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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:

${z}^{\text{'}}=A\left(t\right)z+B\left(t\right)u\left(t\right),$

$z\left(T\right)\in Z$, $u\left(t\right)\in U$, $t>0$, where $Z,U$ are infinite-dimensional Banach spaces, $U$ is reflexive, $u\in {L}^{P}\left(\left[0,{t}_{1}\right],U\right)$, ${t}_{1}>0$, $1, $B\in {L}^{\infty }\left(\left[0,{t}_{1}\right],L\left(U,Z\right)\right)$ and $A\left(t\right)$ generates a strongly continuous evolution operator $U\left(t,s\right)$. 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 $U\left(t,s\right)$ is compact for $0\le s, then the system can never be exactly controllable on $\left[0,{t}_{1}\right]$. This class is so large that includes diffusion equations, damped flexible beam equation, some thermoelastic equations, strongly damped wave equations, etc.

##### MSC:
 93B05 Controllability 34G10 Linear ODE in abstract spaces 93C05 Linear control systems 93C25 Control systems in abstract spaces