PDEs with Distributed Control and Delay in the Loop: Transfer Function Poles, Exponential Modes and Robustness of Stability

In this paper we consider a large class of partial diierential equations (PDEs) with distributed control and with a time-delay in the feedback loop. We analyze the relationship between the poles of the closed-loop transfer function and the exponential modes of the underlying retarded PDE in order to derive internal stability properties from external ones. Our approach is based on a combination of input-output methods and modal analysis. We give a number of suucient conditions for robustness/nonrobustness of closed-loop modal stability with respect to delays. The theory is illustrated by two examples.