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A broad class of evolution equations are approximately controllable, but never exactly controllable. (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 ' =A(t)z+B(t)u(t),

z(T)Z, u(t)U, t>0, where Z,U are infinite-dimensional Banach spaces, U is reflexive, uL P ([0,t 1 ],U), t 1 >0, 1<p<, BL ([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 0s<tt 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.

MSC:
93B05Controllability
34G10Linear ODE in abstract spaces
93C05Linear control systems
93C25Control systems in abstract spaces