The Discrete-Time Geometric Maximum Principle

TL;DR

This paper extends the Pontryagin maximum principle to discrete-time control problems on smooth manifolds, introducing new geometric results and techniques for handling constraints and sensitivity analysis.

Contribution

It presents a new theorem on critical points for discrete-time geometric control systems and applies it to derive maximum principles and integrators on Lie groups.

Findings

Derived Lie group variational integrators in Hamiltonian form.

Established maximum principles without state constraints.

Provided conditions for exact penalization with constraints.

Abstract

We establish a variety of results extending the well-known Pontryagin maximum principle of optimal control to discrete-time optimal control problems posed on smooth manifolds. These results are organized around a new theorem on critical and approximate critical points for discrete-time geometric control systems. We show that this theorem can be used to derive Lie group variational integrators in Hamiltonian form; to establish a maximum principle for control problems in the absence of state constraints; and to provide sufficient conditions for exact penalization techniques in the presence of state or mixed constraints. Exact penalization techniques are used to study sensitivity of the optimal value function to constraint perturbations and to prove necessary optimality conditions, including in the form of a maximum principle, for discrete-time geometric control problems with state or mixed constraints.