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:
, , , where are infinite-dimensional Banach spaces, is reflexive, , , , and generates a strongly continuous evolution operator . 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 is compact for , then the system can never be exactly controllable on . This class is so large that includes diffusion equations, damped flexible beam equation, some thermoelastic equations, strongly damped wave equations, etc.