State constrained $L^\infty$ optimal control problems interpreted as differential games

Piernicola Bettiol

doi:10.3934/dcds.2015.35.3989

Article Contents

2015, Volume 35, Issue 9: 3989-4017. Doi: 10.3934/dcds.2015.35.3989

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State constrained $L^\infty$ optimal control problems interpreted as differential games

Piernicola Bettiol^1,

1.
Laboratoire de Mathematiques, Université de Bretagne Occidentale, 6 Avenue Victor Le Gorgeu, 29200 Brest

Received: March 31, 2014

Revised: September 30, 2014

Published: May 2017

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Abstract

We consider state constrained optimal control problems in which the cost to minimize comprises an $L^\infty$ functional, i.e. the maximum of a running cost along the trajectories. In absence of state constraints, a new approach has been suggested by a recent paper [9]. The main purpose of the present paper is to extend this approach and the related results to state constrained $L^\infty$ optimal control problems. More precisely, using the $(L^\infty, L^1)$-duality, the reference optimal control problem can be seen as a static differential game, in which an extra variable is introduced and plays the role of an opponent player who wants to maximize the cost. Under appropriate assumptions and employing suitable Filippov's type results, this static game turns out to be equivalent to the corresponding dynamic differential game, whose (upper) value function is the unique viscosity solution to a constrained boundary value problem, which involves a Hamilton-Jacobi equation with a continuous Hamiltonian.

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Mathematics Subject Classification: Primary: 49K35, 49N70, 49L25.

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