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Constrained controllability in Banach spaces. (English) Zbl 0612.49026

Controllability (both exact and approximate) of the linear infinite dimensional control system \(x'(t)=Ax(t)+Bu(t)\) has been extensively studied in the unconstrained case where \(u\in L^ 1(0,T)\). The constrained problem, where u is required to take values in a preassigned subset \(\Omega\) of the control space U is much more difficult and has received less attention, although it is important in applications: for instance, in some problems described by the heat equation, the physical situation requires that controls be nonnegative. The authors produce here workable solutions to the problem of constrained null controllability under various assumptions on \(\Omega\) (among them, closedness and compactness).
In addition to earlier results in contrained controllability included in the bibliography, the following two are of some relevance: [H. Antosiewicz, Arch. Ration Mech. Anal. 12, 313-324 (1963; Zbl 0112.058)] and the reviewer [Comput. Methods Optim Probl., 2nd int. Conf. San Remo/Italy 1968, Lect. Notes Oper. Res. Math. Econ. 14, 92-100 (1969; Zbl 0185.241)].
Reviewer: H.O.Fattorini

MSC:

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
49J27 Existence theories for problems in abstract spaces
93C25 Control/observation systems in abstract spaces
49J20 Existence theories for optimal control problems involving partial differential equations
93B05 Controllability
93C05 Linear systems in control theory
93C20 Control/observation systems governed by partial differential equations
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