Exponential stabilization on infinite dimensional system with impulse controls

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.