In this section, we propose new synchronization criteria for the system (6). For the sake of simplicity on matrix representation, {e}_{i}\in {\mathbb{R}}^{5\kappa \times \kappa}, where \kappa =(N-1)n, are defined as block entry matrices, *e.g.*, {e}_{2}={[{0}_{\kappa},{I}_{\kappa},{0}_{\kappa},{0}_{\kappa},{0}_{\kappa}]}^{T}. The notations of several matrices are defined as:

Now, the following theorem is given for synchronization stability of the model of discrete-time CDNs with interval time-varying delays in the coupling term (6).

**Theorem 1** *For given positive integers* {h}_{m}, {h}_{M}, *l* *and positive scalars* *c*, {\rho}_{0}<1, *the system* (6) *is asymptotically synchronous for* {h}_{m}\le h(k)\le {h}_{M}, *if there exist positive matrices* P\in {\mathbb{R}}^{n\times n}, {Q}_{i}\in {\mathbb{R}}^{n\times n}, {R}_{j}\in {\mathbb{R}}^{n\times n}, *any symmetric matrices* {S}_{i}\in {\mathbb{R}}^{n\times n}, *where* i=1,2 *and* j=1,\dots ,4, *and any matrix* M\in {\mathbb{R}}^{n\times n} *satisfying the following LMIs*:

*where* Φ *and* {\mathrm{\Upsilon}}_{[{\rho}_{m}]} *are defined in* (10).

*Proof* Let us consider the following Lyapunov-Krasovskii functional candidate as

V(k)={V}_{1}(k)+{V}_{2}(k)+{V}_{3}(k)+{V}_{4}(k),

(14)

where

The mathematical expectation of the \mathrm{\Delta}{V}_{1}(k) and \mathrm{\Delta}{V}_{2}(k) are calculated as

By calculating the \mathbb{E}\{\mathrm{\Delta}{V}_{3}(k)\}, we get

\begin{array}{rcl}\mathbb{E}\{\mathrm{\Delta}{V}_{3}(k)\}& =& \mathrm{\Delta}{x}^{T}(k)({I}_{N-1}\otimes ({h}_{m}^{2}{R}_{1}+{({h}_{M}-{h}_{m})}^{2}{R}_{2}))\mathrm{\Delta}x(k)\\ -{h}_{m}\sum _{s=k-{h}_{m}}^{k-1}\mathrm{\Delta}{x}^{T}(s)({I}_{N-1}\otimes {R}_{1})\mathrm{\Delta}x(s)\\ -({h}_{M}-{h}_{m})\sum _{s=k-h(k)}^{k-{h}_{m}-1}\mathrm{\Delta}{x}^{T}(s)({I}_{N-1}\otimes {R}_{2})\mathrm{\Delta}x(s)\\ -({h}_{M}-{h}_{m})\sum _{s=k-{h}_{M}}^{k-h(k)-1}\mathrm{\Delta}{x}^{T}(s)({I}_{N-1}\otimes {R}_{2})\mathrm{\Delta}x(s).\end{array}

(17)

Inspired by the work of [29], the following two zero equalities hold with any symmetric matrices {S}_{1} and {S}_{2}:

and

Here, Eqs. (18) and (19) still hold even when we multiply both sides by ({h}_{M}-{h}_{m}). So, by adding the results into Eq. (17), we get

\begin{array}{rcl}\mathbb{E}\{\mathrm{\Delta}{V}_{3}(k)\}& =& \mathrm{\Delta}{x}^{T}(k)({I}_{N-1}\otimes ({h}_{m}^{2}{R}_{1}+{({h}_{M}-{h}_{m})}^{2}{R}_{2}))\mathrm{\Delta}x(k)\\ +({h}_{M}-{h}_{m}){x}^{T}(k-{h}_{m})({I}_{N-1}\otimes {S}_{1})x(k-{h}_{m})\\ -({h}_{M}-{h}_{m}){x}^{T}(k-h(k))({I}_{N-1}\otimes ({S}_{1}-{S}_{2}))x(k-h(k))\\ -({h}_{M}-{h}_{m}){x}^{T}(k-{h}_{M})({I}_{N-1}\otimes {S}_{2})x(k-{h}_{M})+\mathrm{\Sigma}+{\mathrm{\Theta}}_{1}\\ =& {\zeta}^{T}(k){\mathrm{\Xi}}_{3}\zeta (k)+\mathrm{\Sigma}+{\mathrm{\Theta}}_{1},\end{array}

(20)

where

By Lemma 3, the term Σ in (20) can be estimated as

\begin{array}{rcl}\mathrm{\Sigma}& \le & -{(\sum _{s=k-{h}_{m}}^{k-1}\mathrm{\Delta}x(s))}^{T}({I}_{N-1}\otimes {R}_{1})(\sum _{s=k-{h}_{m}}^{k-1}\mathrm{\Delta}x(s))\\ -{(\sum _{s=k-h(k)}^{k-{h}_{m}-1}\mathrm{\Delta}x(s))}^{T}({I}_{N-1}\otimes ({R}_{2}+{S}_{1}))(\sum _{s=k-h(k)}^{k-{h}_{m}-1}\mathrm{\Delta}x(s))\\ -{(\sum _{s=k-{h}_{M}}^{k-h(k)-1}\mathrm{\Delta}x(s))}^{T}({I}_{N-1}\otimes ({R}_{2}+{S}_{2}))(\sum _{s=k-{h}_{M}}^{k-h(k)-1}\mathrm{\Delta}x(s))\\ =& -{\zeta}^{T}(k)({e}_{1}-{e}_{2})({I}_{N-1}\otimes {R}_{1}){({e}_{1}-{e}_{2})}^{T}\zeta (k)\\ -{\zeta}^{T}(k)\mathrm{\Pi}\left[\begin{array}{cc}\frac{1}{1-\alpha (k)}({I}_{N-1}\otimes ({R}_{2}+{S}_{1}))& {0}_{\kappa}\\ \star & \frac{1}{\alpha (k)}({I}_{N-1}\otimes ({R}_{2}+{S}_{2}))\end{array}\right]{\mathrm{\Pi}}^{T}\zeta (k),\end{array}

(21)

where \alpha (k)=({h}_{M}-h(k))/({h}_{M}-{h}_{m}).

Here, when {h}_{m}<h(t)<{h}_{M}, since \alpha (k) satisfies 0<\alpha (t)<1, by a reciprocally convex approach [24], the following inequality for any matrix *M* holds:

which implies

Also, when h(k)={h}_{m} or h(k)={h}_{M}, we get

\begin{array}{rcl}\sum _{s=k-h(k)}^{k-{h}_{m}-1}\mathrm{\Delta}x(s)& =& \sum _{s=k-h(k)}^{k-{h}_{m}-1}(x(s+1)-x(s))\\ =& x(k-{h}_{m})-x(k-h(k))\\ =& x(k-{h}_{m})-x(k-{h}_{m})={0}_{\kappa \times 1}\end{array}

or

\begin{array}{rcl}\sum _{s=k-{h}_{M}}^{k-h(k)-1}\mathrm{\Delta}x(s)& =& \sum _{s=k-{h}_{M}}^{k-h(k)-1}(x(s+1)-x(s))\\ =& x(k-h(k))-x(k-{h}_{M})\\ =& x(k-{h}_{M})-x(k-{h}_{M})={0}_{\kappa \times 1},\end{array}

(23)

respectively.

Thus, if the inequality (12) holds, then from Eqs. (22) and (23), the following inequality still holds:

\begin{array}{rcl}\mathrm{\Sigma}& \le & -{\zeta}^{T}(k)({e}_{1}-{e}_{2})({I}_{N-1}\otimes {R}_{1}){({e}_{1}-{e}_{2})}^{T}\zeta (k)\\ -{\zeta}^{T}(k)\mathrm{\Pi}\left[\begin{array}{cc}\frac{1}{1-\alpha (k)}({I}_{N-1}\otimes ({R}_{2}+{S}_{1}))& {0}_{\kappa}\\ \star & \frac{1}{\alpha (k)}({I}_{N-1}\otimes ({R}_{2}+{S}_{2}))\end{array}\right]{\mathrm{\Pi}}^{T}\zeta (k)\\ \le & -{\zeta}^{T}(k)({e}_{1}-{e}_{2})({I}_{N-1}\otimes {R}_{1}){({e}_{1}-{e}_{2})}^{T}\zeta (k)\\ -{\zeta}^{T}(k)\mathrm{\Pi}\left[\begin{array}{cc}{I}_{N-1}\otimes ({R}_{2}+{S}_{1})& {I}_{N-1}\otimes M\\ \star & {I}_{N-1}\otimes ({R}_{2}+{S}_{2})\end{array}\right]{\mathrm{\Pi}}^{T}\zeta (k)\\ =& {\zeta}^{T}(k){\mathrm{\Xi}}_{4}\zeta (k),\end{array}

which means

\mathbb{E}\{\mathrm{\Delta}{V}_{3}(k)\}\le {\zeta}^{T}(k)({\mathrm{\Xi}}_{3}+{\mathrm{\Xi}}_{4})\zeta (k)+{\mathrm{\Theta}}_{1}.

(24)

Lastly, the \mathbb{E}\{{V}_{4}(k)\} is calculated as

\begin{array}{rcl}\mathbb{E}\{\mathrm{\Delta}{V}_{4}(k)\}& =& {({h}_{M}-{h}_{m})}^{2}({x}^{T}(k)({I}_{N-1}\otimes {R}_{3})x(k)+\mathrm{\Delta}{x}^{T}(k)({I}_{N-1}\otimes {R}_{4})\mathrm{\Delta}x(k))\\ -({h}_{M}-{h}_{m})\sum _{s=k-h(k)}^{k-{h}_{m}-1}({x}^{T}(s)({I}_{N-1}\otimes {R}_{3})x(s)+\mathrm{\Delta}{x}^{T}(s)({I}_{N-1}\otimes {R}_{4})\mathrm{\Delta}x(s))\\ -({h}_{M}-{h}_{m})\sum _{s=k-{h}_{M}}^{k-h(k)-1}({x}^{T}(s)({I}_{N-1}\otimes {R}_{3})x(s)+\mathrm{\Delta}{x}^{T}(s)({I}_{N-1}\otimes {R}_{4})\mathrm{\Delta}x(s))\\ =& {\zeta}^{T}(k){\mathrm{\Xi}}_{5}\zeta (k)+{\mathrm{\Theta}}_{2},\end{array}

(25)

where

\begin{array}{rcl}{\mathrm{\Theta}}_{2}& =& -({h}_{M}-{h}_{m})\sum _{s=k-h(k)}^{k-{h}_{m}-1}{\xi}^{T}(s)\left[\begin{array}{cc}{I}_{N-1}\otimes {R}_{3}& {0}_{\kappa}\\ \star & {I}_{N-1}\otimes {R}_{4}\end{array}\right]\xi (s)\\ -({h}_{M}-{h}_{m})\sum _{s=k-{h}_{M}}^{k-h(k)-1}{\xi}^{T}(s)\left[\begin{array}{cc}{I}_{N-1}\otimes {R}_{3}& {0}_{\kappa}\\ \star & {I}_{N-1}\otimes {R}_{4}\end{array}\right]\xi (s).\end{array}

Furthermore, if the inequalities (13) hold, then the \mathbb{E}\{\mathrm{\Delta}{V}_{3}(k)\}+\mathbb{E}\{\mathrm{\Delta}{V}_{4}(k)\} has an upper bound as follows:

\begin{array}{rcl}\mathbb{E}\{\mathrm{\Delta}{V}_{3}(k)\}+\mathbb{E}\{\mathrm{\Delta}{V}_{4}(k)\}& \le & {\zeta}^{T}(k)({\mathrm{\Xi}}_{3}+{\mathrm{\Xi}}_{4}+{\mathrm{\Xi}}_{5})\zeta (k)+({\mathrm{\Theta}}_{1}+{\mathrm{\Theta}}_{2})\\ =& {\zeta}^{T}(k)({\mathrm{\Xi}}_{3}+{\mathrm{\Xi}}_{4}+{\mathrm{\Xi}}_{5})\zeta (k)\\ -({h}_{M}-{h}_{m})\sum _{s=k-h(k)}^{k-{h}_{m}-1}{\xi}^{T}(s)\left[\begin{array}{cc}{I}_{N-1}\otimes {R}_{3}& {I}_{N-1}\otimes {S}_{1}\\ \star & {I}_{N-1}\otimes {R}_{4}\end{array}\right]\xi (s)\\ -({h}_{M}-{h}_{m})\sum _{s=k-{h}_{M}}^{k-h(k)-1}{\xi}^{T}(s)\left[\begin{array}{cc}{I}_{N-1}\otimes {R}_{3}& {I}_{N-1}\otimes {S}_{2}\\ \star & {I}_{N-1}\otimes {R}_{4}\end{array}\right]\xi (s)\\ \le & {\zeta}^{T}(k)({\mathrm{\Xi}}_{3}+{\mathrm{\Xi}}_{4}+{\mathrm{\Xi}}_{5})\zeta (k).\end{array}

(26)

Therefore, from Eqs. (15)-(26) and by application of the *S*-procedure [25], the mathematical expectation on \mathrm{\Delta}V(k) has a new upper bound as

\mathbb{E}\{\mathrm{\Delta}V(k)\}\le \mathbb{E}\{{\zeta}^{T}(k)\underset{\mathrm{\Phi}}{\underset{\u23df}{({\mathrm{\Xi}}_{1}+{\mathrm{\Xi}}_{2}+{\mathrm{\Xi}}_{3}+{\mathrm{\Xi}}_{4}+{\mathrm{\Xi}}_{5})}}\zeta (k)\}.

(27)

Also, the system (6) with the augmented vector \zeta (k) can be rewritten as

\mathbb{E}\{{\mathrm{\Upsilon}}_{[{\rho}_{m}]}\zeta (k)\}={0}_{\kappa \times 1},

(28)

where {\mathrm{\Upsilon}}_{[{\rho}_{m}]} is defined in (10).

Then a delay-dependent stability condition for the system (6) is

\mathbb{E}\{{\zeta}^{T}(k)\mathrm{\Phi}\zeta (k)\}<0

(29)

subject to

\mathbb{E}\{{\mathrm{\Upsilon}}_{[{\rho}_{m}]}\zeta (k)\}={0}_{\kappa \times 1}.

From Lemma 4(iii) and Assumption 1, the inequality (29) is equivalent to the following condition:

\begin{array}{rcl}\mathbb{E}\{\mathrm{\Phi}+F{\mathrm{\Upsilon}}_{[{\rho}_{m}]}+{(F{\mathrm{\Upsilon}}_{[{\rho}_{m}]})}^{T}\}& =& \underset{{\tilde{\mathrm{\Phi}}}_{[l{\rho}_{0}]}}{\underset{\u23df}{\mathrm{\Phi}+F{\mathrm{\Upsilon}}_{[l{\rho}_{0}]}+{(F{\mathrm{\Upsilon}}_{[l{\rho}_{0}]})}^{T}}}\\ <& {0}_{5\kappa},\end{array}

(30)

where *F* is any matrix with appropriate dimension.

Here, by utilizing Lemma 4(ii), the condition (30) is equivalent to the following inequality:

{\left({\mathrm{\Upsilon}}_{[l{\rho}_{0}]}^{\perp}\right)}^{T}\mathrm{\Phi}\left({\mathrm{\Upsilon}}_{[l{\rho}_{0}]}^{\perp}\right)<{0}_{4\kappa}.

(31)

From the inequality (31), if the LMIs (11)-(13) are satisfied, then the synchronization stability condition (29) holds by Definition 1. This completes our proof. □

As a special case, consider the following discrete-time CDNs with only interval time-varying delays in nodes and randomly changing coupling strength:

{y}_{i}(k+1)=f({y}_{i}(k),{y}_{i}(k-h(k)))+{\rho}_{m}c\sum _{j=1}^{N}{g}_{ij}\mathrm{\Gamma}{y}_{j}(k),\phantom{\rule{1em}{0ex}}i=1,2,\dots ,N.

(32)

By use of the similar method in the driven procedure of the model (6), a model of discrete-time CDNs (32) can be obtained as

x(k+1)=({I}_{N-1}\otimes J+{\rho}_{m}c(\mathrm{\Lambda}\otimes \mathrm{\Gamma}))x(k)+({I}_{N-1}\otimes {J}_{d})x(k-h(k)).

(33)

The following is given for synchronization stability of the model of discrete-time CDNs with only interval time-varying delays in nodes (33).

**Theorem 2** *For given positive integers* {h}_{m}, {h}_{M}, *l*, *and positive scalars* *c*, {\rho}_{0}<1, *the system* (33) *is asymptotically synchronous for* {h}_{m}\le h(k)\le {h}_{M}, *if there exist positive matrices* P\in {\mathbb{R}}^{n\times n}, {Q}_{i}\in {\mathbb{R}}^{n\times n}, {R}_{j}\in {\mathbb{R}}^{n\times n}, *any symmetric matrices* {S}_{i}\in {\mathbb{R}}^{n\times n}, i=1,2, j=1,\dots ,4, *and any matrix* M\in {\mathbb{R}}^{n\times n} *satisfying the following LMIs with* (12) *and* (13):

{\left({\stackrel{\u02c6}{\mathrm{\Upsilon}}}_{[l{\rho}_{0}]}^{\perp}\right)}^{T}\mathrm{\Phi}\left({\stackrel{\u02c6}{\mathrm{\Upsilon}}}_{[l{\rho}_{0}]}^{\perp}\right)<{0}_{4\kappa}.

(34)

*Proof* The above criterion is derived in the similar method as the proof of Theorem 1, instead of the matrix {\mathrm{\Upsilon}}_{[{\rho}_{m}]}, using the following matrix:

{\stackrel{\u02c6}{\mathrm{\Upsilon}}}_{[{\rho}_{m}]}=[({I}_{N-1}\otimes (J-{I}_{n})+{\rho}_{m}c(\mathrm{\Lambda}\otimes \mathrm{\Gamma})),{0}_{\kappa},({I}_{N-1}\otimes {J}_{d}),{0}_{\kappa},-{I}_{\kappa}].

The other procedure is straightforward from the proof of Theorem 1, so it is omitted. □

**Remark 3** The systems (6) and (33) with randomly changing coupling strength and the switched systems [30–38] are similar in the concept of changing parameters. In [30–38], the various problems for the switched neural networks with time-invariant delay were addressed. However, since time delay has not only a fixed value in a practical system [39], the concerned systems with interval time-varying delays were considered in this paper. Moreover, the changing information of a parameter was considered with the probabilistic rule; that is, the Bernoulli sequence.