Volume 26, 2020
|Number of page(s)||38|
|Published online||10 December 2020|
Local feedback stabilization of time-periodic evolution equations by finite dimensional controls
Institut de Mathématiques de Toulouse, Université Paul Sabatier & CNRS,
Toulouse Cedex, France.
2 Department of Mathematics, Indian Institute of Technolodgy Bombay, Powai, Maharashtra - 400076, India.
3 T.I.F.R Centre for Applicable Mathematics, Post Bag No. 6503, GKVK Post Office, Bangalore - 560065, India.
* Corresponding author: firstname.lastname@example.org
Accepted: 6 July 2020
We study the feedback stabilization around periodic solutions of parabolic control systems with unbounded control operators, by controls of finite dimension. We prove that the stabilization of the infinite dimensional system relies on the stabilization of a finite dimensional control system obtained by projection and next transformed via its Floquet-Lyapunov representation. We emphasize that this approach allows us to define feedback control laws by solving Riccati equations of finite dimension. This approach, which has been developed in the recent years for the boundary stabilization of autonomous parabolic systems, seems to be totally new for the stabilization of periodic systems of infinite dimension. We apply results obtained for the linearized system to prove a local stabilization result, around periodic solutions, of the Navier-Stokes equations, by finite dimensional Dirichlet boundary controls.
Mathematics Subject Classification: 93B52 / 93D15 / 35B10 / 34H15 / 76D55
Key words: Periodic parabolic systems / feedback stabilization / finite dimensional controls / Navier-Stokes equations
© EDP Sciences, SMAI 2020
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