Continuity of the Value Function in Sparse Optimal Control

TL;DR

This paper proves the continuity of the value function in sparse optimal control problems, showing it aligns with the L1-optimal control value function under certain conditions, which enhances understanding of control support minimization.

Contribution

It establishes the continuity of the value function for sparse optimal control, linking it to L1-optimal control and providing theoretical insights into control support minimization.

Findings

Value function of sparse optimal control is continuous.

Sparse optimal control coincides with L1-optimal control under normality.

Continuity is proven by relating to the L1-optimal control problem.

Abstract

We prove the continuity of the value function of the sparse optimal control problem. The sparse optimal control is a control whose support is minimum among all admissible controls. Under the normality assumption, it is known that a sparse optimal control is given by L^1 optimal control. Furthermore, the value function of the sparse optimal control problem is identical with that of the L1-optimal control problem. From these properties, we prove the continuity of the value function of the sparse optimal control problem by verifying that of the L1-optimal control problem.