ISNM International Series of Numerical Mathematics Vol. 133 Managing Editors: K.-H. Hoffmann, MOnchen D. Mittelmann, Tempe Associate Editors: R. E. Bank, La Jolla H. Kawarada, Chiba R. J. LeVeque, Seattle C. Verdi, Milano Honorary Editor: J.Todd,Pasadena Optimal Control of Partial Differential Equations International Conference in Chemnitz, Germany, April 20-25, 1998 Edited by K.-H. Hoffmann G. Leugering F. Troltzsch Springer Basel AG Editors: Karl-Heinz Hoffmann Giinter Leugering Zentrum Mathematik Mathematisches Institut Technische Universitat Miinchen Universitat Bayreuth ArcisstraBe 21 95440 Bayreuth 80333 Miinchen Germany Germany and Stiftung Caesar Fredi Troltzsch Friedensplatz 16 Fakultat fiir Mathematik 53111 Bonn TUChemnitz Germany 09107 Chemnitz Germany 1991 Mathematics Subject Classification 49-06, 93-06 A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Deutsche Bibliothek Cataloging-in-Publication Data Optimal control of partial differential equations : international conference in Chemnitz, Germany, April20 -25, 1998/ ed. by K.-H. Hoffmann ... -Basel ; Boston; Berlin: Birkhauser, 1999 (International series of numerica! mathematics ; VoI. 133) ISBN 978-3-0348-9731-0 ISBN 978-3-0348-8691-8 (eBook) DOI 10.1007/978-3-0348-8691-8 This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of iIIustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained. © 1999 Springer Basel AG Origina11y published by Birkhăuser Verlag in 1999 Softcover reprint of the hardcover Ist edition 1999 Printed on acid-free paper produced of chIorine-free pulp. TCF 00 Cover design: Heinz Hiltbrunner, Basel ISBN 978-3-0348-9731-0 987654321 Preface The international Conference on Optimal Control of Partial Differential Equations was held at the Wasserschloss Klaffenbach, a recently renovated moated castle near Chemnitz (Germany), from April 20 to 25,1998. This conference has been an activ ity of the IFIP working group WG7.2. The organizers great fully acknowledge the generous financial support by the Deutsche Forschungsgemeinschaft (DFG), the Siichsisches Staatsministerium fur Wissenschaft und Kunst, the European Com munity, the LUK GmbHf3 Co. (Buhl, Germany), and the Technische Universitiit Chemnitz. The conference has been attended by more than 60 scientists from 15 countries. The scientific program included 40 invited talks in the area of optimization, optimal control and controllability, shape optimization and related mechanical modelling. The underlying idea was to enhance the flow of information in this vast area of important industrial applications between theorists and numerical analysts, and, hence, the topics of the talks have been selected towards a balanced presentation of the overall field. This volume contains 27 refereed original papers which can be classified as follows: 9 papers in controllablity, stabilizability and identifiability, 7 papers in optimal control, 6 papers in shape optimization and 5 papers on general modelling and qualitative issues related to partial differential equations. About one half of the papers can be classified as theoretical work, while the other part is devoted to algorithmic procedures and their numerical implementation. It has become apparent that the theory of optimal control and optimal shapes as well as a concise numerical analysis in this area is extremely important for industrial applications. Moreover, complex processes such as fluid-structure in teractions, noise-reduction, structural mechanics, smart materials etc., in many cases imply a modelling based on partial differential equations. Typically, as large rather than infinitesimal motion becomes more and more relevant, those equations have to be taken nonlinear. It is fair to say that a concise unifying mathematical theory in this area is still lacking. N ervertheless, in recent years there has been a tremendous effort to go beyond the more classical 'linearized theories'. This is particularily true for the problem of optimal control of fluids and the control of fluid-structure interactions, where one tries e.g. to reduce the drag or the ten dency of transitions from laminar to turbulent flow regimes, or for noise-reduction problems and problems of optimal control of vibrations in large flexible struc tures in general, where one wants to actively extinguish unwanted oscillations, or for problems related to opimal melting and cristalization procedures. In many of these applications the optimization starts right on the design level, giving rise to problem of finding shapes of structures that are optimal in some sense. Moreover, in order to handle complicated dynamical behaviour numerically it is sometimes necessary to reduce the complexity of the underlying structures e.g. by applying the technique of homogenisation or to reduce the dimension of its representation e.g. by using Karhunen-Loeve approximations. vi Preface It is apparant that problems in this area constitute a grand challenge for mathe maticians, both for theorists and numerical analysts. The papers collected in this volume provide a competent 'state of the art' presentation. The organizers hope that this book will contribute to the exciting ongoing discussion and express their gratitude to the authors, to the publisher and last but not least all who have worked hard to make the conference happen and to turn it into a success, notably A. Rosch, A. Unger, and G. Weise (Chemnitz). The organizers also greatfully ac knowledge the work ofW. Rathmann (Bayreuth) who was involved in the technical preparation of this volume. Miinchen, Bayreuth and Chemnitz, 20.02.1999 K.-H. Hoffmann G. Leugering F. Troltzsch Contents Well-posedness of Semilinear Heat Equations with Iterated Logarithms Paolo Albano, Piermarco Cannarsa and Vilmos Komornik ............... 1 Uniform Stability of Nonlinear Thei'moelastic Plates with Free Boundary Conditions George Avalos, Irena Lasiecka and Roberto Triggiani .................... 13 Exponential Bases in Sobolev Spaces in Control and Observation Problems Sergei A. A vdonin, Sergei A. Ivanov and David L. Russell ............... 33 Sampling and Interpolation of Functions with Multi-Band Spectra and Controllability Problems Sergei Av donin and William Moran ..................................... 43 Discretization of the Controllability Grammian in View of Exact Boundary Control: the Case of Thin Plates Frederic Bourquin, Jose Urquiza and Rabah Namar ..................... 53 Stability of Holomorphic Semigroup Systems under Nonlinear Boundary Perturbations Francesca Bucci ......................................................... 63 Shape Control in Hyperbolic Problems John Cagnol and Jean-Paul Zolesio ...................................... 77 Second Order Optimality Conditions for Some Control Problems of Semilinear Elliptic Equations with Integral State Constraints Eduardo Casas .......................................................... 89 Intrinsic P(2, 1) Thin Shell Models and Naghdi's Models without A Priori Assumption on the Stress Tensor Michel C. Delfour ....................................................... 99 On the Approximate Controllability for some Explosive Parabolic Problems J.I. Diaz and J.L. Lions ................................................. 115 Frechet-Differentiability and Sufficient Optimality Conditions for Shape Functionals Karsten Eppler .......................................................... 133 State Constrained Optimal Control for some Quasilinear Parabolic Equations Luis A. Fernandez ....................................................... 145 Controllability property for the Navier-Stokes equations Andrei V. Fursikov ...................................................... 157 viii Contents Shape Sensitivity and Large Deformation of the Domain for Norton-Hoff Flows Nicolas Gomez and Jean-Paul Zolesio .................................... 167 On a Distributed Control Law with an Application to the Control of Unsteady Flow around a Cylinder Michael Hinze and Andreas Kauffmann .................................. 177 Homogenization of a Model Describing Vibration of Nonlinear Thin Plates Excited by Piezopatches K.-H. Hoffmann and N. D. Botkin ...................................... 191 Stabilization of the Dynamic System of Elasticity by Nonlinear Boundary Feedback Mary Ann Horn ........................................................ 201 Griffith Formula and Rice-Cherepanov's Integral for Elliptic Equations with Unilateral Conditions in Nonsmooth Domains A.M. Khludnev and J. Sokolowski ....................................... 211 A Domain Optimization Problem for a Nonlinear Thermoelastic System A. Myslinski and F. Tr6ltzsch ........................................... 221 Approximate Controllability for a Hydro-Elastic Model in a Rectangular Domain Axel Osses and Jean-Pierre Puel ......................................... 231 Noncooperative Games with Elliptic Systems Tomas Roubicek ......................................................... 245 Incomplete Indefinite Decompositions as Multigrid Smoothers for KKT Systems Volker H. Schulz ......................................................... 257 Domain Optimization for the Navier-Stokes Equations by an Embedding Domain Method Thomas Slawig .......................................................... 267 On the Approximation and Optimization of Fourth Order Elliptic Systems J. Sprekels and D. Tiba ................................................. 277 On the Existence and Approximation of Solutions for the Optimal Control of Nonlinear Hyperbolic Conservation Laws Stefan Ulbrich ........................................................... 287 Identification of Memory Kernels in Heat Conduction and Viscoelasticity L. v. Wolfersdorf and J. Janno ........................................... 301 Variational Formulation for Incompressible Euler Equation by Weak Shape Evolution Jean-Paul Zolesio ........................................................ 309 International Series of Numerical Mathematics Vol. 133, © 1999 Birkhiiuser Verlag BaseVSwitzerland Well-posedness of Semilinear Heat Equations with Iterated Logarithms Paolo Albano, Piermarco Cannarsa and Vilmos Komornik Abstract. A global existence and uniqueness result is obtained for a semilinear parabolic equation of the form Ut = D.u + feu) + h. The nonlinear term f is assumed to satisfy a suitable growth condition at 00, that allows superlinear growth of If I· An example is given to show the optimality of our growth assumption on f. 1. Introduction Consider the problem ut=°Ll u+f(u)+h in Ox (O,T), { u = on r x (0, T), (1) u(O) = Uo in 0, where • 0 is a nonempty bounded open domain in ]RN having a boundary r of class C2; • f:]R -+ ]R is a function of class C1; • T is a positive number. We study the existence of a unique global solution u E C([O, T]; L2(0)) (2) for every given (3) It is well known that, if f has sublinear growth at 00, then problem (1) has a unique global solution u. If, on the other hand, f has a faster growth at 00, then the solution of (1) may blow up in finite time, unless suitable sign conditions are assumed on f. Such conditions are usually of the form 8f(8) ~ C(l + 82), (4) for all 8 E ]R and some C > o. This paper aims to obtain a global existence and uniqueness result for problem (1), allowing the right-hand side of condition (4) 2 P. Albano, P. Cannarsa and V. Komornik to grow faster than quadratically at 00. For this purpose, we will use a method introduced in [2] for hyperbolic problems. To be more precise, let us introduce the iterated logarithm functions logj defined by the formulae logo S := sand logj S := 10g(lOgj_l s), j = 1,2, .... Let us also define the iterated exponential functions eXPj : IR --> IR as expo s = sand eXPj s = exp(eXPj_l s), j = 1,2, .... Given an integer k 2:: 0, we set ek = eXPk 1. Moreover, for any k 2:: 0, we define k II Lk(S) = logj(ej + lsI), (s E 1R). (5) j=o Then, we will show that problem (1) has a unique global solution u, provided that an integer k 2:: 0 exists such that sf(s) :s; C(l + IsI)Lk(s), 'Vs E 1R, (6) for some C 2:: o. Moreover, if N > 2 we will also assume the following growth condition on 1': II'(s)1 :s; C(l + IsliT), 'Vs E R (7) We observe that no growth condition is required on f' in the case of N = 1. If it N = 2, then the exponent in (7) can be replaced by any positive number. We will also prove that the growth conditions (6)"-(c7) are optimal in the sense that, if an integer k 2:: 1 and real numbers p > 1 and > 0 exist such that f(s) 2:: '£k-l(S)log~s for all s > c, (8) then one can find an initial condition Uo E £2(0) for which (1) has no global solution. Finally, we would like to add that, using the well-posedness result proved in this paper, one can try to obtain null and approximate controllability for equation (1), even in the case of a locally distributed control, i.e. when h has support in a sub domain w c O. These controllability issues have already been addressed in [5] if f grows slower than s log lsi at 00. We leave such a topic to a future work. 2. Abstract Formulation and Preliminaries In this section we recast problem (1) as a semilinear Cauchy problem in the Hilbert space X = L2(0), denoting by 11·11 and (., .J the usual norm and the scalar product in £2(0), respectively. Let us consider the evolution equation { u'(t) = Au(t) + F(u(t)) + h(t), t E (0, T) (9) u(O) = Uo EX,