Boundary Control and Boundary Variation: Proceedings of IFIP WG 7.2 Conference Sophia Antipolis, France, October 15 – 17, 1990 PDF
Preview Boundary Control and Boundary Variation: Proceedings of IFIP WG 7.2 Conference Sophia Antipolis, France, October 15 – 17, 1990
R.J Zol6sio ).dE( yradnuoB lortnoC dna yradnuoB noitairaV Proceedings ofIFIP WG 7.2 Conference Sophia Antipolis, France, October 51 - ,71 1990 galreV-regnirpS Berlin NewYork Heidelberg London Paris Tokyo HongKong Barcelona tsepaduB AdvisoBroya rd L.D. Davisson-A.G.J. MacFarlane • H. Kwakernaak J.L. Massey "YaZ . Tsypkin" A. J. Viterbi Editor Jean Paul Zoldsio Institut Non Lindaire de Nice Universitd de Nice Sophia Antipolis Facultd des Sciences, B.P. No. 71 06108 Nice Cedex 2, France and Cenlxe de Mathdmatiques Appliqudes Ecole des Mines, B.P. 207 06904 Sophia Antipolis, France ISBN 3-540-55351-7 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-55351-7 Springer-Verlag New York Berlin Heidelberg 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 illustrations, recitation, broadcastirnegp,r oduction on microfilms or in other ways, and storage in bdaantkas . Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © International Federation for Information Processing, Geneva, Switzerland, 1992 Printed in Germany Thues e of registerenda mes, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exemptf rom the relevant protective laws and regulations and thereforef ree for generauls e. Typesetting: Camera ready by authors Offsetprinting: Mercedes-Druck, Berlin; Bookbinding: B. Hell, Berlin 60/3020 5 4 3 2 0 1 Printed on acid-free paper PREFACE This volume comprises thep roceedings of the Working Conference "Boundary Control and Boundary Variation" held in Sophia- Antipolis ( France ), october 1990. The conference was organized for the Working Group 7.2 (Computational methods for control systems described by partial differential equations) of Technical Committe 7 (Modelling and Optimization Techniques ) of the international Federation for Information Processing (IFIP). That conferencwea s the fourth one organized in south of France and sponsored by the following French institutions : Centre National de la Recherche Scientifique( CNRS) Centre de Mathrmatiques Appliqures (CMA) in Sophia Antipolis A6rospatiale, Cannes-la-Bocca. These conferences were: - june 68 IFIP 2.7 in Nice, Variation "Boundary proc. dna Control", Boundary no Veriag, Springer L.N.C.I.S. I00, .7891 december - 78 NOCMOC, conference ni ,reilleptnoM .corp noitazilibatS" of Flexible noitazimitpO,"serutcurtS .A.L,.cnlerawtfoS 8891 -january 98 proc. Montpellier, in noitazilibatS" of Flexible "erutcurtS .S.I.C.N.L, no Verlag, Springer ,741 .9891 -october 09 IFIP 2.7 proc. these Sophia-Antipolis, in The aim of these conferences is to stimulate exchange of ideas between the group working on Shape Optimization (including free boundary problems) and the group working on boundary control of hyperbolic systems ( including Stabilization ). J was helped on the organizing committee by J.P. Marmorat , CMA J. Leblond , CMA & My thanks go to the 35 authors of the contributions contained in this volume. Sophia-Antipolis, july 1991 LP. Zolrsio CONTENTS Andr6a-Novel, F. Boustany, E Conrad Control of An Overhead Crane : Stabilization of Flexibilities A. Bensoussan Exact Controllability For Linear Dynamic Systems WSiktehw Symmetric Operators 27 A. Brillard Boundary Homogeneization and Boundary Shape Optimization for a Cylinder 37 G. Buttazzo Relaxed Formulation For a Class of Shape Optimization Problems 50 P. Cannarsa, F. Gozzi On the Smoothness of the Value FunctionA long Optimal Trajectories 60 G. Chen, J. Zhou Some Boundary Control Problems and Computation For The Linear Elastotatic Kirchoff Plate OnA n ExterioDro main 82 D. Cioranescu, P. Donato, E. Zuazua Approximate Controllability for Wave Equation With Coefficients Oscillating 811 D. Cioranescu, .J Saint Jean-Paulin Truss Structures: Fourier Conditions and Eigenvalue Problem 521 F. Conrad, J. Leblond, J.P. Marmorat Boundary Control and Stabilization of the One-dimensional Wave Equation 241 G. Da Prato, J.P. Zol6sio Boundary Control for Inverse Free Boundary Problems 361 M. Delfour, J. Morgan Differentiability of Min Max and Saddle Points under relaxed Assumptions 471 .M Delfour, A. Ouansafi New Noniterative Approximations of the Old Riccati Equation Diffemtial 681 viii A. Eljendy Numerical Approach to the Exact Controllability of Systems Hyperbolic 202 D.Gatarek, J. Sokolowski Shape Sensitivity Analysis for Stochastic Evolution Equations 512 R. Glowinski Boundary Controllability Problems for Wave the and Heat snoitauqE 122 A. Henrot, M. Pierre About Critical Points of the Energy in an Electromagnetic Problem Shaping 832 I. Lasiecka Finite Dimensional compensator for Flexible Structures 352 W. Littman Remarks On Boundary Control For Polyhedral Domains And Related stluseR 272 A. Lunardi Stability of the Travelling Waves a Class in of Free Boundary Problems Arising in Theory Combustion a 285 J.P. Marmorat, G. Payre, J.P. Zolrsio A Convergent Finite Element Scheme for a Wave Equation Boundary a Moving with 297 R. Makinen, P. Neittaanmaki, D. Tiba A Boundary Controllability Approach in Optimal Problems Design Shape 309 R. Triggiani Regularity With InteriPoori nt Control . Part I : Equations Euler-Bemoulli and Wave 123 P. Villaggio, J.P. Zolrsio New Results in Shape .noitazimitpO 356 I.P. ois6loZ Shape Formulation of Free BoundaPrryo blems With Non Condition Bernoulli Linearized 362 CONTROL OF AN OVERHEAD CRANE:STABILIZATION OF FLEXIBILITIES * B. d'Andrea-Novel, F. Boustany t F. Conrad t Abstract This paper deals with the feedback stabilization of the cable of an overhead crane, by the means of the position of the platform. The well-posedness of the closed-loop PDE system with boundary control and ltomogeneous Neumann condition on part of the boundary is esta- blished, the asymptotic stabilization is proved by Lasalle's Invariance Principle for a class of simple feedbacks and decay estimates are given. Illustrative simulations are displayed. Keywords : Asymptotic Stabilization, Boundary control, Distributed Parameter Systems, Flexibilities, Decay Estimates. *Supported by a grant from EDF-DER, Chatou lCentred 'Automatique, Ecole des Mines, 35, rue St Honord 77305 Fontainebleau, Fax : 64 96 74 10 lDSpartement de Math5matiques, Universit6 Nancy I, and U.A. CNRS 750, B.P. 932 60545 Vandoeuvre les Nancy 1 Statement of the problem. We consider an overhead crane, consisting of a motorized platform moving along an horizontal beam i and equipped with a winch, around which a cable is enrolled, holding a load. (see Figure 1, and [3]) Several studies ([3, 5, 7, 8, 20]) dealt with the "Rigid Case", that is the case where the cable is supposed to be rigid (and generally with negligible mass). The system can then be seen as a pendulum with variable length and mobile basis. The aim is to stabilize the load as quickly as possible, or to make it follow a given trajectory as precisely as possible. The actuators are the force xu applied by the motor to the platform, and the torque ~u applied on the winch. When using the non-linear approach, one must make the assumption that the whole state is observed, i.e. : the position of the platform, the length of the cable and the angle of the cable with the vertical axis, as well as their derivatives. Xp X Ul P Z (X, )Z gm Figure 1 : The ov!rhead crane t In [3, ]5 it has been proved that this mechanical system, with less actuators than degrees of freedom, can be completely linearized by using dynamic non linear state feedback. IThis beam si assumed here to be fixed, os ew are concerned with a lanoisnemid-owt problem 3 In this work, we focus on the cable. We take into account its mass and its flexibilities. We make the following assumptions : Assumptions H1 : • The cable is completely flexible and non-stretching. • The length of the cable is constant. • Displacements are small. • The angle of the cable with the vertical axis is small everywhere along the cable. • Dynamics of the platform is ignored, that is, the control is supposed to be the position or equivalently, the velocity of the platform, and not the force .xu Remark. For large displacements and length variation we can make use of the dynamic feedback linearization result for the rigid case ,3[ .]5 Then, close to the final desired position, and for the adequate length of the cable, assumptions H1 make sense and the objective is to stabilize the flexible modes of the cable. The equation of heavy cables. We denote by s the curvilinear abscissa, x(s, t) and by the position at time t of the point which has curvilinear abscissa s. (see Figure 2) 4 8=L s, t) e2 0 re1 Figure 2 : Motion of the cable Let r(s, t) be the tension along the cable and p the lineic mass of the cable; the equation of heavy cables is the following ([17]) : pxtt(s,t) = pg + (r(s,t)x,), (1) where ~z (resp. xt) denotes the derivative of z with respect to s (resp. to t). The non-stretching condition gives : (2) x,(s,t).x,(s,t) = 1 Assumptions H1 enable us to study the tangent linearization of the sys- tem. To do this, we take the following equilibrium as reference : Xref -" .5e2 • ot(s) = (m + sp)g (3) and we set : x(s,t) = x,e.~ + y (4) r(s,t) rye!+6 Replacing x and r in (1) and (2), keeping only first-order terms and project- ing upon the horizontal (el) axis (longitudinal vibrations are second-order quantities), we obtain the following system : yu - (aye), = 0 y.(0, t)=0 (5) y(L,t) = Xp(t,) where a(~) = g(~ + ~) (6) The first boundary condition expresses the verticality of the cable at its lowest end, due to the load; the second one means that the cable is, at its highest end, fixed to the platform. Remark. Given Xp, the solution of system (5) can be explicitely written, using the eigenfunctions of the associated homogeneous problem, which are Bessel functions. The equation describing the movement of the platform completes the dynamical model: MXp = (m + pL)gO + uz (7) where (rn + pL)g si an approximation of the tension, and O is an approxima- tion of sinO. System (5) with (7) leads to a hybrid PDE-ODE system (see [2]). Here (see Assumption H1), we study the simpler problem of stabilizing the cable, ignoring the dynamics of the platform. Consider the following energy function : E(,) = ~Jol V,ty,+~ ay~)d~ +k~y(L,t)~ , k > 0 (8) The integral is the internal energy of the cable, and the last factor could be replaced by (y(L, t) - xc) ,2 so that the platform could be steered to any equilibrium position x¢. A formal computation of the time derivative of E gives : dE [ay.(L,t) + ky(L,t)] (9) dt = y,(i,t)
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