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Exact and approximate controllability for distributed parameter systems

Published online by Cambridge University Press:  07 November 2008

R. Glowinski  and
J.L. Lions
R. Glowinski
Affiliation:
University of Houston, Houston, Texas, USAUniversité Pierre et Marie Curie, Paris, FranceC.E.R.F.A.C.S., Toulouse, France
J.L. Lions
Affiliation:
Collège de France, Rue d'Ulm, 75005 Paris, France
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Extract

This is the second part of an article which was started in the previous volume of Acta Numerica. References in the text to Section 1 refer to the preceding article.

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Type
Research Article
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Acta Numerica , Volume 4 , January 1995 , pp. 159 - 328
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Copyright © Cambridge University Press 1995

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References

REFERENCES

Abergel, F. and Temam, R. (1990), ‘On some control problems in fluid mechanics’, Theoret. Comput. Fluid Dyn. 1, 303–326.CrossRefGoogle Scholar
Auchmuty, G., Dean, E.J., Glowinski, R. and Zhang, S.C. (1987), ‘Control methods for the numerical computation of periodic solutions of autonomous differential equations’, in Control Problems for Systems Described by Partial Differential Equations and Applications (Lasiecka, I. and Triggiani, R., eds.) Lecture Notes in Control and Information, vol. 97, Springer (Berlin) 64–89.CrossRefGoogle Scholar
Bamberger, A. (1977), private communication.Google Scholar
Bardos, C., Lebeau, G. and Rauch, J. (1988), ‘Contrôle et stabilisation dans les problèmes hyperboliques’, in Contrôlabilité Exacte, Perturbation et Stabilisation de Systèmes Distribués, vol. 1 (Lions, J.L. (ed.), Masson (Paris) Appendix 2, 492–537.Google Scholar
Bardos, C. and Rauch, J. (1994), ‘Mathematical analysis of control-type numerical algorithms for the solution of the Helmholtz equation’. To appear.Google Scholar
Berggren, M. (1992), ‘Control and simulation of advection-diffusion systems’, MSc Thesis, Department of Mechanical Engineering, University of Houston.Google Scholar
Berggren, M. and Glowinski, R. (1995), ‘Numerical solution of some flow control problems’, Int. J. Fluid Dyn. To appear.Google Scholar
Bensoussan, A. (1990), Acta Appl. Math. 20, 197.CrossRefGoogle Scholar
Bourquin, F. (1993), ‘Approximation theory for the problem of exact controllability of the wave equation with boundary control, in Mathematical and Numerical Aspects of Wave Propagation (Kleinman, R., Angell, Th., Colton, D., Santosa, F. and Stakgold, I., eds.) SIAM (Philadelphia, PA) 103–112.Google Scholar
Brezzi, F. and Fortin, M. (1991), Mixed and Hybrid Finite Element Methods, Springer (New York).CrossRefGoogle Scholar
Bristeau, M.O., Erhel, J., Glowinski, R. and Periaux, J. (1993), ‘A time dependent approach to the solution of the Helmholtz equation at high wave numbers’, in Proc. Sixth SIAM Conf. on Parallel Processing for Scientific Computing, (Sincovec, R.F., Keyes, D.E., Leuze, M.R., Petzold, L.R. and Reed, D.A., eds.) SIAM (Philadelphia, PA).Google Scholar
Bristeau, M.O., Glowinski, R. and Periaux, J. (1993a), ‘Scattering waves using exact controllability methods’, 31st AIAA Aerospace Sciences Meeting, Reno, Nevada, AIAA Paper 93–0460.Google Scholar
Bristeau, M.O., Glowinski, R. and Periaux, J. (1993b), ‘Using exact controllability to solve the Helmholtz equation at high wave numbers’, in Mathematical and Numerical Aspects of Wave propagation (Kleinman, R., Angell, Th., Colton, D., Santosa, F. and Stakgold, I., eds.) SIAM (Philadelphia, PA) 113–127.Google Scholar
Bristeau, M.O., Glowinski, R. and Periaux, J. (1993c), ‘Numerical simulation of high frequency scattering waves using exact controllability methods’, in Nonlinear Hyperbolic Problems: Theoretical, Applied, and Computational Aspects (Donato, A. and Oliveri, F., eds.) Notes in Numerical Fluid Mechanics, vol. 43, Vieweg (Branschweig) 86–108.CrossRefGoogle Scholar
Burns, J.A. and Kang, S. (1991), ‘A stabilization problem for Burgers' equation with unbounded control and observation’, International Series of Numerical Mathematics, vol. 100, Birkhauser (Basel).Google Scholar
Burns, J.A. and Marrekchi, H. (1993), ‘Optimal fixed-finite-dimensional compensator for Burgers' equation with unbounded input/output operators’, ICASE Report 93–19, ICASE.Google Scholar
Buschnell, D.M. and Hefner, J.N. (1990), Viscous Drag Reduction in Boundary Layers, American Institute of Aeronautics and Astronautics (Washington, DC).CrossRefGoogle Scholar
Carthel, C. (1994), ‘Numerical methods for the boundary controllability of the heat equation’, PhD Thesis, Department of Mathematics, University of Houston.Google Scholar
Carthel, C., Glowinski, R. and Lions, J.L. (1994), ‘On exact and approximate boundary controllabilities for the heat equation: a numerical approach’, J. Opt. Theory Appl. 82, 3, 429–484.CrossRefGoogle Scholar
Chiara, A. (1993), ‘Equation des ondes et régularité sur un ouvert lipschitzien’, C.R. Acad. Sci. I 316, 33–36.Google Scholar
Ciarlet, P.G. (1978), The Finite Element Method for Elliptic Problems, North-Holland (Amsterdam).Google Scholar
Ciarlet, P.G. (1989), Introduction to Numerical Linear Algebra and Optimization, Cambridge University Press (Cambridge).CrossRefGoogle Scholar
Ciarlet, P.G. (1990a), ‘A new class of variational problems arising in the modeling of elastic multi-structures’, Numer. Math. 57, 547–560.CrossRefGoogle Scholar
Ciarlet, P.G. (1990b), Plates and Junctions in Elastic Multi-Structures: An Asymptotic Analysis, Masson (Paris).Google Scholar
Ciarlet, P.G. (1991), ‘Basic error estimates for elliptic problems’, in Handbook of Numerical Analysis, vol. II (Ciarlet, P.G. and Lions, J.L., eds.) North-Holland (Amsterdam) 17–351.Google Scholar
Ciarlet, P.G., Le Dret, H. and Nzengwa, R. (1989), ‘Junctions between three-dimensional and two-dimensional linearly elastic structures’, J. Math. Pures et Appl. 68, 261–295.Google Scholar
Crandall, M.G. and Lions, P.L. (1985), ‘Hamilton-Jacobi equations in infinite dimensions, Part I’, J. Funct. Anal. 62, 379–396.CrossRefGoogle Scholar
Crandall, M.G. and Lions, P.L. (1986a), ‘Hamilton-Jacobi equations in infinite dimensions, Part II’, J. Funct. Anal. 65, 368–405.CrossRefGoogle Scholar
Crandall, M.G. and Lions, P.L. (1986b), ‘Hamilton-Jacobi equations in infinite dimensions, Part III’, J. Funct. Anal. 68, 214–247.CrossRefGoogle Scholar
Crandall, M.G. and Lions, P.L. (1990), ‘Hamilton-Jacobi equations in infinite dimensions, Part IV’, J. Funct. Anal. 90, 237–283.CrossRefGoogle Scholar
Crandall, M.G. and Lions, P.L. (1991), ‘Hamilton-Jacobi equations in infinite dimensions, Part V’, J. Funct. Anal. 97, 417–465.CrossRefGoogle Scholar
Daniel, J. (1970), The Approximate Minimization of Functionals, Prentice Hall (Englewood Cliffs, NJ).Google Scholar
Dean, E.J. and Gubernatis, P. (1991), ‘Pointwise control of Burgers' equation – a numerical approach’, Comput. Math. Appl. 22, 93–100.CrossRefGoogle Scholar
Dean, E.J., Glowinski, R. and Li, C.H. (1989), ‘Supercomputer solution of partial differential equation problems in computational fluid dynamics and in control’, Comput. Phys. Commun. 53, 401–439.CrossRefGoogle Scholar
Dennis, J. and Schnabel, R.B. (1983), Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall (Englewood Cliffs, NJ).Google Scholar
Diaz, J.I. (1991), ‘Sobre la controlabilidad aproximada de problemas no lineales disipativos’, Jornadas Hispano-Francesas Sobre Control de Sistemas Distribuidos, Universidad de Malaga 41–48.Google Scholar
Diaz, J.I. and Fursikov, (1994), To appear.Google Scholar
Downer, J.D., Park, K.C. and Chiou, J.C. (1992), ‘Dynamics of flexible beams for multibody systems: a computational procedure’, Comput. Meth. Appl. Mech. Engrg 96, 373–408.CrossRefGoogle Scholar
Dupont, T., Glowinski, R., Kinton, W. and Wheeler, M.F. (1992), ‘Mixed finite element methods for time-dependent problems: application to control’, in Finite Element in Fluids, vol. 8 (Chung, T., ed.) Hemisphere (Washington, DC) 137–163.Google Scholar
Ekeland, I. and Temam, R. (1974), Analyse Convexe et Problèmes Variationnels, Dunod (Paris).Google Scholar
Engquist, B., Gustafsson, B. and Vreeburg, J. (1978), ‘Numerical solution of a PDE system describing a catalytic converter’, J. Comput. Phys. 27, 295–314.CrossRefGoogle Scholar
Fabre, C., Puel, J.P. and Zuazua, E. (1993), ‘Contrôlabilité approchée de l'équation de la chaleur linéaire avec des contrôles de norme L ∞ minimale’, C.R. Acad. Sci. I 316, 679–684.Google Scholar
Friedman, A. (1988), ‘Modeling catalytic converter performance’, in Mathematics in Industrial Problems, Part 4 (Friedman, A., ed.) Springer (New York) Ch. 7, 70–77.CrossRefGoogle Scholar
Friend, CM. (1993), ‘Catalysis and surfaces’, Scientific American (04) 74–79.CrossRefGoogle Scholar
Fujita, H. and Suzuki, T. (1991), ‘Evolution problems’, in Handbook of Numerical Analysis vol. II (Ciarlet, P.G. and Lions, J.L., eds.) North-Holland (Amsterdam) 789–928.Google Scholar
Fursikov, V. (1992) ‘Lagrange principle for problems of optimal control of ill posed or singular distributed systems’, J. Math. Pures et Appl. 71(2), 139–194.Google Scholar
Gabay, D. (1982), ‘Application de la méthode des multiplicateurs aux inéquations variationnelles’, in Méthodes de Lagrangien Augmenté (Fortin, M. and Glowinski, R., eds.) Dunod-Bordas, (Paris).Google Scholar
Gabay, D. (1983) ‘Application of the method of multipliers to variational inequalities’, in Augmented Lagrangian Methods (Fortin, M. and Glowinski, R., eds.) North-Holland (Amsterdam).Google Scholar
George, J.A. (1971), ‘Computer implementation of the finite element method’, PhD Thesis, Computer Sciences Department, Stanford University.Google Scholar
Glowinski, R. (1984), Numerical Methods for Nonlinear Variational Problems, Springer (New York).CrossRefGoogle Scholar
Glowinski, R. (1991), ‘Finite element methods for the numerical simulation of incompressible viscous flow. Introduction to the control of the Navier-Stokes equations’, in Lectures in Applied Mathematics, vol. 28, American Mathematical Society (Providence, RI) 219–301.Google Scholar
Glowinski, R. (1992a), ‘Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation’, J. Comput. Phys. 103, 189–221.CrossRefGoogle Scholar
Glowinski, R. (1992b), ‘Boundary controllability problems for the wave and heat equations’, in Boundary Control and Boundary Variation (Zolesio, J.P., ed.) Lecture Notes in Control and Information Sciences, vol. 178, Springer (Berlin) 221–237.CrossRefGoogle Scholar
Glowinski, R. and Le Tallec, P. (1989), Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, SIAM (Philadelphia, PA).CrossRefGoogle Scholar
Glowinski, R. and Li, C.H. (1990), ‘On the numerical implementation of the Hilbert uniqueness method for the exact boundary controllability of the wave equation’, C.R. Acad. Sci. I 311, 135–142.Google Scholar
Glowinski, R. and Li, C.H. (1991), ‘On the exact Neumann boundary controllability of the wave equation’, in Mathematical and Numerical Aspects of Wave Propagation Phenomena (Cohen, G., Halpern, L. and Joly, P., eds.) SIAM (Philadelphia, PA) 15–24.Google Scholar
Glowinski, R., Kinton, W. and Wheeler, M.F. (1989), ‘A mixed finite element formulation for the boundary controllability of the wave equation’, Int. J. Numer. Meth. Engrg 27, 623–635.CrossRefGoogle Scholar
Glowinski, R., Li, C.H. and Lions, J.L. (1990), ‘A numerical approach to the exact boundary controllability of the wave equation (I) Dirichlet controls: description of the numerical methods’, Japan. J. Appl. Math. 7, 1–76.CrossRefGoogle Scholar
Glowinski, R., Lions, J.L. and Tremolières, R. (1976), Analyse Numérique des Inéquations Variationnelles, Dunod (Paris).Google Scholar
Glowinski, R., Lions, J.L. and Tremolières, R. (1981), Numerical Analysis of Variational Inequalities, North-Holland (Amsterdam).Google Scholar
Golub, G.H. and Van Loan, C. (1989), Matrix Computations, Johns Hopkins Press (Baltimore, MD).Google Scholar
Hackbush, W. (1985), Multigrids Methods and Applications, Springer (Berlin).Google Scholar
Haraux, A. and Jaffard, S. (1991), ‘Pointwise and spectral control of plate vibrations’, Revista Matematica Iberoamericana 7, 1–24.CrossRefGoogle Scholar
Henry, J. (1978), ‘Contrôle d'un Réacteur Enzymatique à l'Aide de Modèles à Paramètres Distribués. Quelques Problèmes de Contrôlabilité de Systèmes Paraboliques’, Thèse d'État, Université P. et M. Curie.Google Scholar
Hörmander, L. (1976), Linear Partial Differential Operators, Springer (Berlin).Google Scholar
Joó, I. (1991), ‘Contrôlabilité Exacte et propriétés d'oscillations de l'équation des ondes par analyse non harmonique’, C.R. Acad. Sci. I 312, 119–122.Google Scholar
Komornik, V. (1994), Exact Controllability and Stabilization. The Multiplier Method, Masson (Paris).Google Scholar
Ladyzenskaya, O.A., Solonnikov, V.A. and Ural'ceva, N.N. (1968), Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society (Providence, RI).CrossRefGoogle Scholar
Lagnese, J.E. (1989), ‘Exact boundary controllability of Maxwell's equations in a general region’, SIAM J. Control Optimiz. 27, 374–388.CrossRefGoogle Scholar
Lagnese, J.E., Leugering, G. and Schmidt, G. (1992), ‘Modelling and controllability of networks of thin beams’, in System Modelling and Optimization, Lecture Notes in Control and Information, vol. 97, Springer (Berlin), 467–480.CrossRefGoogle Scholar
Lagnese, J.E., Leugering, G. and Schmidt, G. (1994), Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures, Binkhäuser (Boston, Basel, Berlin).CrossRefGoogle Scholar
Lasiecka, I. (1992), Exponential decay rates for the solutions of Euler Bernoulli equations with boundary dissipation occurring in the moments only. J. Diff. Equations 95, 169–182.CrossRefGoogle Scholar
Lasiecka, I. and Tataru, R. (1994), Uniform boundary stabilization of semilinear wave equations with non linear boundary conditions, Diff. and Integral Equations.Google Scholar
Laursen, T.A. and Simo, J.C. (1993), ‘A continuum-based finite element formulation for the implicit solution of multi-body, large deformation frictional contact problems’, Int. J. Numer. Meth. Engrg 36, 3451–3486.CrossRefGoogle Scholar
Lax, P.D. and Phillips, R.S. (1989), Scattering Theory, Academic (New York).Google Scholar
Lions, J.L. (1961), Equations Différentielles Opérationnelles et Problèmes aux Limites, Springer (Heidelberg).CrossRefGoogle Scholar
Lions, J.L. (1968), Contrôle Optimal des Systèmes Gouvernés par des Equations aux Dérivées Partielles, Dunod (Paris).Google Scholar
Lions, J.L. (1969), Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod (Paris).Google Scholar
Lions, J.L. (1983), Contrôle des Systèmes Distribués Singuliers, Gauthier-Villars (Paris) (English translation: Gauthier-Villars (Paris) 1985).Google Scholar
Lions, J.L. (1986), ‘Contrôlabilité exacte des systèmes distribués’, C.R. Acad. Sci. I 302, 471–475.Google Scholar
Lions, J.L. (1988a), ‘Exact controllability, stabilization and perturbation for distributed systems’, SIAM Rev. 30, 1–68.CrossRefGoogle Scholar
Lions, J.L. (1988b), Contrôlabilité Exacte, Perturbation et Stabilisation des Systèmes Distribués, vols. 1 and 2, Masson (Paris). (English Translation, Springer 1994).Google Scholar
Lions, J.L. (1990), El Planeta Tierra, Instituto de España, Espasa Calpe, S.A., Madrid.Google Scholar
Lions, J.L. (1990b), ‘On some hyperbolic equations with a pressure term’, Proc. CIRM Conf. Trento, Italy, October.Google Scholar
Lions, J.L. (1991a), ‘Exact controllability for distributed systems: some trends and some problems’, in Applied and Industrial Mathematics (Spigler, R., ed.) Kluwer (Dordrecht) 59–84.CrossRefGoogle Scholar
Lions, J.L. (1991b), ‘Approximate controllability for parabolic systems’, Harvey Lectures, Haiffa.Google Scholar
Lions, J.L. (1993), ‘Quelques remarques sur la contrôlabilité en liaison avec des questions d'environnement’, in Les Grands Systèmes des Sciences et de la Technologie (Horowitz, J. and Lions, J.L., eds.), Masson (Paris) 240–264.Google Scholar
Lions, J.L. and Magenes, E. (1968), Problèmes aux Limites Non Homogènes, vol. 1, Dunod (Paris). (English Translation, Springer, 1972).Google Scholar
Lions, P.L. and Mercier, B. (1979), ‘Splitting algorithms for the sum of two nonlinear operators’, SIAM J. Numer. Anal. 16, 964–979.CrossRefGoogle Scholar
Marini, G., Testa, P. and Valente, V. (1994), ‘Exact controllability of a spherical cap: numerical implementation of HUM’, J. Opt. Theory and Appl. To appear.CrossRefGoogle Scholar
McManus, K., Poinsot, Th. and Candel, S. (1993), ‘Review of active control of combustion instabilities’, Prog. Energy and Combustion Sci. 19, 1–29.CrossRefGoogle Scholar
Meyer, Y. (1989), Private communication.Google Scholar
Mizohata, S. (1958), ‘Unicité du prolongement des solutions pour quelques opérateurs différentiels paraboliques’, Mem. Coll. Sci. Univ. Kyoto A 31, 219–239.Google Scholar
Narukawa, J. (1983), ‘Boundary value control of thermo-elastic systems’, Hiroshima J. Math. 13, 227–272.CrossRefGoogle Scholar
Nečas, J. (1967), Les Méthodes Directes en Théorie des Équations Elliptiques, Masson (Paris).Google Scholar
Nocedal, J. (1992), ‘Theory of algorithms for unconstrained optimization’, Acta Numerica 1992, Cambridge University Press (Cambridge) 199–242.Google Scholar
Park, K.C., Chiou, J.C. and Downer, J.D. (1990), ‘Explicit-implicit staggered procedures for multibody dynamics analysis’, J. Guidance Control Dyn. 13, 562–570.CrossRefGoogle Scholar
Peaceman, D. and Rachford, H. (1955), ‘The numerical solution of parabolic and elliptic differential equations’, J. Soc. Ind. Appl. Math. 3, 28–41.CrossRefGoogle Scholar
Polack, E. (1971), Computational Methods in Optimization, Academic (New York).Google Scholar
Powell, M.J.D. (1976), ‘Some convergence properties of the conjugate gradient method’, Math. Program. 11, 42–49.CrossRefGoogle Scholar
Raviart, P.A. and Thomas, J.M. (1988), Introduction à l'Analyse Numérique des Équations aux Dérivées Partielles, Masson (Paris).Google Scholar
Roberts, J.E. and Thomas, J.M. (1991), ‘Mixed and hybrid methods’, in Handbook of Numerical Analysis, vol. II (Ciarlet, P.G. and Lions, J.L., eds.) North-Holland (Amsterdam) 523–639.Google Scholar
Rockafellar, A.T. (1970), Convex Analysis, Princeton University Press (Princeton, NJ).CrossRefGoogle Scholar
Russel, D.L. (1978), ‘Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions’, SIAM Rev. 20, 639–739.CrossRefGoogle Scholar
Samaniego, J.M., Yip, B., Poinsot, Th. and Candel, S. (1993), ‘Low frequency combustion instability mechanisms inside dump combustor’, Combustion and Flame 94, 363–381.CrossRefGoogle Scholar
Sanchez Hubert, J. and Sanchez Palencia, E. (1989), Vibrations and Coupling of Continuous Systems (Asymptotic Methods), Springer (Berlin).CrossRefGoogle Scholar
Saut, J.C. and Scheurer, B. (1987) ‘Unique continuation for some evolution equations’, J. Diff. Eqns 66, 118–139.CrossRefGoogle Scholar
Sellin, R.H. and Moses, T. (1989), Drag Reduction in Fluid Flows, Ellis Horwood (Chichester).Google Scholar
Sritharan, S.S. (1991a), ‘Dynamic programming of Navier-Stokes equations’, Syst. Control Lett. 16, 299–307.CrossRefGoogle Scholar
Sritharan, S.S. (1991b), ‘An optimal control problem for exterior hydrodynamics’, in Distributed Parameter Control Systems: New Trends and Applications (Chen, G., Lee, E.B., Littman, W. and Markus, L., eds.) Marcel Dekker (New York) 385–417.Google Scholar
Taflove, A. (1992), ‘Re-inventing electromagnetics, supercomputing solution of Maxwell's equations via direct time integration on space grids’, 30th AIAA Aerospace Sciences Meeting, Reno, Nevada, AIAA Paper 92–0333.Google Scholar
Tataru, D. (1994a), Boundary controllability for conservative P.D.E. To appear.Google Scholar
Tataru, D. (1994b), Decay rates and attractors for semilinear P.D.E. To appear.Google Scholar
Thomée, V. (1990), ‘Finite difference methods for linear parabolic equations’, in Handbook of Numerical Analysis, vol. I (Ciarlet, P.G. and Lions, J.L., eds.) North-Holland (Amsterdam) 5–196.Google Scholar
Yserentant, H. (1993) ‘Old and new convergence proofs for multigrid methods’, Acta Numerica 1993, Cambridge University Press, (Cambridge) 285–326.Google Scholar
Zuazua, E. (1988), ‘Contrôlabilité exacte en un temps arbitrairement petit de quelques modèles de plaques’, Appendix 1 of Contrôlabilité Exacte, Pertubations et Stabilisation de Systèmes Distribués, vol. I, by Lions, J.L., Masson (Paris), 465–491.Google Scholar
Zuazua, E. (1993), ‘Contrôlabilité du système de la thermo-élasticité’, C.R. Acad. Sci. Paris I 317, 371–376.Google Scholar