Smooth Indirect Solution Method for State-constrained Optimal Control Problems with Nonlinear Control-affine Systems

This paper presents an indirect solution method for state-constrained optimal control problems to address the long-standing issue of discontinuous control and costate under state constraints. It is known in optimal control theory that a state inequality constraint introduces discontinuities in control and costate, rendering the classical indirect solution methods ineffective. This study re-examines the necessary conditions of optimality for a class of state-constrained optimal control problems, and shows the uniqueness of the optimal control that minimizes the Hamiltonian under state constraints, which leads to a unifying form of the necessary conditions. The unified form of the necessary conditions opens the door to addressing the issue of discontinuities in control and costate by modeling them via smooth functions. This paper exploits this insight to transform the originally discontinuous problems to smooth two-point boundary value problems that can be solved by existing nonlinear root-finding algorithms. This paper also shows the formulated solution method to have an anytime algorithm-like property, and then numerically demonstrates the solution method by an optimal orbit control problem.