Switching Controls for Analytic Semigroups and Applications to Parabolic Systems

Abstract

In this work, we extend the analysis of the problem of switching controls proposed in [E. Zuazua, J. Eur. Math. Soc. (JEMS), 13 (2011), pp. 85--117]. The problem asks the following question: Assuming that one can control a system using two or more actuators, does there exist a control strategy such that at all times, only one actuator is active? We answer positively when the controlled system corresponds to an analytic semigroup spanned by a positive self-adjoint operator which is null-controllable in arbitrary small times. Similarly to [E. Zuazua, J. Eur. Math. Soc. (JEMS), 13 (2011), pp. 85--117], our proof relies on analyticity arguments and will also work in finite dimensional settings and under some further spectral assumptions when the operator spans an analytic semigroup but is not necessarily self-adjoint.