  Controllability maximization problem under sparsity constraints is a node
selection problem that selects inputs that are effective for control in order
to minimize the energy to control for desired state. In this paper we discuss
the equivalence between the sparsity constrained controllability metrics
maximization problems and their convex relaxation. The proof is based on the
matrix-valued Pontryagin maximum principle applied to the controllability
Lyapunov differential equation.
