Contractivity of the Method of Successive Approximations for Optimal Control

TL;DR

This paper investigates the contractivity properties of strongly contracting dynamical systems and demonstrates that the Method of Successive Approximations (MSA) converges under certain conditions, ensuring uniqueness of optimal control solutions.

Contribution

It reveals that the adjoint system inherits strong contraction properties and establishes new convergence criteria for the MSA in optimal control.

Findings

MSA is a contraction mapping for short intervals or high contraction rates.

New convergence conditions for MSA are derived.

These conditions imply uniqueness of optimal control and sufficiency of Pontryagin's principle.

Abstract

Strongly contracting dynamical systems have numerous properties (e.g., incremental ISS), find widespread applications (e.g., in controls and learning), and their study is receiving increasing attention. This work starts with the simple observation that, given a strongly contracting system, its adjoint dynamical system is also strongly contracting, with the same rate, with respect to the dual norm, under time reversal. As main implication of this dual contractivity, we show that the classic Method of Successive Approximations (MSA), an indirect method in optimal control, is a contraction mapping for short optimization intervals or large contraction rates. Consequently, we establish new convergence conditions for the MSA algorithm, which further imply uniqueness of the optimal control and sufficiency of Pontryagin's minimum principle under additional assumptions.