In this paper, the problem of finite-time H 1 control is addressed for a class of discrete-time switched nonlinear systems with time delay. The concept of H 1 finite-time boundedness is first introduced for discrete-time switched delay systems. Next, a set of switching signals are designed by using the average dwell time approach, under which some delay-dependent sufficient conditions are derived to guarantee the H 1 finite-time boundedness of the closed-loop system. Then, a finite-time H 1 state feedback controller is also designed by solving such conditions. Furthermore, the problem of uniform finite-time H 1 stabilization is also resolved. All the conditions are cast into linear matrix inequalities, which can be easily checked by using recently developed algorithms for solving linear matrix inequalities. A numerical example and a water-quality control system are provided to demonstrate the effectiveness of the main results.FINITE-TIME H 1 CONTROL 915 its system state does not exceed some prescribed bound for bounded initial states. It is known that finite-time stability and Lyapunov asymptotic stability are independent concepts; a finite-time stable system may not be Lyapunov asymptotically stable and vice versa.In the presence of exogenous disturbances, the concept of finite-time boundedness was introduced in [13]. In [14], the problem of finite-time state feedback stabilization for discrete-time linear system was investigated. Sufficient conditions ensuring finite-time stability for nonlinear quadratic systems were derived in [15]. For a class of linear singular impulsive systems with or without exogenous disturbances, the finite-time impulsive stabilization problem was considered in [16][17][18] via the matrix inequality method. In [19], the concepts of finite-time stability and finite-time boundedness were extended to switched linear systems, and some sufficient conditions were provided such that the switched linear systems were finite-time bounded and uniformly finite-time bounded. Recently, the problems of finite-time weighted L 2 gain for switched delay systems with disturbances were handled in [20]. Furthermore, in [21], the finite-time H 1 control was considered for continuous-time switched linear systems with time-varying delay. Using the switched Lyapunov function approach proposed in [4], solutions to the problem of finite-time H 1 control was presented in [22] for a class of discrete-time switched nonlinear systems. More recently, the problem of finite-time dynamic output feedback stabilization was addressed in [23]. It should be pointed out that so far, no attention has been paid to the H 1 finite-time control problem of switched delay systems, especially in a discrete-time context.In this paper, we focus upon the problem of finite-time H 1 control for discrete-time switched nonlinear systems with time delay. Because of the existence of time delay and nonlinear disturbances, this problem is apparently hard to tackle. The main contributions in this paper lie in three aspects. First, the concepts o...

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