Regularity and necessary conditions for a Bolza optimal control problem

We consider a Bolza optimal control problem whose Lagrangian, possibly extended valued, may be discontinuous in the state and control variable so that optimal solutions are not supposed to necessarily satisfy the Maximum Principle. Given an optimal trajectory-control pair, we prove that it satisfies a new Erdmann – Du Bois-Reymond type condition, and show that, from this condition, it is possible to derive boundedness properties of the optimal control and a Lipschitz regularity result for the optimal state arc, just imposing general growth assumptions (allowing some almost linear growth behaviors).