Robustness With Respect To Variable Small Delays For Exponential Stability Of Infinite-dimensional Linear Systems And Unbounded Perturbation

Robustness of stability with respect to small delays, motivated by feedback systems in the control theory is of great theoretically and practically important, has attracted much attention in recent years. Much work has been done in thisfield. In this thesis we concentrate on the following problems:1. Wellposedness and robustness to variable small delays of linear systems in continuous functional phase spaces with unbounded operator in the delay derm;2. Wellposedness and robustness to variable small delays of partial linear functional differential equations in Lp phase spaces;3. On the regularity and stability of Pritchard-Salamon systems;4.On unbounded perturbations of semigroups: compactness and norm continuity.We study the Wellposedness and robustness to variable small delays of partial linear functional differential equations in two phase spaces with unbounded operator in the delay term. We find that the nonsmoothness of delays has great influence to the Wellposedness and stability, and also we give some examples. Moreover, we consider the regularity and stability of Pritchard-Salamon systems, we obtain that perturbation semigroups persist norm continuity, compactness and analyticity of unperturbation semigroups. From this we give some relations between the spectrum bounds and growth bounds of perturbation semigroups and unperturbation semigroups, so we get some new stability results.At last, we study the unbounded perturbation of semigroups, give a sufficient condition for the compactness of the difference of two semigroups. In the case of Hilbert spaces, we can characterize the norm continuity by the resolvents of generators.