A broad class of evolution equations are approximately controllable, but never exactly controllable

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), t > 0, z is an element of Z, where Z, U are Banach spaces, the control function u belong to L-p(0, t(1); U), t(1) > 0, 1 < p < infinity, B is an element of L-infinity (0, t(1); L(U, Z)) and A(t) generates a strongly continuous evolution operator U(t, s) according to Pazy ( 1983; Semigroups of Linear Operators with Applications to Partial Differential Equations). Specifically, we prove the following statement: If U(t, s) is compact for 0 <= s < t <= 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.