Constrained Feedback Stabilization for Bilinear Parabolic Systems
Affiliation(s)
TSI Team, MACS Laboratory, Faculty of Sciences, Moulay Ismail University, Meknes, Morocco.
TSI Team, MACS Laboratory, Faculty of Sciences, Moulay Ismail University, Meknes, Morocco.
ABSTRACT
In this paper, we shall study the stabilization and the robustness of a constrained feedback control for bilinear parabolic systems defined on a Hilbert state space. Then, we shall show that stabilizing such a system reduces stabilization only in its projection on a suitable subspace. For this purpose, a new constrained stabilizing feedback control that allows a polynomial decay estimate of the stabilized state is given. Also, the robustness of the considered control is discussed. An illustrating example and simulations are presented.
In this paper, we shall study the stabilization and the robustness of a constrained feedback control for bilinear parabolic systems defined on a Hilbert state space. Then, we shall show that stabilizing such a system reduces stabilization only in its projection on a suitable subspace. For this purpose, a new constrained stabilizing feedback control that allows a polynomial decay estimate of the stabilized state is given. Also, the robustness of the considered control is discussed. An illustrating example and simulations are presented.
Cite this paper
Tsouli, A. , Boutoulout, A. and El Alami, A. (2015) Constrained Feedback Stabilization for Bilinear Parabolic Systems. Intelligent Control and Automation, 6, 103-115. doi: 10.4236/ica.2015.62011.
Tsouli, A. , Boutoulout, A. and El Alami, A. (2015) Constrained Feedback Stabilization for Bilinear Parabolic Systems. Intelligent Control and Automation, 6, 103-115. doi: 10.4236/ica.2015.62011.
References
[1] Mohler, R.R. (1973) Bilinear Control Process. Academic Press, New York. [2] Tsouli, A. and Boutoulout, A. (2014) Controllability of the Parabolic System via Bilinear Control. Journal of Dynamical and Control Systems, 20, in press.
http://dx.doi.org/10.1007/s10883-014-9247-2 [3] Ball, J. and Slemrod, M. (1979) Feedback Stabilization of Distributed Semilinear Control Systems. Applied Mathematics and Optimization, 5, 169-179.
http://dx.doi.org/10.1007/BF01442552 [4] Jurdjevic, V. and Quinn, J.P. (1978) Controllability and Stability. Journal of Deferential Equations, 28, 381-389.
http://dx.doi.org/10.1016/0022-0396(78)90135-3 [5] Quinn, J.P. (1980) Stabilization of Bilinear Systems by Quadratic Feedback Control. Journal of Mathematical Analysis and Applications, 75, 66-80.
http://dx.doi.org/10.1016/0022-247X(80)90306-6 [6] Ouzahra, M. (2008) Strong Stabilization with Decay Estimate of Semilinear Systems. Systems and Control Letters, 57, 813-815.
http://dx.doi.org/10.1016/j.sysconle.2008.03.009 [7] Ouzahra, M., Tsouli, A. and Boutoulout, A. (2012) Stabilization and Polynomial Decay Estimate for Distributed Semilinear Systems. International Journal of Control, 85, 451-456.
http://dx.doi.org/10.1080/00207179.2012.656144 [8] Bounit, H. and Hammouri, H. (1999) Feedback Stabilization for a Class of Distributed Semilinear Control Systems. Nonlinear Analysis, 37, 953-969.
http://dx.doi.org/10.1016/S0362-546X(97)00577-4 [9] Tsouli, A., Ouzahra, M. and Boutoulout, A. (2014) A Decay Estimate for Constrained Semilinear Systems. Information Sciences Letters, 3, 77-83. http://dx.doi.org/10.12785/isl/030206 [10] Chen, M.S. (1998) Exponential Stabilization of a Constrained Bilinear System. Automatica, 34, 989-992.
http://dx.doi.org/10.1016/S0005-1098(98)00037-5 [11] Kato, T. (1980) Perturbation Theory for Linear Operators. Springer, New York. [12] Ouzahra, M. (2011) Feedback Stabilization of Parabolic Systems with Bilinear Controls. Electronic Journal of Differential Equations, 38, 1-10. [13] Triggiani, R. (1975) On the Stabilizability Problem in Banach Space. Journal of Mathematical Analysis and Applications, 52, 383-403.
http://dx.doi.org/10.1016/0022-247X(75)90067-0 [14] Pazy, A. (1983) Semi-Groups of Linear Operators and Applications to Partial Deferential Equations. Springer Verlag, New York.
http://dx.doi.org/10.1007/978-1-4612-5561-1 [15] Ammari, K. and Tucsnak, M. (2001) Stabilization of Second Order Evolution Equations by a Class of Unbounded Feedbacks. ESAIM: Control, Optimisation and Calculus of Variations, 6, 361-386.
http://dx.doi.org/10.1051/cocv:2001114 [16] Ouzahra, M. (2010) Exponential and Weak Stabilization of Constrained Bilinear Systems. SIAM Journal on Control and Optimization, 48, 3962-3974.
http://dx.doi.org/10.1137/080739161
[1] Mohler, R.R. (1973) Bilinear Control Process. Academic Press, New York. [2] Tsouli, A. and Boutoulout, A. (2014) Controllability of the Parabolic System via Bilinear Control. Journal of Dynamical and Control Systems, 20, in press.
http://dx.doi.org/10.1007/s10883-014-9247-2 [3] Ball, J. and Slemrod, M. (1979) Feedback Stabilization of Distributed Semilinear Control Systems. Applied Mathematics and Optimization, 5, 169-179.
http://dx.doi.org/10.1007/BF01442552 [4] Jurdjevic, V. and Quinn, J.P. (1978) Controllability and Stability. Journal of Deferential Equations, 28, 381-389.
http://dx.doi.org/10.1016/0022-0396(78)90135-3 [5] Quinn, J.P. (1980) Stabilization of Bilinear Systems by Quadratic Feedback Control. Journal of Mathematical Analysis and Applications, 75, 66-80.
http://dx.doi.org/10.1016/0022-247X(80)90306-6 [6] Ouzahra, M. (2008) Strong Stabilization with Decay Estimate of Semilinear Systems. Systems and Control Letters, 57, 813-815.
http://dx.doi.org/10.1016/j.sysconle.2008.03.009 [7] Ouzahra, M., Tsouli, A. and Boutoulout, A. (2012) Stabilization and Polynomial Decay Estimate for Distributed Semilinear Systems. International Journal of Control, 85, 451-456.
http://dx.doi.org/10.1080/00207179.2012.656144 [8] Bounit, H. and Hammouri, H. (1999) Feedback Stabilization for a Class of Distributed Semilinear Control Systems. Nonlinear Analysis, 37, 953-969.
http://dx.doi.org/10.1016/S0362-546X(97)00577-4 [9] Tsouli, A., Ouzahra, M. and Boutoulout, A. (2014) A Decay Estimate for Constrained Semilinear Systems. Information Sciences Letters, 3, 77-83. http://dx.doi.org/10.12785/isl/030206 [10] Chen, M.S. (1998) Exponential Stabilization of a Constrained Bilinear System. Automatica, 34, 989-992.
http://dx.doi.org/10.1016/S0005-1098(98)00037-5 [11] Kato, T. (1980) Perturbation Theory for Linear Operators. Springer, New York. [12] Ouzahra, M. (2011) Feedback Stabilization of Parabolic Systems with Bilinear Controls. Electronic Journal of Differential Equations, 38, 1-10. [13] Triggiani, R. (1975) On the Stabilizability Problem in Banach Space. Journal of Mathematical Analysis and Applications, 52, 383-403.
http://dx.doi.org/10.1016/0022-247X(75)90067-0 [14] Pazy, A. (1983) Semi-Groups of Linear Operators and Applications to Partial Deferential Equations. Springer Verlag, New York.
http://dx.doi.org/10.1007/978-1-4612-5561-1 [15] Ammari, K. and Tucsnak, M. (2001) Stabilization of Second Order Evolution Equations by a Class of Unbounded Feedbacks. ESAIM: Control, Optimisation and Calculus of Variations, 6, 361-386.
http://dx.doi.org/10.1051/cocv:2001114 [16] Ouzahra, M. (2010) Exponential and Weak Stabilization of Constrained Bilinear Systems. SIAM Journal on Control and Optimization, 48, 3962-3974.
http://dx.doi.org/10.1137/080739161