Infinite horizon sparse optimal control

TL;DR

This paper investigates infinite horizon optimal control problems with $L^p$-type costs promoting sparsity, analyzing existence and structure of solutions, and proposes a dynamic programming method for numerical approximation.

Contribution

It introduces a unified analysis of existence and sparsity in $L^p$-cost optimal controls for both convex and nonconvex cases, and develops a dynamic programming approach for their approximation.

Findings

Existence of optimal controls for $p=1$ and $0<p<1$ cases.

Sparsity structure of optimal controls analyzed.

Dynamic programming method proposed for numerical approximation.

Abstract

A class of infinite horizon optimal control problems involving $L^p$-type cost functionals with $0<p\leq 1$ is discussed. The existence of optimal controls is studied for both the convex case with $p=1$ and the nonconvex case with $0<p<1$, and the sparsity structure of the optimal controls promoted by the $L^p$-type penalties is analyzed. A dynamic programming approach is proposed to numerically approximate the corresponding sparse optimal controllers.