×

Output controllability for semi-linear distributed parabolic systems. (English) Zbl 1119.93022

Summary: The purpose of this paper is to explore regional controllability developed for linear system [5, 15] to the semi-linear case. Under the hypothesis that the approximate regional controllability holds, we find a control which steers a system to a desired state in a subregion of the system domain and the approach is based on an extension of the Hilbert uniqueness method and the Schauder fixed-point theorem. The analytical case is then discussed using generalized inverse techniques and successfully illustrated by simulations.

MSC:

93B05 Controllability
93C10 Nonlinear systems in control theory
93C20 Control/observation systems governed by partial differential equations
47H10 Fixed-point theorems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Brezis, H., Analyse fonctionnelle: théorie et application (1983), Paris: Masson, Paris · Zbl 0511.46001
[2] R. F. Curtain and H. Zwart, An introduction to infinite-dimensional linear systems theory. Springer-Verlag (1995). · Zbl 0839.93001
[3] El Jai, A.; Pritchard, A. J., Capteurs et actionneurs dans l’analyse des systèmes distribués (1986), Paris: Masson, Paris · Zbl 0627.93001
[4] _____, Sensors and actuators in distributed systems analysis. Wiley (1988).
[5] El Jai, A.; Pritchard, A. J.; Simon, M. C.; Zerrik, E., Regional controllability of distributed systems, Int. J. Control, 62, 6, 1351-1365 (1995) · Zbl 0844.93016
[6] Fabre, C.; Puel, J. P.; Zuazua, E., Approximate controllability of the semilinear heat equation, Proc. Royal Soc. Edinburgh, 125A, 31-61 (1995) · Zbl 0818.93032
[7] J. A. M. Felippe De Souza, Control of nonlinear distributed parameter systems. In: Proc, VIII Congreso Chileno De Ingenieria Electrica, Chile (1989), pp. 370-375.
[8] Felippe De Souza, J. A. M.; Pritchard, A. J., Control of semilinear distributed parameter systems, Telecomunication and Control, 202-207 (1985), Sao Josè dos Campos, Brasil: INPE Press, Sao Josè dos Campos, Brasil
[9] Fernàndez-Cara, E., Null controllability of the semilinear heat equation, ESAIM Control Optim. Calc. Variations, 2, 87-103 (1997) · Zbl 0897.93011 · doi:10.1051/cocv:1997104
[10] Kassara, K.; El Jai, A., Algorithme pour la commande d’une classe de systèmes á paramètres repartis non linèaires, Rev. Mar. d’aut. d’inf. de Trait. de signal, 1, 1, 3-24 (1993)
[11] Lions, J. L., Contrôlabilité exacte. Perturbations et stabilisation des systèmes distribués (1988), Paris: Masson, Paris · Zbl 0653.93002
[12] J. L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vols. 1, 2. Dunod (1968). · Zbl 0165.10801
[13] Pazy, A., Semigroups of linear operators and applications to partial differential equations (1983), New York: Springer-Verlag, New York · Zbl 0516.47023
[14] Zeidler, E., Applied functional analysis: Applications to mathematical physics (1995), New York: Springer-Verlag, New York · Zbl 0834.46002
[15] Zerrik, E.; Boutoulout, A.; El Jai, A., Actuators and Regional Boundary Controllability of parabolic systems, Int. J. Systems Sci., 31, 1, 73-82 (2000) · Zbl 1080.93651 · doi:10.1080/002077200291479
[16] Zuazua, E., Contrôlabilité exacte d’un modèle de plaques vibrantes en un temps arbitrairement petit (1987), Paris: C.R.A.S, Paris · Zbl 0611.49028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.