Exponential stabilization on infinite dimensional system with impulse controls

TL;DR

This paper investigates exponential stabilization of infinite dimensional systems with periodic impulse controls, establishing equivalences with observability inequalities and Riccati equations, and applies results to coupled heat equations.

Contribution

It introduces new equivalence results linking exponential stabilizability to observability and Riccati equations in infinite dimensional systems with impulse controls.

Findings

Exponential stabilizability is equivalent to weak observability inequalities.

Stabilizability is characterized by the solvability of a Riccati-type equation.

Conditions for stabilization of impulse-controlled coupled heat equations are provided.

Abstract

This paper studies the exponential stabilization on infinite dimensional system with impulse controls, where impulse instants appear periodically. The first main result shows that exponential stabilizability of the control system with a periodic feedback law is equivalent to one kind of weak observability inequalities. The second main result presents that, in the setting of a discrete LQ problem, the exponential stabilizability of control system with a periodic feedback law is equivalent to the solvability of an algebraic Riccati-type equation which was built up in [Qin, Wang and Yu, SIAM J. Control Optim., 59 (2021), pp. 1136-1160] for finite dimensional system. As an application, some sufficient and necessary condition for the exponential stabilization of an impulse controlled system governed by coupled heat equations is given.