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 ∈ Z where Z, U are Banach spaces, the control function u belong to Lp(0, t1; U), t1 > 0, 1 < p < ∞, B ∈, L∞ (0, t1; 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 ⩽ t1, then the system can never be exactly controllable on [0, t1]. This class is so large that includes diffusion equations, damped flexible beam equation, some thermoelastic equations, strongly damped wave equations, etc.