Global Exponential Stability for DCNNs with Impulses on Time Scales

A class of delayed cellular neural networks (DCNNs) with impulses on time scales is considered. By using the topological degree theory, and the time scale calculus theory some sufficient conditions are derived to ensure the existence, uniqueness, and global exponential stability of equilibria for this class of neural networks. Finally, a numerical example illustrates the feasibility of our results and also shows that the continuous-time neural network and the discrete-time analogue have the same dynamical behaviors. The results of this paper are completely new and complementary to the previously known results.


Introduction
Chua and Yang [1] proposed a novel class of informationprocessing systems called cellular neural networks (CNNs) in 1988.The CNNs can be applied in signal processing and can also be used to solve some image processing and pattern recognition problems [2].Since time delays are unavoidable due to finite switching speeds of the amplifiers, delayed cellular neural networks (DCNNs) have been widely studied and successfully applied to pattern recognition, associative memories, and signal processing and optimization, especially in image processing.The dynamic behavior of the networks plays an important role in such applications [3][4][5][6][7][8].Therefore, there are many works on the stability of equilibrium point of delayed cellular neural networks (DCNNs) [5][6][7][8][9][10][11][12][13].
Most neural networks can be classified into two types: continuous or discrete.However, many real-world systems and natural processes cannot be categorized into one of them.They display characteristics of both continuous and discrete styles.For instance, some biological neural networks in biology, bursting rhythm models in pathology, and optimal control models in economics are characterized by abrupt changes of state.These are the familiar impulsive phenomena.Other examples can also be found in information science, electronics, automatic control systems, computer networking, artificial intelligence, robotics, telecommunications, and so forth.Such a kind of phenomena, in which sudden and sharp changes often occur in a continuous process, cannot be well described by pure continuous or pure discrete models.Therefore, it is important and, in effect, necessary to study a new type of neural networks-impulsive neural networks-as an appropriate description of these phenomena of abrupt qualitative dynamical changes of essentially continuous systems.The fundamental theory of impulsive differential equations has been developed in [14].Since delays and impulses can affect the dynamical behaviors of the system, it is necessary to investigate both delay and impulsive effects on the stability of neural networks.For more details, one can refer to [10,13,[15][16][17][18][19][20][21][22][23].
The theory of time scale was initiated by Hilger in 1988, which has recently received a lot of attention [24][25][26].The field of dynamic equations on time scale contains links and extends the classical theory of differential and difference equations.It is well known that both continuous and discrete systems are very important in implementation and applications (see [27][28][29][30]).But it is troublesome to study the stability for continuous and discrete systems, respectively.
Motivated by above, in this paper, we are concerned with the following impulsive DCNN on time scales: where  corresponds to the numbers of units in a neural network;   () corresponds to the state of the th unit at time ;   (  ()) denotes the output of the th unit at time .T + 0 is the T-interval { ∈ T,  ≥ 0}, and T denotes a time scale, which is an arbitrary nonempty closed subset of the real number R and with bounded graininess .For the simplicity, we assume that 0 ∈ T and T is unbounded above; that is, sup T = +∞.Further,   ,   ,   , and   are constants.  ,   denote the strength of the th unit at time  and   (,   ()), respectively.  denotes the external bias on the th unit and   represents the rate with which the th unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs.  ,  = 1, 2, . . .are the moments of impulsive perturbations and satisfy 0 =  0 <  1 <  2 < ⋅ ⋅ ⋅ and lim  → ∞ = ∞, (  ) = 0 (see Definition 3).  (  (  )) represents the abrupt change of the state   () at the impulsive moment   .To the best of our knowledge, this is first paper to study DCNNs with impulses on time scales.
Remark 1.The neural network (1) is a system of differential equations with state-dependent deviating arguments and from (H1), one can see that deviating arguments in (1) may be delayed type, advanced type, or mixed type.
Our main purpose of this paper is to study the existence and global exponential stability of the equilibria of (1) by using the topological degree theory and the time scale calculus theory.The results of this paper are completely new and complementary to the previously known results.
The organization of this paper is as follows.In the next section, some notations, definitions, and lemmas are presented.Section 3 addresses the existence and uniqueness of equilibria of system (1) by using the method of topological degree theory.In Section 4, we give the criteria of global exponential stability of the equilibrium point of system (1).In Section 5, an example is also provided to illustrate the effectiveness of the main results in Sections 3 and 4.

Notations and Preliminaries
In this section, we will first recall some basic definitions and lemmas which will be useful for the proof of our main results.Definition 2 (see [33,34]).A time scale T is arbitrary nonempty closed subset of the real set R with the topology and ordering inherited from R. Definition 3 (see [33,34]).On any time scale T, we define the forward and backward jump operators by Definition 4 (see [33,34]).For a function  : T → R (the range R of  may be actually replaced by Banach space), the (delta) derivative is defined by if  is continuous at  and  is right-scattered.If  is not rightscattered, then the derivative is defined by provided this limit exists.
Definition 6 (see [33,34]).A function  : T  → R is called a delta-antiderivative of  : T → R provided  Δ =  holds for all  ∈ T  .In this case, we define the integral of  by and we have the following formula: Definition 7 (see [33,34]).A function  : T → R is called right-dense continuous (rd-continuous) provided it is continuous at right-dense points of T and the left-sided limit exists (finite) at left-dense point of T. The set of all right-dense continuous functions on T is defined by  rd =  rd (T) =  rd (T, R).If  is continuous at each right-dense point and each left-dense point, then  is said to be continuous function on T. We define (, R) = {() is continuous on }.
Lemma 8 (see [33,34]).If ,  ∈ T, ,  ∈ R and ,  ∈ (T, R), then one has Definition 9 (see [33,34]).A function  : If  is regressive function, then the generalized exponential function   is defined by with the cylinder transformation Let ,  : T → R be two regressive functions; we define Then, the generalized exponential function has the following properties.

Existence and Uniqueness of Equilibrium Point
In this section, we will discuss the existence and uniqueness of equilibria of the DCNN with impulses on time scales and give their proofs.

Theorem 14. Under assumptions (H1) and (H2), if the following condition is satisfied
. ., , then there is exactly one equilibrium point of model (1).
Remark 15.From Lemma 13, we can easily prove that (H) holds implying that the following condition is true: (H0) there exists a vector  = ( 1 ,  2 , . . .,   )  > 0 such that For convenience, we set  = diag( which implies that  is a nonsingular -matrix.So we know that   is a nonsingular -matrix.Hence, there exists a vector  = ( 1 ,  2 , . . .,   )  > 0 such that It follows that (H0) holds.Now, we prove our theorem.
From the homotopy invariance theorem, we obtain where  is the identity operator.By topological degree theory, we can easily know that system (11) Then, = 1, 2, . . ., .
From Theorem 17, we can immediately derive the following result.
Corollary 20.Suppose that system (49) satisfies condition (H2) and (H), and the following assumptions hold: Then, the equilibrium of system (49) is globally exponentially stable.
Remark 21.In [42], by utilizing the time scale calculus theory, topological degree theory, and Hölder's inequality on time scales, authors studied the existence and the global exponential stability of equilibrium point to a class of impulsive BAM neural networks with distributed delays on time scales.But, results obtained in [42] cannot be applied to (1).Also, for establishing the global exponential stability of equilibrium point to (1), our method used in this paper is totally different from that used in [42].

An Example
In this section, an example is given to show the effectiveness of the result obtained in the previous section.Because the condition (4.2) is not dependent on the impulses, we just need to check it with the nonimpulsive system.
We have that which imply that the assumption (H) of Theorem 14 holds.Thus, it follows from Theorems 14 and 17 that system (50) has a unique equilibrium point which is globally exponentially stable (see Figure 1).Since () ≡ 0 for  ∈ T = R and () ≡ 1 for  ∈ T = Z, from the discussion above one can easily see that for T = R or T = Z, (50) always has a unique equilibrium point which is globally exponentially stable.That is, the following continuous-time system have the same dynamical properties, where   ,   ,   , and   are the same as those in (50) (see Figures 2 and 3).

Conclusion
Using the topological degree theory and the time scale calculus theory, some sufficient conditions are obtained to ensure the existence and the global exponential stability of equilibria for DCNNs neural networks with impulses on time scales.This is the first time to apply the time scale calculus theory to unify the study of the stability of the equilibrium for DCNNs with impulses on time scales under the same framework.The results obtained in this paper possess highly important significance and are easily checked in practice.
In addition, the method in this paper may be applied to some other systems such as the BAM and Cohen-Grossberg systems with impulses and so on.

Figure 3 :
Figure 3: Transient responses of states  1 ,  2 in Example when T = Z.