Boundary Stabilization of Thin Plates
SIAM Studies in Applied Mathematics MARSHALL SLEMROD, Managing Editor This series of monographs focuses on mathematics and its applications to problems of current concern to industry, government, and society. These monographs will be of interest to applied mathematicians, numerical analysts, statisticians, engineers, and scientists who have an active need to learn useful methodology.
Series List Vol. 1. Lie-Backlund Transformations in Applications Robert L. Anderson and Nail H. Abragimov Vol. 2. Methods and Applications of Interval Analysis Ramon E. Moore Vol. 3. Ill-Posed Problems for Integrodifferential Equations in Mechanics and Electromagnetic Theory Frederick Bloom Vol. 4. Solitons and the Inverse Scattering Transform Mark J. Ablowitz and Harvey Segur Vol. 5. Fourier Analysis of Numerical Approximations of Hyperbolic Equations Robert Vichnevetsky and John B. Bowles Vol. 6. Numerical Solution of Elliptic Problems Garret Birkhoff and Robert E. Lynch Vol. 7. Analytical and Numerical Methods for Volterra Equations Peter Linz Vol. 8. Contact Problems in Elasticity; A Study of Variational Inequalities and Finite Element Methods N. Kikuchi and J. T. Oden Vol. 9. Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics Roland Glowinski and P. Le Tallec Vol. 10. Boundary Stabilization of Thin Plates John E. Lagnese
John E. Lagnese
Boundary Stabilization of Thin Plates
Society for Industrial and Applied Mathematics
Siam Philadelphia/1989
Copyright © 1989 by the Society for Industrial and Applied Mathematics. All rights reserved. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the Publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, Pennsylvania 19104-2688.
Library of Congress Cataloging-in-Publication Data Lagnese, J. Boundary stabilization of thin plates / John E. Lagnese. p. cm. — (SIAM studies in applied mathematics; 10) Bibliography: p. Includes index. ISBN 0-89871-237-8 1. Elastic plates and shells. 2. Differential equations, Partial. I. Title. II. Series. QA935.L327 1989 531'.382—dc20
89-11316 CIP
Contents
Preface
vii
Chapter 1. Introduction 1.1. 1.2. 1.3. 1.4.
1
An example Background Connection with exact controllability Overview of the remaining chapters
Chapter 2. Thin Plate Models 1.1. 1.2. 1.3. 1.4. 1.5. 1.6.
1 4 7 8
13
Kirchhoff model Mindlin-Timoshenko model von Karman model A viscoelastic plate model A linear thermoelastic plate model Comments
Chapter3. Boundary Feedback Stabilization of Mindlin-Timoshenko Plates
14 16 17 20 22 24
25
1. Orientation
25
2.Existence,uniqueness, and properties of solution
28
3. Uniform asymptotic stability of solutions Appendix
40 57
V
VI
CONTENTS
Chapter 4. Limits of the Mindlin-Timoshenko System and Asymptotic Stability of the Limit Systems 1. 2. 3. 4. 5. 6.
Orientation The limit of the M-T system as K -»0+ The limit of the M-T system as K-> oo Study of the Kirchhoff system Uniform asymptotic stability of solutions Limit of the Kirchhoff systems as y -> 0+
Chapters. Uniform Stabilization in Some Nonlinear Plate Problems
61 61 62 65 69 80 88 99
1. Uniform stabilization of the Kirchhoff system by nonlinear feedback 99 2. Uniform asymptotic energy estimates for a von Karman system 110 Chapter 6. Asymptotic Energy Estimates for Kirchhoff Plates Subject to Weak Viscoelastic Damping 119 1. Formulation of the boundary value problem 2. Existence, uniqueness, and properties of solutions 3. Asymptotic energy estimates
119 122
130
Chapter?. Uniform Asymptotic Energy Estimates for Thermoelastic Plates 151 1. Orientation 2. Existence, uniqueness, regularity, and strong stability 3. Uniform asymptotic energy estimates
151
154
161
Bibliography
171
Index
175
Preface
In recent years important progress has been made in the design and analysis of feedback stabilizers for distributed parameter systems of importance in structural dynamics. Research in this area has been driven, to a large extent, by problems in modern structural engineering which require active feedback control mechanisms to stabilize structures that may be inherently unstable in the absence of control or that may possess only very weak natural damping. Much of the progress recently seen is due to the development of new methods to analyze the stabilizing effects of specific feedback mechanisms. The purpose of this book is to present one of the main directions of current research in this area in the context of an important structural entity, the thin plate. Specifically, this book is intended to provide a comprehensive and unified treatment of asymptotic stability of the motion of a thin plate when appropriate stabilizing feedback mechanisms acting through forces and moments are introduced along a part of the edge of the plate. In particular, primary emphasis is placed in the derivation of explicit estimates of the asymptotic decay rate of the energy of the plate that are uniform with respect to the initial energy of the plate, that is, on uniform stabilization results. The method that is systematically employed throughout this book is the use of multipliers as the basis for the derivation of a priori asymptotic estimates on plate energy. This technique, insofar as it is used here, traces its origins to early work on scattering of an electromagnetic or acoustic wave from a reflecting obstacle. However, it is only in recent years that the power of the multiplier method in the context of boundary stabilization of hyperbolic partial differential equations came to be realized. One of the more interesting applications of the method appears in Chapter 5, where it is used to derive asymptotic decay rates for the energy of the nonlinear von Karman plate, even though the technique is ostensibly a linear one. vii
Vlll
PREFACE
The contents of this book will now be briefly described. Some background material is presented in Chapter 1. Chapter 2 is devoted to heuristic derivations of the various plate models which are studied in subsequent chapters. Asymptotic stability of solutions of the Mindlin-Timoshenko plate model under appropriate velocity feedback at the boundary is considered in Chapter 3. This model is a hyperbolic system of three equations which are coupled through a positive parameter called the shear modulus. Chapter 4 is concerned with the limits of the Mindlin-Timoshenko system as the shear modulus approaches either zero or infinity. The former limit leads to the system of linear plane elasticity, the latter to the Kirchhoff plate model, each with certain dissipative boundary conditions. Uniform asymptotic energy estimates for each system are obtained also in Chapter 4. Uniform stabilization in nonlinear plate problems is the subject of Chapter 5. Two situations are considered: that of a nonlinear feedback law coupled to linear dynamics; and a linear feedback law coupled to the von Karmen model. Of course, the two types of nonlinearities may be coupled, but at the expense of additional technical complications. The final two chapters are devoted to obtaining asymptotic energy estimates for viscoelastic plates with long-range memory and thermoelastic plates, respectively. It is with gratitude that I acknowledge the support and encouragement of my wife Karen during the preparation of this book. The research reported in this book was partially funded by the Air Force Office of Scientific Research under grants AFOSR86-0162 and AFOSR 88-0337. It is a pleasure to acknowledge their support. JOHN E. LAGNESE Georgetown University
Chapter 1
Introduction
Research in the stabilization of mathematical models of vibrating, flexible structures has been considerably stimulated in recent years by an increasing number of questions of practical concern. Of particular interest is the design and analysis of feedback controls that stabilize an otherwise unstable system or increase the margin of stability of a system that is already stable but may possess modes that decay at an unacceptably slow rate. Because structures of interest are spatially distributed and are often extremely flexible, models that mathematically describe their motion of necessity involve partial differential equations (pde's), partial integrodifferential equations, etc. (distributed parameter systems). However, for complex structures, the issue of modeling is far from settled. Modeling of "structural" damping and damping at joints, appropriate interface conditions for structures composed of different types of structural members, modeling of composite materials, etc., are very much at the forefront of current research. Even in cases where a "reasonable" mathematical model can be found, it may involve a large number of coupled dependent variables, nonlinearities, complicated boundary and interface conditions, and the like. A full stability analysis of such a complex mathematical system is out of reach. Consequently, research on stabilization of distributed parameter systems has largely focused on the stabilization of dynamic models of individual structural members such as strings, membranes, and beams. The main purpose of this monograph is to present a systematic study of uniform stabilization of the motion of a thin plate through the action of forces and moments applied at the edge (or on a part of the edge) of the plate. 1.1. An example. We now present a specific example of the sort of problem to be studied. Consider the classical dynamical Kirchhoff plate equation
1
2
CHAPTER 1
In (1.1) H> represents vertical deflection of the "middle surface" of the plate, h is the (uniform) plate thickness, p its (constant) mass density per unit of surface area, and D its (constant) flexural rigidity. (This and other plate models will be derived in Chapter 2.) The total energy E(t) at time ( associated with a solution of (1.1) is given by
where /* is Poisson's ratio (0<ju oo, one finds that the decay rate o)K -> 0, so that uniform asymptotic stability of the limit system cannot be deduced from the established uniform stability results for the M-T system but, rather, must be derived by a separate argument. This is also carried out in Chapter 4. Uniform stabilization in nonlinear plate problems is the subject of Chapter 5. We first consider a problem of stabilizing a linear Kirchhoff plate by the use of nonlinear velocity feedback f ( d w / d t ) in the shear force at the boundary. This is motivated by the fact that certain types of physical dampers may respond in a nonlinear way to the velocity of the object whose motion is to be stabilized. Under appropriate growth assumptions on the function /(£) at £ = 0 and at £ = oo, uniform energy decay estimates are established. We find that, in general, the energy will decay like t~" for some a > 0 depending only on the behavior of /(f) at £ = 0. There are, however, situations in which an exponential decay rate will hold, even though / is not necessarily linear. The second topic considered in Chapter 5 is boundary stabilization of the von Karman plate model. Implicit in the von Karman model is the assumption that the plate is nowhere clamped (in order to have Dirichlet boundary conditions on the Airy stress function); therefore, control is applied over the entire boundary. It is shown that the same type of linear feedbacks in shear force and moments at the edge of the plate that uniformly stabilize a nowhereclamped Kirchhoff plate also lead to uniform asymptotic energy estimates for regular solutions of the von Karman plate model. Chapter 6 deals with the question of uniform stabilization of a viscoelastic plate with long-range memory. It is well known that even quite weak viscoelastic damping induces strong asymptotic stability of the "viscoelastic energy," and that strong viscoelastic damping (characterized by an exponential decay of the "relaxation kernel") will produce uniform asymptotic stability even in the
10
CHAPTER 1
absence of any other damping mechanisms (Desch and Miller [1]). However, we shall consider situations in which the viscoelastic effects are too weak to induce uniform energy decay. We then seek to uniformly stabilize the system by the introduction of additional dissipation at the boundary acting through feedbacks of the same type that uniformly stabilize the purely elastic Kirchhoff plate. It turns out, however, that it is not possible to induce uniform viscoelastic energy decay through the use of feedback involving only the current state of the system. One may, on the other hand, derive nommiform, algebraic decay rates of the type
under asymptotic assumptions on the relaxation kernel that are much weaker than an exponential decay assumption. (Estimates like (1.17) have also been obtained by Leugering [2] in the context of a viscoelastic membrane with boundary damping.) The presence of positive boundary dissipation is crucial to our derivation of (1.17). However, it remains an interesting open question whether the asymptotic assumptions on the relaxation kernel are by themselves sufficient to infer (1.17). In Chapter 7, a uniform decay rate is established for the thermoelastic energy of a Kirchhoff plate which is subject to thermal forces that contribute to plate deformation. Energy dissipation is effected through shear force and moments applied at the boundary. However, it is apparent that such mechanisms can dissipate only mechanical energy, not thermal energy. Therefore, in order to obtain our estimates, we need to assume that the plate is thermally dissipative, a requirement which seems rather restrictive from a practical point of view. An open problem is to determine feedback laws which uniformly (or even strongly) stabilize the elastic energy (without attempting to stabilize the thermal energy) in situations where the plate is not thermally dissipative. There are many other questions related to plate stabilization that could be posed but are not studied here. (i) We consider below only homogeneous, isotropic plates. What is the situation of nonhomogeneous and/or nonisotropic plates? In particular, what can be said regarding boundary stabilization in the case of discontinuous, but piecewise constant, coefficients? (For exact boundary controllability in such "problems of transmission" see Lions [4].) (i) The geometric conditions (1.16) will be used in an essential way in the proofs of our stabilization results, but they are undoubtedly far from necessary conditions and exclude many configurations of interest for which uniform energy decay estimates should be obtainable. (See Bardos, Lebeau, and Rauch [1] where necessary and sufficient conditions on the geometry are given in the case of the wave equation, stabilized through a velocity feedback in the Neumann boundary condition.)
INTRODUCTION
11
(iii) We assume throughout that there are no external forces acting on the plate, except for the control forces. Can one still uniformly stabilize from the boundary in the presence of distributed forces (e.g., thermal forces)? In the presence of boundary forces on I"1, (e.g., a distribution of masses along Fj)? (See Littman and Markus [1], [2] where the question of stabilization of a Euler beam with an attached point mass at the free end is studied.) (iv) It has been shown for the one-dimensional wave equation that a velocity feedback acting in the Neumann boundary condition which uniformly stabilizes the system can suddenly destabilize the system if a time delay is present in the feedback law (Datko, Lagnese, and Polis [1]). Additional examples of this phenomenon have been given by Datko [2], [3] and by Desch et al. [1]. In general, the destabilized modes have a frequency of the order of the reciprocal of the delay. This effect seems to be ameliorated if there is damping (viscous, viscoelastic) present in the system (Hannsgen and Wheeler [1]). However, it is not known if the destabilizing effect of time delays is a general phenomenon, precisely how damping affects the situation, how feedback laws can be changed to moderate or eliminate such destabilizing effects, etc. (v) The issue of stabilizing the motion of composite structures (e.g., a rigid body-beam-plate configuration) has yet to be systematically studied from a distributed parameter systems point of view. Of course, the first question that must be addressed is that of modeling such structures. (A systematic attempt in this direction has been conducted by Ciarlet [2].) There are clearly many other questions which could be added to the above list. For example, there are other boundary conditions of physical interest which could be studied besides those treated in the present investigation. It may be that the methods used here carry over to other situations, but the technical "details" are far from clear. If damping (e.g., viscous, viscoelastic, Kelvin-Voight) is already present in the system, how may one quantify the increase in the margin of stability due to the introduction of dissipative effects at the boundary? What is the situation for nonplanar thin plates, i.e., thin shells? For laminates? As discussed in § 1.3, the determination of optimal stabilizing feedbacks (with respect to appropriate cost functionals) for moments and forces at the boundary remains an open question. Remark 1.2. Some of the results presented in Chapter 3 were previously announced in Lagnese [5].
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Chapter 2
Thin Plate Models
In this chapter we shall describe the thin plate models that will be studied in Chapters 3-6 and briefly discuss how they may be derived. The models are all "classical" (some more classical than others!) and have been derived in greater detail in Chapter I of Lions and Lagnese [1], to which we refer the reader for further information about these and some other plate models. Let us consider a thin plate of uniform thickness h. Points within the body will be represented by rectangular coordinates {x, y, z} or {x1; x2, x3} as convenient. It is assumed that the plate has a middle surface midway between its faces that, in equilibrium, occupies a region fl of the plane z = 0. We denote by {t/,(x, y, z, t), U2(x, y, z, t), U3(x, y, z, t)} (respectively, {u,(x, y, t), "2(x, y, t), u3(x, y, t)}) the displacement vector at time t of the particle which, when the plate is in equilibrium, occupies position {x, y, z} (respectively, position {x, y, 0}). Let Ey, cry denote the strain and stress tensors, respectively. The strain energy 3P of the plate is defined to be
(summation convention assumed), and its kinetic energy 3f is denned by
where ' = d/dt and p is the mass density per unit volume. We shall assume the plate to be homogeneous and isotropic. For an isotropic material the stress-strain relations are
where E is Young's modulus and /j, is Poisson's ratio (0 < /u. < \ in physical situations). It is customary in thin plate theory to assume that the transverse u
14
CHAPTER 2
normal stress b of the strain energy due to bending is given by
where we have set w = u3 and where D = Eh3/[12(1-p2)] is the modulus of flexural rigidity. Similarly, substitution of (1.6) into (1.2) followed by integration in z shows that X splits into Xs and 3Cb, where the kinetic energy in bending 3(b is given by
where p is the (constant) mass density per unit of volume.
THIN PLATE MODELS
15
Remark 1.1. For the Kirchhoff model and the M-T model discussed below, we shall usually treat only the problem of stabilizing the energy 8Ph + Jtb due to bending since, from a physical point of view, it is the asymptotic behavior of the vertical deflection w which is of primary interest. (See, however, Chapter 4, § 2, where the uniform stability of the equations of linear plane elasticity subject to certain dissipative boundary conditions is proved by taking the limit of a uniformly stable M-T system.) However, for the von Karman model we shall have to consider the total energy 0* + ffl since, for this model, the stretching components u l 9 u2 do not uncouple from the bending component w in the strain energy. The equations of motion for w are obtained by setting to zero the first variation of the Lagrangian
(the "Principle of Virtual Work"), where °W is the work done on the plate by external forces and Wb is the part of W that contributes to bending. We shall assume that there is no external loading on the faces of the plate but that the lateral edge is subject to a distribution of forces and moments. The variation of (1.9) is taken with respect to kinematically admissible displacements. Consider a plate that is clamped along a portion F 0 x(-/i/2, h/2) of its edge, where r 0 0. Since F is determined only up to a rigid motion a(t)x + f$(t)y + y(t), we may select {«,/3} so that VF = 0 on S, and thus F = const, on Y, for all f > 0 . We then choose y so that F = 0 on T. Therefore, we may take as boundary conditions for F
The von Karman model, which will be considered in Chapter 5, consists of equations (1.26) and (1.28) and boundary conditions (1.12) and (1.29). 1.4. A viscoelastic plate model. We consider a material occupying a volume V c |R3 and whose stress-strain law is given by
The strains eti are defined in terms of the displacements by the linear relations (1.5). The coefficients aijk, comprise a fourth-order tensor of relaxation functions such that aijkl(t) = 0 for t (t) and 3C(t), the strain energy and kinetic energy, respectively, are defined by
MINDLIN-TIMOSHENKO PLATES
27
The first step is to introduce dissipation into the system by an appropriate choice of the boundary data m l s m 2 , and m3 in (1.3). We have from (1.1)-(1.3) and Green's formula
If we define feedbacks by setting with (1.9)
matrix of real
functions satisfying
F is symmetric and positive semidefinite on Yl,
then the resulting closed-loop system is dissipative in the sense that E ( t ) is nonincreasing. The main purpose of this chapter is to study the asymptotic behavior of E ( t ) under the feedback controls (1.8) and under various conditions on the gains /-,- and the geometry of F0 and Fj. Remark 1.1. There are other choices for dissipative feedbacks besides (1.8), (1.9). For example, we could choose in place of (1.8) where Ar is the surface Laplacian on F and F = [/j,] has the same properties as F, or one might use a linear combination of (1.8) and (1.10). (Feedbacks of the latter type will be employed in Chapter 4 in order to stabilize the Kirchhoff plate.) We might even choose nonlinear feedbacks (important in applications) such as where the functions / (which may be multivalued) satisfy (for example) and are such that the resulting closed-loop system has a global solution. Stabilization under nonlinear feedback at the boundary will be considered in Chapter 5. Remark 1.2. As we shall see below, the feedbacks (1.8) are appropriate for stabilizing the system (1.1)-(1.3) only when F 0 ^0. In fact, when F0 = 0, the closed-loop system (1.1)-(1.3), (1.8) is not even strongly stable; that is, it is not true, in general, that E(t)^>0 as f-»oo. Other, less obvious, feedbacks involving both displacement {«/>, 0) on Y. We set
LEMMA 2.1. Assume
all
Then (a) there exists
such that, for all
(b) For every K0>0 there exists a(K0)>Q such that for all K
LEMMA 2.2. Assume such that, for all (b) For every and all
Then (a) for every
and
there exists
K0 and
there exists
such that for all
The estimates (2.4) and (2.6) are versions of Korn's Lemma, and (2.5) and (2.7) are simple consequences of (2.4) and (2.6), respectively. For a proof of Korn's Lemma, see Gobert [1]. A simple proof of Korn's Lemma may be found in Duvaut and Lions [1, Chap. 3], but requires more smoothness of F than we have assumed. Remark 2.1. The constants a 0 , a(K0), a(A), and a(A, K0) in (2.4)-(2.7) also depend on the flexural rigidity D, Poisson's ratio /JL, and on the region fl. With Lemmas 2.1 and 2.2 and the previously mentioned properties of the forms b and c, the well-posedness of the problem (2.1)-(2.3) is more or less standard. We may use, for example, a Galerkin approach as in Lions and Magenes [1, Chap. 3], or a semigroup approach as in Showalter [1]. Let us outline the semigroup approach. For definiteness, assume that F0 ^ 0The forms c( • ; • ) and a( •; • ) define scalar products on %C and Y, respectively, which are equivalent to the ones previously introduced in these spaces. Let C (respectively, A) denote the canonical isomorphism of 3if (respectively, Y), endowed with the scalar product c( •; • ) (respectively, a( •; •)), onto "3C (respectively, Y'). Then
30
CHAPTER 3
Further, there is a nonnegative operator B&$(Y, V) such that We identify % with its dual W. We then have the dense and continuous embeddings and we write (2.3) as
Let us set
and formally rewrite (3.5) as the system
or
where
We wish to solve (2.10) in the space Yx. 3C. In order to make sense of (2.10) in that space it is natural to introduce Then si: D(s4) -^Y'xSf. Since
Note that In particular, if we have corresponds 11 to the C ' situation.) Next, we set w = 0 in (2.18). Then we see that the {ty, 5 depends on the angle 0 0 e (0, 2 IT). The value of z is determined from a transcendental equation obtained by substituting {$, 0. It is shown in the Appendix that (a) if 0 < 60 < IT, (2.27) has no solution in the strip 0 < R e z < l ; (b) if ir oo for every solution of (2.1), (2.2).) If the resolvent of -c$~lsd is compact, application of this theory amounts to showing that -m^jtf has no spectrum on the imaginary axis. (Since —^~ls^ is dissipative, it has no spectrum in the half-plane Re (A) > 0.) We shall assume that (O, F01^) are such that - = 0, clearly u = 0 since F 0 ^ 0. Suppose that o} is a fixed point of that
We shall assume that
and
are such
for some choice of the point {x0, y0}. Assuming (3.2), we now choose the gains /-,- in (1.8) as where
Denote by G the 3 x 3 matrix G = [gy]. We assume G is positive definite on
The following result will be required in what follows. LEMMA 3.1. Let
for some
Then
42
CHAPTER 3
In (3.5) we have written i/»x for dfy/dx, etc. Remark 3.1. Each of the integrals on the right is well defined for {(/»,
0 we have
hence,
The fraction on the right assumes its minimum at r= 1/w, and for this value of T we have Remark 3.2. From (3.32) and (3.46) we see that &>-»0 as K -»oo. However, as we shall show in Chapter 4, as K^>oo the system (2.1)-(2.3) converges to a dissipative Kirchhoff system which one can prove has a positive decay rate does not depend on the individual gains gtj but rather only on the lower and upper bounds g0 and G0, respectively, we will maximize a> with respect to these quantities over the region (Kg0^G0. In addition to g0 and G0, w depends on the material parameters p, h, p, D, and K, on ft, and also on the "artificial" parameter e introduced in the course of the proof of Theorem 3.1. Therefore, we first maximize a> with respect to e (the other parameters being held fixed) and then maximize with respect to g0, G0. Let us examine the dependence of w on e. We recall that the constants e, 8 in (3.46) satisfy
where
and where Ce does not depend on g0 or G0. Since Ce is an increasing function of e for e in the interval specified in (3.50), we see that the upper bound for 77 is an increasing function of e. Also, 8 is an increasing function of 17 and the factor 5/[2(l + 5C)] in (3.46) is an increasing function of 5. It follows that, as a function of e, the largest value of CD will occur when we choose in (3.50), (3.51) the values
where C\ = C, (e) > 0 and C2 = C2( e) > 0 depend on e (and elastic and geometric constants) but not on g0 or G0. Set G 0 =rg 0 , /"=£!, so that For r fixed, we choose g0 in order to maximize 8 (and therefore ta). It is easily seen that 8 has a unique maximum at g 0 =VC 2 /> and for this g0, 8 = Ci/(2*JrC2). Finally, 8 is maximized at r = l. Therefore, the largest value of a) occurs when g 0 = G0 = -v/C^, that is, when the gain gi};=vT^ fiy, and is given by
MINDLIN-TIMOSHENKO PLATES
55
3.5. Uniform decay of solutions in other norms. 3.5.1. Strengthening the norm. Suppose that the conditions of Theorem 3.1 are met, and assume that the initial data for (1.1)-(1.3), (1.8) satisfy that is to say, {u°, u1}eD(^) in the notation of § 2.2. If u = {$, 0 and G0 > 0 be the largest and smallest numbers, respectively, such that
Since a(w; w) is strictly coercive on Hr0(O), we have from (5.21), (5.22), and trace theory
where A 0 is a constant depending on O and /x. (The last estimate uses 0 < ^ < 1; in fact, O < / A 0 so small that 2i7G0A0^|, that is,
Then from (5.26) and assumption (5.2b) we obtain the estimate
where
in view of (5.11) and the geometric assumptions (5.2a), (5.2b). Let e > 0 and introduce the function From (5.28) we have
But
provided that y = h2/l2^l, as we may assume. Use of (5.30) in (5.29) yields
provided 1 - e/2rj > 0. We choose e so small that
LIMITS OF THE MINDLIN-TIMOSHENKO SYSTEM
87
that is,
Then
where
The estimate (5.33) is analogous to (3.43) of Chapter 3. From the definition (5.18) of p ( t ) we see that
for some constant C depending on ^ and fl but not on y if 0< y ^ 1. We may now proceed exactly as in Chapter 3 and conclude that
with
As a consequence of (5.35) we obtain the estimates
Let us note that 0+ remains in a bounded set reamins in a bounded set o/ remains in a bounded set o/ where we recall the notation
Therefore, choosing a subnet of {wy \ y > 0} (still denoted by {vvy}) we obtain
From (4.7) we have
LIMITS OF THE MINDLIN-TIMOSHENKO SYSTEM
89
Using (6.2c), (6.3), and (6.4) we may pass to the limit as y-»0+ in (6.5) (in the sense of distributions on (0, T)) and we obtain
In addition, If {w°, vv^e Wx H, the problem (6.6), (6.7) admits a unique solution with Equation (6.6) indicates that w is a weak solution of the "classical" Kirchhoff plate equation and satisfies the boundary conditions
Remark 6.1. The above convergence result tacitly assumes that the form b(w\ w) is independent of y. In the context of Remark 5.4, let us assume that the matrix G satisfies (5.39), so that b = by. We then obtain from (6.1) remain in a bounded set of and, therefore, remain in a bounded set of Then, passing to the limit in (6.5) as y-»0+ through an appropriate subsequence yields
where
90
CHAPTER 4
/33 = (w v)g33. The variational equation (6.11) signifies that w is a solution of the plate equation (6.8) with the Dirichlet boundary conditions (6.9) on r0 x (0, oo) and
6.1. Uniform asymptotic stability of the system (6.8)-), a(m- *>),and)8(m-i>), respectively, where a, ft, and A are positive constants. The boundary conditions (6.16) then take the form
From (4.48) we have
6.2.1. An energy identity. To obtain uniform asymptotic energy estimates for (6.16), (6.18), we proceed along the lines of § 5. We start with initial data {w°, wl}e H4(fl) x H2(O) that satisfy the boundary relations (6.18) with w, w' replaced by w° and w 1 , respectively. Since F0 = 0, the solution w will be classical: w e C([0, oo); H4(O)), w' e C([0, oo); H2(H)), w"e C([0, oo); L2(H)). We apply the identity (5.5) to this solution and integrate (5.5) in t from 0 to T to obtain
92
CHAPTER 4
where ar(w) is defined by
Integrating by parts in t in the first term on the left and rewriting the last term on the right using (6.19), we obtain
where By the same computation as in (5.13), we have
which, when substituted into (6.22) yields
Next, substitute w = w into Green's formula (3.20) and integrate over [0, T] to obtain
LIMITS OF THE MINDLIN-TIMOSHENKO SYSTEM
93
An integration by parts in the first term on the left allows (6.25) to be rewritten
where
Multiply (6.26) by ^ and add the product to (6.24) to obtain
Uniform asymptotic energy estimates will be deduced from (6.28). 6.2.2.
A priori estimates. Define the "energy" E ( t ) by
and set
where e > 0 and
From (6.23), (6.27) we have It is clear that
94
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for a suitable constant C which depends on ft, p, a, /3, and A. We wish to establish the estimate
for some positive constant c and for all e < e0 for some e0> 0- It will be proved below that (6.34) is indeed true provided also that a and ft are sufficiently small. More precisely, it will be proved that there is a number a0 > 0, and positive numbers j80 and e0 depending on a, such that (6.34) holds if a g a 0 , p^/30, and eg e0. To establish (6.34) we first calculate p ' ( t ) using (6.28) and (6.32). The result is
Let us calculate the term (writing {x, y} = {x^, xj and using standard summation convention)
Using (6.36) in (6.35) leads to
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We estimate the last three terms on the right-hand side of (6.37) as follows. For arbitrary S > 0, 17 > 0, and cr > 0 we have
where the C/'s are independent of the parameters 5,17, a: Using (6.38)-(6.40) in (6.37) yields the estimate
We choose
Then
Next, choose
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Then if 0 < a S a 0 a n d 0 0 satisfies
where
The estimate (6.45) is analogous to the estimate (3.43) of Chapter 3, and by the same procedure employed there we obtain the following uniform asymptotic energy estimates. THEOREM 6.1. Assume that F is smooth and star-shaped with respect to some point in R2. Let w be a solution of (6.8), (6.18) with (w(0), w'(0)}e H2(fl) x L 2 (ft), where a >0, /3 >0, and A >0. There is a number a0>0 and, for each a, a number /3 0 («) > 0 such that ifa^a0 and j8 ^ /30(a) the following estimates hold:
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97
for some u> > 0. Remark 6.2. ta is given by
with C defined by (6.33) and where e satisfies (6.46). From (6.38) and (6.39) it is easy to see that the constants C, and C2 in (6.46) and (6.42), respectively, may be chosen as
Remark 6.3. The requirement of Theorem 6.1 that a and /3 be small is probably an artifact of the proof and not essential to the validity of the estimates.
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Chapter 5
Uniform Stabilization in Some Nonlinear Elastic Plate Problems
1.
Uniform stabilization of the Kirchhoff system by nonlinear feedback.
1.1. Setting the problem. We study the question of uniformly stabilizing the solution w of a Kirchhoff system by means of nonlinear feedback laws for the controls u, and v2 acting in the two boundary conditions on F,. To simplify matters, we will consider only the model equation (4.6.8), and we assume that the bending moment vt on r\ vanishes (cf. (4.6.12)). Thus the problem to be considered is
The energy of the system (1.1)-(1.3) is where a(w) = a(w; w), and where a(w; w) is given by (4.4.4). By Green's formula (4.3.20)
We set 99
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where / is a single-valued, real continuous function defined on U. Then
Let B be the nonlinear operator from W = Hr0(Q) into W defined by
Note that since dimension = 2, Hr0(il) Bw is continuous from V^ into W. In addition, the operator B is monotone if and only if / is nondecreasing, since
If / is continuous and nondecreasing on R and /(O) = 0, the problem (1.1)-(1.4) may be solved by nonlinear semigroup methods. Indeed, we may proceed as in § 4.4 of Chapter 4 and write (1.1)-(1.4) in the form where
with One verifies that si is maximal monotone on WxH. In fact if U t = {M, , uj and Uz — ("2. v2} we have
so that s4 is monotone. To see that Range (I + sg)=WxH, let {f, g}e WxH and consider the equation that is,
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Inserting «=/+i> into the second equation in (1.10) gives Since A is strictly coercive and (Bv, v)^0 on W, it is easily seen that (A + B + IY1 (defined on Range (A + B + I)) maps bounded sets in W to bounded sets in W. Since, moreover, A + B is monotone and continuous from Winto W, it follows that Range (A + B+I)= W (see, e.g., Browder[l, Chap. 1]). If v solves (1.11) and u =f+v, then from (1.11) we have Au + Bv = g~ve H. Thus{u, v}zD(d). Since s£ is maximal monotone, it follows that — si generates a continuous semigroup of (nonlinear) contractions on D(s4). Therefore, if [w°, w1} e D(stf), there is a unique function U = {H>, w'} which satisfies t^{w(t), w'(t)} is Lipschitz continuous from [0,oo) into WxH; w'(t)eW, Aw(t) + Bw'(t)eH, t>0; t^{w(t), w'(t)} is strongly right differentiable and weakly differentiable from (0,oo) into WxH; t-> Aw(t) + Bw'(t) is weakly continuous and strongly right continuous from (0, oo) into H;
w"(t) + (Bw'(t) + Aw(t)) = 0,
t>0.
For proofs of the above properties see, e.g., Brezis [1, Chap. 3]. Remark 1.1. Let us assume that/(O) = 0. Then D(jit) is dense in WxH. In fact, suppose {w, v}£ D(st). Then we W, v e W, and for some F e H. Since/(0) = 0, (1.12) signifies that w is a weak solution of the boundary value problem
Since/(0) = 0, it follows from (1.13), (1.14) that, in particular, D(s4) => W 0 x Ho(n), where we W0 means that w e //4(O) fl Wf/fl) and satisfies homogeneous boundary conditions
But W0x H20(£l) is obviously dens_e in Hr 0 (fl) x L2(fl). Remark 1.2. Suppose that r0 H Tj = 0, /(O) = 0, and / is differentiable with /'eLJUR). Then {w, v} 1. Therefore,
We apply (1.39) to the first two terms on the right-hand side of (1.38) and obtain
It follows from (1.38) and (1.40) that
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Replacing the last term in (1.33) by (1.41) gives the estimate
for some constants C, C, and C2. The last term in (1.42) is bounded above as follows. We have from (1.16)
Also, by Holder's inequality
Use of Holder's inequality once again gives
for some constant C4. Therefore, from (1.43), (1.44) we have
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Inserting the last inequality back into (1.42) gives an estimate of the form
for some constants C, through C4 and for e > 0, S > 0, and, say, 8^1. Choose 8 > 0 so small that
Then choose e > 0 so small that
We then obtain from (1.45)
as claimed. Let )3>0, multiply (1.46) by e'1", and integrate from t to oo. After an integration by parts we obtain
In view of (1.27) and (1.29) we have
Therefore, F E ( f ) g O if
Letting /3^0 in (1.47) and using (1.48) yields
where
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Proof of Corollary 1.1. We consider only the case p> 1. From (1-46), (1.48) we have
where
Upon integrating (1.50) we obtain (with a = ( p - l ) / 2 )
provided F E .(0)>0. Assume that
Then
It follows from (1.51) and (1.52) that
where
Remark 1,5. The calculations leading to the estimate (1.18) do not utilize the assumption that g ( - ) is nondecreasing. Indeed, everything formally goes through assuming that g ( - ) satisfies (1.16), (1.17), is locally Lipschitz continuous, and g(0)=0. On the other hand, the growth restriction (1.17), which was essential to the proof, should be unnecessary. An interesting question is whether utilizing monotonicity of g in a suitable manner would allow (1.17) to be dropped. Remark 1.6. Stability results of a different nature, using nonlinear feedback at the boundary, were obtained by Wang and Chen [1] in the case of the one-dimensional wave equation. Their results were greatly generalized by Lasiecka [1] to include a variety of second-order systems of Petrowsky type, including wave equations in R" and the plate equation (1.1), and a number of different possible boundary conditions. In both papers, monotone and possibly multivalued feedbacks/fw') (important for applications) were considered. The point of both papers is to characterize the w-limit set of the initial data and to establish strong convergence of solutions to the cu-limit set.
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Remark 1.7. Although we have assumed that r05* 0, the opposite case may also be handled by changing the boundary conditions (1.3), (1.5) to
and combining the techniques of § 6.2 of Chapter 4 with those of the present section. In this way one obtains estimates of the form (1.18)-(1.20) for the "augmented energy" E ( t ) defined in (4.6.29). 2.
Uniform asymptotic energy estimates for a von Karman system.
2.1. Setting the problem. We consider in this section the von Karman system (2.1.26)-(2.1.28). Upon making the change t^t^D/ph in the time scale and the notational change F -» DF, this system is brought to the form
where y = 2D/(Eh) and where the bracket [i/>, tp] denotes the bilinear expression The boundary conditions associated with (2.1) are taken to be
(Thus we assume that F0 = 0 and that Fj = F, referring to the notation used throughout these notes.) The initial conditions are No initial conditions are prescribed for F. As usual, v1 and v2 are the controls through which the energy of the system (2.1)-(2.5) is to be stabilized. For this system the energy is defined as
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The first two terms on the right-hand side are, respectively, the kinetic and strain energies of the plate in bending in the Kirchhoff theory. The remaining term is due to the fact that in-plane stress resultants contribute to the total energy in large deflection theory. Let us calculate £'(0- We have
Using Green's formula (4.3.20) and (2.1) we obtain
the last equality being due to the following lemma. LEMMA 2.1. The trilinear form ty, ([i/f, Symmetry of the form ([i/f, 0,
Remark 2.1. From (2.11) it follows that
Since the dimension is two, we have (Peetre [1])
Therefore, the trilinear terms in (2.13) and (2.14) have a sense (recall that (O, F], w) = (|>, w], F)). It also follows from (2.14) and (2.15) that w"eL°°(0, T;(H 2 (a))') (since Ll(fl)c (H2(fl))'), hence t^w'(t) is weakly continuous into L 2 (ft) (Lions and Magenes [1, Lemma 3.8.1]). Therefore, (2.12) has a sense. Moreover, from (2.11), (2.13), (2.15), and (2.16) we have
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and therefore
if e S 1, e ?4 (Lions and Magenes [1, Thm. 2.8.3]). THEOREM 2.1. // {w°, w1} e H2(fl) x L2(O), there exists a solution of (2.11)(2.14). Remark 2.2. Solutions of (2.11)-(2.14) are not, in general, unique (see Knightly and Sather [1]). The proof will be omitted since it is more or less identical to that given in Lions [1, Chap. 1, § 4], where the von Karman system (2.5) is considered with Dirichlet boundary conditions on both F and w. Remark 2.3. Regularity of solutions is (apparently) an open question. More precisely, what are conditions on the initial data {w°, w1} that guarantee that every solution of (2.11)-(2.14) is classical, that is,
If (2.17) should hold, then F is even more regular:
Inclusions (2.18a), (2.18b) are arrived at in the following manner. One has (Peetre [1])
and therefore
which implies (2.18a). Moreover,
which implies (2.18b). (For regularity of solutions of the stationary von Karman system, see Lions [1], or Ciarlet and Rabier [1].) 2.3. An energy identity. Let w, F be a solution of (2.1), (2.10) which has the regularity (2.17). We apply (4.5.5) to w and integrate the resulting identity in t from 0 to T. The result may be written
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By what is now a familiar calculation, we have
where Next, let us calculate the expression ([w, F], m • Vw) that appears in (2.19). According to Lemma 2.1, we may write But
Therefore, from (2.22), (2.23), and (2.1) we obtain
We may apply the identity (4.5.5) with p = 1 to the first term on the right-hand side of (2.24). Since F = dF/dv = 0 on T we have m - V F = 0 and d(m • VF)/dv = (m • »/)AF on T and so
On the other hand,
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Substitution of (2.25) and (2.26) into (2.24) yields
Use of (2.20) and (2.27) in (2.19) results in the identity
Let us next apply Green's formula (4.3.20) with w = w. After an integration in / from 0 to T we obtain
The first term on the left-hand side of (2.29) may be rewritten as
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Substituting (2.30) for the corresponding term in (2.29) gives
where
Multiply (2.31) by \ and add the product to (2.28) to obtain
2.4. A priori estimates. Having established the identity (2.33), the derivation of suitable a priori estimates proceeds along the lines of § 6.2.2 of Chapter 4. One introduces p(t) by
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so that
We also introduce
where E ( t ) is defined by (2.6), and we set
From (2.9) we have
and from (2.33) we have
The last three terms are bounded from above using (4.6.38)-(4.6.40). The following estimate is thereby obtained:
where ij, ) and D'x = D'(oc) both exist, that D^Q, and D'x, = 0. We shall suppose that Assumption (1.6) means that the material behaves like an elastic solid at t = +00. We now seek feedback laws for the controls vl, v2 that will induce further dissipation in the system. First, the "energy" of the system must be properly denned. The total energy can be expected to consist of two parts. One part involves the current kinetic and strain energies, and the other will involve the past history of strains. To obtain the appropriate expression, we first rewrite (1.1) and (1.3) in the form
where in (1.7) and (1.8) we have introduced the notation w,(s) = w(t-s), We multiply (1.7) by w' and integrate the product over ft. With the aid of Green's formula (4.3.20) we obtain (proceeding formally; everything will be justified in the next section)
ENERGY ESTIMATES FOR VISCOELASTIC PLATES
We have
The first term on the right-hand side of (1.10) equals
The second term on the right-hand side of (1.10) equals
Therefore,
Consequently, we have
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Substituting (1.11) into (1.9) and using the boundary conditions (1.8) yields
We define the total energy of the system (1.1)-(1.5) as
Then (1.12) may be written
The last term on the right-hand side of (1.14) represents viscoelastic damping. We may introduce additional damping through an appropriate choice of u, and v2 in the first integral on the right-hand side of (1.14). It is reasonable to choose Vi and v2 as we did in Chapter 4, § 3: where ^i,^ 2 are defined in (4.3.23), (4.3.24), and where the gain matrix F = \-ftj] is positive semidefinite on F,. We then have
Therefore, the initial-boundary value problem to be considered is (1.1)-(1.4) with the feedbacks v\, v2 defined by (1.15). 2. Existence, uniqueness, and properties of solutions. 2.1. Function spaces and variational formulation. We will adhere to the definitions of Chapter 4, § 4 of the function spaces W( = Hr0(fl)) and
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V( = Hf- 0 (ft)) and the bilinear forms a(w; w), b(w; tv), and c(w; w) (see (4.4.4)_(4.4.6)). We also define the Hilbert space of W-valued functions
