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Existence and structure of optimal solutions of infinite-dimensional control problems. (English) Zbl 0966.49002

Summary: We analyze the structure of optimal solutions for a class of infinite-dimensional control systems. We are concerned with the existence of an overtaking optimal trajectory over an infinite horizon. The existence result that we obtain extends the result of D. A. Carlson, A. B. Haurie and A. Jabrane [SIAM J. Control Optimization 25, No. 6, 1517-1541 (1987; Zbl 0658.49003)] to a situation where the trajectories are not necessarily bounded. Also, we show that an optimal trajectory defined on an interval \([0,\tau]\) is contained in a small neighborhood of the optimal steady-state in the weak topology for all \(t\in [0,\tau]\setminus E\), where \(D\subset[0,\tau]\) is a measurable set such that the Lebesgue measure of \(E\) does not exceed a constant which depends only on the neighborhood of the optimal steady-state and does not depend on \(\tau\).

MSC:

49J27 Existence theories for problems in abstract spaces
93C25 Control/observation systems in abstract spaces
49J15 Existence theories for optimal control problems involving ordinary differential equations

Citations:

Zbl 0658.49003
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