Exact and approximate controllability for distributed parameter systems

Acta Numerica

Volumn 4, Issue 2, 1995, Pages 159-328

  Glowinski, R ^a   Lions, J L ^b  

^a UNIVERSITY OF HOUSTON   (United States)

^b l'alimentation et l'Environnement   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84959656664     PISSN: 09624929     EISSN: 14740508     Source Type: Journal    
DOI: 10.1017/S0962492900002543     Document Type: Article

References (117)

1. Abergel F. and Temam R. (1990), ‘On some control problems in fluid mechanics’, Theoret. Comput. Fluid Dyn. 1, 303–326
2. 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. 10.1007/BFb0038743
3. Bamberger A. (1977), private communication.
4. 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.
5. Bardos C. and Rauch J. (1994), ‘Mathematical analysis of control-type numerical algorithms for the solution of the Helmholtz equation’. To appear.
6. Berggren M. (1992), ‘Control and simulation of advection-diffusion systems’, MSc Thesis, Department of Mechanical Engineering, University of Houston.
7. Berggren M. and Glowinski R. (1995), ‘Numerical solution of some flow control problems’, Int. J. Fluid Dyn. To appear.
8. Bensoussan A. (1990), Acta Appl. Math. 20, 197
9. 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.
10. Brezzi F. and Fortin M. (1991), Mixed and Hybrid Finite Element Methods, Springer (New York).10.1007/978-1-4612-3172-1
11. 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).
12. 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.
13. 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.
14. 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. 10.1007/978-3-322-87871-7_12
15. 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).
16. 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.
17. Buschnell D.M. and Hefner J.N. (1990), Viscous Drag Reduction in Boundary Layers, American Institute of Aeronautics and Astronautics (Washington, DC).10.2514/4.865978
18. Carthel C. (1994), ‘Numerical methods for the boundary controllability of the heat equation’, PhD Thesis, Department of Mathematics, University of Houston.
19. 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
20. Chiara A. (1993), ‘Equation des ondes et régularité sur un ouvert lipschitzien’, C.R. Acad. Sci. I 316, 33–36.
21. Ciarlet P.G. (1978), The Finite Element Method for Elliptic Problems, North-Holland (Amsterdam).
22. Ciarlet P.G. (1989), Introduction to Numerical Linear Algebra and Optimization, Cambridge University Press (Cambridge).
23. Ciarlet P.G. (1990a), ‘A new class of variational problems arising in the modeling of elastic multi-structures’, Numer. Math. 57, 547–560
24. Ciarlet P.G. (1990b), Plates and Junctions in Elastic Multi-Structures: An Asymptotic Analysis, Masson (Paris).1071376
25. 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.
26. 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.
27. Crandall M.G. and Lions P.L. (1985), ‘Hamilton-Jacobi equations in infinite dimensions, Part I’, J. Funct. Anal. 62, 379–396
28. Crandall M.G. and Lions P.L. (1986a), ‘Hamilton-Jacobi equations in infinite dimensions, Part II’, J. Funct. Anal. 65, 368–405
29. Crandall M.G. and Lions P.L. (1986b), ‘Hamilton-Jacobi equations in infinite dimensions, Part III’, J. Funct. Anal. 68, 214–247
30. Crandall M.G. and Lions P.L. (1990), ‘Hamilton-Jacobi equations in infinite dimensions, Part IV’, J. Funct. Anal. 90, 237–283
31. Crandall M.G. and Lions P.L. (1991), ‘Hamilton-Jacobi equations in infinite dimensions, Part V’, J. Funct. Anal. 97, 417–465
32. Daniel J. (1970), The Approximate Minimization of Functionals, Prentice Hall (Englewood Cliffs, NJ).
33. Dean E.J. and Gubernatis P. (1991), ‘Pointwise control of Burgers’ equation – a numerical approach’, Comput. Math. Appl. 22, 93–100
34. 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
35. Dennis J. and Schnabel R.B. (1983), Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall (Englewood Cliffs, NJ).
36. 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.
37. Diaz J.I. and Fursikov (1994), To appear.
38. 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
39. 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.
40. Ekeland I. and Temam R. (1974), Analyse Convexe et Problèmes Variationnels, Dunod (Paris).
41. Engquist B., Gustafsson B. and Vreeburg J. (1978), ‘Numerical solution of a PDE system describing a catalytic converter’, J. Comput. Phys. 27, 295–314
42. 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.
43. Friedman A. (1988), ‘Modeling catalytic converter performance’, in Mathematics in Industrial Problems, Part 4 ( Friedman A., ed.) Springer (New York) Ch. 7, 70–77. 10.1007/978-1-4615-7399-9
44. Friend CM. (1993), ‘Catalysis and surfaces’, Scientific American (04) 74–79
45. 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.
46. 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.
47. 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).
48. 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).
49. George J.A. (1971), ‘Computer implementation of the finite element method’, PhD Thesis, Computer Sciences Department, Stanford University.
50. Glowinski R. (1984), Numerical Methods for Nonlinear Variational Problems, Springer (New York).10.1007/978-3-662-12613-4
51. 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.
52. Glowinski R. (1992a), ‘Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation’, J. Comput. Phys. 103, 189–221
53. 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. 10.1007/BFb0006698
54. Glowinski R. and Le Tallec P. (1989), Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, SIAM (Philadelphia, PA).10.1137/1.9781611970838
55. 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.
56. 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.
57. 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
58. 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
59. Glowinski R., Lions J.L. and Tremolières R. (1976), Analyse Numérique des Inéquations Variationnelles, Dunod (Paris).
60. Glowinski R., Lions J.L. and Tremolières R. (1981), Numerical Analysis of Variational Inequalities, North-Holland (Amsterdam).
61. Golub G.H. and Van Loan C. (1989), Matrix Computations, Johns Hopkins Press (Baltimore, MD).
62. Hackbush W. (1985), Multigrids Methods and Applications, Springer (Berlin).
63. Haraux A. and Jaffard S. (1991), ‘Pointwise and spectral control of plate vibrations’, Revista Matematica Iberoamericana 7, 1–24
64. 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.
65. Hörmander L. (1976), Linear Partial Differential Operators, Springer (Berlin).
66. 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.
67. Komornik V. (1994), Exact Controllability and Stabilization. The Multiplier Method, Masson (Paris).1359765
68. 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).
69. Lagnese J.E. (1989), ‘Exact boundary controllability of Maxwell's equations in a general region’, SIAM J. Control Optimiz. 27, 374–388
70. 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. 10.1007/BFb0113314
71. Lagnese J.E., Leugering G. and Schmidt G. (1994), Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures, Binkhäuser (Boston, Basel, Berlin).10.1007/978-1-4612-0273-8
72. 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
73. Lasiecka I. and Tataru R. (1994), Uniform boundary stabilization of semilinear wave equations with non linear boundary conditions, Diff. and Integral Equations.
74. 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
75. Lax P.D. and Phillips R.S. (1989), Scattering Theory, Academic (New York).
76. Lions J.L. (1961), Equations Différentielles Opérationnelles et Problèmes aux Limites, Springer (Heidelberg).
77. Lions J.L. (1968), Contrôle Optimal des Systèmes Gouvernés par des Equations aux Dérivées Partielles, Dunod (Paris).
78. Lions J.L. (1969), Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod (Paris).
79. Lions J.L. (1983), Contrôle des Systèmes Distribués Singuliers, Gauthier-Villars (Paris) (English translation: Gauthier-Villars (Paris) 1985).
80. Lions J.L. (1986), ‘Contrôlabilité exacte des systèmes distribués’, C.R. Acad. Sci. I 302, 471–475.
81. Lions J.L. (1988a), ‘Exact controllability, stabilization and perturbation for distributed systems’, SIAM Rev. 30, 1–68
82. Lions J.L. (1988b), Contrôlabilité Exacte, Perturbation et Stabilisation des Systèmes Distribués, vols. 1 and 2, Masson (Paris). (English Translation, Springer 1994).
83. Lions J.L. (1990), El Planeta Tierra, Instituto de España, Espasa Calpe, S.A., Madrid.
84. Lions J.L. (1990b), ‘On some hyperbolic equations with a pressure term’, Proc. CIRM Conf. Trento, Italy, October.
85. 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. 10.1007/978-94-009-1908-2_7
86. Lions J.L. (1991b), ‘Approximate controllability for parabolic systems’, Harvey Lectures, Haiffa.
87. 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.
88. Lions J.L. and Magenes E. (1968), Problèmes aux Limites Non Homogènes, vol. 1, Dunod (Paris). (English Translation, Springer, 1972).
89. Lions P.L. and Mercier B. (1979), ‘Splitting algorithms for the sum of two nonlinear operators’, SIAM J. Numer. Anal. 16, 964–979
90. 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.1276174
91. McManus K., Poinsot Th. and Candel S. (1993), ‘Review of active control of combustion instabilities’, Prog. Energy and Combustion Sci. 19, 1–29
92. Meyer Y. (1989), Private communication.
93. Mizohata S. (1958), ‘Unicité du prolongement des solutions pour quelques opérateurs différentiels paraboliques’, Mem. Coll. Sci. Univ. Kyoto A 31, 219–239.
94. Narukawa J. (1983), ‘Boundary value control of thermo-elastic systems’, Hiroshima J. Math. 13, 227–272.
95. Nečas J. (1967), Les Méthodes Directes en Théorie des Équations Elliptiques, Masson (Paris).
96. Nocedal J. (1992), ‘Theory of algorithms for unconstrained optimization’, Acta Numerica 1992, Cambridge University Press (Cambridge) 199–242.
97. 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
98. Peaceman D. and Rachford H. (1955), ‘The numerical solution of parabolic and elliptic differential equations’, J. Soc. Ind. Appl. Math. 3, 28–41
99. Polack E. (1971), Computational Methods in Optimization, Academic (New York).
100. Powell M.J.D. (1976), ‘Some convergence properties of the conjugate gradient method’, Math. Program. 11, 42–49
101. Raviart P.A. and Thomas J.M. (1988), Introduction à l'Analyse Numérique des Équations aux Dérivées Partielles, Masson (Paris).
102. 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.
103. Rockafellar A.T. (1970), Convex Analysis, Princeton University Press (Princeton, NJ).
104. Russel D.L. (1978), ‘Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions’, SIAM Rev. 20, 639–739
105. 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
106. Sanchez Hubert J. and Sanchez Palencia E. (1989), Vibrations and Coupling of Continuous Systems (Asymptotic Methods), Springer (Berlin).10.1007/978-3-642-73782-4 0996423
107. Saut J.C. and Scheurer B. (1987) ‘Unique continuation for some evolution equations’, J. Diff. Eqns 66, 118–139
108. Sellin R.H. and Moses T. (1989), Drag Reduction in Fluid Flows, Ellis Horwood (Chichester).
109. Sritharan S.S. (1991a), ‘Dynamic programming of Navier-Stokes equations’, Syst. Control Lett. 16, 299–307
110. 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. 1108853
111. 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.
112. Tataru D. (1994a), Boundary controllability for conservative P.D.E. To appear.
113. Tataru D. (1994b), Decay rates and attractors for semilinear P.D.E. To appear.
114. 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.
115. Yserentant H. (1993) ‘Old and new convergence proofs for multigrid methods’, Acta Numerica 1993, Cambridge University Press, (Cambridge) 285–326.
116. 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.
117. Zuazua E. (1993), ‘Contrôlabilité du système de la thermo-élasticité’, C.R. Acad. Sci. Paris I 317, 371–376.