Switching Controls for Analytic Semigroups and Applications to Parabolic Systems

TL;DR

This paper extends the analysis of switching control strategies for systems governed by analytic semigroups, demonstrating the existence of controls activating only one actuator at a time under certain spectral and controllability conditions.

Contribution

It provides new results on switching controls for analytic semigroups, including non-self-adjoint cases, broadening the applicability of previous control strategies.

Findings

Existence of switching controls for analytic semigroup systems.

Extension to non-self-adjoint operators under spectral assumptions.

Applicable to finite-dimensional and infinite-dimensional systems.

Abstract

In this work, we push further the analysis of the problem of switching controls proposed in [E. Zuazua, Switching control, J. Eur. Math. Soc. (JEMS), 13(1): 85--117, 2011]. The problem consists in the following one: 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 to this question 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 as the argument of E. Zuazua, our proof relies on analyticity arguments and will also work in finite dimensional setting and under some further spectral assumptions when the operator spans an analytic semigroup but is not necessarily self-adjoint.