Robust maximum hands-off optimal control: existence, maximum principle, and $L^{0}$-$L^1$ equivalence
Siddhartha Ganguly, Kenji Kashima
TL;DR
This work addresses robust sparse optimal control under parametric uncertainty by formulating a robust $L^{0}$ objective and relaxing it to a convex $L^{1}$ surrogate. It develops a robust, nonsmooth Pontryagin maximum principle and proves that the $L^{0}$ and $L^{1}$ formulations share the same set of optimal solutions, enabling a convex semi-infinite programming approach for synthesis. The algorithmic contribution parametrizes controls with piecewise-constant bases and solves a finite-dimensional SIP exactly via robust optimization, yielding sparse, robust controls that satisfy all terminal constraints for every uncertain parameter. The results provide a principled, computationally viable framework for robust hands-off control with potential applications in attitude control, networked systems, and energy-efficient design.
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.