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 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 \(0\leq s<t\leq t_1\), 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.


93B05 Controllability
34G10 Linear differential equations in abstract spaces
93C05 Linear systems in control theory
93C25 Control/observation systems in abstract spaces


Zbl 0516.47023
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