Dynamic and static feedback control for second order infinite‐dimensional systems

Funding information Project of Department of Education of Guangdong Province, Grant/Award Number: 2017KZDXM087; National Natural Science Foundation of China, Grant/Award Number: 61873153, 11671240 Abstract This paper considers dynamic feedback stabilization for abstract second-order systems, where the dynamic feedback controller is designed as another abstract second order infinite/finite-dimensional system. This makes the closed-loop system PDE-PDE or PDE-ODE coupled. The stability of the closed-loop system is found to have three different cases. We first consider the dynamic feedback control in a general Hilbert space which is usually different from the control space. It is shown that the stability of the closed-loop systems under the dynamic and static feedbacks are usually not equivalent. However, if the dynamic control law is a copy of the original system, we deduce, under some conditions, that the coupled system is exponentially stable if and only if the static feedback closed-loop system is exponentially stable. When the dynamic feedback is designed in the control space, the closed-loop system is asymptotically stable if and only if static feedback closed-loop system is asymptotically stable.

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