---

Discontinuity, Nonlinearity, and Complexity 10(3) (2021) 369--380 | DOI:10.5890/DNC.2021.09.002

A. El Alami$^1$ , M. Chqondi$^2$, Y. Akdim$^2$

$^1$ Research Center STIS, Team M2CS, Department of Applied Mathematics and Informatics, ENSAM, Mohammed V University in Rabat, Madinat Al Irfane, Rabat, Morocco

$^2$ Laboratory of Mathematical Analysis & Application, Department of Mathematics and Informatics, Faculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University, FES, Morocco

Download Full Text PDF

 

Abstract

This work addresses the idea of optimal stabilization, namely robustness of optimal stabilization with nonlinearity Lipschitz of distributed semilinear systems using bounded control. This problem is treated under the condition of the unbounded operator, we show that the system is stable once the exact observability assumption is executed together with a Lipschitz property of the nonlinear operator. The concept of bounded control is also investigated in realistic domain. The stabilizing feedback is characterized by the minimization of the problem of cost. We also give different applications to parabolic and hyperbolic equations.

References

1. [1]	Harraki, I.E., Alami, A.E., Boutoulout, A., and Serhani, M. (2016), Regional stabilization for semilinear parabolic systems, IMA Journal of Mathematical Control and Information, 2015-197.

2. [2]	Ouzahra, M. (2010), {Exponential and weak stabilization of constrained bilinear systems}, ph {SIAM J. Control Optim.}, 48, Issue 6, 3962-3974.

3. [3]	Gugat, M. and Troltzsch, F. (2015), Boundary feedback stabilization of the Schlglsystem, Automatica, 51, 192-199.

4. [4]	Ouzahra, M. (2009), {Stabilization of infinite-dimensional bilinear systems using a quadratic feedback control}, International Journal of Control., 82, 1657-1664.

5. [5]	El-Farra, N.H. and Christofides, P.D. (2004), Coordinating feedback and switching for control of spatially distributed processes, Computers and Chemical Engineering, 28, 111-128.

6. [6]	Gugat, M. and Sigalotti, M. (2010), Stars of vibrating strings: switching boundary feedback stabilization, Networks and Heterogeneous Media, 5, 299-314.

7. [7]	Gugat, M. (2008), Optimal switching boundary control of a string to rest in finite time, ZAMM Journal of Applied Mathematics and Mechanics, 88, 283-305.

8. [8]	Gugat, M. and Tucsnak, M. (2011), An example for the switching delay feedback stabilization of an infinite dimensional system: the boundary stabilization of a string, Systems and Control Letters, 60, 226-233.

9. [9]	Ball, J. and Slemrod, M. (1979), {Feedback stabilization of distributed semilinear control systems}, Appl. Math. Opt., 5, 169-179.

10. [10]	Zine, A. and Alami, A.E. (2018), Strong and weak stabilization of semi-linear parabolic systems, IMA Journal of Mathematical Control and Information, Oxford, dnx022, https://doi.org/10.1093/imamci/dny038, 2018.

11. [11]	Chen M.S. (1998), ph{Exponential stabilization of a constrained bilinear system}, Automatica, 34, 989-992.

12. [12]	Quinn, J.P. (1980), {Stabilization of bilinear systems by quadratic feedback control}, ph {J. Math. Anal. Appl.}, 75, 66-80.

13. [13]	Ball, J. (1977), Strongly continuous semi-groups, weak solutions, and the variation of constants formula, Proe. Amer. Math. Soc., 63, 370-373.

14. [14]	Ball, J. (1978), On the asymptotic behaviour of generalized processes, with applications to nonlinear evolution equations, Journal of Differential Equations, 27, 224-265.

15. [15]	Curtain, R.F. and Zwart, H.J. (1991), An introduction to infinite dimensional linear systems theory, Springer-Verlag.

16. [16]	Bardos, C., Lebeau, G., and Rauch, J. (1992), Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim., 30, 1024-1065.

17. [17]	Lebeau, G. (1992), Contr\^{o}le de l\{e}quation de Schrodinger, J. Math. Bures Appl, 71, 267-291.

18. [18]	Alami, A.E. , Harraki, I.E., and Boutoulout, A. (2018), Regional stabilization feedback for infinite semilinear hyperbolic systems, J. Dyn. Control Syst., 24, 343-354. https: doi.org/10.1007/s10883-017-9386-3.

19. [19]	Tsouli, A., Boutoulout, A., and Alami, E.E. (2015), ph{Constrained Feedback Stabilization for Bilinear Parabolic Systems}, Intelligent Control and Automation, 6, 103-115.

20. [20]	Ball, J. and Slemrod, M. (1979), Feedback stabilization of distributed semilinear control systems, J. Appl. Math. Opt. 5.