Abstract
We consider control systems described by a semilinear abstract equation $y' + Ay = F(y,u) + Bu$ in Hilbert space. General conditions for exact reachability and approximate controllability are given which are related with two families of associated quadratic optimal control problems. The minimum norm optimal control and the construction of the reachable set of the corresponding linear system $y' + Ay = Bu$ are characterized respectively. Exact reachability for a class of semilinear control systems is obtained under some assumptions on the range of a nonlinear operator $\mathcal{F}_{(t_0 ,T)} $) (see definition in §4) generated by the nonlinear function F.
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