On Weak Topology for Optimal Control of Switched Nonlinear Systems

TL;DR

This paper introduces a unified topological framework for embedding-based optimal control algorithms in switched nonlinear systems, ensuring convergence to local optima and encompassing many existing methods.

Contribution

It presents a novel topological perspective that generalizes and unifies various embedding-based algorithms for switched optimal control, with convergence guarantees.

Findings

Provides a general procedure for constructing switched optimal control algorithms.

Guarantees convergence to a local optimizer.

Unifies existing embedding-based methods under a common framework.

Abstract

Optimal control of switched systems is challenging due to the discrete nature of the switching control input. The embedding-based approach addresses this challenge by solving a corresponding relaxed optimal control problem with only continuous inputs, and then projecting the relaxed solution back to obtain the optimal switching solution of the original problem. This paper presents a novel idea that views the embedding-based approach as a change of topology over the optimization space, resulting in a general procedure to construct a switched optimal control algorithm with guaranteed convergence to a local optimizer. Our result provides a unified topology based framework for the analysis and design of various embedding-based algorithms in solving the switched optimal control problem and includes many existing methods as special cases.