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

Abstract

AbstractThis 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.