Mixed Use of Pontryagin’s Principle and the Hamilton-Jacobi-Bellman Equation in Infinite- and Finite-Horizon Constrained Optimal Control

Abstract

This paper proposes a framework for solving a class of nonlinear infinite- and finite-horizon optimal control problems with constraints. Establishment of existence and uniqueness of solutions to the Hamilton-Jacobi-Bellman (HJB) equation plays a crucial role in verifying well-posedness of a given problem and in streamlining numerical solutions. The proposed framework revolves around infinite-horizon Bolza-type cost functions with running costs exponentially decaying in time. We show $$\varGamma $$ -convergence of solutions with such cost functions to the solutions of initial constrained (in)finite-horizon problems (that is, without running costs exponentially decaying in time). Basically, we demonstrate how to approximate solutions of (in)finite-horizon constrained optimal problems using our framework. Employing a solver based on the Pontryagin’s Principle, we efficiently obtain optimal solutions for finite- and infinite-horizon problems. Efficiency of the proposed framework is demonstrated in simulation by solving a 3D path planing problem with obstacles for a full nonlinear model of an autonomous underwater vehicle (AUV).