On the Global Practical Exponential Stability of h-Manifolds for Impulsive Reaction–Diffusion Cohen–Grossberg Neural Networks with Time-Varying Delays

Recently, the study of the qualitative behavior of Cohen–Grossberg neural networks has attracted wide attention due to implementing of its ample applications. In fact, this important class of neural networks introduced in 1983 [1] has been widely used as mathematical models in engineering, neuroscience, biology and medicine. As such, the stability, periodicity, almost periodicity and other qualitative properties of Cohen–Grossberg neural networks have been intensively investigated [2,3,4]. Note that this class of neural network systems is quite general and includes several well-known neural networks, such as Hopfield neural networks, Lotka–Volterra systems, Kolmogorov systems and cellular neural networks. It is also worth mentioning that some entropy phenomena are related to efficient neural network training, and entropy is applied to measure the complexity in neural network architectures [5,6].

The effects of time delays on the performance of neural network models is very well known; hence, great progress in the study of delayed Cohen–Grossberg neural network models has been made. Models with constant delays, as well as more general cases involving time-varying delays, have been investigated [7,8,9,10].

The fact that reaction–diffusion effects can disrupt the high performance of neural networks has greatly enhanced the study of the dynamic behavior of neural networks with reaction–diffusion conditions [11,12,13]. The type of Cohen–Grossberg neural networks with reaction–diffusion terms has also been intensively investigated [14,15,16,17]. Indeed, reaction–diffusion neural networks constitute a suitable framework to study dynamics in neural network processes that not only dependent on the evolution time but also dependent on space.

Due to the development of impulsive control strategies, impulsive neural networks have drawn more attention recently, including impulsive Cohen–Grossberg neural networks [18,19,20,21,22]. Such models include impulsive conditions that describe the behavior of states at some discrete instances. In the study of their properties, the theories and apparatus of impulsive differential equations and impulsive control dynamical systems are used [23,24,25,26]. In fact, through suitable short-term perturbations, researchers can control some qualitative properties of neural network models, such as stability, stabilization, synchronization, periodicity, almost periodicity and some others. Hence, the investigations on impulsive Cohen–Grossberg reaction–diffusion neural networks have also increased rapidly. For example, in [27], the existence, uniqueness and global exponential stability of the equilibrium state for an impulsive Cohen–Grossberg neural network with distributed delays and reaction–diffusion terms have been investigated. The uncertain case was also considered, and criteria for global exponential stability have been established. The paper by [28] studies the existence, uniqueness and global exponential stability of an equilibrium point for impulsive reaction–diffusion Cohen–Grossberg neural networks with time-varying delays. The authors in [29] considered impulsive reaction–diffusion Cohen–Grossberg neural networks with time-varying delays and Neumann boundary conditions. They established efficient sufficient conditions to guarantee the existence, uniqueness and global exponential stability of the equilibrium state. The global exponential stability of the equilibrium point of an impulsive reaction–diffusion delayed Cohen–Grossberg neural network has been considered in [30]. In [31], the impulsive control method is applied to a class of delayed Cohen–Grossberg neural networks with reaction–diffusion terms and Dirichlet boundary conditions. The authors proposed sufficient conditions to ensure the global exponential stability of the equilibrium point. The global exponential stability of the equilibrium point for a class of impulsive Cohen–Grossberg neural networks with time-varying delays and reaction–diffusion terms is an object of the investigation in [32]. The extensive work carried out on this class of neural networks shows their importance for theoretical avenues and practical applications.

It is clear from the cited results that the global exponential stability is an essential qualitative behavior for the class of impulsive Cohen–Grossberg neural networks with time-varying delays and reaction–diffusion terms. However, all above publications considered only single neuronal states which, in most of the cases, are equilibrium points. There are many cases when the behavior of a set of states is important. To this end, the concepts of the stability of sets [33] and manifolds [34,35] for different nonlinear problems have been introduced. Both concepts generalize the classical notion of stability of a separate state trajectory. Due to their importance in many applied models that have more than one equilibrium of interest, these concepts have been applied to some neural networks [36,37,38], including impulsive reaction–diffusion delayed Cohen–Grossberg neural network models [39,40]. In fact, there are many experiments that indicate the heterogeneity of neuronal populations growing on sets or low-dimensional manifolds [41].

However, in many practical problems, including neural network models, the stability of states depends on some constrains and restrictions. These constrains can be represented by an appropriate function h, which defines a specific manifold of solutions. Hence, in such cases, the concept of stability of h-manifolds is more appropriate, instead of considering sets or low-dimensional manifolds of solutions of a very general nature [42,43,44]. Recently, a new notion has been applied to some Cohen–Grossberg neural networks [45].

However, the important notion of the stability of h-manifolds is not considered for the class of impulsive reaction–diffusion Cohen–Grossberg delayed neural networks. Our aim is to contribute to the development of this direction. Also, we will extend the concept by considering the practical exponential stability case. In fact, the notions of practical stability are very applicable when a system may be unstable in the classical Lyapunov sense or h-manifolds sense, but its performance can be allowable from an applied point of view. Due to its importance, the concept has been applied to numerous systems [46,47,48], including neural network models [49,50,51,52]. Note that the notion of practical stability is also very useful in engineering design problems [53,54,55,56]. In addition, the exponential stability guarantees there to be a fast convergence rate for neural network models. Our goal is to study the notion of h-manifold practical global exponential stability in comparison with the class of impulsive reaction–diffusion Cohen–Grossberg delayed neural network models and obtain efficient criteria. These results complement some earlier publications of the h-manifold case [39,40] using a newly constructed Lyapunov function.

We summarize the main contributions of our paper as follows:

1. The hybrid practical stability concept with respect to h-manifolds is introduced to an impulsively generalized reaction–diffusion Cohen–Grossberg neural network model with time-varying delays. The introduced notion extends the concepts of impractical stability of a single state applied in [27,28,29,30,31,32] and is also well suited in applied models with several equilibrium points whose state dynamics depend on some constraints modeled by h-manifolds. The consideration of the notion of practical stability brings another layer of improvement as it is applicable in practical cases where ideal mathematically stable behavior is impossible.

2. The impulsive generalization of the developed model allows for the application of impulsive control strategies. Also, in contrast to the impulsive control techniques applied to similar neural network models [27,28,29,31,32], our results are not restrictive regarding the length of impulsive intervals. The upper bounds of the impulsive interval which exists in [27,28,29,31,32] are not constrained, and restrictions on the distance between impulsive moments are removed.

3. By constructing a new Lyapunov function, efficient sufficient conditions are establishes, which guarantee the practical global exponential stability with respect to a h-manifold. The proposed results contribute to the development of the practical stability theory for different classes of differential equations. Also, the obtained criteria are less restrictive than the stability criteria obtained in [27,28,29,30,31,32,39,40] with respect to the variable domain and diffusion coefficients.

4. A robust practical stability analysis is conducted, and criteria for robust global practical exponential stability of the introduced model under some uncertainties are proved.

5. The obtained practical global exponential stability with respect to a h-manifold are applied to the BAM case of the model under consideration. Thus, our results extend the results in [45] to the reaction–diffusion case.

The rest of this paper is organized according to the following scheme: In Section 2, the impulsive reaction–diffusion Cohen–Grossberg neural network model with time-varying delays is introduced. Several main definitions and lemmas on the Lyapunov direct method for impulsive reaction–diffusion delayed systems are presented. We establish our main global practical exponential stability criteria in Section 3. Due to the newly constructed Lyapunov function, the derived criteria are more relaxed with respect to the domain of the space variable and diffusion coefficients. Also, the robust practical global stability of the uncertain system is investigated. Indeed, numerous neural network models exist under the conditions of structural uncertainty due to modeling errors, measurement inaccuracy and mutations in evolutionary processes [57,58]. In addition, we apply these in Section 3 to an impulsive reaction–diffusion Cohen–Grossberg BAM neural network with time-varying delays. Some examples are presented through this Section. Finally, the conclusions are outlined in Section 4.

Let ${\mathbb{R}}^n$ denote the n-dimensional Euclidean space and ${\mathbb{R}}_{+}=[0,\infty )$.

In this paper, we consider the following impulsive reaction–diffusion Cohen–Grossberg neural network model with reaction–diffusion terms and time-varying delays: where $i=1,2,\dots ,l$, $l\ge 2,k=1,2,\dots$, $t>0$, $z={(z_1,z_2,\dots ,z_n)}^T\in \mathrm{\Omega }$, $\mathrm{\Omega }$ is an open and bounded set in ${\mathbb{R}}^n$ with a smooth boundary $\partial \mathrm{\Omega }$, and the measure is expressed by $\mathrm{mes}\mathrm{\Omega }>0$,

In the above impulsive Cohen–Grossberg-type neural network model, ${\theta }_i(t,z)$ denotes the state of the i-th neural unit at time t ($t>0$) and space z ($z\in \mathrm{\Omega }$), l corresponds to the number of neurons in the neural network, the smooth functions $D_{iq}=D_{iq}(t,z)\ge 0$ are the diffusion coefficients along the i-th neuronal node, $a_i\left({\theta }_i(t,z)\right)$ denotes a continuous amplification function, $b_i\left({\theta }_i(t,z)\right)$ is a continuous on its domain appropriately behaved function, $v_{ij}\left(t\right)$ and $w_{ij}\left(t\right)$ are, respectively, the connection weight and time-varying delay connection weight matrices of the j-th neuron on the i-th neuron and are assumed to be continuous functions on ${\mathbb{R}}_{+}$, $F_j\left({\theta }_j(t,z)\right)$ and $G_j\left(u_j(t-{\eta }_j\left(t\right),z)\right)$ are the activation functions of the j-th neuron, ${\eta }_j\left(t\right)$ is the transmission time-varying delay and ${\eta }_j\in C[\mathbb{R},{\mathbb{R}}_{+}],t>{\eta }_j,j=1,\dots ,l$, ${\displaystyle 0\le {\eta }_j\left(t\right)\le \eta ,\frac{d{\eta }_j\left(t\right)}{dt}<{\delta }_j}$$(\eta >0,0<{\delta }_j<1)$, and $I_i(t,z)$ is the external input of the i-th neural unit. The points $\left\{t_k\right\},k=1,2,\dots$ denote the moments impulsive perturbations at which abrupt changes of the state ${\theta }_i(t,z)$ from positions ${\theta }_i(t_k^{-},z)={\theta }_i(t_k,z)$ into the positions ${\theta }_i(t_k^{+},z)$ are observed, and $J_{ik}\left({\theta }_i(t,z)\right)$ are impulsive functions that measure the impulsive control effects on the neuronal state ${\theta }_i(t,z)$ at the instants $t_k$ and space z. Also, we denote by $\mathrm{\Delta }{\theta }_i(t_k,z)={\theta }_i(t_k^{+},z)-{\theta }_i(t_k,z)$, $z\in \mathrm{\Omega }$, $i=1,2,\dots ,l$, $k=1,2,\dots$.

We assume that the sequence of discrete impulsive times $\left\{t_k\right\},k=1,2,\dots$ is such that

We will consider Model (1) under the following boundary and initial conditions: where ${\phi }_0$ is the initial function, ${\phi }_0={({\phi }_{01},{\phi }_{02},\dots ,{\phi }_{0m})}^T$; for any $i=1,2,\dots ,l$, the functions ${\phi }_{0i}(\xi ,z)$ are real-valued, defined on $[-\eta ,0]\times \mathrm{\Omega }$, piecewise continuous with respect to $\xi$ with possibly finite number of discontinuity points of the first kind $\xi \in [-\eta ,0]$ such that ${\phi }_{0i}({\xi }^{+},z)$ and ${\phi }_{0i}({\xi }^{-},z)$ exist, and ${\phi }_{0i}({\xi }^{-},z)={\phi }_{0i}(\xi ,z)$, $z\in \mathrm{\Omega }$, $i=1,2,\dots ,l$. The class of all such functions will be denoted by ${\mathcal{PC}}_{\eta }^{\mathrm{\Omega }}$. By ${\mathcal{PCB}}_{\eta }^{\mathrm{\Omega }}$, we denote the class of all functions $\phi \in {\mathcal{PC}}_{\eta }^{\mathrm{\Omega }}$ that are bounded.

The solution of the initial boundary value problem (IBVP) (1)–(3) will be denoted by

We will also denote by

In order to apply the h-manifold exponential stability concept, we consider a continuous function $h=h(t,\theta ),h:[-\eta ,\infty )\times {\mathbb{R}}^l\to {\mathbb{R}}^p$, $l\le p$, and define the following sets related to h: where ${\displaystyle {\left|\right|h(\xi ,\phi )\left|\right|}_{\eta }=\underset{-\eta \le \xi \le 0}{sup}{\left|\right|h(\xi ,\phi (\xi ,·))\left|\right|}_1}$.

We also assume the following assumptions to ensure the existence and uniqueness of the nodes and our findings [27,28,29,30,31,32,39,40]:

The concept of h-manifold practical global exponential stability [42,43,44,45] will be adopted to the impulsive reaction–diffusion Cohen–Grossberg model (1) as follows:

In order to establish the practical global exponential stability results, we will apply the method of Lyapunov piecewise continuous functions [25].

Define ${\mathcal{W}}_k$ as

Let $t\in {\mathbb{R}}_{+}$, $t\ne t_k$, $k=1,2,\dots$ and $\overline{\phi }\in {\mathcal{PC}}_{\eta }^{\mathrm{\Omega }}$. We will use the following derivative of a function $\mathcal{L}\in {\mathcal{L}}_0$ with respect to a system (1) given by

The proof of the following lemma is similar to the proof of Lemma 2.1 in [51].