Abstract
The present article is concerned with the numerical implementation of the Hilbert uniqueness method for solving exact and approximate boundary controllability problems for the heat equation. Using convex duality, we reduce the solution of the boundary control problems to the solution of identification problems for the initial data of an adjoint heat equation. To solve these identification problems, we use a combination of finite difference methods for the time discretization, finite element methods for the space discretization, and of conjugate gradient and operator splitting methods for the iterative solution of the discrete control problems. We apply then the above methodology to the solution of exact and approximate boundary controllability test problems in two space dimensions. The numerical results validate the methods discussed in this article and clearly show the computational advantage of using second-order accurate time discretization methods to approximate the control problems.
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References
Céa, J.,Optimisation: Théorie et Algorithmes, Dunod, Paris, France, 1971.
Céa, J.,Optimization: Theory and Algorithms, Springer-Verlag, Berlin, Germany, 1978.
Céa, J., andGlowinski, R.,Méthodes Numériques pour l'Ecoulement Laminaire d'un Fluide Rigide Visco-Plastique Incompressible, International Journal of Computational Mathematics, Vol. B3, pp. 225–255, 1972.
Céa, J., Glowinski, R., andNedelec, J. C.,Minimisation de Fonctionnelles Non Différentiables, Conference on Application of Numerical Analysis, Lecture Notes in Mathematics, Springer Verlag, Berlin, Germany, Vol. 228, pp. 19–38, 1971.
Céa, J., Glowinski, R., andNedelec, J. C.,Application des Méthodes d'Optimisation, de Différences et d'Eléments Finis à l'Analyse Numérique de la Torsion Elasto-Plastique d'une Barre Cylindrique, Approximation et Méthodes Itératives de Résolution d'Inéquations Variationnelles et de Problèmes Non Linéaires, Cahier de l'IRIA, No. 12, pp. 7–138, 1974.
Céa, J., andGlowinski, R.,Sur des Méthodes de Minimisation par Relaxation, Revue Française d'Automatique, Informatique, Recherche Opérationnelle, Série Rouge, Vol. R3, pp. 5–32, 1973.
Glowinski, R.,Boundary Controllability Problems for the Wave and Heat Equations, Boundary Control and Boundary Variation, Edited by J. P. Zolesio, Lecture Notes in Control and Information Sciences, Springer Verlag, Berlin, Germany, Vol. 178, pp. 221–237, 1992.
Lions, J. L.,Controlabilité Exacte des Systèmes Distribués, Comptes Rendus de l'Académie des Sciences, Paris, France, Série I, Vol. 302, pp. 471–475, 1986.
Lions, J. L.,Exact Controllability, Stabilization and Perturbation for Distributed Systems, SIAm Review, Vol. 30, pp. 1–68, 1988.
Lions, J. L.,Controlabilité Exacte, Perturbation et Stabilisation des Systèmes Distribués, Vols. 1 and 2, Masson, Paris, France, 1988.
Lions, J. L.,Optimal Control of Systems Governed by Partial Differential Equations, Springer Verlag, New York, New York, 1971.
Glowinski, R.,Numerical Methods for Nonlinear Variational Problems, Springer Verlag, New York, New York, 1984.
Lions, J. L.,Exact Controllability for Distributed Systems: Some Trends and Some Problems, Applied and Industrial Mathematics, Edited by R. Spigler, Kluwer, Dordrecht, Holland, pp. 59–84, 1991.
Ekeland, I., andTemam, R.,Convex Analysis and Variational Problems, North-Holland, Amsterdam, Holland, 1976.
Dean, E. J., Glowinski, R., andLi, C. H.,Supercomputer Solutions of Partial Differential Equation Problems in Computational Fluid Dynamics and in Control, computer Physics Communications, Vol. 53, pp. 401–439, 1989.
Glowinski, R., Li, C. H., andLions, J. L. A Numerical Approach to the Exact Boundary Controllability of the Wave Equation, (I) Dirichlet Controls: Description of the Numerical Methods, Japan Journal of Applied Mathematics, Vol. 7, pp. 1–76, 1990.
Glowinski, R., andLi, C. H.,On the Numerical Implementation of the Hilbert Uniqueness Method for the Exact Boundary Controllability of the Wave Equation, Comptes Rendus de l'Académie des Sciences, Paris, France, Serie I, Vol. 311, pp. 135–142, 1990.
Glowinski, R.,Ensuring Well-Posedness by Analogy: Stokes Problem and Boundary Control for the Wave Equation, Journal of Computational Physics, Vol. 103, pp. 189–221, 1992.
Lions, J. L., andMercier, B.,Splitting Algorithms for the Sum of Two Nonlinear Operators, SIAM Journal on Numerical Analysis, Vol. 16, pp. 964–979, 1979.
Glowinski, R., andLe Tallec, P.,Augmented Lagrangian and Operator Splitting Methods in Nonlinear Mechanics, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1989.
Peaceman, D., andRachford, H.,The Numerical Solution of Parabolic and Elliptic Differential Equations, SIAM Journal on Applied Mathematics, Vol. 3, pp. 28–41, 1955.
Carthel, C.,Numerical Methods for Some Exact and Approximate Controllability Problems for the Heat Equation, PhD Dissertation, Department of Mathematics, University of Houston, 1993.
Dean, E. J., andGubernatis, P.,Pointwise Control of Burgers' Equation: A Numerical Approach, Computer Mathematics and Applications Vol. 22, pp. 93–100, 1991.
Berggren, M.,Control and Simulation of Advection—Diffusion Systems, MS Thesis, Department of Mechanical Engineering, University of Houston, 1992.
Glowinski, R., andLions, J. L.,Exact and Approximate Controllability for Distributed Parameter Systems, Acta Numerica 1994 (to appear).
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Dedicated to J. Céa on his 60th birthday
This work was partially supported by the National Science Foundation Grant INT-8612680 the Texas Board of Higher Education Grant ARP-003652156, and the Faculty Development Program, University of Houston (special thanks are due to Glenn Aumann and Garret J. Etgen). The authors also thank J. A. Wilson for diligently processing the manuscript of this article.
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Carthel, C., Glowinski, R. & Lions, J.L. On exact and approximate boundary controllabilities for the heat equation: A numerical approach. J Optim Theory Appl 82, 429–484 (1994). https://doi.org/10.1007/BF02192213
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DOI: https://doi.org/10.1007/BF02192213