The robust minimal controllability problem

In this paper, we address the robust minimal controllability problem, where the goal is, given a linear time-invariant system, to determine a minimal subset of state variables to be actuated to ensure controllability under additional constraints. We study the problem of characterizing the sparsest input matrices that assure controllability, when the autonomous dynamics' matrix is simple when a specified number of inputs fail. We show that this problem is NP-hard, and under the assumption that the dynamics' matrix is simple, we show that it is possible to reduce the problem to a set multi-covering problem. Additionally, under this assumption, we prove that this problem is NP-complete, and polynomial algorithms to approximate the solutions of a set multi-covering problem can be leveraged to obtain close-to-optimal solutions.