4.1 Article

Contractivity of the Method of Successive Approximations for Optimal Control

期刊

IEEE CONTROL SYSTEMS LETTERS
卷 7, 期 -, 页码 919-924

出版社

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/LCSYS.2022.3228723

关键词

Optimal control; contraction theory; iterative methods

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Strongly contracting dynamical systems and their adjoint systems are shown to have the same rate of contraction under time reversal. This duality leads to new convergence conditions for the Method of Successive Approximations (MSA) algorithm and establishes uniqueness of the optimal control and sufficiency of Pontryagin's minimum principle under certain assumptions.
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 letter 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.

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