Robust maximum hands-off optimal control: existence, maximum principle, and $L^{0}$-$L^1$ equivalence

TL;DR

This paper develops a robust control framework for linear systems with uncertainties, establishing an equivalence between $L^{0}$ and $L^{1}$ formulations and proposing an algorithmic solution based on robust optimization principles.

Contribution

It introduces a robust maximum hands-off control approach, proving the $L^{0}$-$L^{1}$ equivalence under uncertainties and providing a practical algorithmic framework.

Findings

The $L^{0}$ and $L^{1}$ formulations have identical optimal solutions in the robust setting.

The proposed algorithm effectively solves the nonconvex robust control problem.

An illustrative example demonstrates the approach's practical effectiveness.

Abstract

This work advances the maximum hands-off sparse control framework by developing a robust counterpart for constrained linear systems with parametric uncertainties. The resulting optimal control problem minimizes an $L^{0}$ objective subject to an uncountable, compact family of constraints, and is therefore a nonconvex, nonsmooth robust optimization problem. To address this, we replace the $L^{0}$ objective with its convex $L^{1}$ surrogate and, using a nonsmooth variant of the robust Pontryagin maximum principle, show that the $L^{0}$ and $L^{1}$ formulations have identical sets of optimal solutions -- we call this the robust hands-off principle. Building on this equivalence, we propose an algorithmic framework -- drawing on numerically viable techniques from the semi-infinite robust optimization literature -- to solve the resulting problems. An illustrative example is provided to demonstrate the effectiveness of the approach.