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

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.