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This paper studies the exponential stabilization on infinite dimensionalsystem with impulse controls, where impulse instants appear periodically. Thefirst main result shows that exponential stabilizability of the control systemwith a periodic feedback law is equivalent to one kind of weak observabilityinequalities. The second main result presents that, in the setting of adiscrete LQ problem, the exponential stabilizability of control system with aperiodic feedback law is equivalent to the solvability of an algebraicRiccati-type equation which was built up in [Qin, Wang and Yu, SIAM J. ControlOptim., 59 (2021), pp. 1136-1160] for finite dimensional system. As anapplication, some sufficient and necessary condition for the exponentialstabilization of an impulse controlled system governed by coupled heatequations is given.

Exponential stabilization on infinite dimensional system with impulse controls