Some properties of the value function and its level sets for affine control systems with quadratic cost

DOI10.1023/A:1009552511132zbMATH Open0964.49021arXivmath/0607424MaRDI QIDQ1591063

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Publication date: 19 December 2000

Published in: (Search for Journal in Brave)

Abstract: Let $T>0$ fixed. We consider the optimal control problem for analytic affine systems: $dsdotx=f_0(x)+su{m}_{i=1}^m{u}_i{f}_i(x)$, with a cost of the form: $dsC(u)=in{t}_0^{T}su{m}_{i=1}^m{u}_i^2(t)dt$. For this kind of systems we prove that if there are no minimizing abnormal extremals then the value function $S$ is subanalytic. Secondly we prove that if there exists an abnormal minimizer of corank 1 then the set of end-points of minimizers at cost fixed is tangent to a given hyperplane. We illustrate this situation in sub-Riemannian geometry.

Full work available at URL: https://arxiv.org/abs/math/0607424

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