Sign stabilizability

In this paper, the notion sign stabilizability for linear time-invariant control systems of the form x(t) = A · x(t) + B · u(t) is introduced. A class of linear time-invariant systems specified by the sign pattern of the matrices A and B is sign stabilizable if all of its members are stabilizable. Sign stabilizability is a natural extension of two common properties, the sign stability and the sign controllability, since all sign stable and all sign controllable systems are sign stabilizable, but not all sign stabilizable systems are sign stable, or sign controllable. We present a necessary condition for sign stabilizability and in our main result, we characterize sign stabilizability for all systems, whose sign pattern of A allows only real eigenvalues. Finally, the results of this paper are applied to a real world example, the roll dynamic of bicycles. Because of the duality principle between stabilizability and detectability, the results of this paper are also valid for the detectability of the dual system. We emphasize that the results of this paper cover the single and the multi-input case.