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Approximate controllability for semilinear systems. (English) Zbl 1108.93302

Summary: The relations between the (strong) reachable sets of the semilinear evolution equation systems

$\begin{array}{cc}\hfill {x}^{\text{'}}\left(t\right)+A\left(t\right)x\left(t\right)& =f\left(t,x\left(t\right),u\left(t\right)\right)+Hu\left(t\right),\hfill \\ \hfill {x}^{\text{'}}\left(t\right)+A\left(t\right)x\left(t\right)& =f\left(t,x\left(t\right),Hu\left(t\right)\right)+Hu\left(t\right)\hfill \end{array}$

on a Banach space, and their corresponding linear systems are studied. Compared with previous results, the systems considered here are more general ($f$ is not independent of the control $u$), no compactness assumptions on $A$ or $f$ are imposed in some of our main results, and we suppose $f$ is a set-contraction rather than Lipschitz and have less restriction on the contraction coefficient. Other kinds of conditions are involved to guarantee the approximate controllability.

MSC:
 93B05 Controllability 93C10 Nonlinear control systems