Stabilization of second-order LTI switched systems with focus equilibrium

In this paper, we consider and solve the stabilizable problem of second-order linear time-invariant (LTI) switched systems consisting of two subsystems with unstable foci. Based on the phase plane techniques, firstly we single out the "most stabilizing" state trajectory corresponding a special switching law. The switched systems would be stabilizable if the "most stabilising" state trajectory converges and vice versa. Then analysis results reveal that the switched system is stabilizable with satisfying a set of inequalities.