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Local controllability of semilinear evolution systems in Banach spaces. (English) Zbl 0927.93014
Semilinear, abstract, continuous-time control systems defined in Banach spaces are considered. Using fixed-point methods a sufficient condition for local exact controllability in a given time interval is formulated and proved. In the proof, the theory of analytic semigroups of linear bounded operators is extensively used. Moreover, some special cases are also investigated. As an illustrative example, local exact controllability in a given time interval for the semilinear heat equation with homogeneous boundary conditions of Dirichlet type is discussed. The relationships to the local controllability results known in the literature are pointed out. Moreover, several remarks and comments on controllability for nonlinear control systems are given. The results of the paper extend controllability conditions presented in the paper [N. Carmichael and M. D. Quinn, “Fixed-point methods in nonlinear control”, IMA J. Math. Control Inf. 5, No. 1, 41-67 (1988; Zbl 0646.93029)].
MSC:
93B05Controllability
93C10Nonlinear control systems
93C25Control systems in abstract spaces