Stabilization of Linear Systems

This paper considers a finite-dimensional linear time-varying system and is concerned with the question: What is the relation between controllability properties of the system and various degrees of stability of the closed loop system resulting from linear feedback of the state variable?The main results are as follows: For any initial time to, and any continuous and monotonically nondecreasing function $\delta ( \cdot ,t_0 )$ such that $\delta (t_0 ,t_0 ) = 0$, the transition matrix $\hat \Phi ( \cdot , \cdot )$ of the closed loop system can be made such that $\| {\hat \Phi (t,t_0 )} \| \leqq a(t_0 )\exp [ - \delta (t,t_0 )]$ for all $t \geqq t_0 $, if and only if the system is completely controllable. Furthermore, in case of a bounded system, for any $m \geqq 0$, a bounded feedback matrix can be found such that $\| {\hat \Phi (t_2 ,t_1 )} \| \leqq a\exp [ - m(t_2 - t_1 )]$ for all $t_1 $ and $t_2 \geqq t_1 $, if and only if the system is uniformly completely controllable.Thus they can be regarded as exten...