---

Journal of Applied Nonlinear Dynamics 13(3) (2024) 439--448 | DOI:10.5890/JAND.2024.09.002

M. Chqondi, M. Baddi, Y. Akdim

Laboratory LAMA, Department of Mathematics and Informatics, Sidi Mohamed Ben Abdellah University, Faculty of Sciences Dhar El Mahraz - FES, Morocco

Download Full Text PDF

 

Abstract

This research paper examines the problem of stabilizing inhomogeneous semilinear control systems in Hilbert state space. The paper proposes a feedback control that can achieve both strong and weak stabilization under certain assumptions related to approximate observability. The provided applications of the proposed method include the nonlinear Schroedinger equation and heat equation.

References

1. [1]	Ball, J.M. and Slemrod, M. (1979), Feedback stabilization of distributed semilinear control systems, Applied Mathematics and Optimization, 5(1), 169-179.

2. [2]	Ouzahra, M. and Tsouli, A. (2012), Stabilization and polynomial decay estimate for distributed semilinear systems, International Journal of Control, 85(4), 451-456.

3. [3]	Zerrik, E. and Ouzahra, M. (2007), Output stabilization for distributed semilinear systems, Control Theory and Applications, 3(1), 838-843.

4. [4]	Ouzahra, M. (2013), Partial stabilization of semilinear systems using bounded controls, International Journal of Control, 86(12), 2253-2262.

5. [5]	Benzaza, A. and Ouzahra, M. (2019), Weak and strong stabilization of bilinear systems in a Banach space, International Journal of Control, 92(12), 2784-2790.

6. [6]	Bounit, H. and Hammouri, H. (1999), Feedback stabilization for a class of distributed semilinear control systems, Nonlinear Analysis: Theory, Methods and Applications, 37(8), 953-969.

7. [7]	Chqondi, M., Alami, A., and Akdim, Y. (2021), Partial strong stabilization of semi-linear systems and robustness of optimal control, Discontinuity, Nonlinearity, and Complexity, 10(03), 369-380.

8. [8]	Ball, J.M. (1978), On the asymptotic behavior of generalized processes, with applications to nonlinear evolution equations, Journal of Differential Equations, 27(2), 224-265.

9. [9]	Hamidi, Z. and Ouzahra, M. (2018), Partial stabilization of non-homogeneous bilinear systems, International Journal of Control, 91(6), 1251-1258.

10. [10]	Martin, Jr.R.H. (1985), A. Pazy, Semigroups of linear operators and applications to partial differential equations. Bulletin (New Series) of the American Mathematical Society, 12(2), 302-305.

11. [11]	Lasiecka, I. and Tataru, D. (1993), Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differential Integral Equations, 6, 507-533.

12. [12]	Berrahmoune, L. (1999), Stabilization and decay estimate for distributed bilinear systems, Systems and Control Letters, 36(3), 167-171.

13. [13]	Bergé, L. (2000), Soliton stability versus collapse, Physical Review E, 62(3), R3071.

14. [14]	Genoud, F. and Stuart, C.A. (2008), Schrödinger equations with a spatially decaying nonlinearity: existence and stability of standing waves, Discrete and Continuous Dynamical Systems, 21(1), 137-187.