Error estimates for the euler discretization of an optimal control problem with first-order state constraints

We propose some error estimates for the discrete solution of an optimal control problem with first-order state constraints, where the trajectories are approximated with a classical Euler scheme. We obtain order 1 approximation results in the L1 norm (as opposed to the order 2/3 results obtained in the literature). We assume either a strong second-order optimality condition or a weaker formulation in the case where the state constraint is scalar and satisfies some hypotheses for junction points, and where the time step is constant. Our technique is based on some homotopy path of discrete optimal control problems that we study using perturbation analysis of nonlinear programming problems.