Article Contents
Article Contents

# Feedback stabilization with one simultaneous control for systems of parabolic equations

Dedicated to Professor Jiongmin Yong on the occasion of his 60th anniversary

The second author was supported by a grant of the Ministry of Research and Innovation, CNCS - UEFISCDI, project number PN-Ⅲ-P4-ID-PCE-2016-0011

• 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.

Mathematics Subject Classification: 35K40, 35K57, 93D15, 93B52, 93B18.

 Citation:

•  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. V. Barbu  and  G. Wang , Feedback stabilization of semilinear heat equations, Abstr. Appl. Anal., 12 (2003) , 697-714.  doi: 10.1155/S1085337503212100. 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. V. Barbu, Partial Differential Equations and Boundary Value Problems, Dordrecht: Kluwer Academic Publishers, 1998. doi: 10.1007/978-94-015-9117-1. A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems. Volume I. Boston: Birkhäuser, 1992. J.-M. Coron , Controllability and nonlinearity, ESAIM, Proc., 22 (2008) , 21-39.  doi: 10.1051/proc:072203. 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. 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. A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Seoul: Seoul National Univ., 1996. 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. C.-G. Lefter , Feedback stabilization of magnetohydrodynamic equations, SIAM J. Control Optim., 49 (2011) , 963-983.  doi: 10.1137/070697124. 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.