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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.

93B05 Controllability
93C10 Nonlinear systems in control theory
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