Notes on Numerical Methods for Solving Optimal Control Problems

Recent advances in theory, algorithms, and computational power make it possible to solve complex, optimal control problems both for off-line and on-line industrial applications. This paper starts by reviewing the technical details of the solution methods pertaining to three general categories: dynamic programming, indirect methods, and direct methods. With the aid of a demonstration example, the advantages and disadvantages of each method are discussed, along with a brief review of available software. The main result that emerges is the indirect method being numerically competitive with the performance of direct ones based on non-linear programming solvers and interior point algorithms. The sec- ond part of the paper introduces an indirect method based on the Pontryagin Minimum Principle (PMP). It also presents a detailed procedure and software tools (named PINS) to formulate the problem, automatically generate the C++ code, and eventually obtain a numerical solution for several optimal control problems of practical relevance. The application of PMP relates to the analytical derivation of necessary conditions for optimality. This aspect—often regarded in the literature as a drawback—is here exploited to build a robust yet efficient numerical method that formally eliminates the controls from the resulting Boundary Value Problem, thus gaining robustness and a high convergence rate. The elimination of the control is obtained either via their explicit formulation function of state and Lagrange multipliers— when possible—or via an iterative numerical solution. The paper closes presenting a minimum time manoeuvre of a car using a fairly complex vehicle model which also includes tyre saturation.