Mathematics

This talk is devoted to regularity of minimizers and adjoint statesfor the Bolza optimal control problem under state constraints. It iswell known that the adjoint state of the Pontryagin maximumprinciple may be discontinuous whenever the optimal trajectory lies partiallyon the boundary of constraints. Still we prove that if the associatedHamiltonian H(t,x,.) is differentiable and the constraints are sleek,then every optimal trajectory is continuously differentiable.Moreover if for all x on the boundary of constraints, the partial derivativeof H with respect to the last variable, H_p(t,x, .) iis strictlymonotone in directions normal at x to the set of constraints, thenthe adjoint state is also continuous on interior of its interval ofdefinition. Finally, we identify a class of constraints for which theadjoint state is absolutely continuous or even Lipschitz on this openinterval. This allows us to derive necessary conditions foroptimality in the form of variational differential inequalities, maximumprinciple and modified transversality conditions. We also providesufficient conditions for Lipschitz continuity of optimal controlsand for normality of the maximum principle.