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Journal of Convex Analysis 27 (2020), No. 1, 139--163
Copyright Heldermann Verlag 2020

Second-Order Analysis for the Time Crisis Problem

Térence Bayen
Laboratoire de Mathématiques, Université d'Avignon, 84018 Avignon, France

Laurent Pfeiffer
Institute of Mathematics, University of Graz, 8010 Graz, Austria

We prove second-order necessary optimality conditions for the so-called time crisis problem that comes up within the context of viability theory. It consists in minimizing the time spent by solutions of a controlled dynamics outside a given subset K of the state space. One essential feature is the discontinuity of the characteristic function involved in the cost functional. Thanks to a change of time and an augmentation of the dynamics, we relate the time crisis problem to an auxiliary Mayer control problem. This allows us to use the classical tools of optimal control for obtaining optimality conditions. Going back to the original problem, we deduce in that way second order optimality conditions for the time crisis problem.

Keywords: Optimal control, Pontryagin maximum principle, second order optimality conditions.

MSC: 49J15, 49K15, 49J52, 34H05.

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