Exact controllability and rapid stabilization of linear difference delay equations and one-dimensional hyperbolic systems

Abstract

This paper addresses exact controllability and rapid stabilization issues for linear difference delay systems with finitely many pointwise delays. We first establish a criterion for exact controllability in $L^r$, $r\in [1,+\infty )$, characterized by a necessary and sufficient Hautus-type frequency-domain condition. The sufficiency of this condition in $L^r$ was already known [https://hal.science/hal-04830757/], whereas its necessity had previously been established only in $L^1$ [arXiv:2210.13590]. We answer [arXiv:2210.13590, Conjecture 3.8] by proving that this condition is also necessary for $r\in (1,+\infty )$. The analysis is based on rewriting the system as an evolution equation in a suitable Banach space, which allows one to apply duality arguments in the spirit of [https://doi.org/10.1137/S036301299119795X]. Building on this controllability analysis and the abstract stabilization framework developed in [https://doi.org/10.1137/S0363012996301609, arXiv:1301.5744], we prove that our exact controllability criterion in $L^r$ is also a sufficient condition for rapid stabilization in $L^2$, i.e., that the system of interest can be stabilized by a family of state feedback control laws ensuring that the closed-loop system is exponentially stable in $L^2$ with an arbitrarily large decay rate. As an application, we extend our results to linear one-dimensional hyperbolic partial differential equations, which can be reduced to linear difference delay systems via the method of characteristics, obtaining also a necessary and sufficient frequency-domain criterion for exact controllability in $L^r$, $r\in [1,+\infty )$, which, as before, is a sufficient condition for rapid stabilization in $L^2$.