Almost periodic synchronization of quaternion-valued shunting inhibitory cellular neural networks with mixed delays via state-feedback control

This paper studies the drive-response synchronization for quaternion-valued shunting inhibitory cellular neural networks (QVSICNNs) with mixed delays. First, QVSICNN is decomposed into an equivalent real-valued system in order to avoid the non-commutativity of the multiplicity. Then, the existence of almost periodic solutions is obtained based on the Banach fixed point theorem. An novel state-feedback controller is designed to ensure the global exponential almost periodic synchronization. At the end of the paper, an example is given to illustrate the effectiveness of the obtained results.

In this paper, the model of the shunting inhibitory cellular neural networks with mixed time delays is defined as follows: x 0 pq ðtÞ ¼ À a pq ðtÞx pq ðtÞ À . . . ; mng :¼ J ; C pq denotes the cell at the position (p, q) of the lattice; the r-neighborhood of C pq is defined as N r ðp; qÞ ¼ fC kl : maxðjk À pj; jl À qjÞ r; pq 2 J g; and N s (p, q), N u (p, q) are similarly specified; x pq 2 Q is the activity of the cell C pq , T pq : Q ! Q is the external input to C pq , a pq (t)>0 represents the passive decay rate of the cell activity; B kl pq ðtÞ; C kl pq ðtÞ; D kl pq ðtÞ ! 0 are the connection or coupling strength of postsynaptic activity of the cell transmitted to C pq , and the activity functions f ; g; h : Q ! Q are the continuous functions representing the output or firing rate of the cell C pq ; τ(t) ! 0 denotes the transmission time varying delay; K pq (t) denotes the transmission delay kernels.
The initial conditions associated with system (1) are of the form where φ pq ðsÞ ¼ φ R pq ðsÞ þ iφ I pq ðsÞ þ jφ J pq ðsÞ þ kφ K pq ðsÞ; φ K pq ; φ I pq ; φ J pq ; φ K pq : ðÀ 1; 0 ! R are bounded continuous functions. Now, we introduce some relevant definitions and basic lemmas. Definition 1. [59] A function x 2 CðR; R n Þ is said to be almost periodic if, for any > 0, it is possible to find a real number l = l() > 0, denoting length l() of an interval, there exists a number τ = τ() in this interval such that |x(t + τ) − x(t)| < for all t 2 R.
Denote the set of almost periodic functions by APðR; R n Þ. Definition 2. A quaternion-valued function x ¼ x R þ ix I þ jx J þ kx K 2 CðR; Q n Þ is called almost periodic if for every ν 2 {R, I, J, K}: = Λ, x n 2 APðR; R n Þ. Definition 3. [59] Let x 2 R n and A(t) be an n × n matrix function on R. Then the linear system is said to admit an exponential dichotomy on R if there exist positive constants k i , α i , i = 1, 2, projection P, and the fundamental solution matrix X(t) of (2), satisfying k XðtÞPX À 1 ðsÞ k 0 k 1 e À a 1 ðtÀ sÞ ; s; t 2 R; t ! s; k XðtÞðI À PÞX À 1 ðsÞ k 0 k 2 e À a 2 ðsÀ tÞ ; s; t 2 R; t s; where k Á k 0 is the matrix norm on R. Let us consider the following almost periodic system where A(t) is an almost periodic matrix function and f(t) is an almost periodic vector function. Lemma 1. [59] If the linear system (2) admits an exponential dichotomy, then system (3) has a unique almost periodic solution where X(t) is the fundamental solution matrix of (2), I denotes the n × n-identity matrix. Lemma 2.
[59] Let a p be an almost periodic function on R and . . . n: Then the linear system x 0 ðtÞ ¼ diagðÀ a 1 ðtÞ; À a 2 ðtÞ; . . . ; À a n ðtÞÞxðtÞ admits an exponential dichotomy on R.
Assume the activity functions f ; g; h : Q ! Q of (1) can be expressed as where f n ; g n ; h n : R 4 ! R, ν 2 Λ, pq 2 J and the external input T n pq : R ! Q can be expressed as where T n pq : R ! R; n 2 L; pq 2 J . In the following, for a bounded continuous function, we denote " f ¼ sup In order to overcome the non-commutativity of the quaternion multiplication, according to Hamilton rules, we decompose system (1) into an equivalent real-valued system: x À g I ½t; x À g J ½t; x À g K ½t; x g I ½t; x g R ½t; x À g K ½t; x g J ½t; x g J ½t; x g K ½t; x g R ½t; x À g I ½t; x Applying (4)- (7), we obtain an equivalent real-valued system of the quaternion-valued system (1) as follows: with the initial conditions: In what follows, we regard (1) as the drive system, and the corresponding response system is expressed as y 0 pq ðtÞ ¼ À a pq ðtÞy pq ðtÞ À where y pq ðtÞ ¼ y R pq ðtÞ þ iy I pq ðtÞ þ jy J pq ðtÞ þ ky K pq ðtÞ denotes the state of the response system, U pq ðtÞ ¼ U R pq ðtÞ þ iU I pq ðtÞ þ jU J pq ðtÞ þ kU K pq ðtÞ is a state-feedback controller, the rest notations are the same as those in system (1) and the initial condition is Denote z pq (t) = y pq (t) − x pq (t), subtracting (1) from (9) yields the following error system: In order to show the almost periodic synchronization of the drive-response system, we design the state-feedback controller as follows: Definition 4. The response system (9) and the drive system (1) are said to be globally exponentially synchronized, if there exist positive constants M > 0 and λ > 0 such that where k yðtÞ À xðtÞ k 0 ¼ max pq2J;n2L fjy n pq ðtÞ À x n pq ðtÞjg; k c À φ k¼ max pq2J;n2L fsup t2R jc n pq ðtÞ À φ n pq ðtÞjg: Analogously, one can decompose (10) into the following real-valued system:  (1). Thus, the problem of finding an almost periodic solution for (1) is reduced to finding it for system (8). For studying the synchronization of (1) and (9), we just need to consider the exponential stability of system (11)- (14).
Throughout the paper, we assume the following conditions: (A 2 ) For ν 2 Λ, f ν , g ν , h ν , p n 2 CðR; RÞ and for any u n ; v n 2 R, there exist positive constants L n f , (A 3 ) For pq 2 J , the delay kernels K pq : ½0; 1Þ ! R are continuous and |K pq (t)|e λt are integrable on [0,1) for certain positive constant λ.

Main results
In this section, we establish the sufficient conditions for the existence of almost periodic solutions of system (1), and the sufficient conditions for the global exponential synchronization of the drive system (1) and the response system (9

system (8) has a unique almost periodic solution in
ðs À tðsÞÞ À c R pq ðs À tðsÞÞj þ L I g jφ I pq ðs À tðsÞÞ À c I pq ðs À tðsÞÞj þ L J g jφ J pq ðs À tðsÞÞ À c J pq ðs À tðsÞÞj þ L K g jφ K pq ðs À tðsÞÞ À c K pq ðs À tðsÞÞjÞðjc R pq ðsÞj þ jc I pq ðsÞj þ jc J pq ðsÞj Hence, we have Similarly, one can obtain jðGφÞ n À ðGcÞ n j m pq a pq k φ À c k Y ; pq 2 J ; n ¼ I; J; K: Therefore, k Gφ À Gc k m k φ À c k; which implies that Γ is a contraction mapping. According to the Banach fixed point theorem, Γ has a unique fixed point in Y Ã , which means that system (8) has a unique almost periodic solution in Y Ã . The proof is complete. Theorem 2. Assume (A 1 )-(A 4 ) hold and suppose further that There exists a positive constant λ such that where Then the drive system (1) and response system (9) are globally exponentially synchronized. proof. Let us construct a Lyapunov function V(t) as follows From (11)- (14), for any t > 0, ν 2 Λ, pq 2 J , we have Almost periodic synchronization of quaternion-valued shunting inhibitory cellular neural networks On the other hand, we have We also have k yðtÞ À xðtÞ k 0 VðtÞe λt Vð0Þe λt M k c À φ k e À λt ; t ! 0; where Therefore, the drive system (1) and the response system (9) are globally exponentially synchronized. The proof is complete. which implies that (A 4 ) is satisfied. Therefore, the drive system (18) has a unique almost periodic solution. Moreover, take λ = 1, we have g 11 % À 0:218; g 12 % À 0:218; g 21 % À 1:218; g 22 % À 3:218; g % À 0:218 < 0: Thus, (A 5 ) is also satisfied. Therefore, (18) and (19) are globally exponentially synchronized (see  Almost periodic synchronization of quaternion-valued shunting inhibitory cellular neural networks

Conclusion
In this paper, a class of QVSICNNs with mixed delays is studied. To the best of our knowledge, this is the first on studying the problem. Since QVSICNNs include RVSICNNs and CVSICNNs as special cases, our method of this paper can be applied to study the almost periodic synchronization problem of other types of neural networks including RVNNs and CVNNs.
In this paper, the almost periodic synchronization of a class of QVSICNNs with mixed delays is studied. To the best of our knowledge, this is the first on studying the problem. Since QVSICNNs include RVSICNNs and CVSICNNs as special cases, our method of this paper can Almost periodic synchronization of quaternion-valued shunting inhibitory cellular neural networks be applied to study the almost periodic synchronization problem of other types of neural networks including RVNNs and CVNNs.