This paper investigates the stabilization problem of an autonomous Linear Time Invariant (LTI) switched system with interval uncertainty and unstable subsystems. It is proved that the system would be stable by using a common Lyapanov function whose derivative is negative and bounded by a quadratic function within activation regions of each subsystem. First, a sufficient condition for the stability of an Linear Time Invariant switched system with interval uncertainty, based on the convex analysis and interval set theoretical approach, is presented and proved. Moreover, conservatism in the stability robustness bound is obtained. Then, a switching control law is designed to shift the Linear Time Invariant switched system among subsystems to ensure the decrease of the Lyapanov function within the state space. Finally, in order to decrease the switching frequency and to avoid chattering, the switching law is modified. Two examples are included to demonstrate the effectiveness of the theoretical findings.