A proximal gradient method for control problems with nonsmooth and nonconvex control cost

TL;DR

This paper analyzes the convergence of a proximal gradient method applied to control problems with nonsmooth, nonconvex costs that promote sparsity, providing new theoretical insights into stationarity properties of solutions.

Contribution

It introduces a convergence analysis for a proximal gradient approach tailored to nonsmooth, nonconvex control costs, including $L^p$-type functionals for $p \,\in\, [0,1)$, and establishes stationarity properties of weak limit points.

Findings

Proves convergence to stationary points under certain conditions.

Establishes weaker stationarity properties than Pontryagin's maximum principle.

Applicable to control costs promoting sparsity, including $L^p$-type functionals.

Abstract

We investigate the convergence of an application of a proximal gradient method to control problems with nonsmooth and nonconvex control cost. Here, we focus on control cost functionals that promote sparsity, which includes functionals of $L^p$-type for $p\in [0,1)$. We prove stationarity properties of weak limit points of the method. These properties are weaker than those provided by Pontryagin's maximum principle and weaker than $L$-stationarity.