# American Institute of Mathematical Sciences

September  2015, 35(9): 3989-4017. doi: 10.3934/dcds.2015.35.3989

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

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

Received  April 2014 Revised  October 2014 Published  April 2015

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.
Citation: Piernicola Bettiol. State constrained $L^\infty$ optimal control problems interpreted as differential games. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 3989-4017. doi: 10.3934/dcds.2015.35.3989
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