The effect of small time-delays on the closed-loop stability of boundary control systems

It has been observed that for many stable feedback systems, the introduction of arbitrarily small time-delays into the loop causes instability. In this paper we present a systematic treatment of this phenomenon for a large class of boundary control systems which allows for in-span control. Our approach is based on a combination of input-output methods and modal analysis. We give a number of sufficient conditions for robustness/nonrobustness of closed-loop input-output stability with respect to delays. Our framework includes a large class of ill-posed systems, i.e., systems whose open-loop transfer function is unbounded on any right half-plane. We then analyze the relationship between the poles of the transfer function and the exponential modes of the underlying boundary-value problem to derive internal stability properties from external ones.This work was supported by the British Council/DAAD (ARC project 464), by the Human Capital and Mobility programme (Project number CHRX-CT93-0402), by the National Science Foundation (Grant DMS-9206986) and by NATO (Grant CRG 950179).