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International audienceWe consider the differential inclusion x˙∈F(t,x) , with F measurable in t and Lipschitz in x, together with the state constraint x(t)∈Ω and prove the existence of neighboring trajectories lying in the interior of Ω. Using this result, we can characterize the Value Function of the Bolza Problem as the unique lower semicontinuous solution to the corresponding Hamilton-Jacobi-Bellman equation

Hölder estimates for trajectories of differential inclusions and HJB equations with state constraints