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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: $$ z'=A(t)z+B(t)u(t), $$ $z(T)\in Z$, $u(t)\in U$, $t>0$, where $Z,U$ are infinite-dimensional Banach spaces, $U$ is reflexive, $u\in L^P([0,t_1],U)$, $t_1>0$, $1<p<\infty$, $B\in L^\infty([0,t_1],L(U,Z))$ and $A(t)$ generates a strongly continuous evolution operator $U(t,s)$. according to {\it 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 $0\le s<t\le t_1$, 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.

34G10Linear ODE in abstract spaces
93C05Linear control systems
93C25Control systems in abstract spaces
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