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Some optimal control applications of real-analytic stratifications and desingularization. (English) Zbl 1050.49500

Jakubczyk, Bronisław (ed.) et al., Singularities symposium – Łojasiewicz 70. Papers presented at the symposium on singularities on the occasion of the 70th birthday of Stanisław Łojasiewicz, Cracow, Poland, September 25–29, 1996 and the seminar on singularities and geometry, Warsaw, Poland, September 30–October 4, 1996. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Cent. Publ. 44, 211-232 (1998).
From the text: Here the author uses the desingularization theorem to give a detailed proof of the following weak regularity theorem:
Theorem: Let \(f: M\times U\rightarrow TM\) be a real-analytic control system (so the state space \(M\) is a finite-dimensional, real-analytic manifold, the space of control values \(U\) is a compact subanalytic set of some other real-analytic manifold, and \(f\) is jointly real-analytic on \(M\times U\)). If a state \(\widehat x\in M\) can be reached from a state \(\overline x\in M\) on the interval \([0,T]\) by a measurable control, then \(\widehat x\) can be reached from \(\overline x\) on the interval \([0,T]\) by a control that is real-analytic on an open dense subset of \([0,T]\).
He also gives an application to the theory of observability and recovers a result of J.-P. Gauthier and I. A. K. Kupka [Math. Z. 223, No. 1, 47–78 (1996; Zbl 0863.93008)] that if a real-analytic system is strongly observable with respect to the class of real-analytic controls, then it is strongly observable with respect to the class of measurable controls.
For the entire collection see [Zbl 0906.00013].

MSC:

49J15 Existence theories for optimal control problems involving ordinary differential equations
32C05 Real-analytic manifolds, real-analytic spaces
93B20 Minimal systems representations
93C10 Nonlinear systems in control theory

Citations:

Zbl 0863.93008
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