Stabilization problem of discrete-time linear switching systems with bounds on the state and control input is solved in this paper.First,the synthesis of state feedback controllers that ensure the stability of closed-loop switching systems is studied under a sufficient condition.By using the idea of positive invariance,a stabilizing controller design methodology is proposed.Based on these results,the convergence rate problem is also discussed.A state feedback controller that guarantees the optimal convergence rate of closed-loop switching systems is obtained via optimization.Finally,an example made up of two subsystems is studied to show the application of our method.