Discretization methods for optimal control problems with state constraints

Abstract

We consider an optimal control problem described by nonlinear ordinary differential equations, with control and state constraints, including pointwise 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, while the controls are approximated by (not necessarily continuous) piecewise linear classical controls. 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 combined discretization-optimization method to the continuous classical problem, thus reducing computing time and memory. We prove that accumulation points of sequences generated by these methods are admissible and extremal in some sense for the corresponding discrete or continuous, classical or relaxed, problem. For nonconvex problems whose solutions are nonclassical, we show that we can apply the above methods to the problem formulated in Gamkrelidze relaxed form. Finally, numerical examples are given.