An Output Controllability Problem for Semilinear Distributed Hyperbolic Systems

E Zerrik; R Larhrissi; H Bourray

doi:10.2478/v10006-007-0035-y

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An Output Controllability Problem for Semilinear Distributed Hyperbolic Systems

International Journal of Applied Mathematics and Computer Science

Volume 17 (2007): Issue 4 (December 2007)

By: E Zerrik, R Larhrissi and H Bourray

Open Access

|Jan 2008

Abstract

The paper aims at extending the notion of regional controllability developed for linear systems to the semilinear hyperbolic case. We begin with an asymptotically linear system and the approach is based on an extension of the Hilbert uniqueness method and Schauder's fixed point theorem. The analytical case is then tackled using generalized inverse techniques and converted to a fixed point problem leading to an algorithm which is successfully implemented numerically and illustrated with examples.

References

Brezis H. (1993): Analyse fonctionnelle. Théorie et applications.—Paris: Masson.
Search in Google Scholar
Da Prato G., Pritchard A. and Zabczyk J. (1991): On minimum energy problems.SIAM Journal on Control and Optimization, Vol. 29, pp. 209-221.10.1137/0329012
Search in Google Scholar
De Souza F., J.A.M., and A.J.Pritchard (1985): Control of semilinear distributed parameter systems.Telecommunication and Control, INPE, Sao José dos Campos, Brazil, pp. 160-164.
Search in Google Scholar
De Souza F. J.A.M (1985): Control of nonlinear distributed parameter systems. In: Proc. IV Coloquio de Control Automatico (Ibarra Zannatha, Ed.), Centro de investigation y Estudios Avanzados del instituto Politecnico Nacional de Mexico, Mexico, Vol. 1, pp. 37-43.
Search in Google Scholar
El Jai A., Zerrik E., Simon M. C. and Pritchard A. J. (1995): Regional controllability of distributed parameter systems.International Journal of Control, Vol. 62, No. 6, pp. 1351-1365.10.1080/00207179508921603
Search in Google Scholar
Fabre C., Puel J. P. and Zuazua E. (1995): Approximate controllability of the semilinear heat equation, Proceedings of the Royal Society of Edinburgh, Vol. 125 A, pp. 31-61.
Search in Google Scholar
E. (1997): Null controllability of the proximate heat equation, ESAIM: Control Optimization and Calculus of Variations, Vol. 2, pp. 87-103.
Search in Google Scholar
Henry D. (1981): Geometric Theory of Semilinear Parabolic Systems.10.1007/BFb0089647
Search in Google Scholar
Kassara K. and El Jaï A. (1983): Algorithme pour la commande d'une classe de systèmes à paramètres répartis non linéaires.Revue marocaine d'automatique, d'informatique et de traitement de signal, Vol. 1, No. 2, pp. 95-117.
Search in Google Scholar
Klamka J. (2002): Constrained exact controllability of semilinear systems.Systems and Control Letters, Vol. 47, No. 2, pp. 139-147.10.1016/S0167-6911(02)00184-6
Search in Google Scholar
Klamka J. (2001): Constrained controllability of semilinear systems.Nonlinear Analysis, Vol. 47, pp. 2939-2949.10.1016/S0362-546X(01)00415-1
Search in Google Scholar
Klamka J. (2000): Schauder's fixed point theorem in nonlinear controllability problem.Control and Cybernetics, Vol. 29, No. 1, pp. 153-165.
Search in Google Scholar
Klamka J. (1999): Constrained controllability of dynamical systems.International Journal of Applied Mathematics and Computer Science, Vol. 9, No. 2, pp. 231-244.
Search in Google Scholar
Klamka J. (1998): Controllability of second order semilinear infinite-dimensional dynamical systems.Applied Mathematics and Computer Science, Vol. 8, No. 3, pp. 459-470.
Search in Google Scholar
Klamka J. (1991): Controllability of Dynamical Systems, Dordrecht: Kluwer Academic Publishers.
Search in Google Scholar
Lions J.L. (1988): Contrôlabilité Exacte. Perturbations et Stabilisation des Systèmes Distribués, Tome 1, Contrôlabilité Exacte. —Paris: Masson.
Search in Google Scholar
Pazy A. (1983): Semigroups of Linear Operators and Applications to Partial Differential Equations.—Berlin: Springer-Verlag.10.1007/978-1-4612-5561-1
Search in Google Scholar
Zeidler E. (1990): Nonlinear Functional Analysis and Its Applications II/A. Linear Applied Functional Analysis, Springer.10.1007/978-1-4612-0981-2
Search in Google Scholar
Zuazua E. (1990): Exact controllability for the semilinear wave equation.Journal de Mathématiques Pures et Appliquées, 69, pp. 1-31.
Search in Google Scholar
Zeidler E. (1999): Applied Functional Analysis. Applications to Mathematical Physics.—Springer.
Search in Google Scholar
Zerrik E., Bourray H. and El Jai A. (2004): Regional observability of semilinear distributed parabolic systems.International Journal of Dynamical and Control Systems, Vol. 10, No. 3, pp. 413-430.10.1023/B:JODS.0000034438.72863.ca
Search in Google Scholar
Zerrik E. and Larhrissi R. (2002): Regional boundary controllability of hyperbolic systems. Numerical approach.International Journal of Dynamical and Control Systems, Vol. 8, No. 3, pp. 293-311.10.1023/A:1016373003163
Search in Google Scholar
Zerrik E. and Larhrissi R. (2001): Regional Target Control of the wave Equation.International Journal of Systems Science, Vol. 32, No. 10, pp. 1233-1242.10.1080/00207720110035264
Search in Google Scholar
Zerrik E., Boutoulout A. and El Jaï A. (2000): Actuators and regional boundary controllability.International Journal of Systems Science, Vol. 31, No. 1, pp. 73-82.10.1080/002077200291479
Search in Google Scholar