Sparse Controllability

In this chapter, we examine how the sparsity constraint on control inputs influences the controllability of linear dynamical systems. Specifically, we characterize the controllability of discrete-time linear dynamical systems with sparse control inputs. We derive algebraic necessary and sufficient conditions for ensuring controllability in systems with arbitrary transfer matrices. These conditions can be verified in polynomial time, in contrast to traditional Kalman-type rank tests. Additionally, we discuss the time-to-control of the system under these sparsity constraints on the inputs.