# American Institute of Mathematical Sciences

March  2015, 20(2): 613-640. doi: 10.3934/dcdsb.2015.20.613

## Uniform controllability of semidiscrete approximations for parabolic systems in Banach spaces

 1 Université d'Orléans, Bâtiment de Mathématiques (MAPMO), B.P. 6759, 45067 Orléans cedex 2

Received  December 2012 Revised  February 2014 Published  January 2015

We address in this work the minimization of the $L^q$-norm $(q>2)$ of semidiscrete controls for parabolic equation. As shown in [15], under the main approximation assumptions that the discretized semigroup is uniformly analytic and that the degree of unboundedness of control operator is lower than 1/2, uniform controllability is achieved in $L^2$ for semidiscrete approximations for the parabolic systems. The main goal of this paper is to overcome the limitation of [15] about the order 1/2 of unboundedness of the control operator. Namely, we show that the uniform controllability property also holds in $L^q \ (q>2)$ even in the case of a degree of unboundedness greater than 1/2. Moreover, a minimization procedure to compute the approximation controls in $L^q\ (q>2)$ is provided. An example of application is implemented for the one-dimensional heat equation with Dirichlet boundary control.
Citation: Thuy N. T. Nguyen. Uniform controllability of semidiscrete approximations for parabolic systems in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 613-640. doi: 10.3934/dcdsb.2015.20.613
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