A feasible directions algorithm for optimal control problems with state and control constraints: Convergence analysis

In this paper we describe an optimization algorithm for the computation of solutions to optimal control problems with control, state, and terminal constraints. Inequality and equality constraints are dealt with by means of feasible directions and exact penalty approaches, respectively. We establish a general convergence property of the algorithm which makes no reference to the existence of accumulation points; in this analysis the compactness of the space of relaxed controls is used only to guarantee boundedness of the sequence of penalty parameters. We also demonstrate that relaxed accumulation points of sequences generated by the algorithm satisfy standard first-order necessary conditions of optimality. The algorithm contains a number of computation saving features, including an epsilon-active strategy for dealing with the "infinite dimensional" inequality constraints. Our convergence analysis provides techniques for studying the convergence properties of related optimization algorithms in which direction-finding subproblems involve the approximation of directional derivatives of the Chebyshev functional associated with state constraints. A companion paper provides details of implementation and numerical examples.