Abstract
Some results are given involving the controllability of a nonlinear evolution equation of the formx′(t)+A(t,u(t))x(t)=B(t)u(t). The underlying space is a real Banach space with uniformly convex dual space. A method involving ranges of sums of two nonlinear operators is employed along with the Leray-Schauder degree theory to establish sufficient conditions for controllability. A result is also given characterizing a type of reachable sets of certain evolutions. Finally, the lack of exact controllability is shown for quite a large class of nonlinear evolutions with compact perturbationsB(t). The responses are actually straight-line segments emanating from the origin. However, several results herein can be extended to cover a variety of control problems with other known types of responses.
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Communicated by G. P. Papavassilopoulos
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Kaplan, D.R., Kartsatos, A.G. Ranges of sums and control of nonlinear evolutions with preassigned responses. J Optim Theory Appl 81, 121–141 (1994). https://doi.org/10.1007/BF02190316
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DOI: https://doi.org/10.1007/BF02190316