*(English)*Zbl 1108.93014

Summary: As we have announced in the title of this work, we show that a broad class of evolution equations are approximately controllable but never exactly controllable. This class is represented by the following infinite-dimensional time-varying control system:

$z\left(T\right)\in Z$, $u\left(t\right)\in U$, $t>0$, where $Z,U$ are infinite-dimensional Banach spaces, $U$ is reflexive, $u\in {L}^{P}([0,{t}_{1}],U)$, ${t}_{1}>0$, $1<p<\infty $, $B\in {L}^{\infty}([0,{t}_{1}],L(U,Z))$ and $A\left(t\right)$ generates a strongly continuous evolution operator $U(t,s)$. according to *A. Pazy* [Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, 44. New York etc.: Springer-Verlag (1983; Zbl 0516.47023)].

Specifically, we prove the following statement: If $U(t,s)$ is compact for $0\le s<t\le {t}_{1}$, then the system can never be exactly controllable on $[0,{t}_{1}]$. This class is so large that includes diffusion equations, damped flexible beam equation, some thermoelastic equations, strongly damped wave equations, etc.