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Cardaliaguet, P.; Quincampoix, M.; Saint-Pierre, P.

Optimal times for constrained nonlinear control problems without local controllability. (English) Zbl 0884.49002

Appl. Math. Optimization 36, No. 1, 21-42 (1997).
In this article the minimal time problem for a closed target \(C\) with state constraints given by a closed set \(K\) without local controllability at the target is considered. The basic difficulty of this setup is given by the fact that the optimal value function in general is discontinuous for this problem.
The authors derive three main results: The epigraph of the optimal value function is shown to be the viability kernel of a suitable extended differential inclusion corresponding to the control system, an algorithm for the computation of viability kernels is adapted to this problem, thus admitting a numerical solution for the problem, and the optimal value function is characterized to be the smallest supersolution of a suitable Hamilton-Jacobi equation.
The results are obtained by a consequent use of viability theory applied to the differential inclusion \(\bigcup_{u\in U}f(x,u)\) where \(f\) is the right hand side of the control system. The main limitation for the applicability of the results is given by the fact that this differential inclusion is supposed to be a Marchaud-set-valued map. However, apart from this restriction the paper gives an elegant approach to deal with both the analytic and numerical difficulties of this problem.
Reviewer: Lars Grüne (Augsburg)

Cited in 26 Documents

MSC:

49J15 Existence theories for optimal control problems involving ordinary differential equations
65K10 Numerical optimization and variational techniques
93C15 Control/observation systems governed by ordinary differential equations
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
49J24 Optimal control problems with differential inclusions (existence) (MSC2000)

Keywords:

time optimal control; viability theory; differential inclusions; Hamilton-Jacobi equation; numerical solution
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References:

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