Robustness With Respect To Small Delays For Exponential Stability Of Distributed Parameter Control Systems

In the implementation of any feedback control system, it is very likely that time delays will occur. It is therefore of vital importance to understand the sensitivity of control system to introduction of small delays in the feedback loop and problems of this type have attracted a lot of attention. This paper is concerned with the robustness with respect to small delays for exponential stability of distributed parameter control systems, which is first characterized by fundamental operators.We first introduce fundamental operators for abstract differential equations with delays in Banach spaces, and characterize the robustness with respect to small delays via the associated fundamental operators. Specially, in Hilbert spaces, some necessary and sufficient conditions are given in terms of the uniformly square integrability and uniform boundness of the resolvent on the imaginary axis respectively. Furthermore, we consider this robustness for abstract differential equations with unbounded operator in the delay term. Next, sufficient conditions are given for the nonautonomous systems to be robust with respect to small time-varing delays, and these results are applied to the nonautonomous parabolic system.Finally, the boundary feedback control of the undamped Euler-Bemoulli beam with both ends free is concerned. We introduce the dynamical state space, combine the operator semigroup technique, the multiplier technique and the contradiction argument of frequency domain to obtain the best possible conditions for the energy of this system to be uniformly exponentially decay.