Research ArticleOpen AccessGoogle Scholar indexed
Regional Controllability of Semi-Linear Distributed Parabolic Systems: Theory and Simulation
TSI Group, Department of Mathematics and Computer Sciences, Moulay Ismail University, Faculty of Sciences, Meknes, Morocco
TSI Group, Department of Mathematics and Computer Sciences, Moulay Ismail University, Faculty of Sciences, Meknes, Morocco
Department of Mathematics, Al Jouf University, College of Sciences, Sakakah, Kingdom of Saudi Arabia
- 1 TSI Group, Department of Mathematics and Computer Sciences, Moulay Ismail University, Faculty of Sciences, Meknes, Morocco
- 2 TSI Group, Department of Mathematics and Computer Sciences, Moulay Ismail University, Faculty of Sciences, Meknes, Morocco
- 3 Department of Mathematics, Al Jouf University, College of Sciences, Sakakah, Kingdom of Saudi Arabia
Intelligent Control and Automation·Volume 03 (2012)·Pages 146–158·Published 23 May 2012·DOI10.4236/ica.2012.32017
Copy link · social · email
Abstract
The aim of this brief paper is to give several results concerning the regional controllability of distributed systems governed by semi-linear parabolic equations. We concentrate on the determination of a control achieving internal and boundary regional controllability. The approach is based on an extension of the Hilbert Uniqueness Method (HUM) and Schauder’s fixed point theorem. We give a numerical example developed in internal and boundary sub region. These numerical illustrations show the efficiency of the approach and lead to conjectures.
KeywordsSemi-Linear Parabolic SystemsRegionalInternal/Boundary ControllabilityFixed-Point TheoremsDistributed SystemHUM Approach
- R. F. Curtain and A. J. Pritchard, “Infinite-Dimensional Linear Systems Theory, Lecture Notes in Control and Information Sciences,” Springer Verlag, Berlin, 1978.
- R. F. Curtain and H. Zwart, “An Introduction to Infinite Dimensional Linear Systems Theory,” Springer Verlag, Berlin, 1995.
- K. Balachandran and J. P. Dauer, “Controllability of Nonlinear Systems via Fixed-Point Theorems”, Journal of Optimization Theory and Applications, Vol. 53, No. 3, 1987, pp. 345-352. doi:10.1007/BF00938943
- H. Zhou, “Approximate Controllability for a Class of Semilinear Abstract Equations,” SIAM Journal on Control and Optimization, Vol. 21, No. 4, 1983, pp. 551-555. doi:10.1137/0321033
- K. Naito, “Approximate Controllability for Trajectories of Semilinear Control Systems,” Journal of Optimization Theory and Applications, Vol. 60, No. 1, 1989, pp. 57-65. doi:10.1007/BF00938799
- X. Li and J. Yong, “Optimal Control Theory for InfiniteDimension a Systems,” Birkhauser, Basel, 1994.
- W. M. Bian, “Constrained Controllability of Some Nonlinear Systems,” Applicable Analysis, Vol. 72, No. 1-2, 1999, pp. 57-73. doi:10.1080/00036819908840730
- N. Carmichael and M. D. Quinn, “Fixed-Point Methods in Nonlinear Control, Lecture notes in Control and Information Sciences,” Springer Verlag, Berlin, 1984.
- X. Zhang, “Exact Controllability of Semilinear Evolution Systems and Its Application”, Journal of Optimization Theory and Applications, Vol. 107, No. 2, 2000, pp. 415432. doi:10.1023/A:1026460831701
- T. I. Seidmann, “Invariance of the Reachable Set under Nonlinear Perturbations,” SIAM Journal on Control and Optimization, Vol. 25, No. 5, 1985, pp. 1173-1191. doi:10.1137/0325064
- J. Klamka, “Constrained Approximate Controllability,” IEEE Transactions on Automatic Control, Vol. 45, No. 9, 2000, pp. 1745-1749. doi:10.1109/9.880640
- J. Klamka, “Schauder’s Fixed-Point Theorem in Nonlinear Controllability Problems,” Control and Cybernetics, Vol. 29, No. 1, 2000, pp. 153-165.
- J. Klamka, “Constrained Controllability of Semilinear Systems,” Nonlinear Analysis, Vol. 47, No. 6, 2001, pp. 2939-2949. doi:10.1016/S0362-546X(01)00415-1
- K. Balachandran and R. Sakthivel, “Controllability of Integrodifferential Systems in Banach Spaces,” Applied Mathematics and Optimization, Vol. 118, No. 1, 2001, pp. 63-71. doi:10.1016/S0096-3003(00)00040-0