Discretization methods for nonconvex optimal control problems with state constraints

We consider an optimal control problem described by nonlinear ordinary differential equations, with control and state constraints, including pointwise state constraints. Because no convexity assumptions are made, the problem may have no classical solutions, and it is reformulated in relaxed form. The relaxed control problem is then discretized by using the implicit midpoint scheme, while the controls are approximated by piecewise constant relaxed controls. We first study the behavior in the limit of properties of discrete relaxed optimality, and of discrete relaxed admissibility and extremality. We then apply a penalized conditional descent method to each discrete relaxed problem, and also a corresponding discrete method to the continuous relaxed problem that progressively refines the discretization during the iterations, thus reducing computing time and memory. We prove that accumulation points of sequences generated by these methods are admissible and extremal for the discrete or the continuous problem. Finally, numerical examples are given.