In this section, we give a novel delay-dependent stability condition for discrete-time systems with interval-like time-varying delays. Now, consider the following system:

where is the state vector, and are known constant matrices, and is a time-varying delay satisfying , where and are positive integers representing the lower and upper bounds of the delay. For (3.1), we have the following result.

Theorem 3.1.

Give integers and . Then, the discrete time-delay system (3.1) is asymptotically stable for any time delay satisfying , if there exist symmetric positive definite matrices satisfying the following matrix inequalities:

where , .

Proof.

Consider the Lyapunov function , where

with being symmetric positive definite solutions of (3.2) and

Then difference of along trajectory of solution of (3.1) is given by

where

where Fact 1 is utilized in (3.6), respectively.

Note that

and hence

Then we have

Using Lemma 2.2, we obtain

From the above inequality it follows that

where , , and

By condition (3.2), is negative definite; namely, there is a number such that and hence, the asymptotic stability of the system immediately follows from Lemma 2.1**.** This completes the proof.

Remark 3.2.

Theorem 3.1 gives a sufficient condition for stability criterion for discrete-time systems (3.1). These conditions are described in terms of certain diagonal matrix inequalities, which can be realized by using the linear matrix inequality algorithm proposed in [14]. But Zhang et al. in [12] proved that these conditions are described in terms of certain symmetric matrix inequalities, which can be realized by using the Schur complement lemma and linear matrix inequality algorithm proposed in [14].