Control of Coupled Partial Differential Equations PDF
Preview Control of Coupled Partial Differential Equations
ISNM International Series of Numerical Mathematics Volume 155 Managing Editors: K.-H. Hoffmann, München 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 Control of Coupled Partial Differential Equations Karl Kunisch Günter Leugering Jürgen Sprekels Fredi Tröltzsch Editors Birkhäuser Basel · Boston · Berlin Editors: Karl Kunisch Günter Leugering Institute for Mathematics Institute for Applied Mathematics University of Graz University of Erlangen-Nürnberg Heinrichstraße 36 Martensstraße 3 A-8010 Graz D-91058 Erlangen Austria Gemany [email protected] [email protected] Jürgen Sprekels Fredi Tröltzsch Institute for Mathematics Institute for Mathematics Humboldt University Berlin Technical University Berlin Unter den Linden 6 Straße des 17 Juni 136 D-10099 Berlin D-10623 Berlin Germany Germany [email protected] [email protected] 2000 Mathematics Subject Classification: Primary 49-XX, 65K10, 93B40; Secondary 00B25, 26E25, 34H05, 35B40, 35D05, 35K45, 35L45, 35Q30, 35R30, 93B05. Library of Congress Control Number: 2007922150 Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de ISBN 978-3-7643-7720-5 Birkhäuser Verlag AG, Basel - Boston - Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustra- tions, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2007 Birkhäuser Verlag AG Basel · Boston · Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Printed in Germany ISBN-10: 3-7643-7720-8 e-ISBN-10: 3-7643-7721-6 ISBN-13: 978-3-7643-7720-5 e-ISBN-13: 978-3-7643-7721-2 9 8 7 6 5 4 3 2 1 www.birkhauser.ch Contents Preface ................................................................... vii K. Ammari, M. Tucsnak and G. Tenenbaum A Sharp Geometric Condition for the Boundary Exponential Stabilizability of a Square Plate by Moment Feedbacks only .......... 1 V. Barbu, I. Lasiecka and R. Triggiani Local Exponential Stabilization Strategies of the Navier-Stokes Equations, d=2, 3, via Feedback Stabilization of its Linearization ..................................... 13 A. Gaevskaya, R.H.W. Hoppe, Y. Iliash and M. Kieweg Convergence Analysis of an Adaptive Finite Element Method for Distributed Control Problems with Control Constraints .......... 47 M. Gugat Optimal Boundary Control in Flood Management .................... 69 V. Heuveline and H. Nam-Dung On Two Numerical Approaches for the Boundary Control Stabilization of Semi-linear Parabolic Systems: A Comparison ........ 95 M. Hintermu¨ller, S. Volkwein and F. Diwoky Fast Solution Techniques in Constrained Optimal Boundary Control of the Semilinear Heat Equation ............................. 119 M. Hinze and U. Matthes Optimal and Model Predictive Control of the Boussinesq Approximation ........................................... 149 K. Ito and K. Kunisch Applications of Semi-smooth Newton Methods to Variational Inequalities .............................................. 175 vi Contents B. Kaltenbacher Identification of Nonlinear Coefficients in Hyperbolic PDEs, with Application to Piezoelectricity .................................. 193 C. Meyer An SQP Active Set Method for a Semilinear Optimal Control Problem with Nonlocal Radiation Interface Conditions ............... 217 P.I. Plotnikov and J. Sokolowski Shape Optimization for Navier-Stokes Equations ..................... 249 J.-P. Raymond A Family of Stabilization Problems for the Oseen Equations ......... 269 G. Turinici Beyond Bilinear Controllability: Applications to Quantum Control .................................................... 293 D. Wachsmuth Optimal Control Problems with Convex Control Constraints ......... 311 J.-P. Zol´esio Control of Moving Domains, Shape Stabilization and Variational Tube Formulations ....................................... 329 Preface The international Conference on Optimal Control of Coupled Systems of Partial Differential Equations was held at the Mathematisches Forschungsinstitut Ober- wolfach(www.mfo.de) fromApril, 17to 23,2005.The scientific programincluded 30talkscoveringvarioustopicsascontrollability,feedback-control,optimalitysys- tems, model-reduction techniques, analysis and optimal control of flow problems and fluid-structure interactions, as well as problems of shape and topology opti- mization. The applications discussed during the conference range from the opti- mization and control of quantum mechanical systems, the design of piezo-electric acoustic micro-mechanical devices, optimal control of crystal growth, the control of bodies immersed into a fluid to airfoil design and much more. Thus the appli- cations are across all time and length scales. Optimization and control of systems governed by partial differential equa- tionsandmorerecentlybyvariationalinequalitiesisaveryactivefieldofresearch in Applied Mathematics, in particular in numerical analysis, scientific comput- ing and optimization. In order to able to handle real-world applications, scalable and parallelizable algorithms have to be designed, implemented and validated. This requires an in-depth understanding of both the theoretical properties and the numerical realization of such structural insights. Therefore, a ‘core’ develop- ment within the field of optimization with PDE-constraints such as the analysis of control-and-state-constrainedproblems, the role of obstacles, multi-phases etc. and an interdisciplinary ‘diagonal’ bridging regarding applications and numerical simulation are most important. The aim of the conference, therefore, was to bring together applied mathe- maticiansandalsoengineersinordertoprovideastate-of-the-artandtoestablish new standards in the field. It became apparent that the analysis of state-con- strained nonlinear optimal control problems, of such problems governedby varia- tionalinequalitiesandtheanalysisoffreeboundaryvalueproblemsareakeyissues. Moreover, shape and topology optimization becomes critical in material sciences, light-weight materials, complex chambers and flexible structures. Shape-calculus in combination with top-level optimization algorithms and in particular the com- binationoftopologicalandshapegradientsaresubjecttoanalysisandsimulation. The editors express their gratitude to the contributors of this volume, the Oberwolfach Institute, and the Birkha¨user-Verlagfor publishing this volume. K. Kunisch, G. Leugering, J. Sprekels and F. Tro¨ltzsch International SeriesofNumericalMathematics, Vol.155,1–11 (cid:1)c 2007Birkh¨auserVerlagBasel/Switzerland A Sharp Geometric Condition for the Boundary Exponential Stabilizability of a Square Plate by Moment Feedbacks only K. Ammari, M. Tucsnak and G. Tenenbaum Abstract. We consider a boundary stabilization problem for the plate equa- tioninasquare.Thefeedbacklawgivesthebendingmomentonapartofthe boundaryasfunctionofthevelocityfieldoftheplate.Themain resultofthe paper asserts that the obtained closed loop system is exponentially stable if andonlyifthecontrolled partoftheboundarycontainsaverticalandahor- izontalpart ofnon-zerolength (thegeometric opticscondition introducedby Bardos, Lebeau and Rauchin [2] for thewave equation is thusnot necessary inthiscase).Theproofofthemainresultusesthemethodologyintroducedin Ammari and Tucsnak [1], where the exponential stability for the closed loop problemisreducedtoanobservabilityestimateforthecorrespondinguncon- trolled system combined to a boundedness property of the transfer function of the associated open loop system. The second essential ingredient of the proof is an observability inequality recently proved by Ramdani, Takahashi, Tenenbaumand Tucsnak [7] Keywords. Boundary stabilization, Dirichlet type boundary feedback, plate equation. 1. Introduction and main results Inthis workwestudy theboundarystabilizationofasquareEuler-Bernoulliplate by means ofa feedback acting on the bending moment ona part ofthe boundary. Let us first describe the open loop control problem. Let Ω ⊂ R2 be an open boundedset representingthe domainoccupiedby the plate.We denote by∂Ωthe boundary of Ω and we assume that ∂Ω=Γ ∪Γ , where Γ , Γ are open subsets 0 1 0 1 of ∂Ω with Γ ∩Γ = ∅. The transverse displacement of the plate at the point x 0 1 and at the moment t will be denoted by w(x,t). We assume that ∂Ω is fixed, that the plate is simply supported on Γ and that a bending moment (the control) is 0 acting on Γ . 1 2 K. Ammari, M. Tucsnak and G. Tenenbaum With the above notation, the system modelling the vibrations of the plate with boundary control acting on the moment can be written as w¨+∆2w=0, x∈Ω, t>0, (1.1) w(x,t)=0, x∈∂Ω, t>0, (1.2) ∆w(x,t)=0, x∈Γ , t>0 (1.3) 0 ∆w(x,t)=u(x,t), x∈Γ , t>0 (1.4) 1 w(x,0)=w (x), w˙(x,0)=w (x), x∈Ω, (1.5) 0 1 where we have denoted by a dot differentiation with respect to the time t and ν stands for the unit normal vector of ∂Ω pointing towards the exterior of Ω. It is known (see, for instance, Lasiecka and Triggiani [4]) that for any input function u ∈ L2 (0,∞;L2(Γ )) the system (1.1)–(1.5) admits a unique solution loc 1 w ∈ C([0,∞);H1(Ω))∩C1([0,∞);H−1(Ω)) (this result has been proved for any 0 smooth domain Ω). The controllability of the dynamical system determined by (1.1)–(1.5) has been investigated in several works such as Krabs, Leugering and Seidman[3],Leugering[6],[4],Lebeau[5]andin[7].In[4]theexactcontrollability has beenestablishedinthe casewhen thecontrolis activeonthe wholeboundary whereas in [5] the controlled part of the boundary was supposed to satisfy the geometric optics condition of Bardos, Lebeau and Rauch. In [7] the exact con- trollability of the system (1.1)–(1.5) has been established under the assumption that Ω is a square and under a much weakerassumption on the controlledpart of the boundary (Γ is only supposed to contain non-empty vertical and horizontal 1 subsets). The mainresultofthe paperconcernsasystemobtainedbygivingthe input u in (1.9) as function of w˙. More precisely, we consider the equations w¨+∆2w=0, x∈Ω, t>0, (1.6) w(x,t)=0, x∈∂Ω, t>0, (1.7) ∆w(x,t)=0, x∈Γ , t>0 (1.8) 0 ∂ ∆w(x,t)=− (Gw˙), x∈Γ , t>0 (1.9) ∂ν 1 w(x,0)=w (x), w˙(x,0)=w (x), x∈Ω. (1.10) 0 1 The operator G in (1.9) is defined as A−1, where A : H1(Ω) → H−1(Ω) is 0 0 0 defined by A ϕ = −∆ϕ for all ϕ ∈ H1(Ω). The system (1.6)–(1.10) is obtained 0 0 from (1.1)–(1.5) by giving the control u in the feedback form ∂ u(x,t)=− (Gw˙), x∈Γ , t>0. ∂ν 1 This choice of the feedback law is the simplest one which makes the mapping t (cid:8)→ (cid:9)(w(·,t),w˙(·,t)(cid:9)2 decreasing. The concept of solution of (1.6)–(1.10) H1×H−1 0 will be made precise in Section 3. In the same section we also give a proof of the following result.
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