Séance Séminaire

Séminaire ACSIOM

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

$L^\infty$ (or minimax) problems consist in minimizing the (pointwise) maximum of a running cost along the trajectories. These control problems have been extensively studied from various points of view, including dynamic programming, numerical approximations, and nec- essary conditions. A new approach suggests to interpret $L^\infty$ optimal control problems as particular differential games. More precisely, using the $(L^\infty,L^1)$-duality, the reference op- timal 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 non-restrictive assumptions, 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. This new perspective permits to overcome some difficulties arisen in the literature: particularly the fact that (even in absence of state constraints) the Hamiltonian employed was discontinuous and depended on the solution of the related Hamilton-Jacobi equation.