v0 + sw , we have vs ∈ N (v0 ) ∩ SL⊥ for |s| small and then kv0 + swk

dvs |s=0 = w. So it follows ds (2.4)

p(vs ) = u0 + sp0 (v0 )(w) + o(|s|).

Combining the above two estimates (2.3) and (2.4), we obtain


241

J(p(vs )) 1 = J(u0 ) + hJ 00 (u0 )(sp0 (v0 )(w) + o(|s|)), sp0 (v0 )(w) + o(|s|)i 2 +o(ksp0 (v0 )(w) + o(|s|)k2 ) 1 = J(u0 ) + s2 hJ 00 (u0 )(p0 (v0 )(w)), p0 (v0 )(w)i + o(s2 ) 2 < J(u0 ), where the last strict inequality holds for |s| sufficiently small, because p0 (v0 )(w) ∈ H − . Since vs ∈ N (v0 ) ∩ SL⊥ and u0 = p(v0 ), the above contradicts the assumption that v0 is a local minimum point of J ◦ p on SL⊥ . Therefore M I(u0 ) 6 dim L + 1. Theorem 2.2. If p is a local peak selection of J w.r.t. L at v0 ∈ SL⊥ and u0 = p(v0 ) is a nondegenerate critical point of J, then M I(u0 ) > dim L + 1. Proof. Assume that k ≡ M I(u0 ) < dim L + 1. By our assumption, u0 is L nondegenerate, i.e., J 00 (u0 ) is invertible, we have H = H + H − where H + is the maximum positive subspace and H − is the maximum negative subspace corresponding to the orthogonal spectral decomposition of J 00 (u0 ). It follows that codim(H + ) = dim(H − ) = k < dim L + 1, so there exists a non-zero vector v ∈ H + ∩ {L, v0 }. When v ∈ H + , for sufficient small t, we have J(u0 + tv) > J(u0 ). But this contradicts to that u0 is a local maximum point of J in the subspace {L, v0 }. Therefore, M I(u0 ) > dim L + 1. Theorem 2.3. Assume that p is a local peak selection of J w.r.t. L at v0 ∈ SL⊥ such that p is differentiable at v0 and u0 = p(v0 ) ∈ / L. If v0 00 is a local minimum point of J ◦ p on SL⊥ , and J (u0 ) is invertible, then M I(u0 ) = dim L + 1. Proof. Since under the conditions, we have proved that u0 = p(v0 ) is a nondegenerate critical point of J. The conclusion follows by combining the last two theorems. Theorem 2.4. Let v0 ∈ SL⊥ be a point. If there exist a neighborhood N (v0 ) of v0 and a locally defined mapping p : N (v0 )∩SL⊥ → H such that p(v) ∈ {L, v}, J 0 (p(v)) ⊥ {L, v}, ∀v ∈ N (v0 ) ∩ SL⊥ and p differentiable at v0 . If v0 ∈ SL⊥ is a local minimum point of J ◦ p on SL⊥ with u0 = p(v0 ) ∈ / L, then u0 is a critical point of J with M I(u0 ) 6 dim L + 1. Proof. We first prove that u0 = p(v0 ) is a critical point of J. The second part of the theorem follows from a similar proof of Theorem 2.1. For any w ∈ {L, v0 }⊥ , denote v(s) =

v0 + sw . kv0 + swk

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We have v(s) ∈ N (v0 ) ∩ SL⊥ for |s| small and

dv(s) ds |s=0

= w. Therefore

dv(s) |s=0 + o(|s|) ds = u0 + sp0 (v0 )(w) + o(|s|).

p(v(s)) = p(v0 ) + sp0 (v0 )

It follows that J(p(v(s))) = J(p(v0 )) + J 0 (p(v0 ))(p(v(s)) − p(v0 )) + o(kp(v(s)) − p(v0 )k) = J(u0 ) + sJ 0 (u0 )p0 (v0 )(w) + o(|s|). If J 0 (u0 )p0 (v0 )(w) 6= 0 for some w ∈ {L, v0 }⊥ , then when |s| is sufficiently small, we can choose either s > 0 or s < 0 such that J(p(v(s))) < J(p(v0 )) which contradicts the assumption that v0 is a local minimum point of J ◦ p on SL⊥ . Thus J 0 (u0 )p0 (v0 )({L, v0 }⊥ ) = 0. Since by our assumption J 0 (u0 )({L, v0 }) = 0 and by Lemma 2.1 p0 (v0 )({L, v0 }⊥ ) + {L, v0 } = H, it follows that J 0 (u0 )u = 0 ∀u ∈ H, i.e., u0 = p(v0 ) is a critical point of J. It is worthwhile indicating that if p is a local peak selection of J at v0 ∈ SL⊥ , then p(v) ∈ [L, v] and J 0 (p(v)) ⊥ [L, v] for all v ∈ N (v0 ) ∩ SL⊥ . If {vn } ⊂ SL⊥ , vn → v0 and un = p(vn ) → u0 , we have u0 ∈ [L, v0 ] and J 0 (u0 ) ⊥ [L, v0 ]. So such a locally defined mapping generalized the notion of a local peak selection and resolved the problem that a limit of a sequence of local maximum points may not be a local maximum point. This generalization has a potential to design a new type of local algorithm for finding multiple saddle points that are not necessarily of a minimax type. Theorem 2.5. If u0 ∈ / L is a non-degenerate critical point of J such that u0 is not a local minimum point of J along any direction v ∈ {L, u0 }, then M I(u0 ) > dim L + 1.


243

Proof. Assume that k = M I(u0 ) < dim L + 1. By our assumption, u0 is nondegenerate, i.e., J 00 (u0 ) is invertible, we have H = H + ⊕ H − where H + is the maximum positive subspace and H − is the maximum negative subspace corresponding to the orthogonal spectral decomposition of J 00 (u0 ). It follows that codim(H + ) = dim(H − ) = k < dim L + 1, so there exists a non-zero vector v ∈ H + ∩ {L, u0 }. When v ∈ H + , for sufficiently small t, we have J(u0 + tv) > J(u0 ). It then contradicts the assumption that u0 is not a local minimum point of J along any direction v ∈ {L, u0 }.

3

Application to Semilinear Elliptic PDEs

Semilinear elliptic boundary-value problems (BVP) are known to be rich in multiple solutions. Some of the solutions are stable and others are unstable. How to measure the instability properties for unstable solutions is an interesting problem, which is actually part of the motivation of this paper. When the profile of a solution shown as in Figure 1 was obtained and presented to the nonlinear partial differential equation community, it generated warm debates about the existence and the Morse index of such a solution. The authors are happy to know that the existence issue has been recently settled [23]. But the Morse index of such a solution is still unsolved. In this section, we try to answer the question. Consider a semilinear elliptic Dirichlet BVP on a piecewise smooth bounded domain Ω in Rn which has many applications in physics, engineering, biology, ecology, geometry, etc ∆u(x) + f (x, u(x)) = 0, x ∈ Ω, (3.1) u(x) = 0, x ∈ ∂Ω, where the function f (x, u(x)) satisfies the following standard hypothesis: ¯ × R; (h1) f (x, u(x)) is locally Lipschitz on Ω (h2) there are positive constants a1 and a2 such that |f (x, ξ)| 6 a1 + a2 |ξ|s

(3.2) where 0 6 s < (3.3)

n+2 n−2

for n > 2. If n = 2, |f (x, ξ)| 6 a1 exp φ(ξ)

where φ(ξ)ξ −2 → 0 as |ξ| → ∞; (h3) f (x, ξ) = o(|ξ|) as ξ → 0;

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(h4) there are constants µ > 2 and r > 0 such that for |ξ| > r, 0 < µF (x, ξ) 6 ξf (x, ξ),

(3.4) where F (x, ξ) =

Rξ 0

f (x, t)dt.

(h4) says that f is superlinear, which implies that there exist positive numbers ¯ and ξ ∈ R a3 and a4 such that for all x ∈ Ω F (x, ξ) > a3 |ξ|µ − a4 .

(3.5)

The generic energy functional associated to the Dirichlet problem (3.1) is Z Z 1 J(u) = (3.6) |∇u(x)|2 dx − F (x, u(x))dx, u ∈ H ≡ H01 (Ω), 2 Ω Ω R where we use an equivalent norm kuk = Ω |∇u(x)|2 dx for the Sobolev space H = H01 (Ω). It is well known [19] that under Conditions (h1) through (h4), J is C 1 and satisfy (PS) condition. A critical point of J is a weak solution, and also a classical solution of (3.1). 0 is a local minimum point of J and so a trivial solution. Moreover, in any finitely dimensional subspace of H, J goes to negative infinity uniformly. Therefore, for any finite dimensional subspace L, the peak mapping P of J w.r.t. L is nonempty. We need one more hypothesis, that is (h5)

f (x,ξ) |ξ|

is increasing w.r.t. ξ, or

(h5’) f (x, ξ) is C 1 w.r.t. ξ and fξ (x, ξ) −

f (x,ξ) ξ

> 0.

It is clear that (h5’) implies (h5). If f (x, ξ) is C 1 in ξ, then (h5) and (h5’) are equivalent. All the power functions of the form f (x, ξ) = |ξ|k ξ with k > 0, satisfies (h1) through (h5’), and so do all the positive linear combinations of such functions. Under (h5) or (h5’), J has only one local maximum point in any direction, or, the peak mapping P of J w.r.t. L = {0} has only one selection. In other words, P = p. The proof can be found in [17] and [14]. Theorem 3.1. Under the hypothesis (h1) through (h5), if the peak mapping P of J w.r.t. a finitely dimensional subspace L is singleton at v0 ∈ SL⊥ and for any v ∈ SL⊥ around v0 , a peak selection p(v) is a global maximum point of J in [L, v], then p is continuous at v0 . Proof. See [11]. Theorem 3.2. Assume that Conditions (h1) – (h5’) are satisfied and that there exist positive constants a5 and a6 s.t. (3.7)

|fξ (x, ξ)| 6 a5 + a6 |ξ|s−1

where s is specified in (h2). Then the only peak selection p of J w.r.t. L = {0} is C 1 .


245

Proof. See [11]. Since w0 = 0 is the local minimum point of J and along each direction v ∈ H, J has only one maximum point p(v), we have J(p(v)) > 0, ∀v ∈ S. If for each v ∈ SL⊥ , p(v) is a local maximum point of J in [L, v], then p(v) is the p(v) only local maximum point of J along the direction u = kp(v)k , Therefore we have J(p(v)) > 0, ∀v ∈ SL⊥ . As a composite function J(p(·)) is bounded from below by 0. So Ekeland’s variational principle can be applied. With the PS condition, existence result can also be established. Theorem 3.3. Under the hypothesis of (h1) through (h5), and that there exist positive constants a5 and a6 such that |fξ (x, ξ)| 6 a5 + a6 |ξ|s−1 , where s is specified in (h2), if v0 = arg minv∈SH J(p(v)) then u0 = p(v0 ) is a critical point with MI(u0 ) = 1. Proof. Assume u0 = p(v0 ) = t0 v0 . Consider the 1-dimensional function Z Z 1 2 2 g(t) = J(tv0 ) = t |∇v0 (x)| dx − F (x, tv0 (x)) dx. 2 Ω Ω We have g0 (t) = t

Z

Z |∇v0 (x)|2 dx −

f (x, tv0 (x))v0 (x) dx.

Ω

Ω

Z

So 1=

Ω

Meanwhile we have 00

f (x, t0 v0 (x)) 2 v0 (x) dx. t0 v0 (x)

Z

Z |∇v0 (x)| dx − 2

g (t) = Ω

Ω

g00 (t0 ) = 1 −

fξ (x, tv0 (x))v02 (x) dx

Z ZΩ

< 1− Ω

fξ (x, t0 v0 (x))v02 (x) dx f (x, t0 v0 (x)) 2 v0 (x) dx t0 v0 (x)

(ref. (h5))

= 0, which implies that H 0 ∩ {L, v0 } = {0}, where H 0 is the nullspace of the linear operator J 00 (u0 ) and L = {0}, i.e., J 00 (u0 ) may be degenerate, however its restriction to the direction of v0 will not be degenerate, where u0 = t0 v0 is a critical point. By Theorem 2.1, we obtain MI(u0 ) = dim(L) + 1 = 1.

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In the rest of this section, we display a numerical solution of (3.1) where f (x, u(x)) = u3 (x) and the domain Ω is the dumbbell shaped domain as shown in the lower part of Figure 1. It can be checked that all the assumptions (h1) − (h5) are satisfied. So the results in this section can be applied. Many unstable solutions to this problem has been numerically computed and documented in [11]. The details of the numerical minimax method and how to compute such solutions are described in [11]. Some convergent properties of this numerical minimax method are established in [12]. Here we only display the profile and the contours the solution concentrated near the center of the corridor of the dumbbell-shaped domain. This solution is also captured by using a different numerical algorithm in [5]. It is this solution that generated warm debates about the existence and the Morse index of such a solution. Since L = {0}, by the results in this section, we can now conclude that the Morse index of this solution is one.

4

Some New Saddle Point Theorems

As our convergence results in [12] indicate that our algorithm can be used to find a non-minimax critical point, e.g., a Monkey saddle point. Thus the argument already exceeded the scope of a minimax approach. So far the only results we found in the critical point literature which are more general than a minimax principle are two theorems proved by S. I. Pohozaev in [10] or [18]. The following results are interesting generalizations. The first one is an embedding result. It is general but lacks of characterization. The second result has potential applications in devising a new numerical algorithm. Theorem 4.1. Given L = span{w1 , ..., wk } in H and let J¯(t, v, t1 , .., tk ) ≡ J(tv + t1 w1 + ... + tk wk ). If (t∗0 , v ∗ , t∗1 , .., t∗k ) is a conditional critical point of J¯ subject to v ∈ SL⊥ with t∗ 6= 0, then t∗0 v ∗ + t∗1 w1 + ... + t∗k wk is a critical point of J. Proof. By the Lagrange Multiplier Theorem, there exist λ, µ, η1 , .., ηk with λ2 + µ2 + η12 + ... + ηk2 6= 0 such that the Lagrange functional ¯ v, t1 , ..., tk ) + µkvk + L(t, v, t1 , ..., tk ) = λJ(t,

k X i=1

has a critical point at (t∗0 , v ∗ , t∗1 , .., t∗k ). So we have

ηi hwi , vi


247

15

w−axis

10

0

−5 −1

y− is

ax

0 1

1

2

3

Z

−1

0

−1.5

x−axis

X Y

1.12.760.551 2. 2

3.86 1 5.5

y−axis

1

0

−1 −1.5

1 6.6 4.9 3.3611.65

−1

0

1

2

3

x−axis

Fig. 1. The profile of a positive solution and its contours with J = 159.0 and umax = 13.63.

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(4.1)

∂L ∂t

(4.2)

∂L ∂v

= 0

∂L ∂ti

= 0

= 0

⇒

λJ¯t (t∗0 , v ∗ , t∗1 , ..., t∗k ) = 0;

⇒

λJ¯v0 (t∗0 , v ∗ , t∗1 , ..., t∗k ) + µkv ∗ k0 +

k X

ηi wi = 0;

i=1

⇒

λJ¯t0i (t∗0 , v ∗ , t∗1 , .., t∗k ) = 0

or

λhJ 0 (t∗0 v ∗ + t∗1 w1 + ... + t∗k wk ), wi i = 0, (i = 1, .., k).

(4.3) From (4.2), we have (4.4)

hλJ¯v (t∗0 , v ∗ , t∗1 , .., t∗k ), vi + µhkv ∗ k0 , vi +

k X

ηi hwi , vi = 0 ∀v ∈ H.

i=1

In particular λhJ¯v (t∗0 , v ∗ , t∗1 , .., t∗k ), v ∗ i + µhkv ∗ k0 , v ∗ i +

(4.5)

k X

ηi hwi , v ∗ i = 0.

i=1

Since hkv ∗ k0 , v ∗ i = 1 and hwi , v ∗ i = 0 for i = 1, ..., k, we obtain λhJ¯v (t∗0 , v ∗ , t∗1 , .., t∗k ), v ∗ i + µ = 0.

(4.6)

So λ = 0 will lead to µ = 0 and then ηi = 0 by choosing v = wi in (4.3). It contradicts to λ2 + µ2 + η12 + ... + ηk2 6= 0. Therefore λ 6= 0 and (4.1) gives J¯t0 (t∗0 , v ∗ , t∗1 , ..., t∗k ) = 0 or

hJ 0 (t∗0 v ∗ + t∗1 w1 + ... + t∗k wk ), v ∗ i = 0.

It leads to hJ¯v0 (t∗0 , v ∗ , t∗1 , ..., t∗k ), v ∗ i = t∗0 hJ 0 (t∗0 v ∗ + t∗1 w1 + ... + t∗k wk ), v ∗ i = 0. (4.6) then yields µ = 0. Taking any v ⊥ [w1 , ..., wk ] in (4.4), we obtain λhJ¯v0 (t∗0 , v ∗ , t∗1 , ..., t∗k ), vi = 0 or

t∗0 hJ 0 (t∗0 v ∗ + t∗1 w1 + ... + t∗k wk ), vi = 0.


249

Since t∗0 6= 0, it leads to hJ 0 (t∗0 v ∗ + t∗1 w1 + ... + t∗k wk ), vi = 0

∀v ⊥ [w1 , ..., wk ].

Taking (4.3) into account, we have hJ 0 (t∗0 v ∗ + t∗1 w1 + ... + t∗k wk ), ui = 0 ∀u ∈ H or

J 0 (t∗0 v ∗ + t∗1 w1 + ... + t∗k wk ) = 0.

So u∗ = t∗0 v ∗ + t∗1 w1 + ... + t∗k wk is a critical point. It is clear that Theorem 4.1 reduces to Pohozaev’s embedding result in [10] or [18] if we set L = {0}, the trivial subspace. Theorem 4.2. Let v ∗ ∈ SL⊥ be a point. If J has a local peak selection p w.r.t. L at v ∗ and u∗ = p(v ∗ ) such that (i)

p is Lipschitz continuous at v ∗ ,

(ii) dis(u∗ , L) > 0, (iv) u∗ is a conditional critical point of J on p(SL⊥ ), then u∗ is a critical point of J. Proof. Suppose that kJ 0 (u∗ )k > 0. Set δ = 12 kJ 0 (u∗ )k. By Lemma 1.1, there exists s0 > 0 such that J(p(v ∗ (s))) − J(p(v ∗ )) < −δdis(u∗ , L)kv ∗ (s) − v ∗ k ∀0 < s < s0 where v ∗ (s) =

v ∗ + sd ∈ N (v ∗ ) ∩ SL⊥ , kv ∗ + sdk

d = −J 0 (p(v ∗ )).

Then we have (4.7)

J(p(v ∗ (s))) − J(p(v ∗ )) kp(v ∗ (s)) − u∗ k < −δdis(u∗ , L), kp(v ∗ (s)) − u∗ k kv ∗ (s) − v ∗ k

∀ 0 < s < s0 ,

where N (v ∗ ) is a neighborhood of v ∗ in which the local peak selection p is defined. Since p is Lipschitz continuous at v ∗ and u∗ is a conditional critical point of J on p(SL⊥ ), kp(v ∗ (s)) − u∗ k kv ∗ (s) − v ∗ k is bounded and

J(p(v ∗ (s))) − J(p(v ∗ )) → 0 as s → 0. kp(v ∗ (s)) − u∗ k

So the left hand side of (4.7) goes to zero, which leads to a contradiction.

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Theorem 4.3. Let v0 ∈ SL⊥ be a point. If J has a local peak selection p w.r.t. L such that p is differentiable at v0 and u0 = p(v0 ) ∈ / L. If u0 is a conditional critical point of J on p(SL⊥ ) and v0 is not a local maximum point of J ◦ p along the projection of any direction v on SL⊥ , then u0 is a critical point of J with M I(u0 ) 6 dim L + 1. In addition, if u0 is nondegenerate, then M I(u0 ) = dim L + 1. Proof. By Theorem 4.2, we obtain that u0 is a critical point of J. Then following a similar argument in the proof of Theorem 2.1, until we have 1 J(p(vs )) = J(u0 ) + hJ 00 (u0 )(sp0 (v0 )(w) + o(|s|)), sp0 (v0 )(w) + o(|s|)i 2 +o(ksp0 (v0 )(w) + o(|s|)k2 ) 1 = J(u0 ) + s2 hJ 00 (u0 )(p0 (v0 )(w)), p0 (v0 )(w)i + o(s2 ) 2 < J(u0 ), where vs =

v0 + sw ∈ N (v0 ) ∩ SL⊥ , kv0 + swk

w ∈ [L, v0 ]⊥ , kwk = 1, and p0 (v0 )(w) ∈ H − .

So the last strict inequality contradicts to our assumption that v0 is not a local maximum point of J ◦ p along the projection of any direction v on SL⊥ . Thus M I(u0 ) 6 dim L + 1. If in addition, u0 is nondegenerate, we can apply Theorem 2.5 to conclude that M I(u0 ) = dim L + 1. Condition (iv) in Theorem 4.2 is clearly satisfied if w∗ is a local minimum point of J on the solution (stable) manifold M = p(SL⊥ ). So it is clear that Theorems 4.1 and 4.2 are indeed more general than a minimax principle. As matter of fact, Condition (iv) in Theorems 4.2 can be weekend as that w∗ is a conditional critical point of J on any subset containing the path p(v ∗ (s)) for small s > 0. If we set L = {0}, the trivial subspace and assume p(v) is the global maximum point of J on [L, v] = {tv : t > 0} for each v ∈ SL⊥ = SH and p is C 1 , then Theorem 4.2 reduces to a result of Pohozaev in [18]. As we did in our algorithm, by gradually expanding the subspace L, Theorem 4.2 can be used to provide us with information on the Morse index of the critical point. For an example, when we solve a semilinear elliptic equation as shown in [11], we start from the trivial solution and set L = {0} to approximate a solution w1 with MI= 1; then we set L = {w1 } to search for a solution w2 with MI= 2,.... This is the advantage of our approach. As we can see in the above theorems, it becomes very important to check whether or not the local peak selection p is continuous or differentiable at v ∗ . This is very difficult at this stage, since an explicit expression of the local peak selection is not available. We are happy that we have found a solution. We


251

can embed a local peak selection into a more general local selection and then use the implicit function theorem to check whether or not the generalized local selection is differentiable at v ∗ . Finally we prove that if a local peak selection p coincides with the more general local selection at v ∗ , then p is also differentiable at v ∗ . To apply the implicit function theorem, we only have to check whether or not the determinant of an nxn matrix, where n = dim(L), is equal to zero. This can be numerically carried out. This study has led to new approach, details will be presented in a future paper.

References [1] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14(1973), 349-381. [2] A. Bahri and P.L. Lions, Morse index of some min-max critical points. I. Application to multiplicity results, Comm. Pure Appl. Math., Vol. XLI (1988) 1027-1037. [3] H. Brezis and L. Nirenberg, Remarks on Finding Critical Points, Comm. Pure Appl. Math., Vol. XLIV, (1991), 939-963. [4] K.C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Boston, 1993. [5] G. Chen, W. Ni and J. Zhou, Algorithms and Visualization for Solutions of Nonlinear Elliptic Equations Part I: Dirichlet Problems, Int. J. Bifurcation & Chaos, to appear. [6] Y. Chen and P. J. McKenna, Traveling waves in a nonlinearly suspended beam: Theoretical results and numerical observations, J. Diff. Equ., 136(1997), 325-355. [7] Y. S. Choi and P. J. McKenna, A mountain pass method for the numerical solution of semilinear elliptic problems, Nonlinear Analysis, Theory, Methods and Applications, 20(1993), 417-437. [8] W.Y. Ding and W.M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rational Mech. Anal., 91(1986) 238-308. [9] Z. Ding, D. Costa and G. Chen, A high linking method for sign changing solutions for semilinear elliptic equations, Nonlinear Analysis, 38(1999) 151-172. [10] I. Kuzin and S. I. Pohozaev, Entire Solutions of Semilinear Elliptic Equations, Birkhauser, Boston, 1997. [11] Y. Li and J. Zhou, A minimax method for finding multiple critical points and its applications to semilinear PDE, SIAM J. on Scientic Computing, to appear. [12] Y. Li and J. Zhou, Convergence Results of a minimax method for finding multiple critical points, submitted. [13] F. Lin and T. Lin, “Minimax solutions of the Ginzburg-Landau equations”, Slecta Math. (N.S.), 3(1997) no. 1, 99-113. [14] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York, 1989. [15] Z. Nehari, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc., 95(19960), 101-123. [16] W.M. Ni, Some Aspects of Semilinear Elliptic Equations, Dept. of Math. National Tsing Hua Univ., Hsinchu, Taiwan, Rep. of China, 1987.

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[17] W.M. Ni, Recent progress in semilinear elliptic equations, in RIMS Kokyuroku 679, Kyoto University, Kyoto, Japan, 1989, 1-39. [18] S. I. Pohozaev, On an approach to nonlinear equations, Dokl. Akad. Nauk SSSR, 247(1979), 1327-1331; [19] P. Rabinowitz, Minimax Method in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conf. Series in Math., No. 65, AMS, Providence, 1986. [20] M. Schechter, Linking Methods in Critical Point Theory, Birkhauser, Boston, 1999. [21] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1982. [22] M. Struwe, Variational Methods, Springer, 1996. [23] J. Wei and L. Zhang, “On the effect of the domain shape on the existence of large solutions of some superlinear problems”, preprint. [24] M. Willem, Minimax Theorems, Birkhauser, Boston, 1996.

Static Buckling in a Supported Nonlinear Elastic Beam

David L. Russell, Virginia Polytechnic Institute and State University, Blacksburg, VA Luther W. White, University of Oklahoma at Norman, Norman, OK Abstract In the present paper we study finite static buckling effects in a nonlinear elastic beam supported on a flat, rigid, inelastic surface and subject to a gravitational force. We obtain necessary conditions characterizing equilibrium states as minima of the corresponding potential energy expression and we show that we can obtain closed form expressions for the displacement of the beam above the supporting surface on intervals where that displacement is positive.

AMS-MOS Classifications: 35J25, 35J50, 35Q72, 49L10; Keywords: nonlinear beam, supported beam, buckling, constrained equilibrium

1

Introduction

In a recent article [6] the authors have studied buckling phenomena in the context of a nonlinear beam model originally introduced by Lagnese [2]. Our purpose in the present article is to revisit this model in a constrained situation wherein the beam is supported on a flat, rigid, inelastic surface – so that the only permissible transverse displacements are positive – and is subject to a uniform negative force, which can be interpreted as gravity. A variety of applications occur in circumstances where a strip of material, e.g., a track or a roadbed, is laid out over a supporting surface and may be subject to buckling away from the supporting surface as a consequence of temperature-induced horizontal stresses, fast moving loads, etc.. We consider, then, an elastic beam of length L, with uniform cross section, in a two dimensional geometric context. The longitudinal extent of the beam corresponds to the interval 0 ≤ x ≤ L and the beam is assumed to have thickness 2h. It will be convenient to suppose that in equilibrium the elastic axis coincides with the x-axis, even though, strictly speaking, that violates the constraints described in the preceding paragraph. The displaced elastic 253

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Russell and White

axis, or ”neutral curve”, admitting both transverse and lateral displacements, is described by x x + ξ(x) −→ , x ∈ [0, L], y η(x) where, minimally, ξ ∈ H 1 [0, L] and η ∈ H 2 [0, L]; in many cases we will need to assume more smoothness than this. The support constraint corresponds to the condition η(x) ≥ 0. Assuming that linear material elements orthogonal to the elastic axis in equilibrium remain so under admitted displacements, such a deformation results in an infinitesimal stretching or contraction of material filaments parallel to the elastic axis having vertical coordinate y in equilibrium, with resultant arclength increment, using 0 to denote differentiation with respect to x, q ds = (1 + ξ 0 )2 + (y η 0 )2 dx. (1.1) 0

With the basic assumption, described in [6], to the effect that ξ and ξ are of 0 00 the same order as η 2 , (η )2 and (η )2 , and, further, that these three are of the same order as the thickness of the beam, discarding terms of higher order one obtains, again as in [6], the potential energy expression for the beam in the form Z Z A L 0 Bh2 L 00 2 0 2 2 V(ξ, η) = 2ξ + η η dx + dx 2 0 6 0 Z (1.2)

L

+g

η(x) dx − κ ξ(L).

0

Here g > 0 is the constant gravitational force acting in the negative y direction and κ > 0 represents a horizontal compressional force acting at the right (horizontally free) end of the beam; the left end of the beam is assumed horizontally fixed so that ξ(0) = 0. The beam is assumed “transversely 0 clamped” at the endpoints x = 0 and x = L in the sense that η(x) and η (x) vanish there; we formalize these requirements in §2 to follow. The positive constants A and B involve the elastic constants of the beam material and its width (in the third dimension which does not concern us here). All constants shown here include, implicitly, a factor 2h corresponding to beam thickness (for example, the gravitational force is, for constant material mass density, proportional to beam thickness). In §2, to follow, we establish the existence of a minimizer of the potential energy functional (1.2) under certain restrictions on the parameters of the problem, as made precise in §6. In §3 we continue to develop necessary conditions characterizing such a minimum. The purpose of §4 is to present two detailed examples to clarify the results presented in §3. In §5 we carry


255

out explicit computations resulting in analytic expressions for the solution in certain restricted circumstances. Finally, §6 serves as an appendix, establishing certain bounds required in the work of §2.

2

Existence of a Potential Energy Minimizer

In this section we study the static equilibrium problem for the supported elastic beam introduced in §1. Such an equilibrium is characterized as a minimizer of the potential energy expression V(ξ, η) given by (1.2) subject to the positivity constraint η(x) ≥ 0, x ∈ [0, L].

(2.1)

We further suppose that the beam is “clamped” at the endpoints x = 0 and 0 x = L; i.e., we have the boundary conditions (using to indicate differentiation with respect to x) 0

0

η(0) = η (0) = η(L) = η (L) = 0.

(2.2)

We assume the left endpoint of the beam is horizontally fixed, corresponding to (2.3)

ξ(0) = 0

but we place no constraint on the horizontal right hand endpoint displacement; thus ξ(L) is free. These general boundary conditions are distinct from certain free boundary conditions which we will need to introduce later in order to characterize intervals in which the minimizer component η(x) > 0. A rigorous study of the potential energy minimization problem requires, first of all, a definition of appropriate “state spaces”. We let V = H02 [0, L]

(2.4)

and define the subset corresponding to the imposed constraint as V+ = { η ∈ V | η(x) ≥ 0 } . We further define (2.5)

U =

and (2.6)

ξ ∈ H 1 [0, L] ξ(0) = 0 }

X = U×V =

χ ≡

ξ η

ξ ∈ U, η ∈ V

X+ = U × V+ .

;

256

Russell and White Expanding the potential energy in the form Z L Z L 2 0 2 Bh2 00 2 0 0 V(ξ, η) = 2A ξ dx + 2A + ξ η dx η 6 0 0

+

(2.7)

A 2

Z

L

Z 0 4 η dx +

0

L

g η dx + κ ξ(L), 0

we designate the first term as the bilinear form a : X × X 7−→ R defined by Z L 0 0 Bh2 00 00 a(χ1 , χ2 ) = (2.8) 4A ξ1 ξ2 + η η dx. 3 1 2 0 It is clear that a satisfies, for some positive numbers γ0 , γ1 , the coercivity and boundedness relations γ0 kχk2X ≤ a(χ, χ), |a(χ1 , χ2 )| ≤ γ1 kχ1 kX kχ2 kX . We further define the nonlinear functional G : X 7−→ R by Z L 2 0 0 G(χ) = 2A (2.9) ξ η dx. 0

Defining W =

W01,4 [0, L]

1

R L 0 4 with norm kηk = 0 η dx

4

; we observe that

0

|G(χ)| ≤ 2A kξ kL2 kηk2W ≤ 2A kξkU kηk2W . Using the definitions of the previous paragraph, we may express the potential energy as Z L 1 A 4 V(χ) = a(χ, χ) + G(χ) + g η dx + κ ξ(L). kηkW + 2 2 0 For ε > 0 we introduce the perturbed form (2.10) 1 A Vε (χ) = (1 + ε) a(χ, χ) + G(χ) + kηk4W + 2 2

Z

L

g η dx + κ ξ(L), 0

for which we have the lower bound Z L 0 2 Bh2 00 2 1 Vε (χ) ≥ (1 + ε) 4A ξ dx + η 2 3 0 Z − 2A

L

ξ 0

0

12 Z

2

L

dx 0

Z L 0 4 12 A 4 η dx + kηkW + g η dx + κ ξ(L) 2 0

Static Buckling in a Supported Nonlinear Elastic Beam ≥

257

Z L 1 0 2 Bh2 00 2 1 0 2 4A ξ dx − A 2 4A kηk2W ξ + η 3 0 0 Z L A 1 + kηk4W + g η dx + κ ξ(L) ≥ (1 + ε) a(χ, χ) 2 2 0

1 (1 + ε) 2

Z

L

1 1 A kηk4W − β0 g L 2 kηkV − κ L 2 kξkU . 2 Rx 0 Here we have used, for ξ(x) = 0 ξ (x) dx, 1

1

− A 2 a(χ, χ) 2 kηk2W +

Z

x

| ξ(x)| ≤

Z

0

1 Z

x

| ξ (s)| dx ≤

dx

0

x

2

0

1

0

| ξ (s)| dx 2

2

0

1

⇒ | ξ(x)| ≤ x 2 kξkU , x ∈ [0, L],

(2.11)

and, also, the estimates Z

L

Z

12 Z

L

η dx ≤

2

dx

0

12

L

η dx

0

0

1

≤ β0 L 2 kηkV ,

kηkL2 [0,l] ≤ β0 kηkV , η ∈ V. As a consequence we have Vε ≥

(2.12)

1

1 1 ε ε a(χ, χ) − β0 g L 2 kηkV − κ L 2 kξkU ≥ a(χ, χ) 2 2

− L 2 β0 g2 + κ2

1 2

kχkX ≥

1 1 ε γ0 kχk2X − L 2 β0 g2 + κ2 2 kχkX . 2

We now consider the regularized problem Minimize Vε (χ); Pε : . χ ∈ X+ . From (2.12) we conclude that there exists an M0 such that (2.13)

Vε (χ) ≥ M0

and, for χ satisfying Vε (χ) ≤ M1 there exists M2 such that (2.14)

kχkX ≤ M2 .

Let {χn } be a minimizing sequence for Vε in X+ ; thus Vε (χn ) −→ d ≡

inf

+

χ∈X

Vε (χ).

258

Russell and White

From (2.14) it is clear that {χn } is bounded in X. Thus there is a subsequence, which we still denote by { χn }, converging weakly in X to an element χ ∈ X+ , since X+ is weakly closed. In particular the corresponding subsequence { ηn } converges weakly to η ∈ V. Since V embeds compactly in W we may extract a further subsequence, which we continue to call { ηn }, converging strongly to η in W which, further, implies kηn kW −→ kηkW . n 0 o We may now conclude, since the corresponding ξn converges weakly to ξ in 0 2 2 0 2 L [0, L] and converges strongly to η ηn in L2 [0, L], that Z G(χn ) = 2A 0

L

Z 0 2 ξn ηn dx −→ 2A 0

L

ξ

0

0 2 η dx = G(χ).

0

It follows that d = lim inf Vε (χn ) ≥ Vε (χ) and we conclude that there exists an element, which we will now call χε , such that Vε (χε ) = min Vε (χ). + χ∈X Now let us suppose that, possibly with some restrictions on the parameters of our problem, we can show that the set of ε-minima χε ε > 0 ⊂ X+ is bounded in X. Then, as ε −→ 0, a subsequence { ηεk } converges weakly in V to some η 0 ∈ V+ and, passing to a further subsequence if necessary, { ξεk } converges weakly in U to an element ξ0 , which implies lim inf a(χε , χε ) ≥ a(χ0 , χ0 ). Let χ ∈ X+ . Then Vε (χ) =

1 A (1 + ε)a(χ, χ) + G(χ) + kηk4W + (g, η)L2 [0,L] + κ ξ(L) 2 2

1 A (1 + ε) a(χε , χε ) + G(χε ) + kηε k4W + (g, ηε )L2 [0,L] + κ ξε (L). 2 2 In the limit as ε −→ 0 we have ≥ Vε (χε ) ≥

V(χ) ≥

1 A a(χ0 , χ0 ) + G(χ0 ) + kη 0 k4W + (g, η 0 )L2 [0,L] + κ ξ0 (L), 2 2

since it is clear that limε → 0 Vε (χ) = V(χ). Then we have the desired result: V(χ) ≥ V(χ0 ), χ ∈ X+ .


259

Thus the existence of a (not necessarily unique) minimizer for V is established. The proof of the boundedness of the χε is somewhat complicated; we leave it for §6, which serves as an appendix, at the end of this paper. That proof requires some restrictions on the parameters of the problem which may or may not be realistic in some cases. The proof given above is, of course, valid whenever we have a uniform bound on the χε , whether or not that boundedness is obtained as in §6.

3

General Conditions Characterizing a Minimizing Element

We begin this section with a choice of some minimizing element for V, as shown to exist in the previous section and there identified as χ0 . Here we will refer to such an element as χ and we will primarily be dealing with its individual components ξ and η. We have the potential energy form (1.2) and the general boundary conditions (2.2) and (2.3). We impose the positivity constraint (2.1) and we continue to use the spaces U, V and X as in (2.5), (2.4) and (2.6), respectively. Let Y = H0p [0, L] for 12 < p ≤ 2; we define Z = U × Y and the mapping F : Z 7−→ Y by F (ξ, η) = − η. The restriction on p guarantees H0p [0, L] ,→ C 0 [0, L] and thus that Y has a positive cone Y+ , corresponding to η(x) ≥ 0, x ∈ [0, L] with non-empty interior corresponding to η(x) > 0, x ∈ [0, L]. Applying an infinite dimensional version of the Kuhn-Tucker theorem [4],[5] there is an element λ ∈ Y∗ = H −p [0, L] such that the Lagrangian, defined by L(χ, λ) = V(χ) + hλ, F (χ)i , has a stationary point at the solution of the minimization problem min V(χ).

F (χ) ≤ 0

Thus, using the symbol DL to denote the Gateaux derivative with respect to the χ variable in X, we have DL(χ, λ) χ ˆ = 0, ∀χ ˆ ∈ X. Hence, for all ξˆ ∈ U and ηˆ ∈ V we have (3.1)

Z

0 = 0

L

Bh2 00 00 0 2 0 0 0 ˆ A 2ξ + η 2ξ + 2η ηˆ + − hλ, ηî . η ηˆ + g ηˆ dx + κ ξ(L) 3 0

Let ηˆ = 0. Then for all ξˆ ∈ U we have Z L 0 2 0 0 ˆ (3.2) 2A 2ξ + η ξ dx + κ ξ(L) = 0. 0

260

Russell and White

From (3.2) we conclude (3.3)

0 2 0 0 2 κ 0 2ξ + η = 0 ⇒ 2ξ + η = − . 2A 0

Setting ξˆ = 0 in (3.1) we have Z L 0 2 0 0 0 Bh2 00 00 2A η 2 ξ + η ηˆ + (3.4) 0 = η ηˆ + g ηˆ dx − hλ, ηî . 3 0 and then substituting (3.3) into (3.4) we obtain Z L 00 00 0 0 α η ηˆ − κ η ηˆ + g ηˆ dx − hλ, ηî = 0, ∀ ηˆ ∈ V, (3.5) 0 2

where α = Bh 3 . From the cited Kuhn - Tucker conditions λ ∈ H −p [0, L], satisfies the complementary slackness condition

< p ≤ 2,

hλ, ηi = 0, λ ≥ 0.

(3.6) Letting p =

1 2

+ ε, 0 < ε ≤ 2, (3.5) yields the equality, in H −( 2 +ε) , 1

1 2

αη

(3.7)

0000

00

+ κ η = − g + λ.

Using the essential boundary conditions (2.2) and applying the Sobolev 7 embedding theorems we conclude from (3.7) that η ∈ H02 ∩ H 2 −ε and, further, that η ∈ C 2 [0, L], η

(3.8)

000

1

∈ L ε [0, L], 0 < ε ≤

3 . 2

We now take Y = H01 [0, L] so that λ ∈ H −1 [0, L] and conclude there is an ˆ ∈ H 1 [0, L] such that, for all φ ∈ H 1 [0, L], element λ 0 0 Z L ˆ 0 φ0 dx = hλ, φi . λ (3.9) 0

From (3.5) we see that for all φ ∈ H02 [0, L] we have Z Ln Z L o 00 00 0 0 ˆ 0 φ0 dx = 0. α η φ − κ η φ + g φ dx − λ (3.10) 0

0

We define O to be the open set n o (3.11) O = x ∈ (0, L) η(x) > 0 .


261

Then λ must vanish on O in the sense that if supp φ ⊂ O and φ(x) ≥ 0, x ∈ [0, L], then Z L ˆ 0 φ0 dx. hλ, φi = 0 = λ (3.12) 0

From (3.10), for all φ ∈ such that supp φ ⊂ O, Z Ln o 00 00 0 0 α η φ − κ η φ + g φ dx = 0. (3.13) H02 [0, L]

0

Let F = [0, L] \ O. Then η(x) = 0, x ∈ F. Suppose φ ∈ H02 [0, L], supp φ ⊂ Int F. Then, from (3.10), Z L ˆ 0 φ0 dx = 0. (3.14) gφ − λ 0

The conclusion η ∈

C 2 [0, L]

in (3.8) implies, in particular, that 00

η (xb ) = 0

(3.15)

at any boundary point xb of O which is also a cluster point for the set F. This constitutes an important free boundary condition characterizing boundary points of isolated open intervals included in O.

4

Illustrative Examples

Since the implications of the results obtained in the previous section may not be completely obvious, we present two examples in this section showing how those conditions apply and, in particular, the role and limitations of the free 00 boundary condition η (xb ) = 0 applying at certain boundary points xb of the set O as described in (3.11). Example 1 Let us consider a situation wherein, for 0 < x1 < x2 < L, we have η(x) > 0, x ∈ (0, x1 ) ∪ (x2 , L); η(x) = 0, x ∈ [x1 , x2 ]. We assume V(χ) = V(ξ, η) is minimized, subject to the constraints described earlier by the pair (ξ, η), that λ, is the corresponding Lagrange multiplier and ˆ is related to λ by (3.9). From the regularity results of the preceding that λ section we conclude that 0

00

0

00

η(x1 ) = η (x1 ) = η (x1 ) = 0; η(x2 ) = η (x2 ) = η (x2 ) = 0. From the minimality of V(χ) = V(ξ, η) and the inactivity of the constraint η(x) ≥ 0 on (0, x1 ) ∪ (x2 , L) we see that if φ ∈ H02 [0, L], supp φ ⊂ (0, x1 ) ∪ (x2 , L), then, in particular, Z x1 00 00 0 0 0 = α η φ − κ η φ + g φ dx 0

262

Russell and White Z 00 0 x1 = αη φ − 0

x1

h 000 0 i 0 α η + κ η φ − g φ dx

0

Z 000 x1 0 = − α η + κ η φ + 0

Z

x1

=

x1

αη

0000

00 + κ η + g φ dx

0

0000 00 α η + κ η + g φ dx = 0;

0

in each case the properties of φ show that the boundary terms vanish. We conclude from the last identity that αη

(4.1)

0000

00

+ κ η + g = 0, in (0, x1 ) .

Comparing with (3.7) we conclude λ = 0 on (0, x1 ). Similarly we conclude that λ = 0 on (x2 , L) and αη

(4.2)

0000

00

+ κ η + g = 0, in (x2 , L) .

From the above it follows, using (3.9), that for φ ∈ H02 [0, L], supp φ ⊂ (0, x1 ) ∪ (x2 , L) we have Z (4.3)

L

ˆ 0 φ0 dx = hλ, φi = 0. λ

0

ˆ ∈ H 1 [0, L] we have the endpoint conditions Since λ 0 ˆ ˆ λ(0) = 0, λ(L) = 0.

(4.4)

Further, since we can require supp φ ⊂ (0, x1 ), we can use the argument of ˆ 0 is constant on (0, x1 ); the classical du Bois - Reymond lemma [3] to see that λ equivalently ˆ 00 = 0, x ∈ (0, x1 ). λ

(4.5)

Now let supp φ ⊂ (0, x2 ). From (3.10) we see that Z x2 Z 00 00 0 0 0 = α η φ − κ η φ + g φ dx − 0

Z

x1

= 0

0

Z 0

00

αη + κη

0

Z

x2

Z g φ dx −

x1

Z 0 φ − g φ dx +

0

x1

= −

x1

ˆ 0 φ0 dx λ

0

00 00 0 0 α η φ − κ η φ + g φ dx +

Z 00 0 x1 = αη φ −

x2

Z 0 0 00 α η + κ η φ − g φ dx +

x2

x1

x2

ˆ 0 φ0 dx λ

x1 x2

x1

g φ dx −

Z

x2 x1

ˆ 0 φ0 dx λ

ˆ 0 φ0 dx = 0, g φ dx − λ

Static Buckling in a Supported Nonlinear Elastic Beam 0

263

00

the last equality being valid because φ (0) = 0 and η (x1 ) = 0. Integrating by parts again we have Z x1 00 0 x1 00 00 − α η + κ η φ + αη + κη + g φ dx 0

x ˆ 0 φ 2 + −λ x1

Z

x2

x1 Z x1

+

ˆ 00 + g φ dx = λ

0

00

− αη + κη

Z 00 00 αη + κη + g dx +

0

x2

0

0 ˆ (x1 −) + λ (x1 +) φ(x1 )

00 ˆ + g φ dx = 0. λ

x1

This being true for all φ ∈ with supp φ ⊂ (0, x2 ) we once again have the earlier result (4.1) for the interval (0, x1 ) together with the “point support condition” at x1 , H02 [0, L]

ˆ 0 (x1 +) − α η 000 (x1 −) = 0, λ

(4.6)

and the adjoint equation in the interval (x1 , x2 ), (4.7)

ˆ 00 (x) + g = 0, x ∈ (x1 , x2 ). λ

Next passing to φ ∈ H02 [0, L] with supp φ ⊂ (0, L) and carrying out computations similar to those performed above we again obtain (4.2), the “support condition” at x2 , (4.8)

ˆ 0 (x2 −) − α η 000 (x2 +) = 0, λ

and the adjoint equation (4.9)

ˆ 0 (x) = 0, x ∈ (x2 , L). λ

Combining the endpoint conditions (4.4), the support conditions (4.6) and (4.8) with the adjoint equations (4.5), (4.9) and (4.7) we have a complete set ˆ in terms of η and ξ. The equation (4.3) identifies of equations determining λ ˆ 00 . Thus λ ≡ g on the interval (x1 , x2 ) λ, in the distributional sense, as − λ 000 000 and includes Dirac delta components of magnitude − α η (x1 −) and α η (x2 ) at the points x1 and x2 , respectively. On (x1 , x2 ) the multiplier λ corresponds to the constraint force required to support the beam on that interval. Support for the beam on the intervals (0, x1 ) and (x2 , L) corresponds to the vertically oriented distributional forces just identified at x1 and x2 together with the constraint forces exerted at x = 0 and x = L, which are not included in the analysis because the conditions η(0) = η(L) = 0 are given a priori. Further analysis shows that  000  x ∈ (0, x1 ),  α η (x1 ) x, 2 g x ˆ λ(x) = − 2 + c1 x + c2 , x ∈ (x1 , x2 ),   α η 000 (x )(x − L), x ∈ (x2 , L), 2

264

Russell and White

ˆ should be continuous at with c1 and c2 determined by the requirement that λ x1 and x2 . Example 2 The situation studied here differs from that in Example 1 in that the middle interval [x1 , x2 ] is collapsed to a single point which we will call x1 . Using (3.10) and integrating by parts we have, for φ ∈ H02 [0, L], Z L Z L 00 00 0 0 ˆ 0 φ0 dx 0 = α η φ − κ η φ + g φ dx − λ 0

00 0 x1 00 0 L αη φ + αη φ − 0

x1

L

Z

0

Z 0 0 00 α η + κ η φ − g φ dx −

0

L

ˆ 0 φ0 dx = 0. λ

0

Collecting boundary terms and integrating by parts again we have h 00 i 0 0 x 00 ˆ 0 φ x1 − α η 00 + κ η φ 1 α η (x1 −) − η (x1 +) φ (x1 ) − λ 0 00 0 L − α η + κ η φ + x1

L

Z

00

00

αη + κη 0

0

Z

L

+ g φ dx +

ˆ 00 φ dx = 0. λ

0 0

Restricting first to φ such that φ(x1 ) = φ (x1 ) = 0 and arguing as we did to obtain (4.1) and (4.2) in Example 1, we obtain those equations again and also ˆ 00 vanishes identically, i.e., λ ˆ 0 is a constant, on (0, x1 ) and x1 , L). conclude that λ 0 Then considering φ with φ (x1 ) = 0 we obtain ˆ 0 (x1 +) − λ ˆ 0 (x1 −) = α η 000 (x1 +) − η 000 (x1 −) . λ (4.10) ˆ 0 is constant on (0, x1 ) and on (x1 , L) Combining (4.10), the result that λ ˆ ˆ ˆ and the conditions λ(0) = λ(L) = 0 following from the original definition of λ ˆ in terms of as an element of H01 [0, L] we have all that is needed to determine λ ξ and η (actually, only η is involved). The discussion of the role of the original Lagrange multiplier λ is much the same as in Example 1 except that in this case λ is zero except for its single distributional component, a multiple ofthe 000 000 Dirac distribution with support at x1 of strength α η (x1 −) − η (x1 +) . It 00

should be noted that in this example there is no requirement on η (x1 ) other 00 00 than the continuity requirement η (x1 −) = η (x1 +).

5

Analytic Discussion of Static Equilibrium States

This section serves as as a somewhat extended third example, supplementary to Examples 1 and 2 of the preceding section. We want to consider the case where O, the support set for η(x), consists of a single interval (x1 , x2 ) ⊂ Int [0, L]. Throughout the interval [0, L] we have the partial differential equation 0 2 0 0 2ξ + η (5.1) = 0.


265

We have η(x) ≡ 0, x ∈ [0, x1 ]∪[x2 , L] and, in the interval (x1 , x2 ), η(x) satisfies, 2 recalling the abbreviation α = B3h introduced in §3, the partial differential equation (5.2)

αη

0000

− 2A η

0

0 2 0 2ξ + η + g = 0. 0

From the first of these equations we have 0 2 0 2ξ + η = C, x ∈ [0, L], 0

for some constant C. Since η vanishes on [0, x1 ] ∪ [x2 , L], (3.3) gives 0

4A ξ + κ = 0, x = 0, L, on that set and we must, in fact, have 0 κ 0 2 A 2ξ + η ≡ − , x ∈ [0, L]. (5.3) 2 Then setting K =

(5.4)

κ g , G = , α α

the equation (5.2) can be rewritten in the form η

(5.5)

0000

00

+ K η + G = 0.

Since this differential equation is autonomous solutions are invariant under translation of the support interval (x1 , x2 ). It will be convenient to assume that this interval is centered at x = L2 . Looking for solutions symmetric with respect to x = L2 , we note that (5.5) is equivalent to 2 !00 G x − L/2 η + Kη + = 0 2 00

from which, taking the symmetry into account again, there is a constant d such that 2 G x − L/2 00 η + Kη + (5.6) + d = 0. 2 2 b x−L/2 Trying for a particular solution of the form η(x) = + c we find that 2 G G b = −d − Kc and Kb = −G. Thus b = − K , d = −b − Kc = K − Kc.

266

Russell and White

Including the solutions of the homogeneous equation symmetric with respect to x = L2 we have 2 G x − L/2 η(x) = a cos ω(x − L/2) − + c, 2K 1

where ω = K 2 . Let ρ = L/2 − x1 = x2 − L/2. Then the “essential” 0 boundary conditions η(x2 ) = η (x2 ) = 0 yield (5.7)

a cos ωρ −

Gρ2 Gρ + c = 0, − aω sin ωρ − 2 = 0, 2 2ω ω 00

while the “free boundary” condition η (x2 ) = 0 gives (5.8)

− aω 2 cos ωρ −

G = 0. ω2

The second equation of (5.7) together with (5.8) give (5.9)

tan ωρ = ωρ.

We initially take ρ to be the smallest positive solution of this equation; the one such that σ ≡ ωρ lies in (π, 3π/2). Then the third equation yields a = −

G ω 4 cos ωρ

,

which is positive since cos ωρ < 0. Finally, from the first equation of (5.7) we have Gρ2 σ2 G Gρ2 G c = − a cos ωρ + = 4 + = 2 1+ . 2ω 2 ω 2ω 2 K 2 Recalling (5.4) the solution η(x) is thus given, in the interval L2 − ρ, L2 + ρ wherein it is positive, as gα cos ω(x − L/2) σ2 g η(x) = 2 1 − (5.10) + − (x − L/2)2 . κ cos ωρ 2 2κ The amplitude at the mid-point x = L/2 is gα 1 1 σ2 gα σ2 η(L/2) = 2 1 − + = 2 1+ + κ cos ωρ 2 κ cos(σ − π) 2 and the second derivative there is 00

η (L/2) =

g α ω2 g − . 2 κ cos σ κ

267

Static Buckling in a Supported Nonlinear Elastic Beam 0.25

beam displacement eta(x)

0.2

0.15

0.1

0.05

0

−0.05

−0.1 0

0.5

1

1.5 2 2.5 longitudinal variable x

3

3.5

4

Fig. 1. Basic Equilibrium Form for η(x); j = 1.

In Figure 1 we show a typical plot of η(x) obtained using the formula (5.10). Computing the third derivative of η(x) in the interval where η(x) > 0 we obtain the expression 000 g α ω3 L L L η (x) = − 2 sin ω x − , −ρ < x < + ρ. κ cos ωρ 2 2 2 Since, as we have noted earlier, cos ωρ < 0 this is a positive multiple of 000 sin x − L2 . Since η(x) and hence η (x) vanish to the left of L2 − ρ and to the right of L2 + ρ, the lateral forces experienced by the beam at these points 000 L L are B η 2 − ρ + and − B 2 + ρ − , respectively. Using (5.9) we see that these both have the positive value g B α ω3 g α ω3 g ρ B α K2 tan ωρ = ωρ = κ2 κ2 κ2 and represent point forces exerted on the beam by the supporting surface at the points indicated. If the supporting surface were endowed with elastic qualities these would be replaced by distributed forces, of course. All of the above assumes that L ≥ 2ρ. If this is not the case then the free boundary condition (5.8) cannot be achieved; we have η(x) > 0 throughout 00 the open interval (0, L) with η (x) > 0 at both of the points x = 0 and x = L. There is, of course, the critical case where L = 2ρ for which η(x) > 0

268

Russell and White 1.4 1.2

beam displacement eta(x)

1 0.8

j=4

0.6 j=3

0.4 0.2

j=2

0 −0.2 −0.4 −0.6 0

0.5

1

1.5 2 2.5 longitudinal variable x

3

3.5

4

Fig. 2. Equilibrium Forms for η(x), j = 2, 3, 4. 00

00

throughout (0, L) and η (0) = η (L) = 0. It is clear that the equation (5.9) has infinitely many solutions ρj tending asymptotically to (2j+1)π as j → ∞, 2 the one just discussed corresponding to j = 1. In Figure 2 we show the equilibrium forms obtained from (5.10) for the cases j = 2, 3, 4. Our conjecture is that the case j = 1 corresponds to a stable equilibrium associated with a minimum of the potential energy form whereas the cases j > 1 are unstable, corresponding to stationary points of the potential energy functional which are not minima of that functional.

Appendix: Proof of Boundedness of the χε We retain the definitions of spaces, etc., introduced in §2. For ξ ∈ U , η ∈ V and for a variation δ to ξ, also in U , we have (cf. (2.10)) Z L 0 2 0 0 0 Dξ Vε (ξ, η) δ = 4A(1 + ε) ξ δ dx + 2A δ η dx + κ δ(L). 0

Applied at the point χε = (ξε , ηε ) and with δ replaced by ξε , this gives Z L Z L 0 2 0 0 2 4(1 + ε) ξε dx = − 2A (A.1) ξε ηε dx − κ ξε (L) 0

0

which yields the estimate, uniform for ε > 0, 1

4A kξε k2U ≤ 2A kξε kU kηε k2W + κ L 2 kξε kU


269

1

1 κ L2 ⇒ kξε k ≤ kηε k2W + . 2 4A

(A.2)

Applying integration by parts to (A.1) we obtain 0 2 0 4A(1 + ε)ξε + 2A ηε (L) + κ δ(L) Z

L

−

0

0 2 0 4A(1 + ε)ξε + 2A ηε δ dx = 0 0

Restricting attention to perturbations δ(x) such that δ(0) = δ(L) = 0 we obtain the equation

0 2 0 −2A 2(1 + ε)ξε + ηε = 0.

(A.3)

0

Our assumptions on ξ give ξε (0) = 0; using (A.3) with variable δ(L) we conclude that 0 2 0 4A(1 + ε)ξε + 2A ηε (A.4) (L) + κ = 0. Since (A.3) implies 0 2 0 2 0 0 2(1 + ε)ξε + ηε (L) = 2(1 + ε)ξε + ηε (x), x ∈ [0, L), we obtain (A.5)

0 2 κ 2(1 + ε)ξε + ηε (x) ≡ − . 2A 0

Integration and the condition ξε (0) = 0 then give Z x 0 2 κx 1 ξε (x) = − ηε ds + 4A(1 + ε) 2(1 + ε) 0 so that, uniformly for ε > 0, (A.6)

κx 1 |ξε (x)| ≤ + 4A 2

Z

x

0

ηε

2 ds.

0

Now considering perturbations υ ∈ V to η ∈ V+ such that η + t υ ∈ V+ 2 for small t > 0 and recalling the abbreviation α = B3h introduced in §3, we have the inequality Dη Vε (ξ, η)υ = Z L Z L 0 2 0 0 0 00 00 00 00 2A 2 ξ + η η υ + α η υ + g υ dx + ε α η υ dx ≥ 0. 0

0

270

Russell and White

Applying this to ξε and ηε we obtain Z L Z L Z 0 2 0 0 00 00 0 (1 + ε)α 2ξε + ηε ηε υ dx + ηε υ dx + 2A 0

0

L

g υ dx ≥ 0.

0

0

Using (A.5) to solve for ξε we have 0

ξε (x) ≡ −

2 0 1 κ ηε (x) − 2(1 + ε) 4A(1 + ε)

so that Z

L

(1 + ε)α

00

Z

00

L

ηε υ dx + 0

0

Z L 2Aε 0 2 κ 0 0 − g υdx ≥ 0. η η υ dx + (1 + ε) ε (1 + ε) ε 0

Using (2.11) we now have 1

1 κ L2 k ξε kU ≤ kηε k2W + 2 4A 1 ⇒ k ξε kU ≤ 2

(A.7)

L3 3

12

1

κ L2 kηε kV + . 4A 2

We clearly have (cf. (2.10)) 0 = Vε (0, 0) ≥

1+ε A a (χε , χε ) + G (χε ) + kηε k4W + (g, ηε ) + κ ξε (L) 2 2

which implies, using (2.8) and (2.9), that − (g, ηε ) − κ ξε (L) ≥ Z

L

+ 2A 0

Z

1+ε 2

0 2 A 0 ξε ηε dx + 2

L

0 2 00 2 4A ξε + α ηε dx

0

Z

L

0

ηε 0

4

dx ≥ 2A(1 + ε)k ξε k2U

α(1 + ε) A + kηε k2V + kηε k4W − 2A k ξε kU kηε k2W . 2 2 α(1 + ε) A A ≥ 2A(1 + ε)k ξε k2U + kηε k2V + kηε k4W − 2A k ξε k2U − kηε k4W 2 2 2 α(1 + ε) α(1 + ε) = 2A ε k ξε k2U + kηε k2V ≥ kηε k2V . 2 2 From this we conclude, using (2.11) again, that 1 α(1 + ε) g L2 kηε k2V ≤ kηε kV + κ L 2 k ξε kU . 2 2


271

Using (A.7) and dropping ε on the left hand side we see that " 1 # 1 2 3 2 2 1 α 1 g L κ L L kηε k2V + kηε k2V ≤ kηε kV + κ L 2 . 2 2 2 3 4A As a result we see that κ L2 κ2 L g L2 α √ − (A.8) kηε k2V + kηε kV + ≥ 0. 2 2 4A 2 3 If κ L2 α √ − < 0, 2 2 3

(A.9)

so that the parabola described by the equation obtained from (A.8) by changing ≥ to = opens downward, then the inequality is valid only to the left of the largest root of that quadratic equation, leading to the conclusion r kηε kV ≤

g L2 2

g L2 2

+

−

2

κ2 L A

−

κ√ L2 2 3

−

α 2

κ√ L2 2 3

−

α 2

≡ K0 .

√

Thus if (A.9) is true, i.e., if κ < ε > 0,

3α , L2

we have the estimate, independent of

0 ≤ kηε kV ≤ K0 .

(A.10) Using this in (A.7) we obtain

1 k ξε kU ≤ 2

L3 3

12

1

κ L2 + (A.11) . 4A n o Combining (A.10) with (A.11) we see that the set χε ε > 0 is bounded in X, as required to complete the existence argument in §2. K02

References [1] Adams, R. A.: Sobolev Spaces, Academic Press, New York, 1975 [2] Lagnese, J. E.: Recent progress in exact boundary controllability and uniform stabilizability of thin beams and plates, in Distributed Parameter Control Systems, G. Chen, E. B. Lee, W. Littman and L. Markus, Eds., Marcel Dekker, New York, 1991, pp. 61-111

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[3] Ewing, G. M.: Calculus of Variations with Applications, W. W. Norton & Co., Inc., New York, 1969 [4] Luenberger, David G.: Optimization by Vector Space Methods, John Wiley & Sons, Inc., New York, 1969 [5] Russell, D. L.: The Kuhn - Tucker conditions in Banach space with an application to control theory. J. Math. Anal. Appl. (1966). [6] Russell, D. L., and L. W. White: An elementary nonlinear beam theory with finite buckling deformation properties, to appear in SIAM J. Appl. Math.

Optimal control of a nonlinearly viscoelastic rod

Thomas I. Seidman, University of Maryland Baltimore County, Baltimore, MD. E-mail: [email protected] Stuart S. Antman1 , University of Maryland College Park, College Park, MD. E-mail: [email protected] Abstract We consider some typical optimal control problems for a nonlinear model of longitudinal vibrations in a viscoelastic rod. In trying to follow the usual pattern of showing that every infimizing sequence of controls contains a subsequence suitably converging to an optimal control, we confront the severe technical difficulty that the constitutive function cannot be uniformly Lipschitzian in its arguments — e.g., it blows up at a ‘total compression.’ One needs to make careful use of the structure of the system to overcome this difficulty.

1

Introduction

We consider a PDE model for the longitudinal motion of a uniform2 viscoelastic rod: (1.1)

wtt = νs + f,

ν = n (ws , wst )

holding on Q := (0, `)×(0, T ). Here w = w(s, t) is the position at time t ∈ [0, T ] of the material point with reference position s ∈ (0, `) so ws gives the strain and wst is the strain rate; ν = ν(s, t) is then the contact force (given by the constitutive function n) and f = f (s, t) is the external body force. One natural set of boundary conditions for this problem consists of the specification of the contact forces at the endpoints. It is plausible to consider either the body force f as a distributed control or the contact force νˆ at s = ` as a boundary control. We take homogeneous boundary conditions at one end, for simplicity, so that ν (1.2) ≡ 0, ν = νˆ(t). s=0

s=`

1

Supported in part by a MURI Grant from the ARO. This uniformity is purely for expository simplicity. There would be no difficulty in permitting the density ρ, here normalized to 1, and the constitutive function n(··) to depend explicitly (piecewise continuously) on s as well. 2

273

274

T.I. Seidman and S.S. Antman

It is convenient to introduce u := ws , v := wt and then to rewrite (1.1) in the form (1.3)

i) ut = vs , ii) hω, vt i + hωs , n(u, ut )i = ω(`)ˆ ν + hω, f i,

with (1.3-ii) holding for t ∈ (0, T ) and for all suitable test functions ω ∈ H 1 (0, `). Here and below we use h·, ·i to denote the L2 (0, `) inner product and comparable duality products. This model was considered in [1] and [3] (and is generalized in the forthcoming paper [2] to a full vector model which considers transverse motion, shear, and torsion as well as longitudinal motion). Suitable assumptions there on the constitutive function n(··) (permitting fully nonlinear dependence on the strain rate) ensure, for suitable data, both well-posedness and the preclusion of ‘total compression’, i.e., u = ws is pointwise bounded away from 0. In Section 3 we adapt those hypotheses, with particular attention to weakening the conditions imposed on the data f, νˆ which we take as possible controls. Our primary objective in this paper is to prove the existence of optimal controls for three closely related and reasonably typical optimal control problems: Problem 1: Taking the boundary condition νˆ (contact force at: s = `) as control, track a target trajectory w(··) ¯ over [0, T ], Problem 2: Taking the external force f as control, approximate a target state [w, ¯ w ¯t ] at the fixed time T > 0, Problem 3: Again with boundary control νˆ, attain a target state in minimum time, subject to constraints. These problems are closely related here in that essentially the same a priori estimates suffice to provide the compactness needed for the arguments. Note that in order to take limits through the nonlinearity of the constitutive relation, the compactness obtained must necessarily involve pointwise bounds and convergence for u, ut , since we impose no growth condition restricting the dependence of n(y, z) on z and since n(y, z) must blow up as y → 0 to penalize ‘total compression’. This lack of regularity in the constitutive function is a principal technical difficulty for our analysis.

2

Compactness and optimal control

Perhaps the principal technical point of this paper is that the same estimates and compactness needed to obtain existence (and well-posedness) for the direct problem (with f, νˆ specified) also suffice to give existence of optimal controls for the particular problems we consider.

Optimal control of a viscoelastic rod

275

We do not attempt to seek especially weak conditions on the initial data but we impose, as sufficient for our arguments, the requirement that the initial data satisfy: (2.1)

◦

◦

w, wt ∈ H 2 (0, `)

◦

with w (s) > c¯ > 0.

On the other hand, we weaken slightly the assumption (used in [1], e.g.,) that ft , νˆt have L2 bounds. Let us define U := W 1,p [0, T ] → L2 (0, `) , V := W 1,4/3 (0, T ). (2.2) where p is an arbitrary fixed number with p > 1. [Note that then U ⊂ 2 C [0, T ] → L (0, `) with a fixed modulus of continuity (depending on the choice of p) and that V ⊂ C[0, T ] with compact embedding.] Our key hypothesis on the cost functionals considered for optimization is an appropriate coercivity condition: f is bounded in U and νˆ is bounded in V (at least when the cost functional is bounded) While deferring the detailed proofs, which constitute the next two sections, we now assert two lemmas which will be fundamental to our arguments: Lemma 2.1. Under the hypotheses of Section 3, the equation ( 1.1 ) , equivalently (1.3), has a unique solution corresponding to any choice of data [f, νˆ] in U × V and any choice of initial data satisfying ( 2.1 ) . Lemma 2.2. Under the hypotheses of Section 3, the solutions of ( 1.1 ) , ( 1.2 ) corresponding to [f, νˆ] bounded in U × V for fixed or suitably bounded initial data as in ( 2.1 ) all lie in a fixed compact set: In particular, there is a compact K such that (2.3)

ut = vs ∈ K ⊂ C(Q)

and, of course, w, u = ws , and v = wt also lie similarly in compact subsets of C(Q). Further, u is uniformly bounded away from 0 and vt = wtt is bounded in L∞ ((0, T ) → L2 (0, `)). We now show how these lemmas may be used to prove the existence of optimal controls for the problems under consideration, beginning with the following corollary. Corollary 2.1. Suppose that {f k , νˆk } are bounded in the sense of the coercivity condition and converge weakly in L2 (Q), ×L2 (0, T ): f k * f¯, νˆk * ν¯. Then the corresponding solutions converge on Q (i.e., wk , uk = [wk ]s , v k = k converge uniformly and v k = wk converges weakly in L2 (Q)) wtk , ukt = vsk = wst t tt to the solution corresponding to the limit data f¯, ν¯.

276


Proof. By Lemma 2.1 there are indeed corresponding solutions wk and by Lemma 2.2 there is a subsequence for which these corresponding solutions k converging weakly in L2 (Q). Since uk , uk converge uniformly with vtk = wtt t k are bounded with u bounded away from 0 by Lemma 2.2, the corresponding functions ν k = n(uk , ukt ) then also converge uniformly to the appropriate limit; we note, in particular, that the limit boundary condition is then necessarily satisfied. With smooth test functions we easily see that the limit of the solutions satisfies (1.3-ii) with the limit data, as desired, and such ω are dense. Since this solution is unique by Lemma 2.1, we have convergence for the full sequence.

Problem 1: Suppose that f ∈ U and initial data as in (2.1) are already specified. We may then consider the determination of the boundary contact force νˆ(·) in (1.1), (1.2) as an optimal control problem once we have specified a cost functional J for optimization. We take J = J1 (ˆ ν ) + J2 (w) (subject to (1.1), (1.2)) with, for example, Z (2.4)

T

J1 (ˆ ν ) := max{|ˆ ν (t)} + a [0,T ]

|ˆ νt (t)|4/3 dt

0

and, tracking a target trajectory w, ¯ (2.5)

J2 (w)

:= b supQ {|wtt |} + χ∗ (|w − w|/c) ¯

where χ∗ (ω) :=

0 if ω 6 1 on Q, +∞ otherwise.

Thus, we wish to control the rod ‘gently’ (so that νˆ stays small and does not change too abruptly) and to keep the rod from accelerating too violently (so that wtt stays small) while demanding that the rod match the specified target trajectory w ¯ = w(s, ¯ t) to within a tolerance c. We do not expect that J is finite for arbitrary boundary data νˆ ∈ V, but assume a priori — presumably as a condition on the target w ¯ under consideration — that the set Vad of admissible controls is nonempty: there is at least one νˆ ∈ V for which (1.1), (1.2) gives a solution w with wtt bounded and |w − w| ¯ 6 c everywhere on Q. Theorem 2.1. With f, w ¯ and the initial data as above and under the hypotheses on the constitutive function n(··) of Section 3, there is an optimal control ν¯ for Problem 1, i.e., the cost functional J given by ( 2.4 ) , ( 2.5 ) attains its minimum. Proof. Given Lemmas 2.1, 2.2 and Corollary 2.1, the argument has a fairly standard pattern. Let (ˆ ν k ) be an infimizing sequence for J with corresponding


277

solutions (wk ) so J k := J1 (ˆ ν k ) + J2 (wk ) & J∗ := inf{J } Note that the assumption Vad 6= ∅ means that J∗ < ∞ and we can assume that J k < [bound] iy for each k. Since this bounds {ˆ ν k } in V, we may assume, without loss of generality, that (ˆ ν k ) converges uniformly on [0, T ] with weak 4/3 k convergence in L (0, T ) of (ˆ νt ), i.e., νˆk * ν¯, νˆtk * ν¯t . Since J1 (·) is lower semicontinuous in this topology, we have J1 (¯ ν ) 6 lim inf J1 (ˆ ν k ). Next, using Corollary 2.1, we have convergence of (wk ) to the solution corresponding to this control ν¯. Since χ∗ = 0 for each wk , the uniform convergence ensures that this also holds in the limit. We only have weak L2 (Q) k , but we note that ω 7→ sup {|ω|} is lower convergence for the accelerations wtt Q semicontinuous with respect to the weak L2 topology (since {ω ∈ L2 (Q) : |ω| 6 α} is convex and strongly closed for each α). Thus, in the limit we have J2 6 lim inf J2 (wk ) so J 6 J∗ and the minimum is attained at ν¯. Some characterization of this optimal control ν¯ through (formal) computation of first-order optimality conditions (expressed in terms of a linear adjoint equation) would certainly be possible, if rather messy for the particular cost functional we have treated here, but we do not pursue this. Problem 2: Now suppose that νˆ and the initial data are specified and that we seek an optimal distributed control f ∈ U . We consider two variants of this problem: we may insist on minimizing the U -norm while exactly matching the target state at t = T or we may penalize deviation from the target as measured in some norm. Thus, we either consider J = J1 (f ) := kf kU subject to: [w, wt ] (2.6) = [w, ¯ w ¯t ] t=T

or, for example, topologizing deviation in × C 1 [0, `] for the second variant, we may take J = J1 (f ) + J2 (w) with J1 (·) as in (2.6) and C 2 [0, `]

(2.7)

J2 (w) := sup{ |w(s, T ) − w(s)|, ¯ |wss (s, T ) − w ¯ 00 (s)|, |wt (s, T ) − w ¯t (s)|, |wts (s, T ) − [w ¯t ]0 (s)| : s ∈ [0, `]}.

[Note that (2.6) is equivalent to introducing J2 (w) := 0 if there is such a match, and J2 (w) := +∞ otherwise.] For neither variant do we expect J to be finite for arbitrary f ∈ U (for (2.7), because our estimates do not bound wss at all), but we do assume a priori that the admissible control set Uad := {f ∈ U : J < ∞} is nonempty for whichever variant is under consideration. This restricts our choice of target states to consider and, in particular, for the first variant it means requiring that the target be exactly reachable. Theorem 2.2. With νˆ, w ¯ and the initial data as above and under the hypotheses on the constitutive function n(··) of Section 3, there is an optimal

278


control ν¯ for Problem 2, i.e., the cost functional J , given by ( 2.6 ) or ( 2.7 ) as appropriate, attains its minimum. Proof. The proof is similar enough to that given for Theorem 2.1 that we only comment on it briefly. We can now begin by finding an infimizing sequence (f k ) for which fk * f¯ in L2 (Q) and ftk * f¯t in Lp [0, T ] → L2 (0, `) , the topology for which J1 (·) is both coercive and lower semicontinuous. As before, we now extract, if necessary, a subsequence for which Lemma 2.2 gives convergence of the corresponding solutions (wk ). For the first variant we then need only note that Corollary 2.1 ensures that the terminal condition in (2.6) holds in the limit since it holds for each wk . For (2.7) we note, much as in the proof of Theorem 2.1, that the convexity of the set {w : J2 (w) 6 α} for each α gives the needed lower semicontinuity of J2 (·). In any case, we have J 6 J∗ in the limit and the minimum is attained at f¯. Problem 3: Finally, we suppose that f ∈ U and the initial data (subject to (2.1)) are specified and that we have also specified a target state [w, ¯ w ¯t ] ∈ C[0, `] × C[0, `] and constraints. These are to include (the possibility of) both control constraints (for which the boundary control νˆ lies in a specified subset Vad ⊂ V) and state constraints (for which the trajectory t 7→ w ¯ determined by using this νˆ in (1.2) lies in a specified subset W ⊂ C(Q)). While one could consider more general examples, we assume, for expository simplicity, that Vad is convex, closed, and bounded in V and that W is defined (almost) pointwise so that ˆ W = {w(·) : [w, wt ] ∈ W(t) (2.8) for 0 6 t 6 τ = τ (ˆ ν )} t

ˆ where each W(t) is closed in [C 1 (0, `)]2 . We also ask, of course, that there be some control νˆ ∈ Vad for which w(·) is not only inW, but also matches the target, i.e., [w, wt ] (2.9) = [w, ¯ w ¯t ] t=τ (ˆ ν)

Our goal is to minimize the control time τ for this match. Theorem 2.3. With f, w, ¯ the initial data, and the constraint sets Vad , W as above and under the hypotheses on the constitutive function n(··) of Section 3, there is an optimal control ν¯ for Problem 3, i.e., there is a control for which the corresponding solution reaches the target state at the minimum control time τ∗ . Proof. As before, we consider an infimizing sequence (ˆ ν k ) with corresponding k solutions w for which τ k = τ (ˆ ν k ) & τ∗ := inf{τ (ˆ ν ) : νˆ is admissible }


279

where ‘admissible’ means that νˆ ∈ Vad , w(·) ∈ W, and ( 2.9 ) . Without loss of generality, since Vad is bounded, we may assume that (ˆ ν k ) is weakly convergent: νˆk * ν¯ and our assumptions on Vad ensure that we also have ν¯ ∈ ˆ Vad . The admissibility of each νˆk ensures that [wk , wtk ] ∈ W(t) for 0 6 t 6 τ∗ t since that τ k > τ∗ . Since Lemma 2.2 ensures (for a subsequence, to which we ¯ we now restrict our attention) that wk and uk = wsk converge uniformly on Q, k 1 have w (·, t) converging in C (0, `) and, similarly we get uniform convergence k , so that wk (·, t) converges in C 1 (0, `). Hence, the limit of v k = wtk and of wts t ∗ solution w , whose existence is given by Corollary 2.1, also satisfies the defining condition of W on [0, τ∗ ] and so will be in W, as desired, once we show that w∗ satisfies (2.9) at t = τ∗ , making τ (¯ ν ) = τ∗ . To verify (2.9), we need only note that [wk , wtk ] k −→ [w∗ , wt∗ ] τ

τ∗

is ensured by the uniform convergence provided by Lemma 2.2.

3

Hypotheses and first estimates

In this section and the next we provide the estimates giving the compactness of Lemma 2.2 and underlying the existence proof (which we will not present here) of Lemma 2.1. We split the constitutive function n(y, z) for the contact force into an elastic part n(y, 0) =: ϕ0 (y) and a dissipative part σ(y, z) (with σ(y, 0) ≡ 0): (3.1)

n(y, z) = ϕ0 (y) + σ(y, z)

for y > 0 and all z.

We assume that ϕ, σ are smooth where defined, e.g., ϕ0 , σ are C 1 . We impose two hypotheses on the constitutive function n(··), i.e., on ϕ, σ: the first is a rather natural assumption that there be some minimal dissipation while the second, formulated in terms of an auxiliary function ψ, is rather technical. Our first hypothesis is that (3.2)

σz (y, z) > µ

(for some fixed µ > 0).

Note that this gives µ|z|2 6 z σ(y, z). Next, we introduce ψ : (0, ∞) → IR+ , requiring that ψ(y) → ∞ as y → 0 to enforce our prohibition against total compression, and then impose our second hypothesis: (a) For each c > 0 there is a constant λ = λ(c) such that (3.3)

[z ny (y, z)]2 6 λσz (y, z) [1 + z σ(y, z) + ϕ(y)] when ψ(y) 6 c

while (b) There are β, c¯ > 0 such that

280


(3.4)

z ψ 0 (y) 6 n(y, z) + β when ψ(y) > c¯.

We include in this section some estimation which requires only L2 bounds for f, νˆ, beginning with the usual energy estimate. Taking ω = v in (1.3-ii) and noting (2.1)) gives Z t 2 1 kvk + hϕ(u)i + hvs σ(u, vs )i 2 (3.5) 0 Rt Rt ◦ ◦ = 12 k v k2 + hϕ(u)i + 0 v(`, ·)ˆ ν + 0 hv, f i R` where hϕ(u)i = 0 ϕ(u(s, t)) ds, etc. We have used (3.1) and the fact that vs ϕ0 (u) = [ϕ(u)]t since vs = ut . Estimating |v(`, t)|2 6 εkvs k2 + Cε kvk2 and recalling that µ|vs |2 6 vs σ(u, vs ) so we may choose ε > 0 small and absorb this term εkvs k2 on the left. We apply the Gronwall Inequality to obtain Z t 2 kvk , hϕ(u)i, (3.6) hvs σi 6 C 1 + kˆ ν k2 + kf k2 6 C 0

with C denoting a positive constant depending only on T, µ and bounds on the data, as indicated. Next we show that our hypotheses, as hoped, ensure the impossibility of total compression: we will bound u away from 0. We first integrate (1.1) over (0, s¯) to obtain Z s¯ n(u, ut ) (3.7) =: ν(¯ s, t) = [vt − f ] ds. (¯ s,t)

0

◦ We may assume in (3.4) that c¯ > maxs {ψ(u)} so if ψ(u(¯ s, t¯)) > c¯, then there is some τ > 0 with ψ(u(¯ s, τ )) = c¯ and ψ(u(¯ s, ·)) > c¯ on (τ, t¯). Thus (3.4) applies 0 to give ut ψ (u) 6 β + n(u, ut ) there, and use (3.7); integrating in t over (τ, t¯) then gives Z s¯ ¯ Z t¯ Z s¯ t ¯ ¯ ψ(u(¯ s, t)) 6 c¯ + (t − τ )β + v − f. 0

τ

τ

0

The terms on the right can all be estimated, by using an L2 (Q) bound on f for the last and (3.6) for the penultimate term; thus, ψ(u) 6 C and we will later be able to apply (3.3). Further, since ψ(y) → ∞ as y → 0, this shows that u = wt is uniformly bounded away from 0.

4

Further estimates and compactness

We continue our estimation, now assuming U , V bounds for f, νˆ and using (3.3) with the appropriate λ. ◦ ◦ We begin by setting ζ := vt = wtt . Note, first, that (2.1) gives u=ws ◦ in H 1 (0, `), so that u is continuous (and bounded pointwise on [0, `]) as well


281

◦

as bounded away from 0; similarly, ut = [wt ]s is bounded. These bounds ensure that the arguments remain in a compact set for which we have bounds ◦

on ny and nz = σz . This will enable us to verify the regularity of ζ:= ζ(·, 0). Using (1.1) at t = 0, we have ◦

◦ ◦

◦

◦

ζ= [n(u, v s )]s + f (·, 0) = ny wss +σz [w t ]ss + f which, with ny , σz bounded, is in L2 (0, `). Differentiating (1.3-ii) in t one obtains hω, ζt i + hωs , σz ζs i = ω(`)ˆ νt + hω, ft i − hωs , ny ut i. Taking ω = ζ and integrating over (0, t) then gives Rt 1 2 2 kζ(t)k + 0 hζs , σz ζs i i1/4 hR ◦ t (4.1) 6 12 k ζ k2 + C 0 |ζ(`, ·)|4 + C z¯(t) Rt + 0 εhζs , σz ζs i + λ4 ε (1 + hvs σ(u, vs )i + hϕ(u)i) where we have set z¯(t) := max{kζ(t¯)k : 0 6 t¯ 6 t}, have applied (3.3) after noting that |ζs ny ut | 6 εσz (ζs )2 + (1/4εσz )|ny ut |2 , have used the relevant bounds on ft , νˆt , and have noted that the final set of terms were already bounded by (3.6). Since |ζ(`, t)|2 6 (1/`)kζk2 + 2¯ z kζs k we have Z t 1/4 4 |ζ(`, ·)| 6 εkζ(t)k2 + εkζs k2 + Cε [1 + z¯(t)] . 0

Using this in (4.1) with ε chosen to absorb those terms on the left then gives kz(t)k2 6 C[1 + z¯(t)] 6 C[1 + z¯(t¯)] with t¯ arbitrarily fixed so t 6 t¯. Taking the maxt over [0, tb] then gives z¯2 6 C[1 + z¯] (uniformly for t¯ ∈ [0, T ]) and we have bounded3 ζ = vt = wtt in L∞ ((0, T ) → L2 (0, `)). In view of (3.7), this gives us also a uniform pointwise bound for ν. Returning to (4.1) we see that we have also bounded the integral on the left and so have bounded ζs = wstt = utt in L2 (Q); integrating this over (0, t), using (2.1), bounds ut = vs = wst in L∞ ((0, T ) → L2 (0, `)). Summarizing, we have by now shown (4.2) kwtt (·, t)k 6 C, 3

|ν(s, t)| 6 C,

kwstt kQ 6 C,

kwst (·, t)k 6 C

This and a density argument show that ζ is actually in C([0, T ] → L2 (0, `)).

282


as well as (3.6) and the bound away from 0 obtained for u = ws . Now consider ϕ0 (u) + σ(u, ut ) =: n(u, ut ) = ν(¯ s, ·) with s = s¯ arbitrarily fixed in (0, `). Multiplying this by ut and integrating the product over [0, T ] gives, using the bound on ν, Z T Z T 2 µkut k[0,T ] 6 ϕ(u) + ut σ(u, ut ) dt = ut ν dt 6 Ckut k[0,T ] 0

0

L2 (0, T )

which bounds ut = vs = wst in on each s = s¯. Integrating over (0, t) and noting (2.1) then uniformly bounds u pointwise on Q and so, with our previous bound away from 0, restricts the argument of ϕ(·) to a compact subinterval of its domain, whence ϕ(u), ϕ0 (u) are necessarily also bounded. Now we apply |z| 6 |σ(y, z)|/µ to bound ut pointwise, noting that σ(u, ut ) = ν−ϕ0 (u) has been bounded. This, of course, only bounds u = ws and ut = vs = wst in L∞ (Q) and, although a density argument would suffice to show that these remain within ¯ we wish to show that they lie in a compact subset of the closed subspace C(Q), that. [Our bound on ft is in Lp [0, T ] → L2 (0, `) , by (2.2), but we note that it would have been sufficient, up to this point, only to have required a bound on ft in the dual space of C([0, T ] → L2 (0, `)).] We prepare for the next step by recalling a compactness lemma appearing in a footnote in [4] (note also [5]): Lemma 4.1. Let X ,→ Y ,→ Z with the embedding X ,→ Y compact. Suppose that F is a set of functions on [0, T ] which is bounded in L∞ ([0, T ] → X ) and for which {ψt : ψ ∈ F} is bounded in Lp ([0, T ] → Z) for some p > 1. Then F is a precompact subset of C([0, T ] → Y). We now apply this to F = {ν} with X = H 1 (0, `), Y = C[0, `], Z = L2 (0, `), which satisfy the embedding requirements. From (3.7) we already have νs = ζ − f bounded in L2 (0, `) so ν is bounded in H 1 (0, `). On the other hand, (4.3)

νt = [n(u, ut )]t = ny ut + σz ζs

on noting that utt = vst = ζs . At this point, since the arguments of n(u, ut ) have been restricted to a fixed compact subset of the domain (0, ∞) × IR we have uniform bounds on the derivatives ny , nz = σz . As we already have bounds on ut in L∞ (Q) and on ζs in L2 (Q), the right hand side of (4.3) is bounded in L2 ([0, T ] → Z) as desired and we have shown that ν lies in a fixed compact ¯ subset of C([0, T ] → Y) = C(Q). Next we notice that the ordinary differential equation: n(u, ut ) = ν (with ◦ specified initial data u∈ C[0, `]) defines a continuous solution map4 ¯ → C(Q) ¯ S : ν 7→ ut : C(Q) 4

Since n(y, z) is not globally smooth and is not defined on all of IR2 (which has forced


283

¯ and the set K of (2.3) is the image under S of the compact subset of C(Q) obtained to contain ν. The compactness of K follows immediately, of course, from the continuity of S. This effectively completes the proof of Lemma 2.2. The estimates here provide the core of the existence proof but we omit the actual proof, referring the reader to [1] for further detail.

References [1] S.S. Antman and T.I. Seidman, Quasilinear hyperbolic-parabolic equations of onedimensional viscoelasticity, J. Diff. Eqns., 124 (1996), pp. 132–185. [2] S.S. Antman and T.I. Seidman, The spatial motion of nonlinearly viscoelastic rods, in preparation. [3] D. French, S. Jensen, and T.I. Seidman, A space-time finite element method for a class of nonlinear hyperbolic-parabolic equations, Appl. Numer. Math., 31 (1999), pp. 429–450. [4] T.I. Seidman, The transient semiconductor problem with generation terms, II, in Nonlinear Semigroups, PDE, and Attractors (LNM #1394; T.E. Gill, W.W. Zachary, eds.), Springer-Verlag, New York, 1989, pp. 185–198. [5] J. Simon, Compact sets in the space Lp (0, T ; B), Ann. Mat. Pura Appl., 146 (1987), pp. 65–96.

us to use the restriction of the arguments which we have obtained), one must redefine S. E.g., suppose the rectangle D is the compact subset of IR2 to which we have restricted values of [u, ut ]. We redefine n(y, z) when |z| is too large for D so as to be smooth (still), uniformly µ; then further redefine n (now for y outside the Lipschitzian in z, and still satisfying nz relevant range) to coincide with the values for the nearest admissible y-value. There is then no obstruction to the global (re)definition of S and we observe that the results must coincide with the original results for all relevant inputs so the redefinition is nugatory.

>

284


Mathematical Modeling and Analysis for Robotic Control

S. Tsui, Oakland University, Rochester, MI. Abstract We present the results of investigation of the torsional elastic robot beams. Next, we study the geometry of the joint space of a multi-joint robot. This opens doors to a new horizon of future research for torsional elastic multi-joint robots.

1

Introduction

With the rapid development of robotics in engineering, the coupled bending and torsional vibrations of elastomers appear frequently in application. Therefore, in this article, we summarize the recent results of the research on two topics: 1. the design of control for a loong and thin flexible robot arm (see Sections 2 and 3); 2. the analysis of joint space of multi-joint robots (see Section 4). In Section 5, we propose to study mathematical modeling and analysis of multi-joint robots with flexible arms. We first describe the flexible robot system as an evolution equation in an appropriate Hilbert space, and then apply functional analysis, spectral theory of linear operators and semigroup theory of linear operators to investigate stability. Then, we design a controller so that the considered system is exponentially stable under this control, and the tip of the arm of the robot can reach any designated point. Related works on the control of beams can be found in [[2], [3], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]]. We also present the results of the study of the geometry of the joint space of multi-joint robots. Selected references for this topics are from [[1], [4], [5], [6]].

2

The model of a beam with a tip body

Consider a long and thin flexible beam which is rotated by a motor in a horizontal plane. The beam is clamped on a vertical shaft of the motor at one end and has a tip body rigidly attached at the free end as shown in Fig. 1. The beam is of length l and with a uniform mass density ρ per unit length, uniform flexural rigidity EI, and uniform torsional rigidity GJ. Let X0 , Y0 , Z0 be the inertial Cartesian coordinate axes, where X0 , Y0 axes span a horizontal 285

286

Tsui

plane, and Z0 axis is the axle of rotation of the motor. Let X1 , Y1 , Z1 with Z1 = Z0 denote coordinate axes rotating with the motor and θ(t) be the angle of rotation of the motor. Let Q be the mass center of the rigid tip body, and P be the intersection of the beam tip’s tangent with a perpendicular plan passing through the Q. Let C denote the distance between the beam’s tip point and P , and C is assumed to be small. It is also assumed that P and Q never coincide and lie on the same vertical line in the equilibrium state. Let e be the distance between P and Q. We take another coordinate axes, X2 , Y2 , Z2 attached to the tip body, where X2 is the beam’s tip tangent and is obtained by rotation X1 axis by θ1 due to the bending of the beam. During the motion the tip body oscillates about a shear-center axis P X2 like a pendulum. Let Φ be the angle of rotation of the tip body about P X2 . The axes Y2 , Z2 also oscillates together with the tip body. Since the tip body is a rigid body, it is characterized by mass m, and two moments of inertia J0 and JE , where J0 is 0 with respect to the line passing through Q and parallel to the axis P Z2 and JE is with respect to the line passing through Q and parallel to the axis P X2 . Now let y(t, x) and φ(t, x) be the transverse displacement of the beam in the rotating frame X1 , Y1 and the angle of twist of the beam, respectively, at position x, 0 < x < l, and at time t. For the transverse vibration we use the Euler-Bernoulli model with internal viscous damping of the Voigt type [14] ( 2 ∂ y(t,x) ∂ 5 y(t,x) EI ∂ 4 y(t,x) ¨ + 2δ EI = −xθ(t) 2 4 + ρ ρ ∂t ∂t∂x ∂x4 (2.1) 0 y(t, 0) = y (t, 0) = 0, where δ > 0 is a small damping constant of the beam material. The initial conditions are due to the fact that the beam is clamped at x = 0. We assume that the beam material is isotropic and the internal damping constant for the torsional vibration is equal to that of the transverse vibration. Therefore the torsional vibration is governed by ( 2 ∂ φ(t,x) GJ ∂ 3 φ(t,x) GJ ∂ 2 φ(t,x) − 2δ ρk =0 2 2 · ∂t∂x2 − ρk 2 · ∂t ∂x2 (2.2) φ(t, 0) = 0, where ρk2 is the polar momentum of inertia mass for per length of beam. Obviously φ(t, l) = Φ(t), yx (t, l) = θ1 (t). Neglecting some nonlinear small quantities, we obtain the total kinetic energy of end body by T =

1 ˙ l)]2 + 1 J0 [θ(t) ˙ + y˙ 0 (t, l)]2 JE [φ(t, 2 2 1 ˙ + y(t, + m[(l + c)θ(t) ˙ l)]2 , ˙ l) + cy˙ 0 (t, l) + eϕ(t, 2


Y1

Y2

Y0

C

Q

(a)

θ1

P

φ y L

X2

eΦ

X1

x

θ

X0

Z0

φ

(b)

e

Fig. 1. Bending and torsion of a flexible beam with a tip body.

287

288

Tsui

where “.” denotes the time derivative, and “0 ” denotes the spatial derivative. We choose y(t, l), y 0 (t, l), φ(t, l) as the generalized coordinates, and f1 , f2 and f3 as the corresponding generalized forces defined by f1 = EIy 000 (t, l) + 2δEI y˙ 000 (t, l) f2 = −EIy 00 (t, l) − 2δEI y˙ 00 (t, l) f3 = −GIφ0 (t, l) − 2δGI ϕ˙ 0 (t, l). ¿From the second class Lagrange’s equation we have d ∂T ∂T − = fi (i = 1, 2, 3), dt ∂ q˙i ∂qi where q1 = y(t, l), q2 = y 0 (t, l), q3 = φ(t, l). We can derive the following boundary equations of coupled bending and torsional vibrations of flexible beam as follows. y 0 (t, l) + eϕ(t, (2.3) m (l + c)θ(t) + y¨(t, l) + c¨ ¨ l) = EIy 000 (t, l) + 2δEI y˙ 000 (t, l) (2.4)

h h i i ¨ + y¨(t, l) + c¨ mc (l + c)θ(t) y 0 (t, l) + eϕ(t, ¨ l) + J0 θ¨0 (t) + y¨0 (t, l) = −EIy 00 (t, l) − 2δEI y˙ 00 (t, l)

(2.5)

h i ¨ + y¨(t, l) + c¨ me (l + c)θ(t) y 0 (t, l) + eϕ(t, ¨ l) + JE ϕ(t, ¨ l) = −GJφ0 (t, l) − 2δGJ ϕ˙ 0 (t, l).

The rigid turning angle θ(t) of the beam is described by: ¨ + µθ(t) ˙ = τc (t) + EIy 00 (t, 0) Jm θ(t) (2.6) (1) ˙ θ(0) = θ , θ(0) = θ (2) , where Jm is the inertia moment of the electrical motor, µ is the viscous-friction coefficient, EIy 00 (t, 0) is the bending moment of the flexible beam, and τc (t) is the torque of the motor. Unlike in equations (2.3), (2.4), and (2.5), there is no damping term for the angle of turning θ(t) at the shaft end in (2.6), for it is negligible in comparison with the damping at the end of the tip body. The motion differential equations of the robot system is described by (2.1)– (2.6). The turning angle θ(t, y(t)) depends on time variable, t, and the bending moment, y 00 (t, 0). We shall choose the space H = L2 (0, l) × L2 (0, l) × R3 as a state space, which is a Hilbert space equipped with the inner product defined as hu, viH = ρ

Z lh 0

5 i X u1 (x)v1 (x) + k2 u2 (x)v2 (x) dx + ui v¯i , i=3


289

where u = (u1 , u2 , . . . , u5 )T , v = (v1 , v2 , . . . , v5 )T , u, v ∈ H, and (. . . )T means the transpose of (. . . ). Let V be a subspace of H defined by V

= {u = (u1 , u2 , . . . , u5 )T | u1 (x) ∈ H 2 (0, l), u2 (x) ∈ H 1 (0, l), u3 = u1 (l),

u4 = u01 (l), u5 = u2 (l), u1 (0) = 0, u01 (0) = 0, u2 (0) = 0}, where H m (0, l) = {f ∈ L2 (0, l) = f 0 , f 00 , . . . , f (m) ∈ L2 (0, l)} is the m-th degree Sobolev space, m = 1, 2. We now define the inner product on V by hu, viV =

Z lh 0

u001 (x)v100 (x)

+

u02 (x)v20 (x)

i dx +

5 X

ui v¯i .

i=3

It is easy to see that V with the inner product h·, ·iV is a Hilbert space. Define an operator ∧ : H → H as follows.   1 0 |  | 0     0 1 |     ∧u =  (u ∈ H)  −− | −−  u,   |    0 | M  | where



 m mc me . M =  mc J0 + mc2 mce 2 me mce JE + me

It is obvious that ∧ and M are symmetric positive operators. Due to the positivity of the operator ∧, we can define another inner product as i Rlh hu, viH0 = h∧u, viH = ρ 0 u1 (x)v1 (x) + k2 u2 (x)v2 (x) dx +(u3 , u4 , u5 )M (¯ v3 , v¯4 , v¯5 )T . We denote the space (H, h·, ·iH0 ) by H0 . It is apparent that there are two constants c1 and c2 such that c1 kukH 6 kukH0 6 c2 kukH . Thus, H0 is also a Hilbert space. Furthermore, we define the operator B : D(B) → H by EI d4 GJ d2 d3 d d Bu = diag , − 2 2 , −EI 3 , EI , GJ u, u ∈ D(B). ρ dx4 ρk dx dx dx dx

290

Tsui

Here D(B) = {u = (u1 , u2 , . . . , u5 )T | u ∈ V, u001 ∈ H 2 (0, l), u02 ∈ H 1 (0, l)} is the domain of B. In the systems (2.1)–(2.6), the turn angle θ is related to time t and the bending vibration displacement y of the beam,i.e., θ = θ(t, y). If we introduce the notation Ω = −(x, 0, m(l + c), J0 + mc(l + c), me(l + c))T , and then the system (2.1)–(2.5) can be described as the following second order homogeneous evolution equation. ¨ y(t)). ∧¨ u(t) + 2δB u(t) ˙ + Bu(t) = Ωθ(t,

(2.7)

Let A = ∧−1 B, D(A) = D(B). Then (2.7) becomes ¨ y(t)) ∧−1 (Ω). u ¨(t) + 2δAu(t) ˙ + Au(t) = θ(t,

(2.8)

The corresponding second order homogeneous evolution equation is as follows (2.9)

u ¨(t) + 2δAu(t) ˙ + Au(t) = 0.

The above setup first appeared in [14]. *

Set u= (u(1) , u(2) )T ,

u(1) = u(t), u(2) =

du dt ,

*

A =

0 I , −A −2δA

* ¨ y(t)))T . Then (2.8) D(A) = D(A) × D(A), and F (t, u) = (0, ∧−1 (Ω)θ(t, becomes *

(2.10)

* d u (t) * * = A u (t)+ u (t, F ). dt

The corresponding homogeneous evolution equation is *

(2.11)

d u (t) * = A u (t). dt

The equations (2.10) and (2.11) first appeared in [9]. Now, we list four theorems, a lemma and a corollary whose proofs can be found in [9]. Theorem 2.1. The operator A is a densely defined, self-adjoint, positive definite operator on V . Theorem 2.2. The inverse of A exists and it is compact. Theorem 2.3. The spectrum σ(A) of A consists of only countable eigenvalues {λn } with finite multiplicity, so that 0 < λ1 < λ2 < · · · < λn < . . . and λn → ∞(n → ∞). Let orthogonal unital eigenvectors of A corresponding to the eigenvalue λn be φnj where (j = 1, 2, . . . , nk ; nk is finite) such that Aφnj = λn φnj , kφnj kH0 = 1. It is known that {φk1 , . . . , φnnj }∞ k=1 form an orthonormal basis 0 for H .

291


Next, we shall discuss the spectral properties of the main operator A in the 1 evolution equation (2.10). Let’s consider a dense subspace E = D(A 2 ) × H0 with a new inner product defined by * *

1

1

h u , v iE = hA 2 u(1) , A 2 v(1) iH0 + hu(2) , v(2) iH0 ,

(2.12) *

*

where u= (u(1) , u(2) )T , v = (v(1) , v(2) )T ∈ E Lemma 2.1. The space E with the inner product defined in (2.12) is a Hilbert space. It is easy to see that E has an orthonormal basis consisting of the following vectors ∞ φknk 0 φk1 0 ,..., . , , φknk 0 φk1 0 k=1 Theorem 2.4. Denote the spectrum of A by σ(A), the point spectrum of A by σp (A), the resolvent of A by ρ(A), then we have the following results: 1 1. σ(A) = σp (A) ∪ {− 2δ },

ξk = −δλk +

σp (A) = {ξk , ηk }∞ k=1 ,where

p

(δλk )2 − λk , λ pk = −δλk − (δλk )2 − λk 1 1 √ = → , −2δ −δ − δ2

ηk = −δλk −

p

(δλk )2 − λk

as k → ∞

and the eigenvectors of A corresponding to ξk and ηk are, respectively, as follows *

φ kj = p

1 λk + | ξk |2

*

φkj ξk φkj

*

,

ψ kj = p

1 λk + | ηk |2

φkj ηk φkj

*

with k φ kj kE = | u ψkj kE = 1. 2. if µ ∈ ρ(A), then (µI − A)−1 =

(µ2 + 2δµA + A)−1 µ2 + 2δµA + A)−1 (µ + 2δA) 2 −1 2 −I + (µ + 2δµA + A) (µ + 2δµA) µ(µ2 + 2δµA + A)−1

Corollary 2.1. The operator A is a closed linear operator.

.

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Stability and Control of the System

In this section we show that the system in the previous section is asymptotically stable, and we design a control for the system. Again, the proofs for the theorems in this section can be found in [9]. First, we show that the real parts of the eigenvalues of A are bounded above, and there exists a constant ω1 > 0 such that sup{Reµn : µn ∈ σp (A)} = −ω1

(ω1 > 0).

In fact, it is easy to see from Theorem 2.5 that Reξk < 0, Reηk < 0. Since 1 1 limk→∞ ξk = − 2δ , limk→∞ ηk = −∞, it follows that limk→∞ Reξk = − 2δ and limk→∞ Reηk = −∞. Hence we have the following theorem. Theorem 3.1. The operator A in (2.10) or (2.11) is the infinitesimal generator of a C0 -semigroup T (t) on Hilbert Space H0 ⊕ H0 , and there are constants M > 0 and ω > 0 such that kT (t)k 6 M e−ωT (t > 0). Theorem 3.2. The first order homogeneous evolution equation (2.11) has * a unique solution u (t). Theorem 3.3. The solution u(t) of the second order evolution equation (2.9) is asymptotically stable. Theorem 3.4. Suppose for every T > 0, θ¨ : [0, T ] × L2 (0, l) → L2 (0, l) is Lipschitz continuous (with constant N ) in y on L2 (0, l), then nonlinear * evolution equation (2.10) has a unique weak solution u∈ C([0, T ]; H). Theorem 3.5. Let T > 0, θ¨ : [0, T ] × L2 (0, l) → L2 (0, l) be continuously * T ∈ D(A), and nonlinear evolution differentiable, then u 0 = (u(0), u(0)) ˙ equation (2.10) has a unique strong solution. In order to investigate the properties of the solution to (2.10), we denote C([0, +∞)) = {f : f is continuous on [0, ∞) and kf k∞ = supt>0 |f (t)| < +∞}. It is clear that the space C([0, +∞)) with norm k · k∞ is a Banach space. We define an operator on C([0, +∞)) by Z t Kg(t) = e−ω(t−s) g(s) ds, g ∈ C([0, +∞)), 0

where ω can be found in Theorem 3.1. Lemma 3.1. The operator K is a linear bounded operator on C([0, +∞)) and kKk∞ 6 1/ω. Theorem 3.6. Suppose θ¨ : [0, T ] × L2 (0, l) → L2 (0, l) is uniformly Lipshitz continuous in y on L2 (0, l) for any T > 0 with a Lipschitz constant N < * √ cρ ω/a0 M k ∧−1 k. Then the solution u (t) to the nonlinear evolution (2.10) decays exponentially so that the solution u(t) to original system (2.1) − −(2.5) is asymptotically stable in exponential form. Theorem 3.7. If we design the following controller τc (t) = −EIy 00 (0, t) − η(θ(t) − θ0 ),


293

where θ0 ∈ [0, 2π], 0 6 η 6 µ2 /4Jm . µ and Jm can be found in (2.6), for the system (2.1)–(2.6), then the bending vibration y(t, x) and torsional vibration φ(t, x) of the robot arm can be suppressed to be exponentially stable, and the elastic arm of robot can be arrived at any designated position, that is, lim θ(t) = θ0 .

t→∞

4

Joint spaces of robots with joints

In the case of robots with joints the dynamics on the robot is known to be derived through one of the following ways; (1) Lagrangian formulation, (2) Newton-Euler formulation, (3) Kane’s partial velocity, and (4) compact dynamic formulation. In the following figure, we demonstrate how to derive an n-dimensional manifold from an n-joint robot. In order to position the endeffector of a robot arm, one needs to specify a position and an orientation for the end-effector, each of which has three degrees of freedom. In general, a complete robot arm has six joints for six degree of freedoms. A joint can be represented as a rotational angle or a length of a segment. Therefore, the motion of the robot arm depends on how one manipulates these six joints, each joint represented by a variable xi , for i = 1, 2, ..., 6. The set of all (x1 , x2 , . . . , x6 ) form a manifold in a Euclidean space of higher dimension. This is called the joint space of the robot. This invisible joint space dictates the dynamics as well as the kinematics of the robot arm. It is the curvature tensor of this joint space that determines the dynamics of a robot arm. On this joint space we define a metric by the Hamiltonian of the system through the Newton-Euler formulation, and calculate the curvature. In case of n = 2, we can determine the Gaussian curvature, and in higher dimensional cases, we find sectional curvatures. With the aid of metric and curvature, one can find the shortest path between two fixed points on the joint space, which is called the geodesic. The geodesics on a joint space are the paths of minimum energy between two positions, and hence the best path for the robot arm to move. Therefore the most efficient robot design should always try to move the arm on a geodesic. The apparent difficulty in this design is that the visible arm with its end-effector give a false image of the position for the robot arm in the invisible joint space. In the case of robots with two planar joints which is described in the following figure. In this case, we can calculate the first and the second fundamental forms of the joint space. By the Rigidity Theorem for hypersurfaces the joint space is locally (metrically) diffeomorphic to a 2-torus. [5], [6]. Based on the classical Chasler’s Theorem, every rigid body motion can be decomposed into two decoupled portion: one is the translation of its mass

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Fig. 2. Six-joint robot

center and the other one is the rotation with respect to the mass center. In other words, this 6-dimensional configuration manifold M 6 can be diffeomorphic to the compact topological space T (3) × SO(3), where T (3) is a compact subspace in Euclidean 3-space 0,

where A : D(A)(⊂ X) → X generates a C0 semigroup of linear operators eAt , t > 0, on X. X is the state-valued space of x0 and x(t). Let Y be another real Banach space and U ⊂ Y be an arbitrary, given subset called the control-valued set. Let T > 0 be a fixed constant. We define (1.2)

U = {u : [0, T ] → Y | u(t) ∈ U and u is strongly measurable}.

We shall make some assumptions on the nonlinear term f and more assumptions on A later. Since (1.1) is usually formulated from many initial-boundary value 299

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problems of PDEs, it can be called a distributed control system. As usual, we shall adopt the mild solution of (1.1), i.e., Zt At

eA(t−s) f (s, x(s), u(s)) ds,

x(t) = x (t, x0 , u) = e x0 +

(1.3)

0

as the state function or called trajectory, where the integral is Bochner integral in X. For a nonlinear system, there is no guarantee in general that for each u ∈ U, the mild solution exists globally on [0, T ]. So we have to define a class of control functions as follows, (1.4)

Uad = {u(·) ∈ U | x (t, x0 , u) exists on [0, T ], for any x0 ∈ W } .

Any u(·) ∈ Uad is an admissible control. Set a cost functional as a criterion, ZT J (x0 , u) =

(1.5)

Q(t, x(t), u(t)) dt, 0

where Q(t, x, u) is a nonlinear functional defined on [0, T ] × X × Y and to be specified in detail later. An optimal control problem is proposed: For any given x0 ∈ W , find a control u∗ (·) ∈ Uad such that J (x0 , u∗ ) = min J (x0 , u) . u∈U ad

(1.6)

We shall call u∗ an optimal control and x∗ (·) = x (·, x0 , u∗ ) an optimal trajectory. We may refer to this optimal control problem described by (1.1), (1.5) and (1.6) briefly as the (OCP). Now we make some specific assumptions on A, f and Q that will enable us to analyze the described optimal control problem concretely. Hypothesis I. Let −A : D(A) → X be a positive sectorial operator. Then eAt is an analytic semigroup. Let W 2α = D ((−A)α ) be the family of interpolation Banach spaces with the norm kxk2α = |(−A)α x|, for α > 0. The basic properties of eAt under Hypothesis I are listed below, cf. [15, 16]: (a) For any α > 0 and t > 0, the operator eAt maps X into W 2α and it is strongly continuous in t > 0. (b) For any α > 0, there are constants a > 0 and Mα > 0 such that (1.7)

At

e

L(X,W 2α )

= (−A)α eAt L(X) 6 Mα e−at t−α , for t > 0.

Optimal Control and Synthesis of Nonlinear Systems

301

(c) For 0 < α 6 1, there is a constant Cα > 0 such that At e w − w 6 Cα tα |(−A)α w| , for t > 0, w ∈ W 2α . (1.8) We shall fix an α > 0 in Hypothesis III. For that fixed α, let W = W 2α . We now introduce some notation. Let E be any real Banach space. We shall use CLip (E) = CLip ([0, T ] × W × Y ; E) to denote the collection of all strongly continuous functions g : [0, T ] × W × Y → E that are uniformly bounded and uniformly Lipschitz continuous with respect to x ∈ W for any bounded set B ⊂ W . That means there exist constants K0 (B) and K1 (B) such that kg(t, x, u)kE 6 K0 (B), for any t ∈ [0, T ] and (x, u) ∈ B × U,

(1.9) and (1.10)

kg (t, x1 , u) − g (t, x2 , u)kE 6 K1 (B) kx1 − x2 kW , for any t ∈ [0, T ], and (xi , u) ∈ B × U, i = 1, 2.

We then define C 1 (E) = C 1 ([0, T ] × W × Y ; E) to be the collection of all functions g : [0, T ] × W × Y → E that are continuously Fréchet differentiable in x ∈ W . The first-order Fréchet derivative of g with respect to x will be denoted by gx or ∂x g which is valued in L(W, E). Let 1 CLip (E) = CLip (E) ∩ C 1 (E). 1 (X) and Q ∈ C 1 (R). Moreover, Hypothesis II. Assume that f ∈ CLip Lip the following assumptions are satisfied:

(i) For any x(·) ∈ C([0, T ], W ) and any u ∈ U , one has (1.11)

|f (t, x(t), u(t))| ∈ Lp [0, T ],

and (1.12)

|Q(t, x(t), u(t))| ∈ L1 [0, T ].

(ii) For any bounded set B ⊂ W and each u(·) ∈ U , there exist scalar functions β(·) ∈ Lp [0, T ] and γ(·) ∈ L1 [0, T ], both may depend on B and u, such that (1.13)

kfx (t, x(t), u(t))kL(W,X) 6 β(t),

and (1.14)

kQx (t, x(t), u(t))kL(W,R)=W 0 6 γ(t).

for all t ∈ [0, T ], provided that x(t) ∈ B, 0 6 t 6 T .

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Hypothesis III. Let the exponent α in Hypothesis I and the exponent p in Hypothesis II be fixed and satisfy the following condition, (1.15)

p−1 , p or p = 1 and α = 0.

either p > 1 and 0 6 α
0. Let us consider a linearized evolutionary equation, (1.17)

dw ˆ(t), u ˆ(t)) w, = Aw + fx (t, x dt

where u ˆ ∈ Uad is given and x ˆ(t) = x (t, x0 , u ˆ). Let F (t) = fx (t, x ˆ(t), uˆ(t)), then by Hypothesis II one has F ∈ Lp ([0, T ], L(W, X)). By the perturbation theory of semigroups, cf. [16], there exists a family of bounded linear operators G(t, s) defined on ΩT = {(t, s) : 0 6 s 6 t 6 T } and valued in L(X) ∩ L(W ), which is strongly continuous in (t, s), uniformly bounded in L(X) and L(W ), and satisfies the following: 0 6 t 6 T;

G(t, t) = IX , G(t, s) = G(t, τ )G(τ, s),

0 6 s 6 τ 6 t 6 T;

and for any w0 ∈ X and s ∈ [0, T ], Zt A(t−s)

(1.18)

G(t, s)w0 = e

eA(t−σ)

w0 + s

· fx (σ, xˆ(σ), uˆ(σ))G(σ, s)w0 dσ,

t ∈ [s, T ].


303

The rest of the paper is outlined as follows. The first part is devoted to proving the maximum principle via a straightforward approach featuring the oscillating variations of control. This part includes Sections 2, 3 and 4. Then the second part is a contribution of a general synthesis result of optimal control processes, based on the proof of Lipschitz continuity of the value functions and utilizing a technique of differential inclusions. That part consists of Sections 5 and 6.

2

Oscillating Variations of Control

There have been many results on the maximum principle for infinite dimensional optimal controls in various cases and shown by various methods. Notably, [3, 14] and [17] make use of vector measures and the Eidelheit separation theorem; [6] and [9] use Ekeland’s variational principle and directional derivatives; [7] takes the approach of nonlinear programming by using the Kuhn-Tucker technique; [8] exploits the relaxation of control and trajectories; [11, 12, 13] cover the area of quadratic optimal control; [19] and [20] directly treat spike perturbations and provide results on nonlinear and nonquadratic applications. Here we shall present a forthright proof of the maximum principle, a necessary condition satisfied by optimal control if it exists, for the described (OCP) by the approach of oscillating variations of control. Since this approach can reach the goal based on essentially two tool lemmas, the perturbation and robustness lemma for Volterra integral equations and the generalized RiemannLebesgue lemma, it seems mathematically simpler in comparison with some other approaches. In the first part, that consists of Sections 2 through 4, all the lemmas and theorems are valid under Hypotheses I, II and III. Sometimes these hypotheses will not be mentioned repeatedly. We start with a given admissible control u∗ (·) ∈ Uad , that may not be an optimal control at this moment. Let x∗ (t) = x (t, x0 , u∗ ), 0 6 t 6 T . For any u ∈ U and any δ ∈ [0, 1], we define a variation of control uδn (t) as follows, ( u(t), for t ∈ Enδ , uδn (t) = u∗ (t), for t ∈ [0, T ]\Enδ ,

(2.1)

where the sets Enδ are determined by (2.2)

Enδ

=

n−1 [ k=0

kT (k + δ)T , n n

,

n = 1, 2, . . . .

Note that the variation of control certainly depends on u ∈ U , which may not be in Uad . This type of variation of control is called an oscillating variation.

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In order to analyze the implications of the oscillating variations of control, let us define Γδn = Γδn (t, s, x) = eA(t−s) f s, x, uδn (s) , (2.3) where uδn is defined by (2.1). This Γδn is a mapping such that Γδn : ΩT × W → X and Γδn : ΩT \{t = s} × W → W. The two important properties of Γδn are stated in the following lemma. Lemma 2.1. The family of mappings Γδn for n = 1, 2, . . . and δ ∈ [0, 1] has the following properties: (P1) Γδn is a strongly measurable function and it satisfies the local Lipschitz condition in the following sense: for any x0 ∈ W , there exist a constant η > 0 and a nonnegative scalar function ρ(·) ∈ L1 [0, T ] such that

δ

(2.4)

Γn (t, s, x1 ) − Γδn (t, s, x2 ) 6 ρ(s) kx1 − x2 k , for (t, s) ∈ ΩT \{t = s}, for any x1 , x2 ∈ Nη (x0 ) = {x ∈ W : kx − x0 k 6 η} and uniformly for all δ ∈ [0, 1] and all n > 1. (P2) For any given x(·) ∈ C([0, T ], W ), one has Γδn (t, ·, x(·)) ∈ L1 ([0, t], W ) Rt and Γδn (t, s, x(s)) ds is strongly continuous in t ∈ [0, T ]. And 0

Zt (2.5)

Zt Γδn (t, s, x(s)) ds

lim

δ→0+ 0

Γ0n (t, s, x(s)) ds,

= 0

in which the convergence is uniform with respect to 0 6 t 6 T and all n > 1. Proof. By the hypothesis, obviously Γδn (t, s, x) is strongly measurable. For x1 , x2 in Nη (x0 ), we have

δ

δ Γ (t, s, x ) − Γ (t, s, x )

n 1 2 n

δ δ 6 eA(t−s) s, x , u (s) − f s, x , u (s) f 1 n 2 n L(X,W )

6

Mα K1 (Nη (x0 )) kx1 − x2 k , (t − s)α

1 and (1.10) here for this f . By taking ρ(s) = M K (N ) (t− because of f ∈ CLip α 1 η −α s) if 0 6 s < t and = 0 elsewhere, we see that (2.4) is valid and (P1) is satisfied. Note that the L1 norm of ρ is independent of t in [0, T ]. In order to verify (P2), we make the observations as follows:


305

i) Let 0 6 t 6 T be relatively fixed. By the strong continuity of f in u, we have Γδn (t, s, x(s)) = eA(t−s) f s, x(s), uδn (s) → eA(t−s) f s, x(s), u0n (s) as δ → 0+ , for almost every s ∈ [0, t]. ii) Define a family of periodic functions θnδ (·) by h  (k+δ)T 1 − δ, if s ∈ kT , , n n h θnδ (s) = (k+1)T −δ, (2.6) if s ∈ (k+δ)T ∪ {T }, , n n for k = 0, 1, . . . , n − 1. It is easy to see that the following identity holds, Γδn (t, s, x) = eA(t−s) f s, x, uδn (s) = eA(t−s) [δf (s, x, u(s)) + (1 − δ)f (s, x, u∗ (s)) i +θnδ (s) (f (s, x, u(s)) − f (s, x, u∗ (s))) ,

(2.7)

s ∈ [0, t], x ∈ W. By Hypothesis I and (1.11), it follows from (2.7) that (2.8)

δ

Γn (t, s, x(s)) 6 eA(t−s)

L(X,W )

6

(|f (s, x(s), u(s))| + |f (s, x(s), u∗ (s))|)

Mα (|f (s, x(s), u(s))| + |f (s, x(s), u∗ (s))|) (t − s)α

, b (s; x, u, u∗ ) , for 0 6 s < t. Extend b(·) by letting b (s; x, u, u∗ ) = 0 for t 6 s 6 T . Then b (·; x, u, u∗ ) ∈ L1 [0, T ] and kb(·)kL1 [0,T ] is independent of δ ∈ [0, 1], n > 1 and t ∈ [0, T ]. The facts shown in i) and ii) above allow us to apply the Lebesgue Dominated Convergence Theorem to get the uniform convergence relation (2.5) with respect to 0 6 t 6 T and n > 1. The rest is easy to check. Therefore, (P2) is satisfied. The proof is completed. Lemma 2.2. Let uδn and Γδn be defined as above. Then there exists a positive δ0 , 0 < δ0 6 1, such that for every δ ∈ [0, δ0 ], one has uδn (·) ∈ Uad ,

(2.9)

and the corresponding mild solution xδn (t) = x t, x0 , uδn exists on [0, T ] and satisfies

sup xδn (·) − x∗ (·) → 0, as δ → 0+ . (2.10)

>

n 1

C

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Proof. We shall use the fixed point theorem to show that for δ > 0 sufficiently small, there exists a unique global solution x t, x0 , uδn of Eq. (1.3) on [0, T ]. By (P1) and the compactness argument, it follows that for any given x ˆ(·) ∈ C([0, T ], W ) and its closed neighborhood Nλ (ˆ x(·)) = {g ∈ C([0, T ], W ) | kg − x ˆkC 6 λ} ,

(2.11)

λ > 0,

there exists a nonnegative scalar function µ(·) ∈ L1 [0, T ] depending on x ˆ and λ only, such that

δ

Γn (t, s, x1 (s)) − Γδn (t, s, x2 (s)) (2.12) 6 µ(s) kx1 (s) − x2 (s)k , a.e. on [0, t], for any x1 (·), x2 (·) ∈ Nλ (ˆ x(·)) and all δ ∈ [0, 1] and n > 1. Now let x∗ (·) = x (·, x0 , u∗ ) take the place of x ˆ(·) in (2.11) and (2.12). Then µ ∈ L1 [0, T ] depending on x∗ and λ is determined. Define mappings Tnδ : Nλ (x∗ (·)) → C([0, T ], W ) by Zt δ At Tn x (t) = e x0 + Γδn (t, s, x(s)) ds,

(2.13)

t ∈ [0, T ],

0

where Γδn is defined by (2.3), for δ ∈ [0, 1] and integers n > 1. By definition, we have x∗ (·) = Tn0 x∗ , for n = 1, 2, . . . .

(2.14)

For any x(·) ∈ Nλ (x∗ (·)), by (2.12) we have

δ

Tn x (t) − Tnδ x∗ (t) Zt 6

µ(s) kx(s) − x∗ (s)k ds

0

Zt =

2

µ(s)e

Rs

µ(τ )dτ −2

e

0

Rs

µ(τ )dτ

0

0

Zt (2.15)

6

2

µ(s)e

Rs

max

0

s∈[0,T ] 

0

=

 

µ(τ )dτ

kx(s) − x∗ (s)k ds

−2

e

Rs

µ(τ )dτ

0

s=t  Rs  −2 Rs µ(τ )dτ 1 2 0 µ(τ )dτ max e 0 e 2 s∈[0,T ]  s=0

kx(s) − x∗ (s)k

 

   kx(s) − x∗ (s)k 

ds

Optimal Control and Synthesis of Nonlinear Systems   Rt Rs   2 µ(τ )dτ −2 µ(τ )dτ 1 0 6 (2.16) max e 0 kx(s) − x∗ (s)k . e  2 s∈[0,T ] 

307

We can define k · kC,µ by (2.17)

kgkC,µ = max

 

t∈[0,T ] 

−2

e

Rt

µ(τ )dτ

0

  kg(t)k , for g ∈ C([0, T ], W ). 

Note that k · kC,µ can be called a Bielecki norm and it is equivalent to k · kC on the space C([0, T ], W ) because −2

0 < c0 (T ) = e

RT 0

µ(τ )dτ

−2

6e

Rt 0

µ(τ )dτ

6 1.

Let Nλ,µ (x∗ (·)) be the closed λ-neighborhood of x∗ (·) in C([0, T ], W ) with respect to the new norm. Then Nλ,µ (x∗ (·)) ⊂ Nλ (x∗ (·)) for any λ > 0. From (2.15), we obtain

1 λ

δ

6 kx − x∗ kC,µ 6 , (2.18)

Tn x − Tnδ x∗ 2 2 C,µ for any x(·) ∈ Nλ,µ (x∗ (·)). By the property (P2) we have shown in Lemma 2.1, there exists a constant δ0 , 0 < δ0 6 1, such that

λ

δ ∗

= Tnδ x∗ − Tn0 x∗ < , (2.19)

Tn x − x∗ 2 C,µ C,µ for all δ ∈ [0, δ0 ] and n > 1. Combining (2.18) and (2.19), we can assert that (2.20)

x(·) ∈ Nλ,µ (x∗ (·)) implies Tnδ x ∈ Nλ,µ (x∗ (·)) .

for any δ ∈ [0, δ0 ] and any integer n > 1. Therefore, Tnδ maps Nλ,µ (x∗ (·)) into itself. Similar to (2.15), one can prove that for any x1 (·), x2 (·) ∈ Nλ,µ (x∗ (·)), the following inequality holds,

1

δ

(2.21) 6 kx1 − x2 kC,µ .

Tn x1 − Tnδ x2 2 C,µ Hence the mapping Tnδ : Nλ,µ (x∗ (·)) → Nλ,µ (x∗ (·)) is a contraction. According to the fixed point theorem, there exists a unique fixed point of Tnδ in Nλ,µ (x∗ (·)). By the definitions (2.13) and (2.3), this fixed point must be xδn (t) = x t, x0 , uδn , t ∈ [0, T ]. (2.22)

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Therefore, we have proved that for δ ∈ [0, δ0 ], (2.9) is true. Finally, let us prove (2.10) by estimating

δ

(2.23) 6 Tnδ xδn − Tnδ x∗ + Tnδ x∗ − Tn0 x∗

xn − x∗ C,µ

C,µ

C

By (P2) and (2.5), we have

δ ∗ 0 ∗ x − T x

Tn

→ 0, as δ → 0+ , n C

where the convergence is uniform with respect to n > 1. Thus, for any ε > 0, there is a δ1 = δ1 (ε) ∈ (0, 1] such that for any 0 6 δ 6 δ1 , one has

δ ∗

δ ∗ 0 ∗ ∗ (2.24)

Tn x − Tn x = Tn x − x < ε/2. C

C

Then, using (2.21), we obtain (2.25)

δ δ

Tn xn − Tnδ x∗

C,µ

6

1

δ

xn − x∗ , 2 C,µ

for any δ ∈ [0, min {δ0 , δ1 }], and uniformly with respect to n > 1. Substituting (2.24) and (2.25) into (2.23), we find that

δ

xn − x∗

C,µ

<

1 ε

δ + ,

xn − x∗ 2 2 C,µ

which implies that for all n > 1,

δ

(2.26) < ε, for 0 6 δ 6 min {δ0 , δ1 } .

xn − x∗ C,µ

Since k · kC,µ and k · kC are equivalent, (2.26) leads to the conclusion (2.10). The proof is completed. The argument we made in Lemma 2.1 and Lemma 2.2 can be generalized to prove a result on perturbation and robustness of nonlinear Volterra integral equations (NVIE). We state this result as the following lemma. Its proof is omitted to avoid any substantial duplication. Lemma 2.3 (Perturbation and Robustness of NVIE). Consider a nonlinear Volterra integral equation in a Banach space E, with parameters δ ∈ [0, δ1 ] and n = 1, 2, . . . , Zt (2.27)

δ

Hnδ (t, s, x(s)) ds,

x(t) = h (t) + 0

Suppose the following conditions are satisfied:

t ∈ [0, T ].


309

(P0) For each δ ∈ [0, δ1 ], hδ (·) ∈ C([0, T ], E), and hδ − h0 C([0,T ],E) → 0, as δ → 0+ . (P1) Hnδ (t, s, x) : ΩT \{t = s} × E → E satisfies the corresponding property (P1) in Lemma 2.1 with Γδn , W replaced by Hnδ , E. (P2) Hnδ (t, s, x) also satisfies the corresponding property (P2) in Lemma 2.1 with Γδn , W replaced by Hnδ , E. (P3) For δ = 0, there exists a common solution x∗ (·) ∈ C([0, T ], E) of Eq. (2.27) for all n > 1. Then there exists a positive δ0 , 0 < δ0 6 δ1 , such that for any δ ∈ [0, δ0 ] and for any n > 1, there exists a unique solution xδn ∈ C([0, T ], E) of Eq. (2.27) and

sup xδn (·) − x∗ (2.28) → 0, as δ → 0+ .

>

C([0,T ],E)

n 1

This lemma will be used in the next section.

3

First Order Variations of Trajectory

Consider the mild solution of an initial value problem of the following variational equation (3.1)

dw = Aw + fx (t, x∗ (t), u∗ (t)) w, dt w(τ ) = w0 ∈ W,

t ∈ [τ, T ],

where u∗ (·) ∈ Uad is given and x∗ (·) = x (·, x0 , u∗ ). Since Eq. (3.1) is exactly Eq. (1.17) with {ˆ x, uˆ} replaced by {x∗ , u∗ }, we shall use the same notation G(t, s) associated with (1.17) to denote the corresponding evolution operators associated with (3.1). Specifically, by the uniqueness of mild solution w(·) of the initial value problem, it can be expressed by w(t) = G(t, τ )w0 ,

(3.2)

t ∈ [τ, T ],

where G(t, s) : ΩT → L(X) ∩ L(W ) is strongly continuous, uniformly bounded, and satisfies the following equations in the strong sense, for (t, s) ∈ ΩT , Zt (3.3)

A(t−s)

G(t, s) = e

+

eA(t−σ) fx (σ, x∗ (σ), u∗ (σ)) G(σ, s) dσ,

s

and Zt (3.4)

A(t−s)

G(t, s) = e

+ s

G(t, σ)fx (σ, x∗ (σ), u∗ (σ)) eA(σ−s) dσ.

310

You

Notice that (3.3) and (3.4) are valid both in X and W . Consequently, we get the commutative formula of the variational perturbation, i.e., Zt (3.5)

eA(t−σ) fx (σ, x∗ (σ), u∗ (σ)) G(σ, s) ds

s

Zt =

G(t, σ)fx (σ, x∗ (σ), u∗ (σ)) eA(σ−s) dσ,

(t, s) ∈ ΩT .

s

Let us define a function q (t, u, u∗ ) by q(t) = q (t, u, u∗ ) Zt = G(t, τ ) [f (τ, x∗ (τ ), u(τ )) − f (τ, x∗ (τ ), u∗ (τ ))] dτ,

(3.6)

0

where u(·) ∈ U and u∗ (·) ∈ Uad are arbitrarily given. Here, x∗ (t) = x (t, x0 , u∗ ), but it is not required in the proof of the next lemma. Lemma 3.1. For any fixed u∗ ∈ Uad and any u ∈ U , the function q(t) is a unique, strongly continuous solution of the following linear Volterra integral equation, Zt y(t) =

eA(t−s) fx (s, x∗ (s), u∗ (s)) y(s) ds

0

(3.7)

Zt +

eA(t−s) [f (s, x∗ (s), u(s)) − f (s, x∗ (s), u∗ (s))] ds,

t ∈ [0, T ].

0

Proof. Since the uniqueness is relatively easy to show, it suffices to prove that q(·) defined by (3.6) satisfies Eq. (3.7). The verification is as follows, Zt

eA(t−s) fx (s, x∗ (s), u∗ (s)) q(s) ds

0

ZtZs =

eA(t−s) fx (s, x∗ (s), u∗ (s)) G(s, τ )

0 0

· [f (τ, x∗ (τ ), u(τ )) − f (τ, x∗ (τ ), u∗ (τ ))] dτ ds

(by the Fubini-Tonelli theorem and Hypotheses to interchange the order of integration) ZtZt = 0 τ

eA(t−s) fx (s, x∗ (s), u∗ (s)) G(s, τ ) ds


311

· [f (τ, x∗ (τ ), u(τ )) − f (τ, x∗ (τ ), u∗ (τ ))] dτ Zt h =

i G(t, τ ) − eA(t−τ ) [f (τ, x∗ (τ ), u(τ )) − f (τ, x∗ (τ ), u∗ (τ ))] dτ

0

Zt = q(t) −

eA(t−s) [f (s, x∗ (s), u(s)) − f (s, x∗ (s), u∗ (s))] ds,

0

where (3.3) is used. The proof is completed. Before investigating the first order variations of trajectory, we now present another key lemma in this oscillating variation approach, which can be called a generalized Riemann-Lebesgue lemma. Let θnδ be given by (2.6). Let uδn and xδn be the same as defined in (2.1) and Lemma 2.2. Lemma 3.2 (Generalized Riemann-Lebesgue Lemma). Let h(·) ∈ L1 ([0, T ], E) with E being any real Banach space and let θnδ be defined by (2.6). Then, ZT h(t)θnδ (t) dt = 0,

lim

(3.8)

n→∞ 0

for each δ ∈ [0, 1]. Proof. By definition, we have t Z θnδ (s) ds 6 δ(1 − δ)T 6 T → 0, as n → ∞, n n

(3.9)

0

uniformly with respect to t ∈ [0, T ] and δ ∈ [0, 1]. Then for any subinterval (a, b) ⊂ [0, T ] and any h0 ∈ E, let h(t) = χ(a,b) (t)h0 where χ∆ is the characteristic function of subset ∆. Then by (3.9), we have ZT

 b  Z χ(a,b) (t)h0 θnδ (t) dt =  θnδ (t) dt h0 → 0, as n → ∞. a

0

It follows that for any finitely many disjoint open intervals (ai , bi ) ⊂ [0, T ], i = 1, . . . , m, and for any hi ∈ E, i = 1, . . . , m, we have ZT X m 0

i=1

χ(ai ,bi ) hi θnδ dt → 0, as n → ∞.

312

You

Then for any countably valued function h ∈ L1 ([0, T ]; E), h(t) =

∞ P

χ∆i (t)hi

i=1

with hi ∈ E and ∆i ’s being mutually disjoint measurable sets in [0, T ], we have ZT h(t)θnδ (t) dt → 0, as n → ∞.

(3.10) 0

Finally, for any general h ∈ L1 ([0, T ], E), by definition of the Bochner integral, there is a sequence of countably valued functions hj ∈ L1 ([0, T ], E) such that ZT kh(t) − hj (t)kE dt → 0, as j → ∞.

(3.11) 0

We can write

T

T

Z

Z

ZT

h(t)θ δ (t) dt 6 kh(t) − hj (t)k dt + hj (t)θ δ (t) dt . n n E

0

E

0

0

E

By (3.11) and (3.10), for any ε > 0, there is an integer j0 such that RT kh(t) − hj0 (t)kE dt < ε/2 and then for that j0 there is a positive integer 0

RT

n0 = n0 (j0 , ε) such that for n > n0 we have hj0 (t)θnδ (t) dt < ε/2. It

0

E follows that

T

Z

h(t)θnδ (t) dt < ε, for any n > n0 .

0

E

Thus (3.8) is proved. The proof is completed. Let δ0 > 0 be the constant stated in Lemma 2.2 and fixed. Lemma 3.3. For every δ ∈ [0, δ0 ], there is a positive integer n(δ) such that i 1h δ lim (3.12) xn(δ) (t) − x∗ (t) = q(t), δ→0+ δ where the convergence is uniform in t ∈ [0, T ] and q(t) is given by (3.6). Moreover, (3.13)

1 lim + δ→0 δ

ZT

δ [f (t, x∗ (t), u(t)) − f (t, x∗ (t), u∗ (t))] θn(δ) (t) dt = 0,

0

where u ∈ U and u∗ ∈ Uad are the same as in Lemma 2.1, and x∗ (t) = x (t, x0 , u∗ ).


313

Proof. By Hypothesis II, we know that f (t, x∗ (t), u(t)) − f (t, x∗ (t), u∗ (t)) ∈ L1 ([0, T ], X). According to Lemma 3.2, it follows that for each δ ∈ [0, 1], ZT (3.14)

lim

n→∞

[f (t, x∗ (t), u(t)) − f (t, x∗ (t), u∗ (t))] θnδ (t) dt = 0.

0

Hence for each 0 < δ 6 δ0 , there exists a positive integer n1 (δ) such that (3.15) T Z [f (t, x∗ (t), u(t)) − f (t, x∗ (t), u∗ (t))] tδn (t) dt < δ2

, for n > n1 (δ).

0

Now consider i 1h δ xn (t) − x∗ (t) δ Zt h i 1 = eA(t−s) f s, xδn (s), uδn (s) − f s, x∗ (s), uδn (s) ds δ 0

(3.16)

1 + δ

Zt

h i eA(t−s) f s, x∗ (s), uδn (s) − f (s, x∗ (s), u∗ (s)) ds,

0

where the two integral terms on the right side of (3.16) are denoted by I1 (n) and I2 (n), respectively. Then we have the expression

(3.17)

1 I2 (n) = δ

Zt

F (t, s) δ + θnδ (s) ds,

0

where (3.18)

F (t, s) = eA(t−s) [f (s, x∗ (s), u(s)) − f (s, x∗ (s), u∗ (s))] .

By Lemma 3.2, for every 0 < δ 6 δ0 , it holds that Zt (3.19)

F (t, s) θnδ (s) ds = 0

lim

n→∞ 0

314

You

uniformly in t ∈ [0, T ]. Hence for every such δ, there exists a positive integer n2 (δ) such that

(3.20)

t

Z

δ 2

sup

F (t, s) θn (s) ds < δ , for n > n2 (δ). t∈[0,T ]

0

Let n(δ) = max {n1 (δ), n2 (δ)} and denote uδn(δ) = uδ , xδn(δ) = xδ . Define (3.21)

∆x(t, δ) =

i 1h δ x (t) − x∗ (t) , for 0 < δ 6 δ0 , δ

and (3.22)

xδ (t, λ) = λxδ (t) + (1 − λ)x∗ (t), for 0 6 λ 6 1.

Note that Lemma 2.2 and (2.10) imply that

sup xδ (·, λ) − x∗ (·) → 0, as δ → 0+ . (3.23)

66

C

0 λ 1

Hence there is δ2 ∈ (0, δ0 ] such that xδ (·, λ) ∈ N1 (x∗ (·)) for 0 6 δ 6 δ2 . Therefore,

n o

sup xδ (s, λ) : 0 6 s 6 T, 0 6 λ 6 1, 0 6 δ 6 δ2 (3.24) 6 kx∗ (·)kC + 1 (a constant depending on x∗ only). On the other hand, we have Zt (3.25)

Z1 A(t−s)

I1 (n(δ)) =

e 0

fx s, xδ (s, λ), uδ (s) ∆x(s, δ) dλ ds.

0

Assembling together (3.16) and (3.17) with n = n(δ), (3.20) and (3.25), we get Zt ∆x(t, δ) = (3.26)

 eA(t−s)

0

Z1

 fx s, xδ (s, λ), uδ (s) dλ ∆x(s, δ) ds

0

Zt +

F (t, s) ds + R(δ), 0

where R(δ) is a term satisfying kR(δ)k < δ, so that R(δ) → 0 uniformly.


315

Here we have a Volterra integral equation with a parameter δ ∈ [0, δ2 ], with unknown y, that is Zt δ

H δ (t, s, y(s)) ds,

y(t) = h (t) +

(3.27)

t ∈ [0, T ],

0

where Zt δ

h (t) = 

F (t, s) ds + R(δ), 0

H δ (t, s, x) = eA(t−s)

Z1

fx

 s, xδ (s, λ), uδ (s) dλ x.

0

Indeed, Eq. (3.27) is a particular case of Eq. (2.27), where δ ∈ [0, δ1 ] is replaced by δ ∈ [0, δ2 ] and no parameter n. We can check that, in this case, all four conditions in Lemma 2.3 are satisfied. (P0) is satisfied because hδ − h0 C = kR(δ)kC 6 δ. (P1) can be verified as follows. Since H δ (t, s, x) is linear in x, and using an identity similar to (2.7) for fx s, xδ (s, λ), uδ (s) , one has

(3.28) H δ (t, s, x1 ) − H δ (t, s, x2 ) 6

6

Mα (t − s)α Mα (t − s)α

Z1

δ δ ∗ f s, x (s, λ), u(s) + f (s, x (s, λ), u (s)

x

x

dλ kx1 − x2 k 0

Z1

(β(s, u) + β (s, u∗ )) dλ kx1 − x2 k = ρ(s) kx1 − x2 k

0

where β(·, u) and β (·, u∗ ) come from (1.13) and depend on the bounded set B which is the closed ball in W of radius kx∗ kC + 1 from (3.24), and ( Mα (t − s)−α [β(s, u) + β (s, u∗ )] , for 0 6 s < t, ρ(s) = (3.29) 0, for t 6 s 6 T. (P2) can be verified as follows. First, by (3.23), (1.13) and β(·, u), β (·, u∗ ) as in (3.28), one can use the Lebesgue Dominated Convergence Theorem to get Z1 (3.30)

lim

δ→0+ 0

fx s, xδ (s, λ), uδ (s) dλ = fx (s, x∗ (s), u∗ (s)) ,

316

You

uniformly in s ∈ [0, T ]. Eq. (3.30) shows that for each s ∈ [0, T ], H δ (t, s, x(s)) → H 0 (t, s, x(s)), as δ → 0+ .

Besides, one has H δ (t, s, x(s) 6 ρ(s)kx(·)kC ∈ L1 [0, T ] and the L1 norm of ρ is independent of t ∈ [0, T ], where ρ(·) is the same function given in (3.29). Thus, one can use the Lebesgue Dominated Convergence Theorem again to obtain the uniform convergence relation (2.5) for H δ . (P3) is also satisfied. For δ = 0, Eq. (3.27) becomes exactly Eq. (3.7) and that equation has a strongly continuous solution q(t) given by (3.6), as shown by Lemma 3.1. Therefore, we can apply Lemma 2.3 to conclude that there is a δ3 ∈ (0, δ2 ] such that for any 0 < δ 6 δ3 , there exists a unique solution of Eq. (3.27), which is exactly ∆x(·, δ) ∈ C([0, T ], W ), such that

h

i

1 δ

∗ +

k∆x(·, δ) − q(·)kC = x (·) − x (·) − q(·) (3.32)

→ 0, as δ → 0 . δ (3.31)

C

Thus (3.12) is proved. Finally, (3.13) follows from (3.15). completed. Lemma 3.3 yields an expression of perturbation, (3.33)

The proof is

xδ (t) = x∗ (t) + δq(t) + o(δ),

that indicates q(t) is the first order variation of the trajectory. We emphasize that only two essential tool lemmas are used so far: the lemma of perturbation and robustness of NVIE (Lemma 2.3) and the generalized Riemann-Lebesgue lemma (Lemma 3.2).

4

Maximum Principle for Optimal Control

The maximum principle is a necessary condition satisfied by an optimal control, if it exists. The usual description of the maximum principle features a local condition that is satisfied pointwise by an optimal control function u∗ (t) except possibly for a set of Lebesgue measure zero. A bridge toward the pointwise result is a global maximality result, or called a variational inequality, as stated in the following theorem. Theorem 4.1. Under Hypotheses I, II and III, if there exists an optimal control u∗ (·) ∈ Uad and let x∗ (·) = x (·, x0 , u∗ ), then the following inequality is satisfied, (4.1) ZT

Qx (t, x∗ (t), u∗ (t)) q(t) dt >

0

where q(·) is defined by (3.6).

ZT 0

[Q (t, x∗ (t), u∗ (t)) − Q (t, x∗ (t), u(t))] dt,


317

Proof. By the optimality of the control process {u∗ (·), x∗ (·)}, one has i 1h 06 (4.2) J x0 , uδ − J (x0 , u∗ ) = J1 + J2 , δ where uδ and xδ are the same as described in the proof of Lemma 3.3, just before (3.21), and ZT h i Q t, xδ (t), uδ (t) − Q t, x∗ (t), uδ (t) dt,

(4.3)

1 J1 = δ

(4.4)

ZT i 1 h ∗ J2 = Q t, x (t), uδ (t) − Q (t, x∗ (t), u∗ (t)) dt. δ

0

0

For J1 part, we have ZT ∗ ∗ (4.5) J1 − Qx (t, x (t), u (t)) q(t) dt 0 Z h i 1 δ ∗ ∗ ∗ 6 δ Q t, x (t), u(t) − Q (t, x (t), u(t)) − Qx (t, x (t), u (t)) q(t) dt δ (δ) En

Z + δ (δ) [0,T ]\En

h i 1 Q t, xδ (t), u∗ (t) − Q (t, x∗ (t), u∗ (t)) δ −Qx (t, x (t), u (t)) q(t) dt. ∗

∗

1 and by (3.12), we have Since Q ∈ CLip h i 1 lim Q t, xδ (t), u∗ (t) − Q (t, x∗ (t), u∗ (t)) δ→0+ δ (4.6) ∗ ∗ −Qx (t, x (t), u (t)) q(t) = 0.

On the other hand, by utilizing (3.24), (1.14) and (3.12) we can get a constant δ4 , 0 < δ4 6 δ3 (6 δ2 6 δ0 6 1), such that h i 1 δ ∗ ∗ ∗ Q t, x (t), u (t) − Q (t, x (t), u (t)) δ 1 Z δ ∗ x (t) − x (t) (4.7) δ ∗ = Qx t, x (t, λ), u (t) dλ · δ 0

6 g1 (t) (kq(·)kC + 1) ,

t ∈ [0, T ], for any δ ∈ (0, δ4 ] ,

318

You

where g1 (·) ∈ L1 [0, T ] depends on N1 (x∗ (·)), cf. (3.24). Based on (4.6) and (4.7), we can apply the Lebesgue Dominated Convergence Theorem to obtain h Z i 1 Q t, xδ (t), u∗ (t) − Q (t, x∗ (t), u∗ (t)) lim δ δ→0+ δ [0,T ]\En(δ) (4.8) ∗ ∗ −Qx (t, x (t), u (t)) q(t) dt = 0 Moreover, similarly we can get a δ5 , 0 < δ5 6 δ3 , such that h i 1 Q t, xδ (t), u(t) − Q (t, x∗ (t), u(t)) − Qx (t, x∗ (t), u∗ (t)) q(t) δ (4.9) 6 g2 (t), t ∈ [0, T ], for any δ ∈ (0, δ5 ] , where g2 (·) ∈ L1 [0, T ] depends on N1 (x∗ (·)) and u as well. Equation (4.9) and the fact that δ meas En(δ) = δT → 0, as δ → 0+ , imply the first integral term on the right side of inequality (4.5) also converges to zero as δ → 0+ . Therefore, we have proved that ZT lim J1 =

(4.10)

δ→0+

Qx (t, x∗ (t), u∗ (t)) q(t) dt.

0

Next we treat the I2 part. In fact, we have Z 1 J2 = [Q (t, x∗ (t), u(t)) − Q (t, x∗ (t), u∗ (t))] dt δ δ En(δ)

ZT =

[Q (t, x∗ (t), u(t)) − Q (t, x∗ (t), u∗ (t))] dt

0

(4.11)

1 + δ

ZT

δ [Q (t, x∗ (t), u(t)) − Q (t, x∗ (t), u∗ (t))] θn(δ) dt

0

ZT →

[Q (t, x∗ (t), u(t)) − Q (t, x∗ (t), u∗ (t))] dt text, asδ → 0+ ,

0

because (4.12)

1 lim δ→0+ δ

ZT 0

[Q (t, x∗ (t), u(t)) − Q (t, x∗ (t), u∗ (t))] θ δ (t) dt = 0,


319

that can be shown similarly as we did for (3.13). Finally, we substitute (4.10) and (4.11) into (4.2) and then obtain

ZT 06

∗

ZT

∗

Qx (t, x (t), u (t)) q(t) dt 0

−

[Q (t, x∗ (t), u∗ (t)) − Q (t, x∗ (t), u(t))] dt,

0

that immediately leads to (4.1). The proof is completed. Now we are going to present the main results in the first part, the maximum principle. Associated with the (OCP), we define a functional H(t, x, ψ, u) to be

(4.13)

H(t, x, ψ, u) = hψ, f (t, x, u)i − Q(t, x, u),

where (t, x, ψ, u) ∈ of W 0 versus W . Theorem 4.2 satisfied. If u∗ (·) x (·, x0 , u∗ ), then it

[0, T ] × W × W 0 × Y and h·, ·i stands for the dual product (Maximum Principle). Let Hypotheses I, II and III be ∈ Uad is an optimal control of the (OCP ) and x∗ (·) = holds that

H (t, x∗ (t), ψ(t), u∗ (t)) = max H (t, x∗ (t), ψ(t), v) , v∈U

(4.14)

for a.e. t ∈ [0, T ],

where the dual function ψ ∈ C ([0, T ], W 0 ) is given by

ZT (4.15)

ψ(t) = −

Qx (s, x∗ (s), u∗ (s)) G(s, t) ds,

t ∈ [0, T ],

t

in which G(·, ·) is defined by (3.2) and satisfies (3.3) and (3.4).

320

You

Proof. We can substitute (3.6) for q(t) in (4.1) and then interchange the order of integration by the Fubini-Tonelli theorem. It yields (4.16) ZT Zt ∗ ∗ Qx (t, x (t), u (t)) G(t, τ ) · [f (τ, u∗ (τ ), u(τ )) − f (τ, x∗ (τ ), u∗ (τ ))] dτ dt 0

0 ZT

=

 T  Z  Qx (t, x∗ (t), u∗ (t)) G(t, τ )dt τ

0 ∗

· [f (τ, u (τ ), u(τ )) − f (τ, x∗ (τ ), u∗ (τ ))] dτ ZT =−

hψ(τ ), f (τ, u∗ (τ ), u(τ )) − f (τ, x∗ (τ ), u∗ (τ ))i dτ

0

ZT >

[Q (t, x∗ (t), u∗ (t)) − Q (t, x∗ (t), u(t))] dt.

0

It follows that for any u(·) ∈ U , the following maximality inequality holds, ZT

∗

ZT

∗

H (t, x (t), ψ(t), u (t)) dt >

(4.17) 0

H (t, x∗ (t), ψ(t), u(t)) dt.

0

Recall that starting from Lemma 2.1 in Section 2 and Lemma 3.1 in Section 3, we only require u(·) ∈ U, that may or may not be in Uad . Therefore, for any v ∈ U and any nontrivial subinterval [a, b] ⊂ [0, T ], the control function u(·) defined by ( v, if t ∈ [a, b], u(t) = (4.18) ∗ u (t), if t ∈ [0, T ]\[a, b], is in U. Then (4.17) is valid for this particular control u(·), which implies that Zb

∗

∗

Zb

H (t, x (t), ψ(t), u (t)) dt >

(4.19) a

H (t, x∗ (t), ψ(t), v) dt.

a

Since a and b, with 0 6 a < b 6 T , are arbitrary, by the Lebesgue differentiation theorem, we can assert that (4.20)

1 lim + r→0 2r

Zt+r H (s, x∗ (s), ψ(s), u∗ (s)) ds = H (t, x∗ (t), ψ(t), u∗ (t)) t−r


321

for almost every t ∈ [0, T ] (called Lebesgue points), and (4.21)

Zt+r H (s, x∗ (s), ψ(s), v) ds = H (t, x∗ (t), ψ(t), v)

1 lim + r→0 2r

t−r

for any t ∈ [0, T ], by taking a = t − r and b = t + r. From (4.19), (4.20) and (4.21) it follows that for any v ∈ U , (4.22)

H (t, x∗ (t), ψ(t), u∗ (t)) > H (t, x∗ (t), ψ(t), v) , for a.e. t ∈ [0, T ].

Thus (4.14) is proved. The proof is completed. Note that in the presentation of Theorem 4.2, the functional H(t, x, ψ, u) defined by (4.13) is in terms of the dual product h·, ·i of W 0 versus W . If we adopt the dual product h·, ·i of W versus W 0 , with this notation unchanged, but the first component is in W and the second component is in W 0 , then the same maximum principle takes the adjoint version, which is the next theorem. Theorem 4.3. Let Hypotheses I, II and III be satisfied. If u∗ (·) ∈ Uad is an optimal control of the (OCP ) and x∗ (·) = x (·, x0 , u∗ ), then it holds that H (t, x∗ (t), ϕ(t), u∗ (t)) = max H (t, x∗ (t), ϕ(t), v) , v∈U

(4.23)

for a.e. t ∈ [0, T ]

where (4.24)

H(t, x, ϕ, u) = hf (t, x, u), ϕi − Q(t, x, u),

with the dual product being W versus W 0 , and ZT (4.25)

ϕ(t) = −

G† (s, t)Q†x (s, x∗ (s), u∗ (s)) ds,

t ∈ [0, T ],

t

that is the mild solution of the “backward” evolution equation in W 0 , dϕ = −A† ϕ − fx† (t, x∗ (t), u∗ (t)) ϕ + Q†x (t, x∗ (t), u∗ (t)) , dt (4.26) t ∈ [0, T ], ϕ(T ) = 0, and G(·, ·) is the same as in Theorem 4.2. Proof. As we mentioned in Section 1, the superscript † is used to denote the adjoint operator. Eq. (4.26) is a linear evolution equation, whose backward mild solution satisfying ϕ(T ) = 0 is given by ZT eA

ϕ(t) = (4.27)

† (s−t)

fx† (s, x∗ (s), u∗ (s)) ϕ(s) ds

t

ZT −

eA t

† (s−t)

Q†x (s, x∗ (s), u∗ (s)) ds,

t ∈ [0, T ].

322

You

From (4.15) and (4.25), we see that exactly ψ † (t) = ϕ(t) for t ∈ [0, T ]. Then (4.23) follows from (4.14). It suffices to show that ϕ(t) given by (4.25) satisfies the equation (4.27). In fact, by (3.4) we have A† (s−t)

G† (s, t) = e

(4.28)

Zs eA

+

† (σ−t)

fx† (σ, x∗ (σ), u∗ (σ)) G† (s, σ) dσ.

t

Substituting (4.28) into (4.25), we find ZT ϕ(t) = −

G† (s, t)Q†x (s, x∗ (s), u∗ (s)) ds

t

ZT = −

eA

† (s−t)

Q†x (s, x∗ (s), u∗ (s)) ds

t

ZTZs (4.29)

−

eA

† (σ−t)

fx† (σ, x∗ (σ), u∗ (σ)) G† (s, σ) · Q†x (s, x∗ (s), u∗ (s)) dσ ds

t t

ZT = −

eA

† (s−t)

Q†x (s, x∗ (s), u∗ (s)) ds

t

ZT −

† eA (σ−t) fx† (σ, x∗ (σ), u∗ (σ))

G† (s, σ) · Q†x (s, x∗ (s), u∗ (s)) ds dσ

σ

t

ZT = −

ZT

eA

† (s−t)

Q†x (s, x∗ (s), u∗ (s)) ds

t

ZT eA

+

† (σ−t)

fx† (σ, x∗ (σ), u∗ (σ)) ϕ(σ) dσ,

t ∈ [0, T ].

t

Therefore, ϕ(·) satisfies (4.27). The proof is completed. Inheriting the terminology of finite dimensional optimal control theory, one can call the function ϕ(·) as the co-state function and call Eq. (4.26) as the adjoint state equation with a terminal value condition. Remark 4.1. The maximum principle is an implicit, nonlinear equality relation satisfied by any optimal control, if it exists. The maximum principle is a necessary condition only. In its descriptions, the dual function ψ in Theorem 4.2 and the co-state function ϕ in Theorem 4.3 depend on the optimal control u∗ and the optimal state trajectory x∗ . Therefore, except for some simpler cases, the maximum principle alone is not enough for solving an optimal control problem.


5

323

Lipschitz Continuity of Value Functions

For infinite dimensional nonlinear optimal control, a weak solution or a viscosity solution of the Hamilton-Jacobi equation associated with the value functions is closely related to the synthesis of an optimal control under certain conditions, cf. [1, 2], and other references on Hamilton-Jacobi equations. However, the value functions may fail to be Gâteaux differentiable almost everywhere, and in the current theory of viscosity solutions the uniqueness and regularity aspects are better established than the existence aspect. The latter issue is usually addressed case-by-case and usually requires sophisticated techniques, even in finite dimensional cases. In short, the optimal feedback control for nonlinear distributed systems remains a largely open issue. In this section, we shall prove the Lipschitz continuity of the value functions for the (OCP) described in Section 1 and the affiliated problems. Based on this crucial property, the next section will provide a general result on the state feedback implementation of an optimal control, assuming it exists. First, let us imbed the (OCP) over the given time interval [0, T ] into a family of optimal control problems over subintervals Iτ = [τ, T ] ⊂ [0, T ], described by (5.1)

(5.2)

dx = Ax + f (t, x, u), t ∈ [τ, T ], dt x(τ ) = x0 ,   ZT   minτ Jτ (x0 , u) = Q(t, x(t), u(t)) dt ,   u(·)∈Uad τ

with (5.3)

τ Uad = {u : [τ, T ] → U | u is strongly measurable and

x (t, τ, x0 , u) exists over [τ, T ]}

Here, the mild solution of the initial value problem (5.1) is denoted by x(t) = x (t, τ, x0 , u). We shall refer to this optimal control problem described by (5.1), (5.2) and (5.3) briefly as the (OCP)τ . Since we do not address the existence of optimal control in this paper, we have to make certain assumptions. Hypothesis IV. Assume that for each x0 ∈ W , there exists an optimal control process denoted by {u∗τ , x∗τ }, where x∗τ (t) = x (t, τ, x0 , u∗τ ), of the (OCP )τ , for τ ∈ [0, T ]. Moreover, for any given bounded set B ⊂ W , there is a constant ρ = ρ(B) > 0 such that kx∗τ kC([τ,T ],W ) 6 ρ(B) for any x0 ∈ B. Below, the norm of C([τ, T ], W ) for any τ ∈ [0, T ] will be denoted by k · kC for simplicity. For convenience, the notation of initial state x0 will be replaced by z in the sequel. We shall denote by Sr (E) the closed ball in a Banach space E centered at the origin and of radius r.

324

You Define the value function V (τ, z) to be τ V (τ, z) = min {Jτ (z, u) | u ∈ Uad }.

(5.4)

We are going to investigate the properties of the value function. Lemma 5.1. Let Hypotheses I–IV be satisfied. Then for any r > 0, there is a constant L1 = L1 (r) > 0 such that |V (τ, z1 ) − V (τ, z2 )| 6 L1 (r) kz1 − z2 k ,

(5.5)

for any τ ∈ [0, T ] and any z1 , z2 in Sr (W ). Proof. By Hypothesis IV, let {u1 , x1 } and {u2 , x2 } be optimal control processes of the (OCP)τ with the initial state being z1 and z2 , respectively. Here we drop the optimal indication, superscript ∗. The optimality condition tells us V (τ, z1 ) − V (τ, z2 ) 6 Jτ (z1 , u2 ) − Jτ (z2 , u2 ) ,

(5.6)

V (τ, z2 ) − V (τ, z1 ) 6 Jτ (z2 , u1 ) − Jτ (z1 , u1 )

Hence we have (5.7)

|V (τ, z1 ) − V (τ, z2 )| 6 max {|Jτ (z1 , u2 ) − Jτ (z2 , u2 )| , |Jτ (z2 , u1 ) − Jτ (z1 , u1 )|} .

By Hypothesis IV, for any given r > 0, there exists a constant ρ(r) > 0 such that (5.8)

kx (·, τ, z, u∗ )kC 6 ρ(r), for any z ∈ Sr (W ).

By Hypothesis II, using the mild solution formula and the Henry-Gronwall formula, cf. [16], one can show that there exists a constant K(r) > 0 such that (5.9)

kx (·, τ, z1 , u) − x (·, τ, z2 , u)kC 6 K(r) kz1 − z2 k , for any z1 , z2 ∈ Sr (W ),

where u = either u1 or u2 , the corresponding optimal controls mentioned above. 1 , (5.8) and (5.9) imply that Then by Hypothesis II again, specifically Q ∈ CLip there exists a constant L1 = L1 (r) > 0 such that for any τ ∈ [0, T ], (5.10)

|Jτ (z1 , u) − Jτ (z2 , u)| 6 L1 (r) kz1 − z2 k , for any z1 , z2 ∈ Sr (W ),

with u = either u1 or u2 as above. Finally, substituting (5.10) into (5.7), we obtain (5.5). The proof is finished.


325

Lemma 5.2. Let Hypotheses I–IV be satisfied. Then for any r > 0, there is a constant L2 = L2 (r) > 0 such that |V (τ1 , z) − V (τ2 , z)| 6 L2 (r) |τ1 − τ2 | , for any τ1 , τ2 ∈ [0, T ] and any z ∈ Sr W 2 = Sr (D(A)). Proof. Let 0 6 τ1 < τ2 6 T and let u2 denote an optimal control corresponding to the (OCP)τ2 over [τ2 , T ] with the initial state x (τ2 ) = z. τ by Define u ∈ Uad u0 , for t ∈ [τ1 , τ2 ) , u(t) = (5.12) u2 (t) for t ∈ [τ2 , T ] , (5.11)

where u0 ∈ U is arbitrarily given. Then one has V (τ1 , z) − V (τ2 , z)

(5.13)

6 Jτ1 (z, u) − Jτ2 (z, u2 ) 6 |Jτ1 (z, u) − Jτ2 (z, u2 )| Zτ2 6 |Q (t, x (t, τ1 , z, u0 ) , u0 )| dt τ1

ZT +

Q t, x1 (t), u2 (t) − Q t, x2 (t), u2 (t) dt,

τ2

where x1 (t) = x (t, τ2 , x (τ2 , τ1 , z, u0 ) , u2 ) ,

x2 (t) = x (t, τ2 , z, u2 ) .

Let the two integral terms on the right side of the last inequality in (5.13) be 1 I1 and I2 , respectively. Since Q ∈ CLip and by (1.9), there exists a constant K0Q = K0 (r, u0 , Q) such that |Q (t, x (t, τ1 , z, u0 ) , u0 )| 6 K0Q ,

for t ∈ [τ1 , τ2 ] and z ∈ Sr W 2 ,

so that I1 6 K0Q |τ1 − τ2 | .

(5.14)

Next, by (1.10), we can get a Lipschitz constant K1Q = K1 (r, u0 , u2 , Q) such that ZT (5.15)

I2 6 τ2

K1Q x1 (t) − x2 (t) dt.

326

You

From the formula of mild solution and by Hypothesis I, for τ2 < t 6 T , we have

1

x (t) − x2 (t)

Zτ2

A(τ2 −τ1 ) A(t−τ2 )

A(t−τ2 ) 6 e e z−e z + eA(t−s) f (s, x (s, τ1 , z, u0 ) , u0 ) ds τ1

Zt +

A(t−s)

e

L(X,W )

f s, x1 (s), u2 (s) − f s, x2 (s), u2 (s) ds

τ2

6 C1 |τ1 − τ2 | (−A)1+α eA(t−τ2 ) z Zτ2

(5.16) + eA(τ2 −s)

L(W )

A(t−τ2 )

e

L(X,W )

|f (s, x (s, τ1 , z, u0 ) , u0 )| ds

τ1

Zt + τ2

6

Mα f s, x1 (s), u2 (s) − f s, x2 (s), u2 (s) ds α (t − s)

C1 Mα |(−A)z| |τ1 − τ2 | + (t − τ2 )α Zt + τ2

Zτ2 τ1

M0 Mα |f (s, x (s, τ1 , z, u0 ) , u0 )| ds (t − τ2 )α

Mα f s, x1 (s), u2 (s) − f s, x2 (s), u2 (s) ds. α (t − s)

Let u0 ∈ U be relatively fixed. Then there is a constant ρ1 (r) > 0 such that kx (s, τ1 , z, u0 )k 6 ρ1 (r), for s ∈ [τ1 , τ2 ] and z ∈ Sr W 2 . Hence, there exist constants K0f = K0 (r, u0 , f ) and K1f = K1 (r, u0 , u2 , f ) such that |f (s, x (s, τ1 , z, u0 ) , u0 )| 6 K0f , for s ∈ [τ1 , τ2 ] and z ∈ Sr W 2 , and

f s, x1 (s), u2 (s) − f s, x2 (s), u2 (s) 6 K f x1 (s) − x2 (s) , 1 for s ∈ [τ2 , T ] and z ∈ Sr W 2 . Then, from (5.16) and the aforementioned, it follows that

1

x (t) − x2 (t) (5.17) 6

C1 Mα r M0 Mα K0f |τ − τ | + |τ1 − τ2 | 1 2 (t − τ2 )α (t − τ2 )α

Optimal Control and Synthesis of Nonlinear Systems Zt + τ2

327

Mα K1f

x1 (s) − x2 (s) ds α (t − s) Zt

= h(t) |τ1 − τ2 | + τ2

Mα K1f

x1 (s) − x2 (s) ds, α (t − s)

where h(t) =

t ∈ (τ2 , T ] ,

C1 Mα r + M0 Mα K0f . (t − τ2 )α

By the Henry-Gronwall inequality, cf. [16, Appendix D], (5.17) implies that

1

x (t) − x2 (t) 6 h(t) |τ1 − τ2 | E (µε , t) , (5.18) where E(µ, t) is a positive, continuous, nondecreasing scalar function of t, defined on [0, ∞) and E(µ, 0) = 1, that satisfies lim

t→∞

1 log E(µ, t) = µ, t

and here µε = Mα K1f Γ(ε), with ε > 0 arbitrarily small and Γ(·) being the Gamma function. Substituting (5.18) into (5.15), we then get 1 I2 6 (5.19) K1Q E (µε , T ) Mα C1 r + M0 K0f T 1−α |τ1 − τ2 | . 1−α From (5.13), (5.14) and (5.19), there exists a constant `1 (r) = `1 (r, u0 , u2 ) > 0 such that V (τ1 , z) − V (τ2 , z) 6 `1 (r) |τ1 − τ2 | , for any τ1 < τ2 in [0, T ] and z ∈ Sr W 2 . On the other hand, let u1 be an optimal control of the (OCP)τ1 over [τ1 , T ] with the initial state x (τ1 ) = z ∈ Sr W 2 . Then (5.20)

V (τ2 , z) − V (τ1 , z) 6 Jτ2 (z, u1 ) − Jτ1 (z, u1 ) 6 |Jτ2 (z, u1 ) − Jτ1 (z, u1 )| Zτ2 6 |Q (t, x (t, τ1 , z, u1 ) , u1 (t))| dt τ1

ZT |Q (t, x (t, τ2 , z, u1 ) , u1 (t)) − Q (t, x (t, τ1 , z, u1 ) , u1 (t))| dt.

+ τ2

A parallel argument of what we have done from (5.13) through (5.20) shows that there exists a constant `2 (r) = `2 (r, u1 ) such that (5.21)

V (τ2 , z) − V (τ1 , z) 6 `2 (r) |τ1 − τ2 | ,

328

You

for any τ1 < τ2 in [0, T ] and z ∈ Sr W 2 . Finally, let L2 (r) = max {`1 (r), `2 (r)} and (5.11) is proved. The proof is completed. Theorem 5.1. Under Hypotheses I–IV, the value function V (τ, z) : [0, T ] × W 2 → R is locally Lipschitz continuous in (τ, z), that is, for any given r > 0, there exists a constant L(r) > 0 such that |V (τ2 , z) − V (τ1 , z)| 6 L(r) (|τ1 − τ2 | + kz1 − z2 k) , for any (τi , zi ) ∈ [0, T ] × Sr W 2 , i = 1, 2. Proof. By Hypothesis I, W 2 has a stronger topology than W . Thus, there is a constant C > 0 such that kwk 6 CkwkW 2 = C|(−A)w|. From Lemmas 5.1 and 5.2, (5.22) is valid by taking (5.22)

L(r) = max {L1 (Cr), L2 (r)} . Certainly in (5.22) one can replace kz1 − z2 k by C kz1 − z2 kW 2 . In order to simplify the presentation of a synthesis result in the next section, we now convert the original (OCP)τ of Lagrange form to its Meyer form, by introducing a new state variable y. Let y(t) = y(t, τ, z, u) be the solution of the following ODE with the initial condtion, (5.23)

dy = Q(t, x(t, τ, z, u), u(t)), dt y(τ ) = ζ.

t ∈ [τ, T ],

Then define the augmented state function and its initial data by x(t) (5.24) ξ(t) = col(x(t), y(t)) = , y(t) z (5.25) ξ(0) = Z = col(z, ζ) = . ζ The augmented value function is defined to be (5.26)

Φ(τ, Z) = minτ {y(T, τ, Z, u)}, u∈Uad

and (5.22) is translated to the Lipschitz property of Φ as follows, (5.27)

6

|Φ (τ1 , Z1 ) − Φ (τ2 , Z2 )| 6 L(r) |τ1 − τ2 | + kZ1 − Z2 kW ×R , for (τi , Zi ) ∈ [0, T ] × Sr W 2 × R.

Synthesis of Optimal Control

As commented earlier, except for quadratic optimal controls, cf. [11, 12, 13], and some nonquadratic optimal controls, cf. [1, 2, 19, 20, 18], etc., the synthesis


329

of a nonlinear distributed optimal control in general is a widely-open issue. Here, synthesis means an implementation of an optimal control by using a state feedback. In this section, a theoretical approach is provided to reach a fairly general result on the synthesis of an optimal control, if it exists, for the described (OCP). This result can be regarded as a generalization of the results on optimal feedback controls in [19] and [20] related to systems governed by specific PDEs. We should also mention that the idea of synthesis via solving a multivalued differential equation is rooted in the finite dimensional theory of optimal control, cf. [4] and [5]. Let us define a set-valued function in X × R, Az + f (t, z, u) Π(t, z) = , for (t, z) ∈ [0, T ] × W 2 . (6.1) Q(t, z, u) For each (t, z), the value Π(t, z) is a set in general neither convex nor compact. Definition 6.1. The lower Dini derivative of a function g(t, Z) : [0, T ] × W 2 × R → R at a point (t, Z) in the direction (1, η) with η ∈ W 2 × R is denoted and defined by (6.2)

1 D− g(t, Z; 1, η) = lim inf [g(t + δ, Z + δη) − g(t, Z)]. + δ δ→0

By Theorem 5.1, since Φ has the shown Lipschitz property, we know D− Φ(t, Z; 1, η) exists as a finite number. We make more assumptions in this section. Hypothesis V. Assume that for any (t, x) ∈ [0, T ]×W , f (t, x, U ) ⊂ X and Q(t, x, U ) ⊂ R are closed sets respectively. Also assume that for any bounded set B ⊂ W , there exist constants K2f (B) > 0 and K2Q (B) > 0 such that (6.3)

|f (t1 , x1 , u) − f (t2 , x2 , u)| 6 K2f (B) (|t1 − t2 | + kx1 − x2 k) , |Q (t1 , x1 , u) − Q (t2 , x2 , u)| 6 K2Q (B) (|t1 − t2 | + kx1 − x2 k) ,

for any t1 , t2 ∈ [0, T ] and any x1 , x2 ∈ B, u ∈ U . For a control function u(·) valued in U , the minimal requirement is just for any x(·) ∈ C([0, T ], W ), both f (t, x(t), u(t)) and Q(t, x(t), u(t)) are measurable. So we adopt this definition for U accordingly. By [16, Lemma 47.2], under Hypotheses I–IV, for any τ ∈ [0, T ], every x0 ∈ W 2 and every constant control function u, the mild solution x(·) is a strong solution in W and satisfies 0,1−θ x(·) ∈ C [0, T ], W 2 ∩ Cloc (6.4) (0, T ], W 2θ , 0 6 θ 6 1. Now we use the method of dynamic programming to prove a key lemma. Lemma 6.1. For any (τ, Z) ∈ [0, T ] × W 2 × R , one has (6.5)

min D− Φ(τ, Z; 1, η) = 0.

η∈Π(τ,z)

330

You

Proof. Take any η ∈ Π(τ, z), then there is a u ˆ ∈ U such that Az + f (τ, z, uˆ) (6.6) η= , p (τ, z, u ˆ) . Q (τ, z, uˆ) Let u(t) ≡ u ˆ, t ∈ [τ, T ], and let ξ(t) be the corresponding augmented trajectory with this u(·) and the initial data ξ(τ ) = Z = col(z, ζ) in W 2 × R. Then for 0 6 δ 6 T − τ , one has τ +δ Z ξ(τ + δ) = Z + p (s, x(s), u ˆ) ds,

(6.7)

τ

where x(·) = x (·, τ, z, uˆ). As we mentioned earlier, this x(·) is a strong solution in W 2 and it satisfies (6.4). Hence, p (s, x(s), u ˆ) is strongly continuous on [τ, T ]. It follows that p (s, x(s), uˆ) = p (τ, z, u ˆ) + o(1), as δ → 0+ , and ξ(τ + δ) = Z + δp (τ, z, uˆ) + o(δ), as δ → 0+ .

(6.8)

By the optimality condition, we have Φ(τ + δ, ξ(τ + δ)) − Φ(τ, Z) ZT = y(τ + δ) + Q t, x t, τ + δ, x(τ + δ), u∗τ +δ , u∗τ +δ (t) dt − Φ(τ, Z) τ +δ τ +δ Z

= y(τ ) +

Q (t, x (t, τ, z, uˆ) , u ˆ) dt τ

ZT (6.9)

+

Q t, x t, τ + δ, x(τ + δ), u∗τ +δ , u∗τ +δ (t) dt − Φ(τ, Z)

τ +δ

ZT Q (t, x (t, τ, z, uτ ) , uτ (t)) dt − Φ(τ, Z) > 0.

= ζ+ τ

where u∗τ +δ stands for an optimal control of the (OCP)τ +δ with the initial data x (τ + δ, τ, z, uˆ), and uτ is given by ( u ˆ, for τ 6 t 6 τ + δ, uτ (t) = ∗ uτ +δ (t), for τ + δ < t 6 T,


331

τ . From (6.9), (6.8), and the Lipschitz continuity of Φ shown which belongs to Uad at the end of Section 5, it follows that

(6.10)

= = =

1 [Φ(τ + δ, ξ(τ + δ)) − Φ(τ, Z)] δ 1 [Φ (τ + δ, Z + δp (τ, z, uˆ) + o(δ)) − Φ(τ, Z)] δ 1 [Φ (τ + δ, Z + δp (τ, z, uˆ)) − Φ(τ, Z)] + o(1) δ 1 [Φ(τ + δ, Z + δη) − Φ(τ, Z)] + o(1) > 0. δ

By taking the infimum limit in (6.10) as δ → 0+ , we obtain (6.11)

inf

η∈Π(τ,z)

D− Φ(τ, Z; 1, η) > 0.

On the other hand, let {u∗τ , x∗τ } be an optimal process of the (OCP)τ with the initial data x(τ ) = z and Z = col(z, ζ) again. Let ξ ∗ (·) be the corresponding augmented trajectory with ξ ∗ (τ ) = Z. Since f (s, x∗τ (t), u∗τ (t)) and Q (s, x∗τ (t), u∗τ (t)) are Bochner and Lebesgue integrable, respectively, almost every s ∈ [τ, T ] is a Lebesgue point. Thus, from (6.7), we get 

(6.12)

 τ +δ Z 1 ξ ∗ (τ + δ) = Z + δ  p (s, x∗τ (s), u∗τ (s)) ds δ τ

= Z + δp (τ, z, u∗τ (τ )) + o(δ), as δ → 0+ . According to dynamic programming, we have (6.13)

Φ (τ + δ, ξ ∗ (τ + δ)) − Φ(τ, Z) = 0

Repeating steps in (6.10) and then taking the infimum limit as δ → 0+ , now with ξ(τ + δ) replaced by ξ ∗ (τ + δ) and (6.12), we end up with (6.14)

D− Φ (τ, Z; 1, p (τ, z, u∗τ (τ ))) = 0,

where u∗τ (τ ) ∈ Π(τ, z). Finally, (6.11) and (6.14) imply (6.5). As a corollary of Lemma 6.1, let {u∗ , x∗ } be an optimal process of the original (OCP) with x0 ∈ W 2 , and let ξ ∗ be the corresponding augmented state trajectory. Then (6.15)

D− Φ (t, ξ ∗ (t); 1, ∂t ξ ∗ (t)) =

min

η∈Π(t,x∗ (t))

D− Φ (t, ξ ∗ (t); 1, η) = 0,

for almost every t ∈ [0, T ], where ∂t represents a strong derivative in time t.

332

You Note that Φ(t, Z) satisfies the following nonlinear equation,

(6.16)

min D− Φ(t, Z; 1, η) = 0, for a.e. t ∈ [0, T ],

η∈Π(t,z)

(6.17)

Φ(T, Z) = ζ.

Define a set-valued function P (t, Z) by (6.18)

P (t, Z) = argη {D− Φ(t, Z; 1, η) = 0} = {η ∈ Π(t, z) : D− Φ(t, Z; 1, η) = 0} .

Then consider a differential inclusion or a multivalued differential equation with the initial value condition as follows,

(6.19)

dξ ∈ P (t, ξ(t)), t ∈ [0, T ], dt ξ(0) = Z0 = col (x0 , 0) , with x0 ∈ W 2 .

Definition 6.2. A function ξ(·) ∈ C [0, T ], W 2 × R is called a strong solution of the initial value problem (6.19) if its strong derivative exists a.e., it satisfies the differential inclusion in (6.19) a.e., and the given initial value condition is satisfied. The following theorem is the main result in the second part of this paper, and it provides a synthesis of an optimal control under the made assumptions. Theorem 6.1. Let Hypotheses I–V be satisfied. Let x0 ∈ W 2 . A strong solution ξ(t) = col(x(t), y(t)), t ∈ [0, T ], of the initial value problem of the differential inclusions (6.19) provides a synthesis (closed-loop) solution to the original optimal control problem in the following sense: 1) its first component x(t) is an optimal trajectory with x(0) = x0 ; 2) there exists a strongly measurable selection b(t) = b(t, x(t)) ∈ P (t, ξ(t)), whose affiliated u(t), t ∈ [0, T ], is a corresponding optimal feedback control; 3) the terminal value of its second component gives the optimum of the criterion functional, that is, y(T ) = V (0, x0 ) = min J (x0 , u). u∈Uad

Proof. Let ξ(t) = col(x(t), y(t)) be a strong solution of the IVP (6.19). It will be shown later in Lemma 6.3 that there exists a control function u ∈ Uad such that the following equation is satisfied for almost every t ∈ [0, T ], (6.20)

dξ = dt

Ax(t) + f (t, x(t), u(t)) . Q(t, x(t), u(t))


333

We now show that this control process {u, x} is optimal in comparison with any other admissible control process denoted by {v, g}, where v(·) ∈ Uad and g(·) = x (·, x0 , v) and whose augmented state trajectory is denoted by λ(·). By the Hypotheses, for any x0 ∈ W 2 and any admissible control, the mild solution must be absolutely strongly continuous. Then, by the shown Lipschitz continuous property of Φ, we can assert that Φ(t, ξ(t)) and Φ(t, λ(t)) are absolutely continuous on [0, T ]. Hence there derivatives in t exist almost everywhere and are Lebesgue integrable over [0, T ]. Thus, by the NewtonLeibniz formula, we have Φ(T, λ(T )) − Φ (0, Z0 ) ZT d = Φ(t, λ(t)) dt dt 0

ZT =

lim

1 [Φ(t + δ, λ(t + δ)) − Φ(t, λ(t))] dt δ

lim

1 [Φ(t + δ, λ(t) + δp(t, g(t), v(t)) + o(δ)) − Φ(t, λ(t))] dt δ

lim

1 [Φ(t + δ, λ(t) + δp(t, g(t), v(t))) − Φ(t, λ(t))] dt δ

δ→0+ 0 ZT

(6.21) =

δ→0+ 0 ZT

=

δ→0+

(by (6.8))

0

ZT D− Φ(t, λ(t); 1, p(t, g(t), v(t))) dt > 0,

= 0

where the penultimate equality follows from the Lipschitz property of Φ and the last inequality comes from Lemma 6.1 and its corollary (6.15). On the other hand, through the steps parallel to (6.21) and by the definition in (6.18), one can show that ZT (6.22)

Φ(T, ξ(T )) − Φ (0, Z0 ) =

D− Φ (t, ξ(t); 1, ∂t ξ(t)) dt = 0, 0

where ∂t ξ(t) = dξ(t)/dt is given by the selection in (6.20). Therefore, we find that by (6.21) and (6.22), (6.23)

J (x0 , v) = Φ(T, λ(T )) > Φ (0, Z0 ) = Φ(T, ξ(T )) = J (x0 , u) ,

for any v(·) ∈ Uad . This proves that u(·) ∈ Uad is an optimal control. Thus, statements 1) and 3) have been proved. Statement 2) will be shown in Lemma 6.3.

334

You

The following lemma, whose proof can be found in [10, Theorem 4.6], is a generalization of the famous Filippov lemma. Lemma 6.2. Let Λ be a measure space with a complete σ-finite nonnegative measure. Let Θ be a Banach space such that there exists a countable subset S in its dual space, which separates points of Θ. Let M be a separable complete metric space. Assume that h : Λ → M is a measurable closed-set-valued function and ρ : Λ × M → Θ is a function such that i) ρ(·, m) is strongly continuous for each m ∈ M, and ii) ρ(t, ·) is demicontinuous for a.e. t ∈ Λ. If κ : Λ → Θ is a strongly measurable function satisfying (6.24)

κ(t) ∈ ρ(t, h(t)), a.e. in Λ,

then there exists a strongly measurable selection m(t) ∈ h(t) such that (6.25)

κ(t) = ρ(t, m(t)), a.e. in Λ.

Lemma 6.3. Under the same assumptions as in Theorem 6.1, for any strong solution ξ(·) of the differential inclusion in (6.19), thre exists a control function u ∈ Uad such that (6.20) is satisfied a.e. on [0, T ]. Proof. Since P (t, Z) ⊂ Π(t, z), any strong solution ξ(t) of the differential inclusion in (6.19) can be viewed as a strong solution of the following differential inclusion, (6.26)

dξ ∈ Π(t, x(t)), a.e. t ∈ [0, T ], dt

ξ(0) = Z0 = col (x0 , 0) ,

in which the set-valued function Π(t, x(t)) is given by Ax(t) + f (t, x(t), u) (6.27) Π(t, x(t)) = :u∈U Q(t, x(t), u) In order to apply Lemma 6.2, let Λ = [0, T ],

Θ = X × R,

and M = X × R.

Note that Θ = M is a separable, reflexive Banach space. Define ρ : Λ×M → Θ by Ax(t) (6.28) ρ(t, m) = + m, 0 that satisfies the two conditions in Lemma 6.2. Let h : Λ → M be defined as f (t, x(t), u) (6.29) h(t) = :u∈U , Q(t, x(t), u)


335

which is a measurable and closed-set-valued function, by Hypothesis V. In this dξ(t) case, κ(t) = : Λ → M is strongly measurable as a strong derivative of an dt almost everywhere differentiable function, and κ(t) satisfies Eq. (6.24) with ρ and h given by (6.28) and (6.29). Therefore, all the conditions in Lemma 6.2 are satisfied. We can apply Lemma 6.2 to this case and conclude that there exists a strongly measurable selection m(t, u) ∈ h(t), which means there exists a control function u(·), such that dξ Ax(t) + f (t, x(t), u(t)) (6.30) = ρ(t, m(t, u)) = Q(t, x(t), u(t)) dt for a.e. t ∈ [0, T ]. It is seen that u(·) ∈ Uad and Eq. (6.20) is satisfied almost everywhere on [0, T ]. The proof is completed. This also completes the proof of Theorem 6.1. Remark 6.1. The results of this paper also demonstrate that it does not require the convexity of the cost functional to establish the maximum principle and the general synthesis for an optimal control. However, nonconvex criteria of optimality may affect the existence and uniqueness theory of optimal control. An example of an optimal control problem related to the diffusion of epidemics that has a nonconvex cost functional can be found in [3, Section 6.1]. Two examples of finding concrete optimal feedback controls by this approach can be found in [19] and [20]. Even though the style of presenting and proving this synthesis result seems abstract, there is actually a potentiality to develop a scheme of approximation for constructing optimal feedback controls based on this method.

References [1] V. Barbu, Optimal feedback controls for a class of nonlinear distributed parameter systems, SIAM Control & Optim., 21 (1983), pp. 871–894. [2] V. Barbu and G. D. Prato, Hamilton-Jacobi equations and synthesis of nonlinear control processes in Hilbert spaces, J. Diff. Eqns., 48 (1983), pp. 350–372. [3] N. Basile and M. Mininni, An extension of the maximum principle for a class of optimal control problems in infinite dimensional spaces, SIAM J. Control & Optim., 28 (1990), pp. 1113–1135. [4] L. Berkovitz, Optimal feedback control, SIAM J. Control & Optim., 27 (1989), pp. 991–1007. [5] V. Dzhafanov, Multivalued synthesis for one class of controllable systems, J. Appl. Math. Mech., 56 (1992), pp. 581–583. [6] H. Fattorini, A unified theory of necessary conditions for nonlinear nonconvex control systems, Appl. Math. & Optim., 15 (1987), pp. 141–185. , Optimal control problems for distributed parameter systems in Banach [7] spaces, Appl. Math. & Optim., 28 (1993), pp. 225–257.

336 [8]

[9] [10] [11]

[12] [13] [14]

[15] [16] [17] [18] [19]

[20]

You , Existence theory and the maximum principle for relaxed infinite dimensional optimal control problems, SIAM J. Control & Optim., 32 (1994), pp. 311– 331. , Optimal control problems with state constraints for semilinear distributed parameter systems, J. Optim. Theory & Appl., 88 (1996), pp. 25–59. S. Hou, Implicit function theorem in topological spaces, Applicable Analysis, 13 (1982), pp. 209–217. I. Lasiecka and R. Triggiani, Differential and Algebraic Riccati Equations With Applications to Boundary/Point Control Problems: Continuous Theory and Approximation Theory, Springer, Berlin, 1991. , Control Theory for Partial Differential Equations: Continuous and Approximation Theories, vol. I, Cambridge University Press, 2000. , Control Theory for Partial Differential Equations: Continuous and Approximation Theories, vol. II, Cambridge University Press, 2000. X. Li and Y. Yao, Maximum principle of distributed parameter systems with time lag, in Lecture Notes in Control and Information Science, F. Kappel, K. Kunish, and W. Schappacher, eds., vol. 75, Springer, New York, 1985, pp. 410–427. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. G. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New York, 2000. (to be published). Y. Yao, Vector measure and maximum principle of distributed parameter systems, Scientia Sinica (Ser. A), 26 (1988), pp. 102–112. Y. You, A nonquadratic Bolza problem and a quasi-Riccati equation for distributed parameter systems, SIAM J. Control & Optim., 25 (1987), pp. 904–920. , Nonlinear optimal control and synthesis of thermal nuclear reactors, in Distributed Parameter Control Systems: New Trends and Applications, C. Chen, E. Lee, W. Littman, and L. Marcus, eds., Marcel Dekker, New York, 1991, pp. 445–474. , Optimal feedback control of Ginzburg-Landau equation for superconductivity via differential inclusion, Discussions Mathematicae Differential Inclusions, 16 (1996), pp. 5–41.

Forced Oscillation of The Korteweg-De Vries-Burgers Equation and Its Stability

Bing-Yu Zhang1 , University of Cincinnati, Cincinnati, Ohio Abstract This paper studies an infinite-dimensional dynamic system described by the Korteweg-de Vries-Burgers equation posed on a finite domain with an external excitation. It shows that if the external excitation is time periodic with small amplitude, then the system admits a unique time periodic solution which, as a limit cycle, forms an inertial manifold for the system.

1

Introduction

An initial and boundary-value problem (IBVP) for a model equation for unidirectional propagation of waves is investigated here. The equation in question, which incorporates nonlinear, dispersive and dissipative effects, is the forced Korteweg-de Vries-Burgers equation (1.1)

ut + ux + uux − uxx + uxxx = f

for x ∈ [0, 1] and t ≥ 0. The equation is subjected to the initial condition (1.2)

u(x, 0) = φ(x),

x ∈ (0, 1),

and the boundary conditions (1.3)

u(0, t) = 0,

u(1, t) = 0,

ux (1, t) = 0

Here f ≡ f (x, t) is a given function. The IBVP (1.1)-(1.3) may be viewed as an infinite-dimensional dynamic system with the forcing f as an external excitation.

1

Supported in part by the Charles P. Taft Memorial Fund. E-mail: [email protected]

337

338

Zhang Our main concern in this paper is the following two questions.

Question 1. If the external excitation f is time periodic with period ω, does it force the equation (1.1) produce a time periodic solution (forced oscillation) of periodic ω satisfying the boundary conditions (1.3)? If such a time periodic solution exists, it forms a limit cycle in the phase space of the dynamic system (1.1)-(1.3). Question 2. What stability does this limit cycle possess? There have been many studies on time periodic solutions of partial differential equations in the literature. For early works on this subject, see Brézis [6], Vejvoda et al. [17], Keller and Ting [9], Rabinowitz [12, 13] and the references therein. For recent works, see Bourgain [5], Craig and Wayne [8], Wayne [19]. In particular, see Wayne [18] for a recent review on time periodic solutions of nonlinear partial differential equations. While there have been many results for parabolic and hyperbolic equations, there are few discussions on time-periodic solutions of nonlinear dispersive wave equations such as the KdV-Burgers equation discussed in this paper. Especially, there is very few discussion on stability of time periodic solutions. In this paper it will be shown that for given time periodic forcing f (of period ω) with small amplitude, the equation (1.1) does possess a unique time periodic solution u∗ (x, t) of period ω satisfying the boundary conditions (1.3). Furthermore, it will be shown that this unique time periodic solution u∗ (x, t), as a limit cycle for the system (1.1)-(1.3), is also a global attractor in the space H j (0, 1) (j = 0, 3); for any φ ∈ H j (0, 1) with φ satisfying φ(0) = φ(1) = φ0 (1) = 0 if j = 3, the corresponding solution u(x, t) of (1.1)(1.3) satisfies (1.4)

ku(·, t) − u∗ (·, t)kH j (0,1) ≤ Ce−µt

for any t ≥ 0 where µ > 0 is independent of φ. In other words, this unique time periodic solution, as a limit cycle, forms an inertial manifold for the dynamic system (1.1)-(1.3). The paper is organized as follows. In section 2 we discuss the global well-posedness of the IBVP (1.1)(1.3). It will be shown that for given T > 0, φ ∈ H j (0, 1) and f ∈ H j/3 (0, T ; L2 (0, 1)) ∩ L2 (0, T ; H j/3 (0, 1)), (1.1)-(1.3) admits a unique solution u ∈ C(0, T ; H j (0, 1)) ∩ L2 (0, T ; H j+1 (0, 1)) with j = 0, 3. The proof is standard; the local well-posedness is established first by using contraction mapping principle and then the global well-posedness is obtained by finding the needed global a priori estimate. In section 3 we discuss large time behavior of the system (1.1)-(1.3) without assuming time periodicity of the forcing. The asymptotic estimates of (1.1)(1.3) in the space H j (0, 1) when j = 0 and j = 3 are established directly as

Forced Oscillation of the KdV-Burgers Equation

339

usual via energy estimate method. In particular, the solution of the system (1.1)-(1.3) tends to zero in the space H j (0, 1) as t → ∞ if the forcing f tends to zero and the solution decays exponentially if the forcing decay exponentially as t → ∞. The results presented in this section are crucial in discussing time periodic solutions of the system (1.1)-(1.3). The existence of the forced oscillation and its stability analysis will be discussed in section 4. The similar results can be also obtained by the same approach for the system with the external excitations acted on the boundaries:  ut + ux + uux − uxx + uxxx = 0,      u(x, 0) = φ(x), (1.5)      u(0, t) = h1 (t), u(1, t) = h2 (t), ux (1, t) = h3 (t) for x ∈ (0, 1) and t ≥ 0.

2

Well-posedness

Consideration is first directed to the homogeneous linear problem  ut + ux + uxxx − uxx = 0, for x ∈ (0, 1) and t ≥ 0,      u(x, 0) = φ(x), (2.1)      u(0, t) = 0, u(1, t) = 0, ux (1, t) = 0. By semigroup theory, its solution is given by u(t) = W (t)φ, where the spatial variable is suppressed and W (t) is the C0 −semigroup in the space L2 (0, 1) generated by the operator Aψ = −ψ 000 − ψ 0 + ψ 00 with the domain of D(A) = {ψ ∈ H 3 (0, 1), ψ(0) = ψ(1) = ψ 0 (1) = 0}. By d’Alembert’s formula, one may use the semigroup W (t) to formally write the solution of the inhomogeneous linear problem  ut + ux + uxxx − uxx = f, for x ∈ (0, 1), t ≥ 0,      u(x, 0) = 0, (2.2)      u(0, t) = 0, u(1, t) = 0, ux (1, t) = 0

340

Zhang

in the form

Z

t

u(t) =

W (t − τ )f (·, τ )dτ.

0

For T > τ ≥ 0, let XTj (j = 0, 3) be the collection of all (φ, f ) ∈ L2 (0, 1) × H j/3 (0, T ; L2 (0, 1)) ∩ L2 (0, T ; H j/3 (0, 1)) with φ satisfying the compatibility condition φ(0) = 0,

(2.3)

φ(1) = 0,

φ0 (1) = 0.

if j = 3 and let j Yτ,T = C([τ, T ]; H j (0, 1)) ∩ L2 (τ, T ; H j+1 (0, 1)).

For (φ, f ) ∈ XTj , its norm k(φ, f )kX j is defined by T

1/2 = kφk2H j (0,1) + kf k2H j/3 (0,T ;L2 (0,1)) + kf k2L2 (0,T ;H j/3 (0,1)

k(φ, f )kX j

T

j and for v ∈ Yτ,T , its norm kvkY j

τ,T

is defined by

kvkY j = kvkC([τ,T ];H j (0,1)) + kvkL2 (τ,T ;H j+1 (0,1)) . τ,T

j If τ = 0, the space Yτ,T will be abbreviated simply YTj . The following two lemmas reveal some smoothing effects for solutions of (2.1) and (2.2), which will also play important roles later in studying stability of time periodic solutions of the nonlinear system (1.1)-(1.3). Lemma 2.1. Let T > 0 be given and u be a solution of (2.1). Then there exists a constant C independent of φ such that

3 kukY 0 ≤ kφkL2 (0,1) T 2

(2.4) if φ ∈ L2 (0, 1) and

kukY 3 + kut kY 0 ≤ CkφkH 3 (0,1)

(2.5)

T

T

φ0 (1)

if φ ∈ with φ(0) = φ(1) = = 0. Lemma 2.2. Let T > 0 be given and u be a solution u of (2.2). Then there exists a constant C independent of f such that H 3 (0, 1)

kukY 0 ≤ Ckf kL2 (0,T ;(0,1))

(2.6) if f ∈ (2.7) if f ∈

T

L2 (0, T ; L2 (0, 1))

and

kukY 3 + kut kY 0 ≤ C kf kL2 (0,T ;H 1 (0,1)) + kft kL2 (0,T ;L2 (0,1)) T

L2 (0, T ; H 1 (0, 1))

T

∩ H 1 (0, T ; L2 (0, 1)).

341


We only present a proof for Lemma 2.1. The proof of Lemma 2.2 is similar. Proof of Lemma 2.1: Multiply both sides of the equation in (2.1) by 2u and integrate over (0, 1) with respect to x. Integration by parts leads to the equality d dt

Z

1

Z 2

u (x, t)dx +

u2x (0, t)

1

+2

0

u2x (x, t)dx = 0

0

for any t ≥ 0, from which (2.5) follows easily if one notes that Z

1

Z u2 (x, t)dx ≤

0

1

u2x (x, t)dx.

0

To prove (2.5), let v = ut , which solves  vt + vx + vxxx − vxx = 0,      v(x, 0) = φ∗ (x),      v(0, t) = 0, v(1, t) = 0,

vx (1, t) = 0

with φ∗ (x) = φ00 (x) − φ000 (x) − φ0 (x). By (2.4), 3 kut kY 0 ≤ kφ∗ kL2 (0,1) ≤ CkφkH 3 (0,1) . T 2

(2.8)

It follows from the equation uxxx = −ut − ux + uxx that kuxx kL2 (0,T ;L2 (0,1)) ≤ Ckuxxx kL2 (0,T ;H −1 (0,1)) ≤ Ckut − ux + uxx kL2 (0,T ;H −1 (0,1)) ≤ C kut kL2 (0,T ;L2 (0,1)) + kukL2 (0,T ;L2 (0,1)) + kux kL2 (0,T ;L2 (0,1))

≤ CkφkH 3 (0,1) . One infers that kuxxx kL2 (0,T ;L2 (0,1)) ≤ Ckut − ux + uxx kL2 (0,T ;L2 (0,1)) ≤ CkφkH 3 (0,1) , kuxxxx kL2 (0,T ;L2 (0,1)) ≤ Ck(ut − ux + uxx )x kL2 (0,T ;L2 (0,1)) ≤ CkφkH 3 (0,1) and sup ku(·, t)kH 3 (0,1) ≤ CkφkH 3 (0,1) .

0≤t≤T

The proof is complete. Attention now is turn to the well-posedness of the nonlinear problem (1.1)(1.3).

342

Zhang

Theorem 2.3. Let T > 0 be given. Then for any (φ, f ) ∈ XTj , j = 0, 3, with φ satisfies (2.3) if j = 3, the IBVP (1.1)-(1.3) admits a unique solution u ∈ YTj satisfying kukY j ≤ γ(k(φ, f )kX 0 )k(φ, f )kX j ,

(2.9)

T

T

T

where γ : R+ → R+ is a nondecreasing continuous function. Moreover, the corresponding solution map is continuous from the space XTj to the space YTj . Remark 2.4. The solution map is in fact real analytic from the space XTj to the space YTj , j = 0, 3 (cf. [20]). Proof: Using the notation of the semigroup W (t), rewrite (1.1)-(1.3) in its integral form: Z t Z t u(t) = W (t)φ + (2.10) W (t − τ )f (·, τ )dτ − W (t − τ )(uux )(·, τ )dτ. 0

0

Let r > 0 and θ > 0 be two constants to be determined and let S denote the set S = {v ∈ Yθ0 ; kvkY 0 ≤ r}. θ

For given r and θ, S is a complete metric space. For given (φ, f ) ∈ XT0 , define a map Γ on S: Z t Z t Γ(v) = W (t)φ + W (t − τ )f (·, τ )dτ − W (t − τ )(vvx )(·, τ )dτ 0

0

for any v ∈ S. By Lemma 2.1 and Lemma 2.2, Z θ kΓ(v)kY 0 ≤ Ck(φ, f )kX 0 + kvvx (·, τ )kL2 (0,1) dτ θ

θ

0

Z ≤ Ck(φ, f )kX 0 + θ

θ

sup |v(x, τ )|kvx (·, τ )kL2 (0,1) dτ

0 x∈(0,1)

Z ≤ Ck(φ, f )kX 0 + C θ

θ 0

≤ Ck(φ, f )kX 0 + C sup θ

1/2

1/2

kvkL2 (0,1) kvx kL2 (0,1) kvx kL2 (0,1) dτ

0≤τ ≤θ

1/2 kv(·, τ )kL2 (0,1)

≤ Ck(φ, f )kX 0 + Cθ 1/4 kvk2Y 0 . θ

θ

If one chooses (2.11)

r = 2Ck(φ, f )kX 0 θ

Z

θ 0

3/2

kvx (·, τ )kL2 (0,1) dτ


343

and Cθ 1/4 r ≤ 1/2,

(2.12) then

kΓ(v)kY 0 ≤ r θ

for any v ∈ S. Thus Γ maps S into S. Similarly, one can show that for r and θ chosen as in (2.11)-(2.12), 1 kΓ(v1 ) − Γ(v2 )kY 0 ≤ kv1 − v2 kY 0 . θ θ 2 In other words, the map Γ is a contraction. Its unique fixed point u = Γ(u) is the unique solution of (1.1)-(1.3) in the space S. We have thus established the local well-posedness of (1.1)-(1.3) in the space XT0 . To obtain the global well-posedness, it suffices now to show that estimate (2.9) with j = 0 holds for any smooth solution u of (1.1)-(1.3). Multiply both sides of the equation (1.1) by 2u and integrate over (0, 1) with respect to x. Integration by parts leads to Z 1 Z 1 Z d 1 2 2 2 u( (x, t)dx + ux (0, t) + 2 ux (x, t)dx = 2 f (x, t)u(x, t)dx. dt 0 0 0 It yields that d dt

Z

1

Z

1

u2 (x, t)dx +

0

Z

1

u2x (x, t)dx ≤

0

f 2 (x, t)dx,

0

from which (2.9) (j = 0) follows. To see that the IBVP (1.1)-(1.3) is well-posed in the space XT3 , let S ∗ denote the set S ∗ = {v ∈ Yθ3 ; vt ∈ Yθ0 , kvkY 3 + kvt kY 0 ≤ r} θ

θ

for given r and θ. A similar argument shows Γ is a contraction mapping from S ∗ to S ∗ if θ and r are chosen appropriately. Thus (1.1)-(1.3) is locally well-posed in the space XT3 . For its globally well-posedness, one needs to show that (2.9) with j = 3 holds for any smooth solution of (1.1)-(1.3). To this end, let h = ut . Then h solves  ht + hx + (uh)x − hxx + hxxxx = ft ,      h(x, 0) = φ∗ (x, 0)      h(0, t) = 0, h(1, t) = 0, hx (1, t) = 0

344

Zhang

with φ∗ (x) = f (x, 0) − φ0 (x) − φ000 (x) + φ00 (x) − φ(x)φ0 (x). As before, it follows that Z 1 Z 1 Z 1 Z d 1 2 h (x, t)dx+ h2x (x, t)dx ≤ ft2 (x, t)dx+2ku(·, t)kL2 (0,1) h2x (x, t)dx, dt 0 0 0 0 which yields that kut kY 0 = khkY 0 ≤ C0 (kukY 0 )k(φ, f )kX 3 T

T

T

T

where C0 : → is a nondecreasing continuous function. The estimate (2.9) (j = 3) then follows from the equation R+

R+

uxxx = f − ut − uux − ux + uxx , the above inequality and the estimate (2.9) with j = 0. The proof is complete.

3

Large time behavior

In this section we view (1.1)-(1.3) as a dynamic system with external forcing. Our main concern is large time behavior of its solutions. First we investigate its large time behavior in the space L2 (0, 1). Theorem 3.1. Let T > 0 and 0 < ε < 1 be given. For f ∈ L2loc (R+ ; L2 (0, 1)) and φ ∈ L2 (0, 1), the solution u of (1.1)-(1.3) satisfies ku(·, t)kL2 (0,1) −(1−ε)t

≤ e

r

kφkL2 (0,1) +

2 −(1−ε)(t−s) e ε

Z 0

s

e−2(1−ε)(s−τ ) kf (·, τ )k2L2 (0,1) dτ

1/2

r Z t 1/2 2 + e−2(1−ε)(t−τ ) kf (·, τ )k2L2 (0,1) dτ ε s

(3.1) and (3.2)

kukL2 (t,t+T ;H 1 (0,1)) ≤ ku(·, t)kL2 (0,1) + kf kL2 (t,t+T ;L2 (0,1))

for any 0 ≤ s ≤ t < +∞. Consequently, assuming f ∈ C(R+ ; L2 (0, 1)), (i) if lim kf (·, t)kL2 (0,1) = 0, then t→+∞

lim ku(·, t)kL2 (0,1) + kukL2 (t,t+T ;H 1 (0,1)) = 0;

t→+∞

(ii) if kf (·, t)kL2 (0,1) ≤ Ce−αt for some α > 0, then ku(·, t)kL2 (0,1) +kukL2 (t,t+T ;H 1 (0,1)) ≤ 2e−(1−ε)t kφkL2 (0,1) +Cε,α Ce− min{1−ε,α}t


345

for any t ≥ 0, where

Cε,α =

q  q 1 1  +  α ε|1−ε−α| 

if α 6= 1 − ε,

q q    2t 1 + ε α

if α = 1 − ε.

Proof: For given φ and f , the solution u of (1.1)-(1.3) satisfies the identity Z 1 Z 1 Z d 1 2 2 2 u (x, t)dx + ux (0, t) + 2 ux (x, t)dx = 2 f (x, t)u(x, t)dx. dt 0 0 0 for any t ≥ 0. Since Z

1

Z u (x, t)dx ≤ 2

0

1

u2x (x, t)dx,

0

one obtains d dt

Z

1

Z

1

u2 (x, t)dx + 2(1 − ε)

0

u2x (x, t)dx ≤

0

2 ε

Z

1

f 2 (x, t)dx

0

and by Gronwall’s inequality, Z 1 Z 2 t −2(1−ε)(t−τ ) u2 (x, t)dx ≤ e−2(1−ε)t kφk2L2 (0,1) + e kf (·, τ )k2L2 (0,1) dτ ε 0 0 for any t ≥ 0. In particular, for any 0 ≤ s ≤ t Z 1 u2 (x, t)dx 0 Z 2 t −2(1−ε)(t−τ ) −2(1−ε)(t−s) 2 ≤ e ku(·, s)kL2 (0,1) + e kf (·, τ )k2L2 (0,1) dτ ε s Z 2 −2(1−ε)(t−s) s −2(1−ε)(s−τ ) −2(1−ε)t 2 ≤ e kφkL2 (0,1) + e e kf (·, τ )k2L2 (0,1) dτ ε 0 Z 2 t −2(1−ε)(t−τ ) + e kf (·, τ )k2L2 (0,1) dτ ε s and Z

t+T t

Z 0

1

Z u2x (x, τ )dxdτ

≤

ku(·, t)k2L2 (0,1)

t+T

+ t

kf (·, τ )k2L2 (0,1) dτ

for any t ≥ 0 and T > 0. Combining the above inequalities yields the estimate (3.1) and (3.2). The proof is complete.

346

Zhang

Next we describes large time behavior of solutions of (1.1)-(1.3) in the space H 3 (0, 1). Theorem 3.2. Let T > 0 be given. Suppose f ∈ C 1 (R+ ; L2 (0, 1)) ∩ L2loc (R+ ; H 1 (0, 1)) and φ ∈ H 3 (0, 1). If φ satisfies (2.3) and f satisfies the condition lim kf (·, t)kL2 (0,1) < 1/2, t→+∞

then for any η with 0 < η < 1 − 2 lim kf (·, t)kL2 (0,1) , there exists s1 > 0 t→+∞

depending only on kφkL2 (0,1) + kf kCb (R+ ;L2 (0,1)) such that the corresponding solution u of (1.1)-(1.3) satisfies

ku(·, t)kH 3 (0,1)

≤ γ kφkL2 (0,1) + kf kCb (R+ ;L2 (0,1)) −η(t−s1 )

·e

Z

t

+

−2η(t−τ )

e s

"

ku(·, s1 )kL2 (0,1) + kut (·, s1 )kL2 (0,1)

kf (·, τ )k2L2 (0,1)

+

kft (·, τ )k2L2 (0,1)

1/2

dτ

+kf kH 1 (t,t+T ;L2 (0,1)) +e−η(t−s)

Z

s

s1

1/2 −2η(s−τ ) 2 2 kf (·, τ )kL2 (0,1) + kft (·, τ )kL2 (0,1) dτ e

#

and

kukL2 (t,t+T ;H 4 (0,1)) ≤ γ kφkL2 (0,1) + kf kCb (R+ ;L2 (0,1)) −η(t−s1 )

·e

Z

t

+

−2η(t−τ )

e s

ku(·, s1 )kL2 (0,1) + kut (·, s1 )kL2 (0,1)

1/2 2 2 kf (·, τ )kL2 (0,1) + kft (·, τ )kL2 (0,1) dτ +kf kH 1 (t,t+T ;L2 (0,1))

+e−η(t−s)

Z

s s1

1/2 e−2η(s−τ ) kf (·, τ )k2L2 (0,1) + kft (·, τ )k2L2 (0,1) dτ +kf kL2 (t,t+T ;H 1 (0,1)) .

for any s1 ≤ s ≤ t, where γ : R+ → R+ is a nondecreasing continuous function. Consequently,

Forced Oscillation of the KdV-Burgers Equation (i) if lim

t→+∞

347

kf (·, t)kL2 (0,1) + kft (·, t)kL2 (0,1) + kf kL2 (t,t+T ;H 1 (0,1)) = 0, then lim kukY 3

t,t+T

t→+∞

= 0;

(ii) if kf (·, t)kL2 (0,1) + kft (·, t)kL2 (0,1) + kf kL2 (t,t+T ;H 1 (0,1)) < Ce−αt , for some α > 0 and any t ≥ 0, then kukY 3 ku(·, s1 )kL2 (0,1) + ≤ γ kφkL2 (0,1) + kf kCb (R+ ;L2 (0,1)) t,t+T

i +kut (·, s1 )kL2 (0,1) e−η(t−s1 ) + Cα,η Ce− min{α,η}t for any t ≥ s1 , where  q 1   |1−ε−α| + Cη,α =

  √t − s + 1

if α 6= η,

√2 α

√2 α

if α = η.

Proof: Let h = ut . Then h solves  ht + hx + (uh)x − hxx + hxxxx = ft ,      h(x, 0) = φ∗ (x, 0)      h(0, t) = 0, h(1, t) = 0, hx (1, t) = 0 with

φ∗ (x) = f (x, 0) − φ0 (x) − φ000 (x) + φ00 (x) − φ(x)φ0 (x).

It holds that d dt Z

Z

1

Z h (x, t)dx + 2

0 1

ft (x, t)h(x, t)dx + 2 0

h2x (x, t)dx + h2x (0, t)

0

Z

1

=2

1

2

u(x, t)hx (x, t)h(x, t)dx. 0

Thus d 2 kh(·, t)k2L2 (0,1) + 2(1 − ε − ku(·, t)kL2 (0,1) )khx (·, t)k2L2 (0,1) ≤ kf (·, t)k2L2 (0,1) , dt ε from which one obtains kh(·, t)k2L2 (0,1)

−2ηs,t (t−s)

≤e

kh(·, s)k2L2 (0,1)

2 + ε

Z

t s

e−2ηs,t (t−τ ) kft (·, τ )k2L2 (0,1) dτ

348

Zhang

and Z t+T

khx (·, τ )k2L2 (0,1) dτ

t

1 ≤ εηs,t

Z

t+T t

kf (·, τ )k2L2 (0,1) dτ +

1 kh(·, t)k2L2 (0,1) 2ηs,t

for 0 ≤ s ≤ t where ηs,t = 1 − ε − sup ku(·, τ )kL2 (0,1) . s≤τ ≤t

Since lim ku(·, t)kL2 (0,1) ≤ 2 lim kf (·, t)kL2 (0,1) < 1

t→+∞

t→+∞

for given 0 < η < 1 − 2 lim kf (·, t)kL2 (0,1) , if let t→+∞

1 ε = (1 − η − 2 lim kf (·, t)kL2 (0,1) ), t→+∞ 2 then according to Theorem 3.1, there exists s1 > 0 depending only on kφkL2 (0,1) + kft kCb (R+ ;L2 (0,1)) such that for any s > s1 , inf

s1 ≤s≤t s1 . In particular, kh(·, s)kL2 (0,1)

r Z 1/2 s 2 −η(s−s1 ) −2η(s−τ ) 2 ≤ e kh(·, s1 )kL2 (0,1) + e kft (·, τ )kL2 (0,1) dτ ε s1 for any s > s1 and kh(·, t)kL2 (0,1) −η(t−s1 )

≤ e

kh(·, s1 )kL2 (0,1) +

r

2 −η(t−s1 ) e ε

Z

r Z s 2 + e−2η(s−τ ) kft (·, τ )k2L2 (0,1) dτ ε s1

s

−2η(s−τ )

e

s1 1/2

1/2 kft (·, τ )k2L2 (0,1) dτ

for any s1 < s < t. By Theorem 2.3, there exists a γ = γ(k(φ, f )kXs0 ) such 1 that kh(·, s1 )kL2 (0,1) ≤ γ(k(φ, f )kXs0 )k(φ, f )kXs3 . 1

1

Forced Oscillation of the KdV-Burgers Equation One arrives that

349

r

Z s 1/2 2 −η(t−s) −η(t−s1 ) −2η(s−τ ) 2 kut (·, t)kL2 (0,1) ≤ e e kft (·, τ )kL2 (0,1) dτ + e ε s1 r Z s 1/2 2 −ηt −2η(s−τ ) 2 +γ(k(φ, f )kXs0 )k(φ, f )kXs3 e (3.3) + e kft (·, τ )kL2 (0,1) dτ 1 1 ε s1 and

r kut kL2 t,t+T ;H 1 (0,1) ≤

(3.4)

1 kut (·, t)kL2 (0,1) + 2η

r

1 kft kL2 (t,t+T ;L2 (0,1)) εη

for any s1 ≤ s ≤ t. Recall that uxxx = f − ut − ux − uux + uxx .

(3.5)

There exists a constant C such that kuxx (·, t)kL2 (0,1) ≤ Ckuxxx (·, t)kH −1 (0,1) ≤ C kf (·, t)kL2 (0,1) + kut (·, t)kL2 (0,1) + ku2 (·, t)kL2 (0,1) + kux (·, t)kL2 (0,1) ≤ C kf (·, t)kL2 (0,1) + kut (·, t)kL2 (0,1) + (1 + ku(·, t)kL2 (0,1) )kux (·, t)kL2 (0,1) . Thus, for given T > 0, kuxx kL2 (t,t+T ;L2 (0,1)) ≤ C kf kL2 (t,t+T ;L2 (0,1)) + kut kL2 (t,t+T ;L2 (0,1)) 1 + ku(·, τ )kL2 (0,1) kux kL2 (t,t+T ;L2 (0,1)) . + sup t≤τ ≤t+T

It follows from the equation (3.5) again that kuxxx kL2 (t,t+T ;L2 (0,1) ≤ kf − ut − ux kL2 (t,t+T ;L2 (0,1)) + kuxx − uux kL2 (t,t+T ;L2 (0,1)) 1 + ku(·, τ )kL2 (0,1) kuxx kL2 (t,t+T ;L2 (0,1)) ≤ kf − ut − ux kL2 (t,t+T ;L2 (0,1)) + sup

6

t τ ≤t+T

≤ C

6

sup

t τ ≤t+T

2 1 + ku(·, τ )kL2 (0,1) kf kL2 (t,t+T ;L2 (0,1)) + kut kL2 (t,t+T ;L2 (0,1)) +kux kL2 (t,t+T ;L2 (0,1)) .

As u ∈ L2 (t, t + T ; H 3 (0, 1)) and ut ∈ L2 (t, t + T ; H 1 (0, 1))), it yields that u ∈ C(t, t + T ; H 2 (0, 1)) by interpolation and that sup kuxx (·, τ )kL2 (0,1)) ≤ C kukL2 (t,t+T ;H 3 (0,1)) + kut kL2 (t,t+T ;H 1 (0,1)) t≤τ ≤t+T

350

Zhang

for some constant C independent of t, T and u. Consequently, using (3.5) yields that kuxxx (·, t)kL2 (0,1)) ≤ kf (·, t) − ut (·, t)kL2 (0,1)) + kuxx (·, t) − ux (·, t) − u(·, t)ux (·, t)kL2 (0,1)) ≤ kf (·, t) − ut (·, t)kL2 (0,1)) + 2 + ku(·, t)kL2 (0,1)) kuxx (·, t)kL2 (0,1)) ≤ kf (·, t) − ut (·, t)kL2 (0,1)) + C 2 + ku(·, t)kL2 (0,1)) kukL2 (t,t+T ;H 3 (0,1)) +kut kL2 (t,t+T ;H 1 (0,1)) 3 ≤ kf (·, t)kL2 (0,1)) + kut (·, t)kL2 (0,1)) + C sup 2 + ku(·, τ )kL2 (0,1) t6τ ≤t+T · kf kL2 (t,t+T ;L2 (0,1)) + kut kL2 (t,t+T ;H 1 (0,1)) + kux kL2 (t,t+T ;L2 (0,1)) 3 ≤ C 1 + ku(·, t)kL2 (0,1) + kf kL2 (t,t+T ;L2 (0,1)) kf kL2 (t,t+T ;L2 (0,1)) + +kut kL2 (t,t+T ;H 1 (0,1)) + kux kL2 (t,t+T ;L2 (0,1)) +kf (·, t)kL2 (0,1)) + kut (·, t)kL2 (0,1)) and kuxxxx kL2 (t,t+T ;L2 (0,1)) ≤ kf − ut kL2 (t,t+T ;H 1 (0,1)) + k(ux + uux )x kL2 (t,t+T ;L2 (0,1)) +kuxxx kL2 (t,t+T ;L2 (0,1)) ≤ kf − ut kL2 (t,t+T ;H 1 (0,1)) + kuxx kL2 (t,t+T ;L2 (0,1)) + kuxxx kL2 (t,t+T ;L2 (0,1)) +2kux kL2 (t,t+T ;L2 (0,1)) sup kuxx (·, τ )kL2 (0,1) t6τ 6t+T ≤ kf kL2 (t,t+T ;H 1 (0,1)) + C 1 + kux kL2 (t,t+T ;L2 (0,1)) kukL2 (t,t+T ;H 3 (0,1)) +kut kL2 (t,t+T ;H 1 (0,1)) . Thus there exists a constant Cε,η depending only on η and ε such that for any t ≥ s1 , ku(·, t)kH 3 (0,1) ≤ Cε,η 1 + ku(·, t)kL2 (0,1) + kf kL2 (t,t+T ;L2 (0,1)) ) kf kH 1 (t,t+T ;L2 (0,1)) +ku(·, t)kL2 (0,1) + kut (·, t)kL2 (0,1) and kukL2 (t,t+T ;H 4 (0,1))

Forced Oscillation of the KdV-Burgers Equation 351 ≤ kf kL2 (t,t+T ;H 1 (0,1)) + Cε,η 1 + ku(·, t)kL2 (0,1) + kf kL2 (t,t+T ;L2 (0,1)) ) · kf kH 1 (t,t+T ;L2 (0,1)) + ku(·, t)kL2 (0,1) + kut (·, t)kL2 (0,1) . Theorem 3.2 follows consequently. The proof is complete.

4

Forced oscillation and its global stability

In this section we assume the forcing f is a time-periodic function of period ω > 0 and study if it generates a time periodic solution for the equation (1.1) with the boundary condition (1.3). Theorem 4.1. If f ∈ Cb1 (R+ ; L2 (0, 1)∩L2loc (R+ , H 1 (0, 1)) is a time periodic function of period ω satisfying 1 sup kf (·, t)kL2 (0,1) < , 4 0≤t≤ω

(4.1)

then the equation (1.1) admits a unique time periodic solution u∗ ∈ Cb (R+ ; H 3 (0, 1)) of period ω satisfying the boundary conditions (1.3). Proof: For the given forcing f satisfying (4.1), choose φ ∈ H 3 (0, 1) satisfying the compatibility condition (2.3). Let u(x, t) be the corresponding solution of the IBVP (1.1)-(1.3). By Theorem 3.2, the set {ku(·, t)kH 3 (0,1) }+∞ t=0 is uniformly bounded. Let tk be a sequence with tk → ∞ as k → ∞ such that u(·, tk ) converges to a function ψ ∈ H 3 (0, 1) weakly in H 3 (0, 1) and strongly in L2 (0, 1) as k → ∞. If one takes ψ as an initial data in the IBVP (1.1)-(1.3) with the given forcing f , then the corresponding solution, named as u∗ (x, t), is a time periodic function of period ω. To see this is true, let v(x, t) = u(x, t+ω)−u(x, t). Because of the periodicity of f , v(x, t) solves the following linear problem with the variable coefficient b(x, t) = u(x, t + ω) + u(x, t):  vt + vx + (bv)x − vxx + vxxx = 0,      v(x, 0) = φ∗ (x),      v(0, t) = 0, v(1, t) = 0, vx (1, t) = 0,

(4.2)

where φ∗ (x) = u(x, ω) − u(x, 0). It leads to d dt

Z

1

Z

1

v 2 (x, t)dx + vx2 (0, t) + 2

0

Z

b(x, t)vx (x, t)v(x, t)dx

0

or d dt

Z

1 0

1

vx2 (x, t)dx = 2 0

Z

1

v (x, t)dx + 2(1 − kb(·)kL2 (0,1) ) 2

0

v 2 (x, t)dx ≤ 0

352

Zhang

for any t ≥ 0. By Gronwall’s inequality, Rt

kv(·, t)kL2 (0,1) ≤ kv(·, τ )kL2 (0,1) e

τ

−2(1−kb(·,s)kL2 (0,1) )ds

for any t ≥ τ ≥ 0. By Theorem 3.1, lim kb(·, t)kL2 (0,1) ≤ 2 lim ku(·, t)kL2 (0,1) ≤ 4 lim kf (·, t)kL2 (0,1) < 1.

t→∞

t→∞

t→∞

One can choose τ > 0 large enough such that 2γ = 2 1 − sup kb(·, t)kL2 (0,1) > 0. t≥τ

Consequently,

kv(·, t)kL2 (0,1) ≤ kv(·, τ )kL2 (0,1) e−γ(t−τ )

for any t ≥ τ . In particular, ku(·, tk + ω) − u(·, tk )kL2 (0,1) ≤ kv(·, τ )kL2 (0,1) e−γ(tk −τ ) for any tk ≥ τ . Note that u(·, tk ) converges to ψ = u∗ (·, 0) strongly in L2 (0, 1) and that u(·, tk + ω) converges to u∗ (·, ω) strongly in L2 (0, 1) as k → ∞. Since ku∗ (·, ω) − u∗ (·, 0)kL2 (0,1) ≤ ku∗ (·, ω) − u(·, tk + ω)kL2 (0,1) + +ku(·, tk + ω) − u(·, tk )kL2 (0,1) + ku(·, tk ) − u∗ (·, 0)kL2 (0,1) for any tk ≥ τ , we conclude that u∗ (x, ω) = u∗ (x, 0) for x ∈ (0, 1) a.e. and that u∗ (x, t) is a time periodic function of period ω. To show the uniqueness, let u1 and u2 be such two time periodic solutions with the given forcing f . Let v = u1 − u2 . Then v solves the linear problem (4.2) with b = u1 + u2 and φ∗ (x) = u1 (x, 0) − u2 (x, 0). By Theorem 3.1, lim kb(·, t)kL2 (0,1) ≤

t→∞

lim ku1 (·, t)kL2 (0,1) + lim ku2 (·, t)kL2 (0,1)

t→∞

t→∞

≤ 4 lim kf (·, t)kL2 (0,1) t→∞

< 1. Consequently, v decays to zero exponentially in the space L2 (0, 1). Therefore u1 (x, t) ≡ u2 (x, t) for any x ∈ (0, 1) and t ≥ 0 because they are time periodic functions. The proof is complete. The time periodic solution u∗ forms a limit circle for the dynamic system described by (1.1)-(1.3). To investigate its stability, let φ be a s−compatible


353

function in the phase space H s (0, 1) and u be the corresponding solution of the IBVP (1.1)-(1.3). If let w = u − u∗ , then w solves the equation wt + wx + wwx + (u∗ w)x − wxx + wxxx = 0 and satisfies the initial condition w(x, 0) = φ − u∗ (x, 0) and the homogeneous boundary condition (1.3). This leads us to considering the large time behavior of the following initial-boundary-value problem  ut + ux + uux + (vu)x + uxxx − uxx = 0,      u(x, 0) = φ(x), (4.3)      u(0, t) = 0, u(1, t) = 0, ux (1, t) = 0 for x ∈ (0, 1) and t ≥ 0 where v ≡ v(x, t) is a given function with v ∈ C(R+ ; H 3 (0, 1)) ∩ L2loc (R+ ; H 4 (0, 1)) and vt ∈ C(R+ ; L2 (0, 1)) ∩ L2loc (R+ ; H 1 (0, 1)). By the same arguments used in the section 2 one can show that (4.3) is globally well-posed in the space H j (0, 1) for j = 0, 3. As for its large time behavior, we have the following result. Proposition 4.2. Let T > 0 and j = 0 or 3 be given. There exist constants η1 ∈ (0, 1) and η2 > 0 such that if (4.4)

lim kvkY 0

t→∞

t,t+T

< η1 ,

then for φ ∈ H j (0, 1) with φ satisfying (2.3) if j = 3, the unique solution u of (4.3) satisfies ku(·, t)kH j (0,1) ≤ γ kφkL2 (0,1) e−η2 t kφkH j (0,1) (4.5) for any t ≥ 0, where γ : R+ → R+ is a nondecreasing continuous function depending on j, T and v. The following stability result for the forced oscillation u∗ (x, t) then follows. Theorem 4.3. There exists an η ∈ (0, 1) such that if f ∈ Cb1 (R+ ; L2 (0, 1) ∩ L2loc (R+ , H 1 (0, 1)) is a time periodic function of period ω satisfying sup kf (·, t)kL2 (0,1) < η,

0≤t≤ω

then the equation (1.1) admits a unique time periodic solution u∗ ∈ Cb (R+ ; H 3 (0, 1)) of period ω satisfying the boundary conditions (1.3). Moreover, for given j = 0, 3, there exists a δ > 0 such that for any compatible φ ∈ H j (0, 1) with φ satisfying (2.3) if j = 3, the corresponding solution u of (1.1)-(1.3) satisfies ku(·, t) − u∗ (·, t)kH j (0,1) ≤ Ce−δt

354

Zhang

for any t ≥ 0. In other words, the set {u∗ (·, t), 0 ≤ t ≤ ω}, as a limit circle, forms an inertial manifold in the space H j (0, 1) for the dynamic system (1.1)(1.3). Indeed, for given compatible φ ∈ H j (0, 1), let u(x, t) be the corresponding solution. Then w(x, t) = u(x, t) − u∗ ((x, t) solves the system (4.3) with v(x, t) = u∗ (x, t) and φ∗ (x) = φ(x) − u∗ (x, 0). By Theorem 3.1, one may choose η small enough such that lim ku∗ kY 0

t→∞

t,t+T

< η1 ,

where η1 is as given in Proposition 4.1. If Proposition 4.2 holds, then there exist τ > 0 and δ > 0 such that kw(·, t)kH s (0,1) ≤ γ(kw(·, τ )kL2 (0,1) )kw(·, τ )kH j (0,1) e−δ(t−τ ) for any t ≥ τ , which yields Theorem 4.3. Thus it remains to prove Proposition 4.2. Proof of Proposition 4.2: For the solution u of (4.3) it holds that Z 1 Z 1 Z d 1 2 u (x, t)dx + u2x (0, t) + 2 u2x (x, t)dx = 2 ux (x, t)u(x, t)v(x, t)dx dt 0 0 0 for any t ≥ 0, which implies that Z 1 Z d 1 2 u (x, t)dx + 2(1 − kv(·, t)kL2 (0,1) ) u2x (x, t)dx ≤ 0 dt 0 0 for any t ≥ 0. By the assumption (4.4), there exists s > 0 such that sup (1 − kv(·, t)kL2 (0,1) ) = η > 0.

s≤t 0, let Q = W (T + s), qk = h(kT + s), Z fk = −

(k+1)T +s

W ((k + 1)T + s − τ )(uvt )x (·, τ )dτ

kT +s

and

Z g(qk ) = −

(k+1)T +s

W ((k + 1)T + s − τ )((u + v)h)x (·, τ )dτ

kT +s

for k = 0, 1, 2, · · · . We have then qk+1 = Qqk + g(qk ) + fk ,

k = 0, 1, 2, · · · .

Note that there exists a constant C1 depending only on T , v and kφkL2 (0,1) such that √ kfk kL2 (0,1) ≤ 2 T kvt kY 0 kukY 0 kT +s,(k+1)T +s

kT +s,(k+1)T +s

and kg(qk )kL2 (0,1) ≤ CT ak kqk kL2 (0,1) , where ak = ku + vkY 0

kT +s,(k+1)T +s

. Since kQk < 1 and

lim ku + vkY 0

t→∞

kT +s,(k+1)T +s

≤ η1

by the estimate (4.5) with j = 0 we have just proved, one can choose η1 and s such that, for any k ≥ 0, kQk + CT ak = α < 1 and kfk k ≤ C2 ku(·, s)kL2 (0,1) β k , where 0 ≤ β < 1 and the constant C2 depends only on kvt kCb (R+ ;L2 (0,1)) and T . As a result, for such chosen η1 and s, kqk+1 kL2 (0,1) ≤ αkqk kL2 (0,1) + C2 β k

356

Zhang

for any k ≥ 0, which leads to kqk+1 kL2 (0,1) ≤ αk kq0 kL2 (0,1) + C2 ku(·, s)kL2 (0,1)

k X

αj β k−j

j=0

or kqk+1 kL2 (0,1) ≤ αk kq0 kL2 (0,1) + C2 kη k ku(·, s)kL2 (0,1) = αk kut (·, s)kL2 (0,1) + C2 kη k ku(·, s)kL2 (0,1) for any k ≥ 0 where η = min{α, β}. Hence there exists a η2 > 0 such that khkY 0

t,t+T

= kut kY 0

t,t+T

≤ γ1 kφkL2 (0,1) kφkH 3 (0,1) e−η2 t

for any t ≥ 0 where γ1 : R+ → R+ is a nondecreasing function depending only on v and T. It follows from the equation uxxx = −uux − ux + uxx − ut − (vu)x by the same argument as that used in the proof of Theorem 3.2 that kukY 3 ≤ γ kφkL2 (0,1) e−η2 t kφkH 3 (0,1) t,T +t

for any t ≥ 0. The proof is complete.

References [1] C. J. Amick, J. L. Bona and M. E. Schonbek, Decay of solutions of some nonlinear wave equations. J. Differential Equations 81 (1989), 1–49. [2] J. L. Bona and L. Luo, Decay of solutions to nonlinear, dispersive wave equations, Diff. & Integral Equ. 6 (1993), 961–980. [3] J. L. Bona, V. A. Dougalis, An initial and boundary value problem for a model equation for propagation of long waves, J. Math. Anal. Appl. 75 (1980), 503–522. [4] J. L. Bona and L. Luo, Initial-boundary-value problems for model equations for the propagation of long waves, Evolution equations (Baton Rouge, LA, 1992), 63, 65–94, Lecture Notes in Pure and Appl. Math., 168, Dekker, New York, 1995. [5] J. Bourgain, Construction of periodic solutions of nonlinear wave equations in higher dimension, Geom. Funct. Anal., 5 (1995), 105 – 140. [6] H. Brézis, Periodic solutions of nonlinear vibrating strings and duality principles, Bull. Amer. Math. Soc., 8 (1983), 409 – 425. [7] P. Constantin, C. Foiacs, B. Nicolaenko and R. Temam, Integral manifolds and inertial manifolds for dissipative partial differential equations, Applied Mathematical Sciences, 70 (1989), Springer-Verlag, New York-Berlin


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[8] W. Craig and C. E. Wayne, Newton’s method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math., 46 (1993), 1409–1498. [9] J. B. Keller and L. Ting, Periodic vibrations of systems governed by nonlinear partial differential equations, Comm. Pure and Appl. Math. (1966), 371 – 420. [10] J. U. Kim, Forced vibration of an aero-elastic plate, J. Math. Anal. Appl. 113 (1986), 454–467. [11] R. M. Miura, The Korteweg-de Vries equation: A survey of results, SIAM review 18 (1976), 412 – 459. [12] P. H. Rabinowitz, Periodic solutions of nonlinear hyperbolic differential equations, Comm. Pure Appl. Math. 20 (1967), 145 – 205. [13] P. H. Rabinowitz, Free vibrations for a semi-linear wave equation, Comm. Pure Appl. Math. 31 (1978), 31 – 68. [14] D. L. Russell and B.-Y. Zhang, Controllability and stabilizability of the thirdorder linear dispersion equation on a periodic domain, SIAM J. Contr. Optim. 31 (1993), 659–676. [15] D. L. Russell and B.-Y. Zhang, Exact controllability and stabilizability of the Korteweg-de Vries equation, Trans. Amer. Math. Soc. 348 (1996), 3643–3672 [16] D. L. Russell and B.-Y. Zhang, Smoothing and decay properties of solutions of the Korteweg-de Vries equation on a periodic domain with point dissipation, J. Math. Anal. Appl. 190 (1995), 449–488. [17] O. Vejvoda, partial differential equations: time-periodic solutions, Mrtinus Nijhoff Publishers, 1982. [18] C. E. Wayne, Periodic solutions of nonlinear partial differential equations, Notices Amer. Math. Soc. 44 (1997), 895–902. [19] C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys. 127 (1990), 479 – 528. [20] B.-Y. Zhang, Analyticity of solutions for the generalized Korteweg-de Vries equation with respect to their initial datum, SIAM J. Math. Anal. 26 (1995), 1488–1513.

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