Discretization-Optimization Methods for Optimal Control Problems

We consider an optimal control problem described by nonlinear ordinary differential equations, with control and state constraints. Since this problem may have no classical solutions, it is also formulated in relaxed form. The classical control problem is then discretized by using the implicit midpoint scheme for state approximation, while the controls are approximated by piecewise constant classical ones. We first study the behavior in the limit of properties of discrete optimality, and of discrete admissibility and extremality. We then apply a penalized gradient projection method to each discrete classical problem, and also a corresponding progressively refining discretization-optimization method to the continuous classical problem, thus reducing computing time and memory. We show that accumulation points of sequences generated by these methods are admissible and extremal for the corresponding discrete or continuous, classical or relaxed, problem. For nonconvex problems whose solutions are non-classical, we show that we can apply the above methods to the problem formulated in the Gamkrelidze form. Finally, numerical examples are given. Key-Words: Optimal control, discretization, midpoint scheme, piecewise constant controls, penalized gradient projection method, relaxed controls.