Robustness For Abstract Differential Equation With Small Time-varied Delay And Unbounded Operator In The Delay Term

In the implementation of any feedback control system, it is very likely that time delays will occur. But time delays are always omitted only because they are too small. 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. The robustness with respect to small delays for exponential stability of infinite-dimensional linear systems has been studied since 1986. In 1986, using two counter-examples Huang[2] showed that, in general, exponential stability is not robust with respect to small delays for infinite-dimensional linear system. In [2], Huang gave a classical sufficient condition of robustness with respect to small time-varied delay for exponential stability. In [3], the author extended the result in [2] to a more general situation. But all the results were established on the basis that the operator in delay term is bounded. For the condition that the operator is unbounded in delay term which is constant, Liu[4] and Guo[8] have made some research and got some sufficient and necessary conditions. Our paper have studied the well-posedness and robustness of the system with small time-varied delay in the same situation of [4]. We have not only obtained robustness with respect to small time-varied delays for exponential stability of the system, but also made some essential improvement to result of [4].