Switching Controls for Analytic Semigroups and Applications to Parabolic Systems

Abstract

In this work, we extend the analysis of the problem of switching controls proposedin [E. Zuazua,J. Eur. Math. Soc. (JEMS), 13 (2011), pp. 85--117]. The problem asks the followingquestion: Assuming that one can control a system using two or more actuators, does there exist acontrol strategy such that at all times, only one actuator is active? We answer positively when thecontrolled system corresponds to an analytic semigroup spanned by a positive self-adjoint operatorwhich 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 infinite dimensional settings and under some further spectral assumptions when the operator spans ananalytic semigroup but is not necessarily self-adjoint.