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Huang and Xiang Advances in Difference Equations 2013, 2013:56 http://www.advancesindifferenceequations.com/content/2013/1/56 RESEARCH Open Access Stability and l-gain analysis for 2D discrete switched systems with time-varying delays in the second FM model Shipei Huang and Zhengrong Xiang* * Correspondence: xiangzr@mail.njust.edu.cn School of Automation, Nanjing University of Science and Technology, Nanjing, 210094, People’s Republic of China Abstract This paper is concerned with the problems of stability and l2 -gain analysis for 2D (two-dimensional) discrete switched systems with time-varying delays described by the second FM state-space model Firstly, we introduce the definition of the average dwell time for a 2D discrete switched system, which is an extension of the ‘average dwell time’ concept of a 1D (one-dimensional) switched system Secondly, based on the average dwell time approach, delay-dependent sufficient conditions for the existence of the exponential stability for the 2D discrete switched system are derived and l2 -gain performance for the considered system is also analyzed All the obtained results are formulated in a set of LMIs (linear matrix inequalities) Finally, a numerical example is given to illustrate the effectiveness of the proposed results Keywords: 2D systems; switched systems; time-varying delays; l2 -gain; average dwell time; linear matrix inequality Introduction D (Two-dimensional) systems, which are a class of multi-dimensional systems, have received considerable attention over the past few decades due to their wide applications in many areas such as multi-dimensional digital filtering, linear image processing, signal processing, and process control [–] The D system theory is frequently used as an analysis tool to solve some problems, e.g., iterative learning control [, ] and repetitive process control [, ] The problems on realization, stability analysis, stabilization, filter design, and so on for D or nD systems have attracted a great deal of interest by many researchers Xu et al [, ] investigated the realization of D systems, and the problems of stability and stabilization for these systems were studied extensively in [–] The observer and filter design problems have also been considered in [–] It is known that modeling uncertainties and disturbances are unavoidable in practical systems, and it is important to investigate the problems of H∞ control and robust stabilization of D systems Recently, many results on H∞ control for D systems have been presented in [–] Because time delays frequently occur in practical systems and are often the source of instability, the H∞ control problem for a class of D discrete systems with state delays has also been investigated in [, ] On the other hand, since switched systems have numerous applications in many fields, such as mechanical systems, automotive industry, switched power converters, this class of © 2013 Huang and Xiang; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Huang and Xiang Advances in Difference Equations 2013, 2013:56 http://www.advancesindifferenceequations.com/content/2013/1/56 systems has also attracted considerable attention during the past several decades [–] Recently, some approaches have been applied widely to deal with these systems; see, for example, [–] and references cited therein As stated in [, ], in many modeling problems of physical processes, a D switching representation is needed One can cite a D physically based model for advanced power bipolar devices and heat flux switching and modulating in a thermal transistor At present, there have been a few reports on D discrete switched systems Benzaouia et al [] firstly considered D switched systems with arbitrary switching sequences, and the process of switch is considered as a Markovian jumping one In addition, the stabilization problem of discrete D switched systems was also studied in [] In [], we extended the concept of average dwell time in D switched systems to D switched systems and designed a switching rule to guarantee the exponential stability of D switched delay-free systems However, to the best of our knowledge, no works have considered the disturbance attenuation property of D switched systems to date Moreover, because of the complicated behavior caused by the interaction between the continuous dynamics and discrete switching, the problem of disturbance attenuation performance analysis for D switched systems is more difficult to study, and the methods proposed in [–] cannot be directly applied to such systems This motivates the present study In this paper, we are interested in investigating the issues of the exponential stability and l -gain analysis for D discrete switched systems with time-varying delays represented by the second FM model The main contributions of this paper can be summarized as follows: (i) Based on the average dwell time approach, a delay-dependent exponential stability criterion for such systems is obtained and formulated in terms of LMIs (linear matrix inequalities); (ii) The Lyapunov-Krasovskii function with exponential term, which is different from the previous ones, is constructed to investigate the stability of the considered systems; and (iii) In order to investigate the disturbance attenuation property of the considered systems, we for the first time introduce the concept of l -gain for a D switched system, which is an extension of the l -gain performance index in the D case The l -gain performance index can characterize the disturbance attenuation property of the underlying systems, and then, based on the established stability results, delay-dependent sufficient conditions for the existence of l -gain performance are derived in terms of LMIs, which can be easily verified by using some standard numerical software The proposed method can also be applied to non-switched D discrete linear systems This paper is organized as follows In Section , problem formulation and some necessary lemmas are given In Section , based on the average dwell time approach, delaydependent sufficient conditions for the existence of the exponential stability and l -gain property are derived in terms of a set of matrix inequalities A numerical example is provided to illustrate the effectiveness of the proposed approach in Section Concluding remarks are given in Section Notations Throughout this paper, the superscript ‘T’ denotes the transpose, and the notation X ≥ Y (X > Y ) means that matrix X – Y is positive semi-definite (positive definite, respectively) · denotes the Euclidean norm I represents an identity matrix with an appropriate dimension diag{ai } denotes a diagonal matrix with the diagonal elements , i = , , , n X – denotes the inverse of X The asterisk ∗ in a matrix is used to denote the term that is induced by symmetry The set of all nonnegative integers is represented Page of 22 Huang and Xiang Advances in Difference Equations 2013, 2013:56 http://www.advancesindifferenceequations.com/content/2013/1/56 Page of 22 by Z+ The l norm of a D signal w(i, j) is given by ∞ w ∞ = w(i, j) i= j= and w(i, j) belongs to l {[, ∞), [, ∞)} if w < ∞ Problem formulation and preliminaries Consider the following D discrete switched systems with time-varying delays described by the second FM model: σ (i,j+) x(i + , j + ) = A σ (i+,j) + Ad σ (i+,j) x(i, j + ) + A σ (i,j+) x(i + , j) + Ad x i – d (i), j + σ (i,j+) σ (i+,j) x i + , j – d (j) + B w(i, j + ) + B w(i + , j), () z(i, j) = H σ (i,j) x(i, j) + Lσ (i,j) w(i, j), where x(i, j) ∈ Rn is the state vector, w(i, j) ∈ Rq is the noise input which belongs to l {[, ∞), [, ∞)}, z(i, j) ∈ Rp is the controlled output i and j are integers in Z+ σ (i, j) : Z+ × Z+ → N = {, , , N} is the switching signal N is the number of subsystems σ (i, j) = k, k ∈ N , denotes that the kth subsystem is active Ak , Ak , Akd , Akd , Bk , Bk , H k , and Lk are constant matrices with appropriate dimensions d (i) and d (j) are delays along horizontal and vertical directions, respectively We assume that d (i) and d (j) satisfy d ≤ d (i) ≤ d , d ≤ d (j) ≤ d , () where d , d , d , and d denote the lower and upper delay bounds along horizontal and vertical directions, respectively In this paper, it is assumed that the switch occurs only at each sampling point of i or j The switching sequence can be described as (i , j ), σ (i , j ) , (i , j ), σ (i , j ) , , (iπ , jπ ), σ (iπ , jπ ) , , () where (iπ , jπ ) denotes the π th switching instant It should be noted that the value of σ (i, j) only depends upon i + j (see the references [, ]) Remark If there is only one subsystem in system (), it will degenerate to the following D system: x(i + , j + ) = A x(i, j + ) + A x(i + , j) + Ad x i – d (i), j + + Ad x i + , j – d (j) + B w(i, j + ) + B w(i + , j), z(i, j) = Hx(i, j) + Lw(i, j) Therefore, the addressed system () can be viewed as an extension of D time-varying delays systems to switched systems Huang and Xiang Advances in Difference Equations 2013, 2013:56 http://www.advancesindifferenceequations.com/content/2013/1/56 Page of 22 For system (), we consider a finite set of initial conditions, that is, there exist positive integers z and z such that ⎧ ⎪ x(i, j) = hij , ∀ ≤ j ≤ z , i = –d , –d + , , , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪x(i, j) = vij , ∀ ≤ i ≤ z , j = –d , –d + , , , ⎪ ⎨ h = v , ⎪ ⎪ ⎪ ⎪ x(i, j) = , ∀j > z , i = –d , –d + , , , ⎪ ⎪ ⎪ ⎪ ⎩ x(i, j) = , ∀i > z , j = –d , –d + , , , () where z < ∞ and z < ∞ are positive integers, hij and vij are given vectors Definition System () with w(i, j) = is said to be exponentially stable under the switching signal σ (i, j) if for a given Z ≥ , there exist positive constants c and ξ such that x(i, j) ≤ ξ e–c(D–Z) x(i, j) C () i+j=Z i+j=D holds for all D ≥ Z, where x(i, j) i+j=Z C x(i – θh , j) , x(i, j – θv ) , sup –d ≤θh ≤, i+j=Z –d ≤θv ≤ η(i – θh , j) , δ(i, j – θv ) , η(i – θh , j) = x(i – θh + , j) – x(i – θh , j), δ(i, j – θv ) = x(i, j – θv + ) – x(i, j – θv ) Remark From Definition , it is easy to see that when Z is given, i+j=Z x(i, j) C will be bounded and i+j=D x(i, j) will tend to be zero exponentially as D goes to infinity, which also means x(i, j) will tend to be zero exponentially Definition [] For any i + j = D > Z = iZ + jZ , let Nσ (i,j) (Z, D) denote the switching number of the switching signal σ (i, j) on an interval (Z, D) If Nσ (i,j) (Z, D) ≤ N + D–Z τa () holds for given N ≥ , τa ≥ , then the constant τa is called the average dwell time and N is the chatter bound Remark Definition is an extension of the ‘average dwell time’ concept in a D switched system, which can be seen in [] In what follows, based on the extended average dwell time concept, we will investigate the problems of stability and l -gain analysis for a D discrete switched system with time-varying delays It should be noted that we have studied the problems of stability analysis and stabilization of delay-free D switched systems using the average dwell time approach in [] Huang and Xiang Advances in Difference Equations 2013, 2013:56 http://www.advancesindifferenceequations.com/content/2013/1/56 Page of 22 Remark Similar to the D switched system case, Definition means that if there exists a positive number τa such that the switching signal σ (i, j) has the average dwell time property, the average time interval between consecutive switching is at least τa The average dwell time method is used to restrict the switching number of the switching signal during a time interval such that the stability or other performances of the system can be guaranteed Definition Consider D discrete switched system () For a given scalar γ > , system () is said to have l -gain γ under the switching signal σ (i, j) if it satisfies the following conditions: () When w(i, j) = , system () is asymptotically stable; () Under the zero boundary condition, it holds that ∞ ∞ ∞ z ∞ w , < γ ∀ = w ∈ l [, ∞), [, ∞) , i= j= i= j= where ∞ w ∞ w(i, j + ) w(i + , j) = i= j= ∞ , z ∞ = i= j= z(i, j + ) z(i + , j) Remark It is not difficult to see that Definition is an extension of the l -gain performance index in the D case γ characterizes the disturbance attenuation performance The smaller γ is, the better performance is Definition Consider D discrete switched system () For a given scalar γ > , system () is said to have weighted l -gain γ under the switching signal σ (i, j) if it satisfies the following conditions: () When w(i, j) = , system () is asymptotically stable; () Under the zero boundary condition, it holds that ∞ ∞ ∞ α i+j z ∞ w , < γ ∀ = w ∈ l [, ∞), [, ∞) i= j= i= j= Remark Similar to the D switched system case, Definition means that system () can also have disturbances attenuation properties when it satisfies conditions () and () in Definition Lemma Consider D discrete switched system () Suppose that there exist a series of C functions Vk : Rn → R (k ∈ N ) and two positive scalars λ and λ for which the following inequality holds: λ x(i, j) ≤ Vk x(i, j) ≤ λ x(i, j) , C () ∀i, j ∈ Z+ , ∀k ∈ N if there exists a number < α < for which Vk (x(i, j)) along with the solution of system () satisfies the inequality Vk x(i, j) , Vk x(i, j) ≤ α i+j=D i+j=D– D > Z = max(z , z ), ∀k ∈ N, () Huang and Xiang Advances in Difference Equations 2013, 2013:56 http://www.advancesindifferenceequations.com/content/2013/1/56 Page of 22 and μ ≥ such that Vσ (iπ – ,jπ – ) x(i, j) , Vσ (iπ ,jπ ) x(i, j) ≤ μ π = , , , () i+j=(mπ )– i+j=mπ then D discrete switched system () is exponentially stable for every switching signal with the average dwell time scheme τa > τa∗ = ln μ – ln α () Proof Let χ = Nσ (i,j) (Z, D) denote the switch number of switching σ (i, j) on an interval [Z, D), and let (iπ –χ+ , jπ –χ+ ), (iπ –χ+ , jπ –χ+ ), , (iπ , jπ ) denote the switching points of σ (i, j) over the interval [Z, D) Denoting mi = ii + ji , i = π – χ + , , π , it follows from () and () that Vσ (iπ ,jπ ) x(i, j) < α D–mπ Vσ (iπ ,jπ ) x(i, j) i+j=mπ i+j=D ≤ μα D–mπ Vσ (iπ – ,jπ – ) x(i, j) i+j=(mπ )– < μ α D–mπ α mπ –mπ – Vσ (iπ – ,jπ – ) x(i, j) i+j=(mπ – )– = μ α D–mπ – Vσ (iπ – ,jπ – ) x(i, j) < · · · i+j=(mπ – )– < μχ α D–Z Vσ (iπ –χ ,jπ –χ ) x(i, j) () i+j=Z According to Definition , one obtains χ = Nσ (i,j) (Z, D) ≤ N + D–Z τa () Then from (), we have Vσ (iπ ,jπ ) x(i, j) i+j=D < μχ α D–Z Vσ (iπ –χ ,jπ –χ ) x(i, j) i+j=Z ln μ = μN e–(– τa –ln α)(D–Z) Vσ (iπ –χ ,jπ –χ ) x(i, j) () i+j=Z From (), we get x(i, j) i+j=D ln μ N –(– τa ≤ λ– λ μ e –ln α)(D–Z) x(i, j) C () i+j=Z Therefore, according to Definition , system () is exponentially stable under the average dwell time scheme () Huang and Xiang Advances in Difference Equations 2013, 2013:56 http://www.advancesindifferenceequations.com/content/2013/1/56 Page of 22 Main results 3.1 Stability analysis Theorem Consider system () with w(i, j) = For a given positive constant α < , if there exist positive definite symmetric matrices Phk , Pvk , Qkh , Qkv , Whk , Wvk , Rkh , Rkv , and matrices Nk = k N k N , Nk = k N , Mk = k N k M k M , Mk = k M k M , Xk = k Xk X k ∗ X > and Y k = k Yk Y k ∗ Y > with appropriate dimensions, k ∈ N , such that T k k (Ph + Pv ) k k –(Ph + Pv ) ⎡ Ŵ ⎢∗ ⎢ ⎢ ⎣∗ ∗ d T k Rh d –Rkh ∗ ∗ ∗ T k Rv –Rkv Xk ∗ Nk ≥ , Rkh Xk ∗ Nk ≥ , Rkh Yk ∗ Mk ≥ , Rkv Yk ∗ Mk ≥ , Rkv ⎤ ⎥ ⎥ ⎥ < , ⎦ () () where = Ak = Ak – I = Ak ⎡ Ŵ ⎢ ⎢∗ ⎢ ⎢∗ Ŵ=⎢ ⎢∗ ⎢ ⎢ ⎣∗ ∗ Ak Akd Akd , Ak Akd Akd , Ak – I Akd Akd , Ŵ ∗ ∗ ∗ ∗ Ŵ Ŵ ∗ ∗ ∗ Ŵ Ŵ ∗ ∗ k –α d N k –α d N –α d Whk ∗ ⎤ k ⎥ –α d M ⎥ ⎥ ⎥ ⎥, d k ⎥ –α M ⎥ ⎥ ⎦ d k –α Wv k Ŵ = –αPhk + Whk + (d – d + )Qkh + α d Nk + NkT + d α d X , k kT + d α d Yk , + M Ŵ = –αPvk + Wvk + (d – d + )Qkv + α d M k k kT Ŵ = –α d Nk – N , – N + d α d X k k k kT + d α d Y , – M – M Ŵ = –α d M k k kT k kT , + N + d α d X – N – N Ŵ = –α d Qkh + α d N k k kT k kT , + M + d α d Y – M – M Ŵ = –α d Qkv + α d M hold, then under the average dwell time scheme τa > τa∗ = ln μ , – ln α () Huang and Xiang Advances in Difference Equations 2013, 2013:56 http://www.advancesindifferenceequations.com/content/2013/1/56 Page of 22 where μ ≥ and satisfies Phk ≤ μPhl , Phl ≤ μPhk , Qkh ≤ μQlh , Qlh ≤ μQkh , Whk ≤ μWhl , Whl ≤ μWhk , Rkh ≤ μRlh , Pvk ≤ μPvl , Rlh ≤ μRkh , Pvl ≤ μPvk , () Qkv ≤ μQlv , Qlv ≤ μQkv , Wvk ≤ μWvl , Rkv ≤ μRlv , Rlv ≤ μRkv , Wvl ≤ μWvk , ∀k, l ∈ N, the system is exponentially stable Proof See the Appendix for the detailed proof, it is omitted here Remark In Theorem , we propose a sufficient condition for the existence of exponential stability for the considered D discrete switched system () It is worth noting that this condition is obtained by using the average dwell time approach 3.2 l2 -gain performance analysis Theorem Consider system () For given positive constants γ and α < , if there exist positive definite symmetric matrices Phk , Pvk , Qkh , Qkv , Whk , Wvk , Rkh , Rkv , and matrices Nk = k N k N , Nk = k N k N , Mk = k M k M , Mk = k M k M , Xk = k Xk X k ∗ X > and Y k = k Yk Y k ∗ Y > with appropriate dimensions, k ∈ N , such that () and the following inequality ⎡ Ŵ ⎢ ⎢∗ ⎢ ⎢∗ ⎢ ⎢ ⎢∗ ⎢ ⎢∗ ⎢ ⎢ ⎣∗ ∗ –γ I ∗ ∗ ∗ ∗ ∗ –γ I ∗ ∗ ∗ ∗ T k k (Ph + Pv ) T k k B (Ph + Pv ) BT (Phk + Pvk ) –(Phk + Pvk ) ∗ ∗ ∗ T k Rh T k d B R h d BT Rkh d –Rkh ∗ ∗ T k Rv T k d B R v d BT Rkv d –Rkv ∗ ⎤ T ⎥ LTk ⎥ ⎥ LTk ⎥ ⎥ ⎥ ⎥ < , ⎥ ⎥ ⎥ ⎥ ⎦ –I () where = Hk Hk , hold, then under the average dwell time scheme (), the system is exponentially stable and has weighted l -gain γ Proof It is easy to get that () can be deduced from (), and according to Theorem , we can obtain that system () is exponentially stable Now we are in a position to consider the l -gain performance of system () under the zero boundary condition Following the proof line of Theorem , we get the following Huang and Xiang Advances in Difference Equations 2013, 2013:56 http://www.advancesindifferenceequations.com/content/2013/1/56 Page of 22 relationship for the kth subsystem: Vkh (i + , j + ) + Vkv (i + , j + ) – αVkh (i, j + ) – αVkv (i + , j) + z – γ w ⎤T ⎛ ⎡ ⎡ kT ⎤ x(i, j + ) A ⎜ ⎥ ⎜ ⎢ ⎢ kT ⎥ x(i + , j) ⎥ ⎜ ⎢ ⎢A ⎥ ⎥ ⎢ ⎢ kT ⎥ ⎢x(i – d (i), j + )⎥ ⎜ ⎢A ⎥ ⎥ ⎜ ⎢ ⎢ d ⎥ ⎢x(i + , j – d (j))⎥ ⎜ ⎢AkT ⎥ ⎥ ⎥ ⎜ ⎢ ˜ +⎢ ≤⎢ ⎥ ⎜ ⎢ d ⎥ Phk + Pvk ⎥ ⎢ x(i – d , j + ) ⎥ ⎜ ⎢ ⎥ ⎥ ⎜ ⎢ ⎢ ⎢ x(i + , j – d ) ⎥ ⎜ ⎢ ⎥ ⎜ ⎥ ⎥ ⎢ ⎢ ⎥ ⎜ ⎢ ⎢ kT ⎥ B ⎣ w(i, j + ) ⎦ ⎜ ⎣ ⎦ ⎝ BkT w(i + , j) ⎡ kT A – I ⎢ kT ⎢ A ⎢ kT ⎢ A ⎢ d ⎢ AkT ⎢ + ⎢ d ⎢ ⎢ ⎢ ⎢ ⎢ kT ⎣ B BkT ⎤ AkT ⎥ AkT – I⎥ ⎥ ⎥ AkT d ⎥ kT ⎥ Ad ⎥ d Rkh ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ BkT ⎦ BkT d Rkv ⎤ x(i, j + ) ⎢ ⎥ ⎢ x(i + , j) ⎥ ⎢ ⎥ ⎢x(i – d (i), j + )⎥ ⎢ ⎥ ⎢x(i + , j – d (j))⎥ ⎢ ⎥ ×⎢ ⎥ ⎢ x(i – d , j + ) ⎥ ⎢ ⎥ ⎢ x(i + , j – d ) ⎥ ⎥ ⎢ ⎢ ⎥ ⎣ w(i, j + ) ⎦ w(i + , j) ⎤T x(i, j + ) ⎢ ⎥ Xk – ⎣(i – d (i), j + )⎦ ∗ r=i–d (i) η(r, j + ) i–d (i)– – r=i–d ⎡ ⎤T x(i, j + ) ⎥ Xk ⎢ ⎣x(i – d (i), j + )⎦ ∗ η(r, j + ) ⎡ ⎤T x(i + , j) ⎢ ⎥ Yk – ⎣x(i + , j – d (j))⎦ ∗ t=j–d (j) δ(i + , t) j– j–d (j)– – t=j–d ⎡ ⎤T x(i + , j) ⎥ Yk ⎢ ⎣x(i + , j – d (j))⎦ ∗ δ(i + , t) ⎡ ⎡ kT ⎤T A ⎢ kT ⎥ ⎢A ⎥ ⎢ kT ⎥ ⎢A ⎥ ⎢ d ⎥ ⎢AkT ⎥ ⎢ d ⎥ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ kT ⎥ ⎣ B ⎦ BkT ⎡ kT A – I ⎢ kT ⎢ A ⎢ kT ⎢ A ⎢ d ⎢ AkT ⎢ d ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ kT ⎣ B BkT ⎡ i– Nk Rkh Nk Rkh Mk Rkv Mk Rkv ⎤T ⎞ AkT ⎥ ⎟ ⎟ AkT – I⎥ ⎟ ⎥ ⎥ ⎟ AkT d ⎥ ⎟ ⎥ ⎟ ⎟ AkT d ⎥ ⎟ ⎥ ⎟ ⎥ ⎥ ⎟ ⎟ ⎥ ⎥ ⎟ ⎟ ⎥ BkT ⎦ ⎟ ⎠ BkT ⎡ ⎤ x(i, j + ) ⎢ ⎥ d ⎣(i – d (i), j + )⎦ α η(r, j + ) ⎤ x(i, j + ) ⎥ d ⎢ ⎣x(i – d (i), j + )⎦ α η(r, j + ) ⎡ ⎡ ⎤ x(i + , j) ⎢ ⎥ d ⎣x(i + , j – d (j))⎦ α δ(i + , t) ⎤ x(i + , j) ⎥ d ⎢ ⎣x(i + , j – d (j))⎦ α , δ(i + , t) ⎡ Huang and Xiang Advances in Difference Equations 2013, 2013:56 http://www.advancesindifferenceequations.com/content/2013/1/56 Page 10 of 22 where ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ˜ =⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ˜ ∗ ∗ ∗ ∗ ∗ ∗ ∗ H kT H k ˜ ∗ ∗ ∗ ∗ ∗ ∗ –α d Qkh ∗ ∗ ∗ ∗ ∗ –α d Qkv ∗ ∗ ∗ ∗ –α d Whk ∗ ∗ ∗ H kT Lk H kT Lk ˜ d –α Wvk ∗ ∗ ∗ ˜ = –αPhk + Whk + (d – d + )Qkh + H kT H k , ⎤ H kT Lk ⎥ H kT Lk ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ LkT Lk ⎦ ˜ ˜ = –αPvk + Wvk + (d – d + )Qkv + H kT H k , ˜ = ˜ = LkT Lk – γ I Then by the Schur complement lemma, we can obtain from () and () that Vkh (i + , j + ) + Vkv (i + , j + ) – αVkh (i, j + ) – αVkv (i + , j) + z –γ w < () Summing up both sides of () from D – to with respect to j and to D – with respect to i and applying the zero boundary condition, one gets Vσ (iπ ,iπ ) (i, j) – Vσ (iπ ,iπ ) (i, j) < α Ŵ(i, j) i+j=D– i+j=D– i+j=D D– < α D–mπ α D––i–j Ŵ(i, j) Vσ (iπ ,iπ ) (i, j) – m=mπ – i+j=m i+j=mπ D– ≤ μα D–mπ α D––i–j Ŵ(i, j) Vσ (iπ – ,iπ – ) (i, j) – i+j=(mπ )– m=mπ – i+j=m μα D–(mπ –) Vσ (iπ – ,iπ – ) (i, j) – μα D–mπ < D– α D––i–j Ŵ(i, j) – m=mπ – i+j=m μNσ (i,j) (i+j,D) α D–(mπ –) Vσ (iπ – ,iπ – ) (i, j) = i+j=mπ – D– μNσ (i,j) (i+j+,D) α D––i–j Ŵ(i, j) – m=mπ – i+j=m μNσ (i,j) (i+j,D) α D–mπ – Vσ (iπ – ,iπ – ) (i, j) < Ŵ(i, j) i+j=mπ – i+j=mπ – i+j=mπ – D– μNσ (i,j) (i+j+,D) α D––i–j Ŵ(i, j) – m=mπ – – i+j=m Huang and Xiang Advances in Difference Equations 2013, 2013:56 http://www.advancesindifferenceequations.com/content/2013/1/56 Page 11 of 22 μNσ (i,j) (i+j–,D) α D–mπ – Vσ (iπ – ,iπ – ) (i, j) ≤ i+j=(mπ – )– D– μNσ (i,j) (i+j+,D) α D––i–j Ŵ(i, j) – m=mπ – – i+j=m < ··· μNσ (i,j) (i+j,D) α D– Vσ (,) (i, j) < i+j= D– μNσ (i,j) (i+j+,D) α D––i–j Ŵ(i, j), – () m= i+j=m where Ŵ(i, j) = z –γ w z(i + , j) z(i, j + ) = –γ w(i + , j) w(i, j + ) Under the zero initial condition, it holds that μNσ (i,j) (i+j,D) α D– Vσ (,) (i, j) = () i+j= Thus we have D– μNσ (i,j) (i+j+,D) α D––i–j Ŵ(i, j) < – m= i+j=m Vσ (iπ ,iπ ) (i, j) < () i+j=D Multiplying the both sides of () by μ–Nσ (i,j) (,D) , we get the following inequality: D– μ–Nσ (i,j) (,i+j+) α D––i–j Ŵ(i, j) < () m= i+j=m That is, D– D– μ–Nσ (i,j) (,i+j+) α D––i–j z μ–Nσ (i,j) (,i+j+) α D––i–j w < m= i+j=m m= i+j=m Note that Nσ (i,j) (, i + j + ) ≤ (i + j)/τa , then using (), we have μ–Nσ (i,j) (,i+j+) = e–Nσ (i,j) (,i+j+) ln μ ≥ e(i+j) ln α () It follows that D– D– e(i+j) ln α α D––i–j z μ–Nσ (i,j) (,i+j+) α D––i–j w < m= i+j=m m= i+j=m D– D– α D– z ⇒ m= i+j=m < γ α D––i–j w m= i+j=m () Huang and Xiang Advances in Difference Equations 2013, 2013:56 http://www.advancesindifferenceequations.com/content/2013/1/56 ∞ D– ∞ D– α D– z ⇒ , the following x(i, j + ) x(i – d (i), j + ) T x(i, j + ) x(i – d (i), j + ) x(i + , j) x(i + , j – d (j)) – T k Xk X () x(i + , j) x(i + , j – d (j)) T Yk x(i + , j) α d x(i + , j – d (j)) Yk x(i + , j) α d x(i + , j – d (j)) T () Adding the terms on the right-hand sides of equations ()-() to (), allows us to write () as Vkh (i + , j + ) + Vkv (i + , j + ) – α Vkh (i, j + ) + Vkv (i + , j) ⎤T ⎛ x(i, j + ) ⎢ ⎥ ⎜ ⎢ x(i + , j) ⎥ ⎜ ⎢ ⎥ ⎜ ⎢x(i – d (i), j + )⎥ ⎜ ⎢ ⎥ ⎜ ≤⎢ ⎥ ⎜ ⎢x(i + , j – d (j))⎥ ⎜ ⎢ ⎥ ⎜ ⎣ x(i – d , j + ) ⎦ ⎝ x(i + , j – d ) ⎡ ⎡ kT ⎤ A ⎢ kT ⎥ ⎢A ⎥ ⎢ kT ⎥ ⎢Ad ⎥ k k ⎥ +⎢ ⎢AkT ⎥ Ph + Pv ⎢ d ⎥ ⎢ ⎥ ⎣ ⎦ ⎤T AkT ⎢ kT ⎥ ⎢A ⎥ ⎢ kT ⎥ ⎢Ad ⎥ ⎥ ⎢ ⎢AkT ⎥ ⎢ d ⎥ ⎥ ⎢ ⎣ ⎦ ⎡ Huang and Xiang Advances in Difference Equations 2013, 2013:56 http://www.advancesindifferenceequations.com/content/2013/1/56 ⎡ kT ⎤ AkT A – I ⎢ kT ⎥ AkT ⎢ A – I⎥ ⎢ kT ⎥ k ⎢ Ad ⎥ AkT d ⎥ d Rh ⎢ + ⎢ kT ⎥ AkT ⎢ Ad d ⎥ ⎢ ⎥ ⎣ ⎦ ⎤ ⎡ x(i, j + ) ⎢ ⎥ ⎢ x(i + , j) ⎥ ⎥ ⎢ ⎢x(i – d (i), j + )⎥ ⎥ ×⎢ ⎢x(i + , j – d (j))⎥ ⎥ ⎢ ⎥ ⎢ ⎣ x(i – d , j + ) ⎦ x(i + , j – d ) ⎡ ⎤T x(i, j + ) i– ⎢ ⎥ – ⎣(i – d (i), j + )⎦ r=i–d (i) η(r, j + ) ⎡ ⎤T x(i, j + ) i–d (i)– ⎢ ⎥ – ⎣x(i – d (i), j + )⎦ r=i–d η(r, j + ) ⎡ ⎤T x(i + , j) j– ⎢ ⎥ – ⎣x(i + , j – d (j))⎦ t=j–d (j) δ(i + , t) ⎡ ⎤T x(i + , j) j–d (j)– ⎢ ⎥ – ⎣x(i + , j – d (j))⎦ t=j–d δ(i + , t) Page 20 of 22 AkT –I ⎢ kT ⎢ A ⎢ kT ⎢ Ad ⎢ ⎢ AkT ⎢ d ⎢ ⎣ ⎡ d Rkv k X ∗ Xk ∗ Yk ∗ Yk ∗ ⎤T ⎞ AkT ⎥ ⎟ ⎟ AkT – I⎥ ⎟ ⎥ kT ⎥ ⎟ Ad ⎥ ⎟ ⎥ ⎟ ⎟ AkT d ⎥ ⎟ ⎥ ⎟ ⎦ ⎠ ⎡ ⎤ x(i, j + ) ⎢ ⎥ d ⎣(i – d (i), j + )⎦ α η(r, j + ) ⎡ ⎤ x(i, j + ) k N ⎢ ⎥ d ⎣x(i – d (i), j + )⎦ α Rkh η(r, j + ) ⎡ ⎤ x(i + , j) k M ⎢ ⎥ d ⎣x(i + , j – d (j))⎦ α Rkv δ(i + , t) ⎡ ⎤ x(i + , j) k M ⎢ ⎥ x(i + , j – d (j))⎦ α d k ⎣ Rv δ(i + , t) Nk Rkh () Thus it follows from ()-() that Vkh (i + , j + ) + Vkv (i + , j + ) < α Vkh (i, j + ) + Vkv (i + , j) () When D ≥ mπ > z = max(z , z ), we have Vkh (, D) = Vkv (D, ) = Then summing up both sides of () from D – to with respect to j and to D – with respect to i, one gets Vk (i, j) = Vkh (, D) + Vkh (, D – ) + Vkh (, D – ) + · · · + Vkh (D – , ) + Vkh (D, ) i+j=D + Vkv (, D) + Vkv (, D – ) + Vkv (, D – ) + · · · + Vkv (D – , ) + Vkv (D, ) < α Vkh (, D – ) + Vkv (, D – ) + Vkh (, D – ) + Vkv (, D – ) + · · · + Vkh (D – , ) + Vkv (D – , ) Vk (i, j) =α i+j=D– () Huang and Xiang Advances in Difference Equations 2013, 2013:56 http://www.advancesindifferenceequations.com/content/2013/1/56 Page 21 of 22 Thus () can be directly obtained Moreover, by (), we can find two positive scalars λ and λ such that () holds, where λ = λmin Phk + λmin Pvk , k∈N λ = max λmax Phk + λmax Pvk + d λmax Qkh + d λmax Qkv + d λmax Whk k∈N + d λmax Wvk + (d – d ) λmax Qkh + (d – d ) λmax Qkv + d λmax Rkh + d λmax Rkv In addition, () can be deduced 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