Sparse optimal stochastic control

TL;DR

This paper develops a framework for sparse optimal control of continuous-time stochastic systems using viscosity solutions to HJB equations, providing conditions for $L^0$ optimality and linking to $L^1$ control for certain systems.

Contribution

It introduces a novel analysis of $L^0$ optimal control via viscosity solutions and establishes an equivalence with $L^1$ control in control-affine systems.

Findings

Characterizes the value function as a viscosity solution to HJB.

Provides necessary and sufficient conditions for $L^0$ optimality.

Shows an equivalence between $L^0$ and $L^1$ control problems for control-affine systems.

Abstract

In this paper, we investigate a sparse optimal control of continuous-time stochastic systems. We adopt the dynamic programming approach and analyze the optimal control via the value function. Due to the non-smoothness of the $L^0$ cost functional, in general, the value function is not differentiable in the domain. Then, we characterize the value function as a viscosity solution to the associated Hamilton-Jacobi-Bellman (HJB) equation. Based on the result, we derive a necessary and sufficient condition for the $L^0$ optimality, which immediately gives the optimal feedback map. Especially for control-affine systems, we consider the relationship with $L^1$ optimal control problem and show an equivalence theorem.