Norm Continuity Of Solution Semigroup And Robust Controllability For Delay Systems With Unbounded Delay Operators

The development of things not only depends on the current state, but also depends on the past history. That is, it is inevitable to exist delay in dynamic systems. Therefore, the properties (for example, stability, controllability) of delay systems have attracted much attention in recent years. The delay systems are as follows, where A is the infinitesimal generator of C 0semigroup (T (t))t≥0 on a Banach space X , A0 is a linear operator defined from X to X . When A0 satisfies some suitable conditions, the above equation has the unique mild solution: Nakagiri S. and Yamamoto M. have proved that the solution semigroup of the above equation is a C 0semigroup under the condtion that A0 is bounded [1 ,2]. Jiang, Guo, Huang have obtained the similar result in the case that A0 is an unbounded operator [ 4]. Fisher A. and Neerven J.M.A.M.Van have proved the solution semigroup of the above equation is norm continuous for t > r if the C 0semigroup (T (t))t≥0 generated by A is norm continuous for t >0[ 3].The results in [ 3]were obtained in the case that A0 is a bounded operator. However, in actual applications, there are many reasons to result in the appearance of unbounded delay operator, for example, adding some special material in the space flight body's elastomer material as the damping, adding some damping in the wave communication medium. Therefore, it is more valuable to consider the delay systems with unbounded delay operator. Much work has been done in this filed, e.g., Liu, Jiang, Huang have showed that robust stability of nonsmooth Pritchard -Salamon systems with small delay and robust stability of dynamic systems with unbouded operator in delay term [ 5,6]. Jiang, Huang considered the robust controllability of linear control systems with unbounded control operators [ 7]. Yu, Liu have studied the robust controllability with respect to small delays for linear dynamic systems in Banach spaces [ 8]. In this thesis we will concentrate on the norm continuity and the robust controllability of linear control systems with unbounded delays operator. We will prove the norm continuity, and then prove the robust controllability of linear control systems.