A necessary algebraic condition for controllability and observability of linear time-varying systems

In this note, we give an algebraic condition which is necessary for the system x´(t)=A(t)x(t)+B(t)u(t), y(t)=C(t)x(t), either to be totally controllable or to be totally observable, where x∈R^d, u∈R^p, y∈R^q, and the matrix functions A, B and C are (d-2), (d-1) and (d-1) times continuously differentiable, respectively. All conditions presented here are in terms of known quantities and therefore easily verified. Our conditions can be used to rule out large classes of time-varying systems which cannot be controlled and/or observed no matter what the nonzero time-varying coefficients are. This work is motivated by the deep result of Silverman and Meadows.