Maximum Hands-off Control without Normality Assumption

TL;DR

This paper analyzes maximum hands-off control for linear systems without assuming normality, introducing Lp-optimal control to explore properties like existence, bang-off-bang behavior, and relations to L1 control, with implications for stability.

Contribution

It extends the analysis of maximum hands-off control to non-normal systems by introducing Lp-optimal control and establishing key properties and relations.

Findings

Existence of maximum hands-off control without normality assumption

Bang-off-bang property of the control under new conditions

Continuity and convexity of the value function

Abstract

Maximum hands-off control is a control that has the minimum L0 norm among all feasible controls. It is known that the maximum hands-off (or L0-optimal) control problem is equivalent to the L1-optimal control under the assumption of normality. In this article, we analyze the maximum hands-off control for linear time-invariant systems without the normality assumption. For this purpose, we introduce the Lp-optimal control with 0<p<1, which is a natural relaxation of the L0 problem. By using this, we investigate the existence and the bang-off-bang property (i.e. the control takes values of 1, 0 and -1) of the maximum hands-off control. We then describe a general relation between the maximum hands-off control and the L1-optimal control. We also prove the continuity and convexity property of the value function, which plays an important role to prove the stability when the (finite-horizon) control is extended to model predictive control.