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Feedback stabilization with one simultaneous control for systems of parabolic equations
Faculty of Mathematics, University "Al. I. Cuza" Iași, Romania, Octav Mayer Institute of Mathematics, Romanian Academy, Iași Branch |
In this work controlled systems of semilinear parabolic equations are considered. Only one control is acting in both equations and it is distributed in a subdomain. Local feedback stabilization is studied. The approach is based on approximate controllability for the linearized system and the use of an appropriate norm obtained from a Lyapunov equation. Applications to reaction-diffusion systems are discussed.
References:
[1] |
F. Ammar Khodja, A. Benabdallah, C. Dupaix and I. Kostin,
Controllability to the trajectories of phase-field models by one control force, SIAM J. Control Optim., 42 (2003), 1661-1680.
doi: 10.1137/S0363012902417826. |
[2] |
V. Barbu and G. Wang,
Feedback stabilization of semilinear heat equations, Abstr. Appl. Anal., 12 (2003), 697-714.
doi: 10.1155/S1085337503212100. |
[3] |
V. Barbu and G. Wang,
Internal stabilization of semilinear parabolic systems, J. Math. Anal. Appl., 285 (2003), 387-407.
doi: 10.1016/S0022-247X(03)00405-0. |
[4] |
V. Barbu,
Partial Differential Equations and Boundary Value Problems, Dordrecht: Kluwer Academic Publishers, 1998.
doi: 10.1007/978-94-015-9117-1. |
[5] |
A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter,
Representation and Control of Infinite Dimensional Systems. Volume I. Boston: Birkhäuser, 1992. |
[6] |
J.-M. Coron,
Controllability and nonlinearity, ESAIM, Proc., 22 (2008), 21-39.
doi: 10.1051/proc:072203. |
[7] |
J.-M. Coron, S. Guerrero and L. Rosier,
Null controllability of a parabolic system with a cubic
coupling term, SIAM J. Control Optim., 48 (2010), 5629-5653.
doi: 10.1137/100784539. |
[8] |
J.-M. Coron and J.-P. Guilleron,
Control of three heat equations coupled with two cubic
nonlinearities, SIAM J. Control Optim., 55 (2017), 989-1019.
doi: 10.1137/15M1041201. |
[9] |
A. V. Fursikov and O. Yu. Imanuvilov,
Controllability of Evolution Equations, Seoul: Seoul National Univ., 1996. |
[10] |
C. Lefter,
Feedback stabilization of 2D Navier-Stokes equations with Navier slip boundary
conditions, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 70 (2009), 553-562.
doi: 10.1016/j.na.2007.12.026. |
[11] |
C.-G. Lefter,
Feedback stabilization of magnetohydrodynamic equations, SIAM J. Control Optim., 49 (2011), 963-983.
doi: 10.1137/070697124. |
[12] |
A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
show all references
References:
[1] |
F. Ammar Khodja, A. Benabdallah, C. Dupaix and I. Kostin,
Controllability to the trajectories of phase-field models by one control force, SIAM J. Control Optim., 42 (2003), 1661-1680.
doi: 10.1137/S0363012902417826. |
[2] |
V. Barbu and G. Wang,
Feedback stabilization of semilinear heat equations, Abstr. Appl. Anal., 12 (2003), 697-714.
doi: 10.1155/S1085337503212100. |
[3] |
V. Barbu and G. Wang,
Internal stabilization of semilinear parabolic systems, J. Math. Anal. Appl., 285 (2003), 387-407.
doi: 10.1016/S0022-247X(03)00405-0. |
[4] |
V. Barbu,
Partial Differential Equations and Boundary Value Problems, Dordrecht: Kluwer Academic Publishers, 1998.
doi: 10.1007/978-94-015-9117-1. |
[5] |
A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter,
Representation and Control of Infinite Dimensional Systems. Volume I. Boston: Birkhäuser, 1992. |
[6] |
J.-M. Coron,
Controllability and nonlinearity, ESAIM, Proc., 22 (2008), 21-39.
doi: 10.1051/proc:072203. |
[7] |
J.-M. Coron, S. Guerrero and L. Rosier,
Null controllability of a parabolic system with a cubic
coupling term, SIAM J. Control Optim., 48 (2010), 5629-5653.
doi: 10.1137/100784539. |
[8] |
J.-M. Coron and J.-P. Guilleron,
Control of three heat equations coupled with two cubic
nonlinearities, SIAM J. Control Optim., 55 (2017), 989-1019.
doi: 10.1137/15M1041201. |
[9] |
A. V. Fursikov and O. Yu. Imanuvilov,
Controllability of Evolution Equations, Seoul: Seoul National Univ., 1996. |
[10] |
C. Lefter,
Feedback stabilization of 2D Navier-Stokes equations with Navier slip boundary
conditions, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 70 (2009), 553-562.
doi: 10.1016/j.na.2007.12.026. |
[11] |
C.-G. Lefter,
Feedback stabilization of magnetohydrodynamic equations, SIAM J. Control Optim., 49 (2011), 963-983.
doi: 10.1137/070697124. |
[12] |
A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
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