An algorithmic guide for finite-dimensional optimal control problems

We survey the main numerical techniques for finite-dimensional nonlinear optimal control. The chapter is written as a guide to practitioners who wish to get rapidly acquainted with the main numerical methods used to efficiently solve an optimal control problem. We consider two classical examples, simple but significant enough to be enriched and generalized to other settings: Zermelo and Goddard problems. We provide sample of the codes used to solve them and make these codes available online. We discuss direct and indirect methods, Hamilton-Jacobi approach, ending with optimistic planning. The examples illustrate the pros and cons of each method, and we show how these approaches can be combined into powerful tools for the numerical solution of optimal control problems for ordinary differential equations.