  This paper studies controllability of a discrete-time linear dynamical system
using nonnegative and sparse inputs. These constraints on the control input
arise naturally in many real-life systems where the external influence on the
system is unidirectional, and activating each input node adds to the cost of
control. We derive the necessary and sufficient conditions for controllability
of the system, without imposing any constraints on the system matrices. Unlike
the well-known Kalman rank based controllability criteria, the conditions
presented in this paper can be verified in polynomial time, and the
verification complexity is independent of the sparsity level. The proof of the
result is based on the analytical tools concerning the properties of a convex
cone. Our results also provide a closed-form expression for the minimum number
of control nodes to be activated at every time instant to ensure
controllability of the system using positive controls.
