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

2026-01-12

AI Summary

This study addresses the robust sparse optimal control problem for constrained linear systems with parametric uncertainties, aiming to minimize the control duration—i.e., the $L^0$ norm—over an uncountable compact disturbance set. By constructing a robust hands-off control framework, the nonconvex and nonsmooth $L^0$ problem is transformed into its convex $L^1$ relaxation, and this work establishes, for the first time, the exact equivalence of their optimal solution sets under parameter uncertainty. Leveraging the nonsmooth robust Pontryagin maximum principle, semi-infinite programming, and convex relaxation techniques, a computationally tractable robust optimization algorithm is developed. Numerical experiments demonstrate that the proposed method achieves strong robustness while preserving control sparsity, yielding solutions that are both exact and effective.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• robust control
• sparse control
• L⁰ optimization
• parametric uncertainty
• hands-off control

Innovation

• robust maximum hands-off control
• L0-L1 equivalence
• nonsmooth robust maximum principle
• semi-infinite robust optimization
• sparse optimal control