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: \[ z'=A(t)z+B(t)u(t), \]
\(z(T)\in Z\), \(u(t)\in U\), \(t>0\), 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 \(A(t)\) 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 \(0\leq s<t\leq t_1\), 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.