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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{aligned} x'(t) + A(t)x(t)& = f(t, x(t), u(t)) + Hu(t),\\ x'(t) + A(t)x(t)& = f(t, x(t),Hu(t)) + Hu(t)\end{aligned} 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 systems in control theory
##### Keywords:
semilinear evolution equation system
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