# On the Exact Solution of the Minimal Controllability Problem

This paper studies the minimal controllability problem (MCP), i.e., the problem of, given a linear time-invariant system, finding the sparsest input vector that ensures system’s controllability. We show that the MCP can be exactly solved by (polynomially) reducing it to the minimum set covering problem, when the system dynamics matrix A is simple and the left-eigenbasis associated to A is known. In addition, we also show that the approximated solutions to the minimum set covering problem lead to feasible (sub-optimal) solutions to the MCP; hence, bounds on the optimality gap can be immediately obtained from existing literature on approximation algorithms to the minimum set covering problem. Further, we prove that under the same assumption the decision version of the minimal controllability problem is NP-complete, i.e., there exists an input vector that ensures system’s controllability and at most a prescribed number of nonzero entries. Another contribution is the fact that we analyze the relation of the MCP with its structural counterpart, the minimal structural controllability problem (MSCP) which is known to admit a polynomial complexity solution procedure. We provide an illustrative example where the solution to the MCP is found using the main results and reductions developed in this paper; in particular, we show that the solution to the MSCP is not a solution to MCP even when the dynamic matrix considered to the MCP is simple; hence, disproving the general belief that a solution to MSCP is a solution to MCP when the dynamic matrix is simple, and the eigenbasis structure known.