Differential Equations & Control Theory

differential equations and control theory

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

differential equations and control theory

edited by Sergiu Aizicovici Nicolae H. Pavel Ohio University Athens, Ohio

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Additional Volumes in Preparation

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Preface

This volume is based on papers presented at the International Workshop on Differential

Equations and Optimal Control, held at the Department of Mathematics of Ohio University in Athens, Ohio. The main objective of this international meeting was to feature new trends in the theory and applications of partial differential and functional-differential equations and their optimal control. The workshop can be viewed as a follow-up to the March 1993 Ohio University International Conference on Optimal Control of Differential Equations, whose proceedings were edited by N. H. Pavel and also published by Marcel Dekker, Inc., as volume 160 of the series: Lecture Note in Pure and Applied Mathematics. A large variety of related topics is covered in this volume, both theoretical and applied, deterministic and stochastic. The topics include: nonlinear programming and control with closed range operators, stabilization of the diffusion equations, flow-invariant sets with respect to the Navier-Stokes equations, numerical approximation of the Riccati equation, telegraph systems, dispersive equations, viable domains for differential equations, almost periodic solutions to neutral functional equations, Wentzell boundary conditions, parabolic phase-field models with

memory, Kato classes of distributions, optimal control and algebraic Riccati equations, identification problems for wave equations via optimal control, integrodifferential and other functional equations, stochastic Navier-Stokes equations, Lavrentiev phenomena, volatility for American options via optimal control, obstacle problems, necessary conditions of optimality for semilinear problems, Lyapunov stability, least action for ,/V-body problems, and more. The workshop was sponsored by the College of Arts and Sciences, the Department of Mathematics, and the Research Office of Ohio University. We gratefully acknowledge their financial support, which made this workshop possible. We are also very indebted to all participants and contributors. Finally, our thanks go also to Marcel Dekker, Inc., for undertaking the publication of this volume. Sergiu Aizicovici Nicolae H. Pavel

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

Contents

Preface Contributors

1.

Existence and Uniqueness of Solutions to a Second Order Nonlinear Nonlocal Hyperbolic Equation Azmy S. Ackleh, Sergiu Aizicovici, Michael Demetriou, and Simeon Reich

2.

Fully Nonlinear Programming Problems with Closed Range Operators Sergiu Aizicovici, D. Motreanu, and Nicolae H. Pavel

3.

Internal Stabilization of the Diffusion Equation Laura-Iulia Anita and Sebastian Ani/a

4.

Flow-Invariant Sets with Respect to Navier-Stokes Equation V. Barbu and Nicolae H. Pavel

5.

Numerical Approximation of the Riccati Equation via Fractional Steps Method Tudor Barbu and Costica Moro§anu

6.

Asymptotic Analysis of the Telegraph System with Nonlinear Boundary Conditions L. Barbu, E. Cosma, Gh. Moroganu, and W. L. Wendland

1.

Global Existence for a Class of Dispersive Equations Radu C. Cascaval

8.

Viable Domains for Differential Equations Governed by Caratheodory Perturbations of Nonlinear m-Accretive Operators Ovidiu Carjd and loan L Vrabie

9.

Almost Periodic Solutions to Neutral Functional Equations C. Corduneanu

10.

The One Dimensional Wave Equation with Wentzell Boundary Conditions Angela Favini, Gisele Ruiz Goldstein, Jerome A. Goldstein, and Silvia Romanelli

11.

On the Longterm Behaviour of a Parabolic Phase-Field Model with Memory Maurizio Grasselli and Vittorino Pata

12.

On the Kato Classes of Distributions and the BMO-Classes Archil Gulisashvili

13.

The Global Solution Set for a Class of Semilinear Problems Philip Korman

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

14.

Optimal Control and Algebraic Riccati Equations under Singular Estimates for eAlB in the Absence of Analyticity. Part I: The Stable Case Irena Lasiecka and Roberta Triggiani

15.

Solving Identification Problems for the Wave Equation by Optimal Control Methods Suzanne Lenhart and Vladimir Protopopescu

16.

Singular Perturbations and Approximations for Integrodifferential Equations J. Liu, J. Sochacki, and P. Dostert

17.

Remarks on Impulse Control Problems for the Stochastic Navier-Stokes Equations J. L. Menaldi and S. S. Sritharan

18.

Recent Progress on the Lavrentiev Phenomenon with Applications Victor J. Mizel

19.

Abstract Eigenvalue Problem for Monotone Operators and Applications to Differential Operators Silviu Sburlan

20.

Implied Volatility for American Options via Optimal Control and Fast Numerical Solutions of Obstacle Problems

Srdjan Stojanovic 21.

First Order Necessary Conditions of Optimality for Semilinear Optimal Control Problems M. D. Voisei

22.

Lyapunov Equation and the Stability of Nonautonomous Evolution Equations in Hilbert Spaces Quoc-Phong Vu and Siu Pang Yung

23.

Least Action for TV-Body Problems with Quasihomogeneous Potentials Shih-liang Wen and Shiqing Zhang

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

Contributors

Azmy S. Ackleh

University of Louisiana at Lafayette, Lafayette, Louisiana

Sergiu Aizicovici

Ohio University, Athens, Ohio

Laura-Iulia Anija

University "ALL Cuza", lasi, Romania

Sebastian Anifa

L. Barbu

University "ALL Cuza", lasi, Romania

Ovidius University, Constanja, Romania

Tudor Barbu Institute of Mathematics of Romanian Academy, lasi, Romania

V. Barbu

University of lasi, lasi, Romania

OvidiuCarja

"ALL Cuza" University of lasi, lasi, Romania

Radu C. Cascaval

C. Corduneanu

University of Missouri, Columbia, Missouri

University of Texas at Arlington, Arlington, Texas

Michael Demetriou Worcester Polytechnic Institute, Worcester, Massachusetts

P. Dostert

James Madison University, Harrisonburg, Virginia

Angelo Favini

Universita di Bologna, Bologna, Italy

Gisele Ruiz Goldstein University of Memphis, Memphis, Tennessee

Jerome A. Goldstein University of Memphis, Memphis, Tennessee Maurizio Grasselli

A. Gulisashvili

Politecnico di Milano, Milan, Italy

Ohio University, Athens, Ohio

Philip Korman University of Cincinnati, Cincinnati, Ohio Irena Lasiecka

University of Virginia, Charlottesville, Virginia

Suzanne Lenhart University of Tennessee, Knoxville, Tennessee J. Liu

James Madison University, Harrisonburg, Virginia

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

J. L. Menaldi

Wayne State University, Detroit, Michigan

Victor J. Mizel

Carnegie-Mellon University, Philadelphia, Pennsylvania

Costica Morosanu

Gh. Moroganu

University "ALL Cuza", lasi, Romania

Stuttgart University, Stuttgart, Germany

D. Motreanu University of lasi, lasi, Romania Vittorino Pata

Politecnico di Milano, Milan, Italy

Nicolae H. Pavel

Ohio University, Athens, Ohio

Vladimir Protopopescu Oak Ridge National Laboratory, Oak Ridge, Tennessee Simeon Reich The Technion-Israel Institute of Technology, Haifa, Israel Silvia Romanclli

Universita di Bari, Bari, Italy

Silviu Sburlan Ovidius University, Constantza, Romania J. Sochacki

James Madison University, Harrisonburg, Virginia

S. S. Sritharan

U.S. Navy, San Diego, California

Srdjan Stojanovic

University of Cincinnati, Cincinnati, Ohio

Roberto Triggiani

University of Virginia, Charlottesville, Virginia

M. D. Voisei

Ohio University, Athens, Ohio

loan I. Vrabie "ALL Cuza" University of lasi, lasi, Romania Quoc-Phong Vu

Ohio University, Athens, Ohio

Shih-liang Wen

Ohio University, Athens, Ohio

W. L. Wendland Siu Pang Yung

Stuttgart University, Stuttgart, Germany University of Hong Kong, Hong Kong, China

Shiqing Zhang Chongqing University, Chongqing, China

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

Existence and Uniqueness of Solutions to a Second Order Nonlinear Nonlocal Hyperbolic Equation

AZMY S. ACKLEH Department of Mathematics, University of Louisiana at Lafayette,

Lafayette, LA 70504, USA

SERGIU AIZICOVICI Department of Mathematics, Ohio University,

Athens, OH 45701, USA

MICHAEL DEMETRIOU Department of Mechanical Engineering, Worcester Polytechnic Institute,

Worcester, MA 01609, USA

SIMEON REICH Department of Mathematics, The Technion-Israel Institute of Technology, 32000 Haifa, ISRAEL

We establish existence and uniqueness of weak solutions to a class of second order distributed parameter systems with sudden changes in the input term. Such systems are often encountered in flexible structures and structure-fluid interaction systems that utilize smart actuators. A Galerkin finite dimensional approximation scheme for computing the solution of these systems is developed and its strong convergence is proved. Numerical results are also presented.

1

Introduction

In this paper we consider the nonlinear, nonlocal partial differential equation wtt + KT.WZXXX + Kzwxxxxt = [/3(x,t)g(y)]xx

+ f(x,t),

with boundary and initial conditions given by

wx(0,t)=w(0,t) = 0,

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

wx(l,t)=w(l,t)=Q,

(1.1)

In equation (1.1) the function y satisfies = \ Jo

where X[xi,x2] denotes the characteristic function on the interval [xi,X2J, with 0 < x\ < £2 < 1. The constants KI, KI and ks are positive and g is a Lipschitz continuous function. There is an extensive literature on linear and semilinear second order (in time) evolution

equations (e.g., [1, 2, 3, 4, 6, 7, 12, 13, 14, 16]). For example, the existence-uniqueness results presented in [4] apply to the system (1.!)-(!.2) for the case g = 0. However, to our knowledge, no existence-uniqueness results for the system (1.1)-(1.2) (with a nontrivial nonlinear function g) are available. Equation (1.1) is a general form of the model developed by Demetriou and Polycar-

pou [9, 10]. Indeed, in the context of the flexible structure encountered in Demetriou and

Polycarpou [9], KI denotes the stiffness

parameter, KZ the damping parameter and ks the

sensor piezoceramic constant; see Banks et al. [5] and Dosch et al. [11]. When the actuator

(input) failure term /3(x,t)g(y) is written as

P(x,t)g(y) = AW (kaX(xim](x)e(t))

g(y)

with the time profile (Polycarpou and Helmicki [15]) of the failure given by

f 0 fl(

" = {l-^»,

ift!-,•

A>

°'

< L3 >

and the nominal forcing (actuator) term given by

f ( x , t ) = [kaX[Xl,X2](x)e(t)]xx,

ka > 0,

then equation (1.1) has exactly the same form as the beam equation considered in Demetriou and Polycarpou [9]. The time T; denotes the unknown instance of the failure occurrence

and the signal e denotes the input voltage to the patch. Similarly, ka denotes the actuator

piezoceramic constant; see Banks et al. [5]. Therefore, this model describes the dynamics

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

of a flexible cantilevered beam before (t < T/) and after (t > Tf) the occurrence of an anticipated actuator failure commencing at an unknown time Tf. In view of the above, the plant equation (1.1) can now be written as follows:

wtt

Our efforts here are a continuation of an earlier work [8]. There, the following Galerkin approximations for solutions of the system (1. !)-(!. 2) were considered:

where {V'j}^! are the eigenfunctions corresponding to the eigenvalues {Xj}^ of the strictly d4 positive self adjoint operator A — -— with the dense domain in L 2 (0, 1) given by ax*

-D(A) = { 6 # 4 (0, 1) : 0'(0) = 0(0) - 0, 0'(1) - 0(1) = 0}. It is well known that the eigenvalues Xj are simple and that the set of eigenfunctions {tjjj} forms a complete orthonormal system in L 2 (0, 1). A priori bounds which are based on energy estimates were established for these Galerkin approximations. Furthermore, in order to detect, diagnose and accommodate the actuator failure, a model-based fault diagnosis scheme was presented. Our scheme consisted of a detection/diagnostic observer and an estimator of the actuator failure term. Since the proposed scheme is infinite dimensional,

a finite dimensional approximation was considered for computational purposes. The present paper is organized as follows. In Section 2 we give the definition of weak solutions to problem (1. !)-(!. 2) and establish existence-uniqueness of such solutions using a Galerkin approximation technique. The strong convergence of this approximation is also proved. In Section 3 we use the Galerkin method to give a numerical solution to a model problem.

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

2

Existence and Uniqueness of Weak Solutions

We begin this section by letting H — L2(0, 1) and V = HQ(O, 1), so we have the Gelfand

triple V ^-> H ^ V with V = #~ 2 (0, 1). We denote by {-,-) the inner product in H, while (•, -}v*,v stands for the usual duality product. Let || • ||, || • ||v, and || • \\y* denote the norms of the spaces H, V, and V , respectively. Assume that the parameters in (1. !)-(!. 2) satisfy the following conditions:

(A/}) The function j3 e L°° (Q,T,H), with ||/?|| LOO(0>T . ff) < L.

(Ag) The nonlinear function g satisfies the following Lipschitz condition: \9 (6) - 5(6)1 < ^ 16 - 61 , for all 6,6 € R, K s

where C\ < K^/L. (A;) The forcing term f e L2 (0, T; V*) . To establish the existence-uniqueness of solutions we use a Galerkin type method which is comparable to the one employed in the study of well-posedness for other second order (in time) evolution equations (see, e.g., [1, 2, 3, 4, 12, 13, 14]). To this end, we define the

space of functions

UT = (u : u 6 Wl>2(0, T; V), utt e L 2 (0, T; V*)} with norm U

UT =

i.2( 0 ,T;V)

We now define the notion of a weak solution to the problem (1. !)-(!. 2).

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

Definition 2.1 We say that a function w G UT is a weak solution of (1. !)-(!. 2) if -it satisfies >)v,V + Ki(wxx(t),(j>xx)

+ K2(wxxt(t),(j>xx)

(2.1)

= + w0 in V, w™ —>• w\ in H as m —>• oo. For each

TO we define an approximate solution to the problem (1. !)-(!. 2) by wm (t) = X^i C™(i)il)i, where wm is the unique solution to the m- dimensional system «(*), ^) + Kl(w?x(t), ^jxx) + «2« S4 (

= {/3(i) 5 (y m W), ^xx> + v-,v,

(2.3)

j = 1, 2, . . . , m,

with initial conditions

iom(0) = 10™,

iotm(0) - wf.

The function ym in equation (2.3) satisfies

ym(t)= f ksX[xi,X2](x)w™t(x,t)dx. Jo

Multiplying the equation (2.3) by ^C*j"(t) and summing up over j we obtain

(f(t),w?(t))v.,v.

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(2.4)

Hence,

Upon integrating this equality we obtain + 2

(2.5)

/o Jo Now, using the assumption (Af), the fourth term on the right hand side of (2.5) can be

bounded as follows: 2 [\f(T),W?(T))V.,VdT 0. Furthermore, note that from assumption (Ag) it follows that

\g(ymm 0. Hence, the third term on the right hand side of (2.5) satisfies the following estimate:

WQ in V, w™ —) w\ in H as m —> oo, we conclude that there exists a positive constant C independent of TO such that + K! |K (t) |2 + v «2 - LC,y

7o

||• 5 weakly in L 2 (0,T).

Note that w(0) = w 0 -

Following [13] we fix j < TO and let r\ £ Cl[0,T] with rj(T) = 0 be arbitrarily chosen.

Set rjj(t) = r)(t)ip,j and multiply both sides of (2.3) by the function rj(t). Integrating over [0, T] we obtain - Ki(w™x(t),r)jxx(t)) mt)g(ym(t)),r]JXX(t))

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

+ K2(w'xlxt(t),rjjxx(t))}dt -

Using integration by parts for the first term and letting f)j = ^TJJ we get •T 0

o

{(/3(t)g(ym(t)},rjJXX(t)}

+ (f(t)^(t))v.y}

dt +

Using the above weak convergences and taking subsequential limits as m = mk -> oo in

the previous equation we get (t),T)j(t)) + Ki(wxx(t),rijXX(t)}

+ K2(wxxt(t),rijxx(t))}dt (2.7)

o

Recalling that r]j(t) = r)(t)wj and further restricting r\ so that 77 6 Q°(0,T), we get

fT / {r, (t) (-w^t),^) + Klri (t) (wxx(t),^jxi} + K2r1(t)(wxxt(t'),tjjjxx)} CT = I rj(t){(/3(t)g(t),^xx) Jo

dt

+(f(t),^j)v,v}dt.

This implies that for each ipj,

— (w^t),^) + Ki(wxx(t),ipjxx) + K2(wxxt(t),t/j:jxx)

= (P(t}g(t),tl)jxx)

+ (f(t),tl)j)v,v. (2.8)

Since ipj is total in V we thus have that wtt e L2(0, T; V) and for all <j> £ V,

(wtt(t), 0) + KiKi(t), 0xx> + K2(wxxt(t), (f>xx) - (/3(t)g(t), 4>xx} + (f(t),4>)v>,v-

(2.9)

We already have w(0) — WQ and to argue that wt (0) = w\, we return to (2.7) which holds

for all rjj(t) = r](t)'ipj, T? e C^O,^, 7/(T) = 0. Integrating by parts the first term in (2.7) and using (2.8) we obtain

|^ = {«;!, 77X0)). From this it follows that wt (0) = wi. To prove that the limit function is indeed a weak

solution left to be shown that g(t) = g(y(t}} for a.e. t 6 [0, T]. Recall that we already

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

proved that g(ym) —> ~g weakly in Z/ 2 (0, T) (along a subsequence). Our next goal is to show

that this weak convergence is actually a strong one. To achieve this step we follow ideas developed in [7] for linear second order (in time) evolution equations, and adopted for other nonlinear second order problems in [1, 12]. we

let zm(t) = wm(t) — w ( t ) , where wm is the unique solution to the finite dimensional system (2.3)-(2.4) and w is the limit function which solves the linear problem (2.9) with w(0) = w0 and Wt(Q) = w\. Now, use the test function wm in (2.3) and the test function w in (2.9) and add and subtract terms to obtain \\z?(t)f + Kl \\z™(t)\\2 + 2K2

\\z^ (r)f dr

+2 / ( f t ( r ) ( g ( y m ( r ) ) - g ( y ( r ) ) , z ^ x ( r ) ) d r Jo. +2 '°,t +2 ( Jo

Here, (t) = 2 \-(Wt(f),W?(t))

I

-

Kl(Wxx(t),W™(t))

~ 2K2 [\Wxxr(T),W™T(T))dT

Jo

m

) + Kl(w0xx,w™x) + I (/3(r)g(y (T)),wXXT(T))dr Jo

(f3(T}g(r}^XT(r)}+'2

I v.,vdr] . ^o J

The third term on the right-hand side of (2.10) satisfies the following estimate: 2

t < 2 f \\/3(r)\\ \g(ym(r) - g(y(r))\ \\z™xx (r)\\ dr Jo _ < 2 T ^ \ym(r) - y(r)\ ||/3(r)|| \\z?xx(r)\\ dr JO

• 0. Recalling

mk

-W-+Q weakly in W ' (0, T; V) and that g ( y ) ->• 5 weakly in L 2 (0, T),

we see that Tmk(t) —> 0 because w satisfies the integrated form of (2.9). Furthermore, we also see that the third and the fourth terms on the right-hand side of the above inequality

converges to 0. Hence, we have that wmk —>• w strongly in C([0,T]; V) and that w™k —)• wt strongly in C([0, T]; H) n L 2 (0, T; V). This implies that T /•T

m

2

/ \9(y "(t))-9(ym dt< Jo

i^\2 rT

(^ \s /

f ^°

E and C : D(C] C X —> E be (possibly unbounded) closed linear operators with dense domains D(A) and D(C) in Y and X, respectively. We also consider a (Gateaux) differentiable map B : V x U —> E, where U and V are open subsets of X and Y, respectively, and a locally Lipschitz function L : W —> IR on an open subset W of Y x X containing V x U . This paper is devoted to the following nonlinear programming problem:

(Locally) Minimize L(y, u) subject to Ay = Cu + B(y, u).

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

The set of constraints for our problem (P) is

M = { ( y , u) e (D(A) n V} x (D(C) nU):Ay = Cu + B(y, u)}.

(1)

We assume M ^ 0. The tangent cone T(y,u)M of the set M in (1) at a point (y, u) G M is given by

T(y,u)M

= {(z,w)£D(A)xD(C):3p(t)-+QmY and q ( t ) -> 0 in X as * -> 0+ such that (y + t(z + p ( t ) ) , u + t(w + q ( t ) ) ) G M}

(2)

(see Aizicovici, Motreanu and Pavel [1] or Motreanu and Pavel [5], Pavel [6]). In what follows, the superscript * denotes the adjoint of a linear operator. The generalized directional derivative of L(y,u), the generalized gradient of L and the partial generalized gradients of L(y,u) with respect to the variables y and u (in the sense of Clarke [3]) are denoted by L°(y,u),dL(y,u), dyL(y,u) and duL(y,u), respectively (unless otherwise specified). Finally, B'(y,u), By(y,u) and Bu(y,u) denote the Gateaux derivative and the partial Gateaux derivatives of B(y,u), respectively. The basic hypotheses are the following:

(#1) If ( y , u ] e M, z e D(A) and w e D(C) satisfy

Az = Cw + B'(y,u)(z,w),

(3)

then (z,w) e T(y>u)M.

(Hi) For all ( y , u ) G M, R(A - By(y,u}} and R(C + Bu(y,u)) are closed in E and either R(A - By(y, u)) C R(C + Bu(y, u)) (4) or

R(C + Bu(y, u ) ) C R(A - By(y, u ) ) .

(5)

Remark 1 For our main result (Theorem 1) it is sufficient to assume that (Hi) and (Hi) hold for optimal pairs ( y , u ) of problem ( P ) , only. The following simple Lemma will be essentially used in the proof of our main result.

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

Lemma 1. Let F : U —* R be a locally Lipschitz function on an open subset of a Banach space X,Xi a subspace of X and let G be a linear functional on X\ such that:

(1°) G(v) 0 define p : [0, e] —> L 2 (Q) by />(0) = 0 and /»( 0 in L2(£l) as t —*• 0 + . This is true because

yl/(y + «) + «/'(y + u)(z

+ w)-f(y

+u + t(z

+ w))f

= \ f ( y + u)(z + w)- -(/(y + u + t(z + w)) - f(y + u))\2 t -> \ f ' ( y + u)(z + w)- f ' ( y + u)(z + w)\2 = 0 for almost all x G O as t -^ 0+

and by the Mean Value Theorem and assumption (i) we have l

-\f(y + u) + t f ' ( y + u)(z

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

+ w)-f(y

+u

+ t(z + w))\2

< ( \ f ( y + u}(z + w)\ + K\z + w\)2 < 4A'2 z + w\2 G Ll(ty. By (32) and the definition of p, we have

-A(y+tz) = f(y+u)+tf(y+u)(z+w)

= f ((y + u) + t(z + w + p(t))) , V* G [0, ,

This is because

(y + u) + i(z + w + p ( t ) ) = f - l [ f ( y + u) + t f ' ( y + u)(z + w)]). Therefore the assertion (Hi) holds true with p ( t ) = 0 and q(t) = p ( t ) . To prove that condition (#2) holds, we first verify that

To this end, we show that the problem on

= 0

has a unique solution z G HQ($I) fl # 2 (Q), for every / G I/ 2 (fi). Actually, since ii M9

— ^ i > "£ G - n 0 ( S i j ,

from assumption (i) we have the estimate

- K - KX?\\Vz\\l,(a) = (1- KX^Vzlfaw,

Vz e

Consequently, we can apply the Lax-Milgram Theorem to deduce the existence and the uniqueness of the solution z as desired. To complete the proof of condition (^2) it remains to justify that R(C +

Bu(y,u)) = R(Bu(y,u)) is closed in L 2 (0). Actually R(Bu(y,u)) = Indeed, for any g G L2(£l), jrr^—^g G L2(£l) as well. This is because 1

f'(y +

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

a.e.

n

Thus,

g = Bu(y,u] (

-i-u)9) '

i-6

' 9 ^ R(Bv(y,u)).

As a specific example of a function / satisfying the requirements in our example, we indicate

f ( t ) = kt + e g ( t ) , VteIR, l

where g (E C (IK) is such that max{|g(£)|, |g'(t)|} < c, Vt G -ffi, with constants 0 < k\ < \i and c > 0. Then, if e > 0 is sufficiently small, the conditions imposed in our example for / are satisfied. Assuming that the hypotheses in our example are fulfilled, Theorem 1 enables us to find necessary optimality conditions for the problem: Minimize /n /Oy(r) g(x, t}dx dt + \ Jn u2dx subject to f -Ay = /(y + u) i n f i \ y =0 on 9H. Here / satisfies (i) and (ii), and g G (7(0 x_R; R) satisfies the growth condition

with CQ >0, 2 2,

if TV = 1,2.

According to Theorem 1, if (y, u] is an optimal pair there exists p G HQ($}) satisfying

We get that

-Ap-u = -g(x,y) f'(y From this we deduce that

and the optimal pair (y , u) must solve the system ( -&y = f ( y + u)

}{ -A (777%^) \f(y+u)J = -g(x,y) -^ i*'

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

References [1] S. Aizicovici, D. Motreanu and N. H. Pavel (1999). Nonlinear programming problems associated with closed range operators. Appl. Math. Optimiz. 40:211-228. [2] H. Brezis (1992). Analyse Fonctionnelle. Theorie et Applications. Masson. Paris. [3] F. H. Clarke (1983). Optimization and Nonsmooth Analysis. John Wiley and Sons, New York.

[4] D.G. De Figueiredo (1989). The Ekeland Variational Principle with Applications and Detours, Springer, Berlin. [5] D. Motreanu and N. H. Pavel (1999). Tangency, Flow-Invariance for Differential Equations and Optimization Problems, Marcel Dekker, Vol. 219, New York. [6] N. H. Pavel (1984). Differential Equations, Flow-Invariance and Appli-

cations. Pitman Res. Notes Math. 113, London.

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

Internal Stabilization of the Diffusion Equation

Laura-Iulia Ani^a1 and Sebastian Ani^a 2

faculty of Physics, University "ALL Cuza", Ia§i 6600, Romania and 2 Faculty of Mathematics, University "ALL Cuza", Ia§i 6600, Romania

ABSTRACT: In this paper we analyze a stabilizability problem for the diffusion model. We provide results of stabilizability based on spatially localized control, i.e., we show that it is possible to diminish exponentially the density of a diffusive gas, by acting in a nonempty and large enough subset of the spatial domain.

1. INTRODUCTION AND SETTING OF THE PROBLEM We consider a general mathematical model describing the diffusion of a gas in a bounded domain fi C Rn (n € N*) with a smooth boundary d£l. Let y(x,t) be the density of the gas in the position x 6 fi at the moment t > 0. The diffusion is described by the following system:

yt — Ay + a(x)y = m(x}u(x,t),

( x , t ) 6 QT = n x (0,T)

y ( x , t ) = Q,

(x,t) € Sr = an x (0,T)

y(x,Q)=y0(x},

x <E n

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

(1.1)

(T €E [0,+00]). Here m is the characteristic function of UJ, where a; is a nonempty open subset satisfying uj CC fi and u(x, i) is a control function. So, this is the case when the control acts only on a subset of fi. We assume that

a 6 L°°(n);

y0 e L°°(fl),

yo(x) > 0 a.e. a: € fi.

It is well known that for any T € (0,+00), there exists u G ^ 2 ( 0. Proof. First we shall prove that A" < 0 implies that (1.1) is not "nonnegative" stabilizable. Indeed, if there exists a control acting in ID such that

and

yu(x, t)>0

a.e. in Q = fi x (0, +00),

then it is obvious that

yu(x,t)>z(x,t) a.e. (x, t) 6 (fi \ w) X (0, +00), where z is the solution to

zt - Az + a(a;)z = 0,

(x, t) 6 (fi \ w) x (0, +00)

(x, t) e (da u aw) x (o, +00)

(z,t) = o, zx,Q = y0x,

(2.2)

x 6 fi.

The solution z to (2.2) is strictly positive a.e.. On the other hand, using the Fourier development for z(t) in L 2 (fi \ w ) we may infer that

>Mie-A',

Vi>0,

where M\ > 0 is a constant. The conclusion is now obvious. If X^ > 0, then we shall prove first that there exists a positive T 6 (0, +00) and a control u €E L?(QU,T) such that

yu(x,t) > 0 a.e. in QT and

yu(x,T) = 0 a.e. x E u. Indeed, if y0(x) = 0 on a subset of cu of positive measure, then we consider the

following system:

yt - Ay + a(x)y = -m(x)p •

x € fi, t > 0

y(x,t) =0,

x<Ed$l, t>Q

y(x,Q) =y0(x),

xeQ

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

(2.3)

where p > 0 will be precised later and

sgnLi(ul}y = \\y\\ll(a}y,

if \\y\\mv) = 0.

In what follows we shall use the next auxiliary result:

Lemma 2.2. For any p > 0 large enough, there exists T G (0, +00) such that (2.3) has a unique and nonnegative solution y on the time interval (0,T) and in

addition y(x,T) — 0

a.e. x € ui.

Proof of Lemma 2.2. The operator defined by

D(A) = {y€Wt'\ty; Ay € L1^)} Ay = Ay - a(-)y,

Vy e D(A),

is the generator of a compact Co-semigroup in L1(fi) (see [6]).

For each e > 0 we consider the approximating system

lit — A.y» +' a(x]ii = —m(x]p • ,,HtfWIIil^+e' ,.?,,' —r-, V '* V /f a

x G fi,' t > 0

y(x, 0 = 0,

x e an, i > o

2/(a;,0) = yo(^),

a; e fi,

Since the application

y

(2-3)'

m-y

(form L 1 (fi) to L1(J7)) is locally Lipschitz we conclude that problem (2.3)' has a unique mild solution ye € C(R+; L J (n)). Using the comparison result for parabolic operators we conclude that if 0 < e\ < £2, then

o 0 and using the Baras compactness theorem we get that

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

for any T > 0. It follows that

inC([0,T]),for any T > 0. Let

f = Sup{T e [0, +00]; |b(£)IU'(-> > 0, Vi e [0,T]}. Passing to the limit (e —»• 0 + ) in (2.3)' we conclude that y is the solution of (2.3) on [0, T] for any 0 < T < T. Multiplying (2.3) by sgn y and integrating over fi we get - tp.

Denote by w(t) = \\y\\L1 (fix (o,t)) follows that:

an

d by a a constant satisfying a > ||a||L°°(n)- It

A*) = ll3/(*)IU'(n) < \\yo\\v(n) + az(t) - tp and so

(e-atz(W < \\y0\\L^)e-at ~ Pte~at,

Vt € [0, T].

In conclusion

-(l - e~at) - p fse^ds Cx

) -(i a

JU

e at - e~at) - 4 az + P ' (~ a +azHF)-

Thus So, for p > 0 large enough, there exists T 6 (0, +00) such that (2.3) has a unique and nonnegative solution on the time interval (0, T) and in addition

y(x,T) = 0 a.e. x € u and

Proof of Theorem 2.1 (continued). We conclude that using the feedback control

u(t) = -p-sgnLl(u}y(t), we get that

yu(x,t) > 0

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

a.e. inQ T

and

u

y

(x, T) = 0 a.e. in u.

Multiplying (2.3) by y and integrating over QT we conclude that

y(T) e L 2 (fi).

For t > T we shall use the following feedback control:

u(t] =

where p,y is the measure defined by

Here

dy , . y(x = lim ^ dv~

— y(x)

MX

where v~ is the outward normal versor to u). So, f a is a measure with support in du C UJ. The following system

' yt-&y + a(x}y = fj,y(t),

(x, t) e fi X (T, +00)

(ar, t) = 0,

(x, t)edttx (T, +00)

(x,T) = h(x)

x<Ett

(h € L 2 (fi)) has a unique weak solution y & C7([T,Ti];L 2 (n)) n J4Cr([T,T1];L2(fi \ aJ))nAC([T ; r 1 ];L 2 (a;))nL 2 (T,T 1 ;^(0))nL 2 oc (T,T 1 ;// 2 (n\cU))nLL(T,T VTj e (T,+oo) (see [3], [4]) and it satisfies

y(x, t) = 0 a.e. x E w, i > 0 and the restriction of y to (O \ a;) X [T, +00) is obviously the solution to ' yt - Ay + a(x)y = 0,

(z, t) 6 (fi \ w) X (T, +00)

y(x, t) = 0,

(x, t) £ (flfi U 0

and we get the exponential stabilization of the solution y.

3. FINAL REMARKS The main conclusion of the previous section is that the feedback control

-p-sgnLl(u})y(t),

te [0,TJ

u(t] = stabilizes the system (1-1) if and only if A" > 0. If we denote by AI the first eigenvalue of the operator

D(A) = # 0 1 (O)n# 2 (n), Ay = -Ay + a(-)y,

Vy € D(A),

then we have

AI < A?. If AI > 0, the system (1.1) can be stabilized by the trivial control u = 0. If A! < 0, then the control u = 0 does not stabilize (1.1) (for related results see [1]), but the system can be stabilized if A" > 0.

In the same manner as in Section 2 can be proved that (1.1) negative" controllable.

is not exact "non-

REFERENCES [ 1 ] L.I. Ani^a, Asymptotic behaviour of the solutions of some reaction-diffusion processes, International ,]. Appl. Math., submitted. [ 2 ] S. Anilja and V. Barbu, Local exact controllability of a reaction-diffusion system, Diff. Integral Eqs., to appear. [ 3 ] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Boston (1993).

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

[ 4 ] V. Barbu, Partial Differential Acad. Publ., Dordrecht (1998).

Equations and Boundary Value Problems. Kluwer

[ 5 ] V. Barbu, Exact controllability of the superlinear heat equation, Appl. Math. Op-

tim., to appear. [ 6 ] H. Brezis and A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl, 62 (1983), 73-97. [ 7 ] E. Fernandez-Cara, Null controllability of the semilinear heat equation, ESAIM: Control, Optim., Gale. Var., 2 (1997), 87-107. [ 8 ] G. Lebeau and L. Robbiano, Controle exact de 1'equation de la chaleur, Comm.

Partial Diff.

Eqs., 30 (1995), 335-357.

[ 9 ] J.L. Lions, Controlabilite exacte, stabilisation et perturbation de systemes distribues, RMA 8, Masson, Paris (1988). [10 ] J. Zabczyk, Mathematical Control Theory: An Introsuction, Birkhauser, Boston (1992).

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

Flow-Invariant Sets with Respect to Navier-Stokes Equation V. BARBU* AND N. H. PAVEL** * University of lasi, Department of Mathematics, 6600 lasi, Romania

**Ohio University, Department of Mathematics, Athens , Ohio 45701, USA

1.

Introduction. A new result (Theorem 2.1) on the flow-invariance of a

closed subset with respect to a differential equation associated with a nonlinear

semigroup generator on Banach spaces is given(the proof will be given in [1]).

Applications to the flow-invariance of controlled flux sets ( the Enstrophy and Helicity sets) with respect to Navier-Stokes equations are presented. Recall that K is said to be a flow invariant set with respect to a differential equation y' = Ay, if every solution y starting in K (i.e. y(0) = x 6 A') remains in K as long as it exists (i.e.

y(t) 6 K. for all t in the domain of y ] . In our

cases here, we are using the strong solutions, so actually we deal with the flow invariance of A* D D(A).

We think that the existing results on this topic are not

applicable to our cases treated here. Indeed, the general result of R.. H. Martin Jr. [8] requires the right hand side A of (2.9) to be continuous and dissipative on K. None of these key conditions (on A) are required here.

Note also that the subsets considered here are closed, but not necessarily convex. A different approach to flow-invariance of such sets with respect to

Navier-Stokes equations was given by Barbu and Sritharan [2].

Our general framework is V 0 (precisely, for \ui < 1).

The fundamental result on the generation of nonlinear semigroups S'A(t) =

S(t) is the following one (known as the exponential formula of Crandall-Liggett

[6])Let A be w-dissipative (i.e. (2.1)

holds) satisfying the range condition (2.3).

Then

lim (/ - -A)~nx = S(i) G D(A)

(2.4)

for all x 6 D(A) and t > 0. Moreover \\S(t)x - S(i)y\\ < etw\\x - y\\, Mt > 0,

x, y 6 D(A), S(t + s) = S(t)S(s), 5(0)

= /—the identity on X, lim uo S(t)x = x,

Vx 6 D(A), \\S(t)x - S(s)x\\ < \t - s \Ax exp(2^-0(t + 5)), t,s > 0, for all x G D(A), where uj0 = max{0,w} and \Ax = inf{||t/||, y e Ax}. If s -^ S(s)x is differentiable at s = t, then u(t) = S(t)x G D(A) and it is the only (strong) solution to the Cauchy problem

u'(f) = Au(t),

u(0) = a;,

x 6 D(A),

t >0

(2.5)

In our example applications here, the following hypotheses are fulfilled.

(HI) For every x £ D(A), t —*• S(t}x is differentiable at every t > 0. So:

lim ^^^ ~ ^ = Aa;,Va; € D(A)J

' (2.5

and u(i) = S(i)x is the only strong solution to the problem (2.5). Let K be a

closed subset of X. The basic hypothese on the relationship between A and K

are given below: (H2)

The projection Pjf(y}

on

K exists for all y 6 D(A). Moreover,

PK(D(X))cD(A)nK.

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

(2.6)

This means, that for every y £ -D(A), there is y0 — Pi{(y} £ D(A) fl K such that the distance d(y; K) from y to K satisfies

d(y- K) = mi{\\y -z\\,z£ K} = \\y - y0\\ = d(y; K n D(A})

(2.7)

for some y0 £ Kr\D(A). Recall also the definition of the tangential (contingent)

cone TK(X) to K at x £ K in the sense of Bouligand [4, Ch.l]

TK(x) = {v £ X, lim \d(x + hv; A') = 0} HO n

(2.8)

= {v £ X; 3r(h) £ X with r(h) -*• 0 as h j 0 and < 0} = M]f if

= 0.

Indeed,