Sign Controllability: Sign Patterns That Require Complete Controllability

We apply tools from the theory of sign-solvable systems and use the directed graph of a matrix in order to obtain sufficient conditions for a linear control system (A,B) to be completely controllable solely due to the sign patterns of the coefficient matrices A and B. We show that such conditions are necessary and sufficient for a particular class of linear control systems. We also consider an alternative approach to controllability, based on a reformulation of the classical condition (that the controllability matrix is of full rank) and obtain equivalent conditions for the general case.