Sparsity-Constrained Controllability Maximization With Application to Time-Varying Control Node Selection

In this letter, we consider the maximization of a quantitative metric of controllability with a constraint of $L^{0}$ norm of the control input. Since the optimization problem contains a combinatorial structure, we introduce a convex relaxation problem for the sake of reducing computation burden. We prove the existence of solutions to the main problem and also give a simple condition under which the relaxed problem gives a solution to the main problem. It should be emphasized that the main problem can formulate time-varying control node selection, which attempts to extract when and where exogenous inputs should be provided in order to achieve high controllability of multi-agent systems.