Global Exponential Robust Stability of High-Order Hopfield Neural Networks with S-Type Distributed Time Delays

By employing differential inequality technique and Lyapunov functional method, some criteria of global exponential robust stability for the high-order neural networks with S-type distributed time delays are established, which are easy to be verified with a wider adaptive scope.


Introduction
Neural networks and their various generalizations have been successfully employed in many areas such as pattern recognition, cognitive modeling, adaptive control, and combinatorial optimization [1][2][3][4][5][6][7].Hopfield neural networks (HNNs), as some forms of recurrent artificial neural networks, have been widely studied in recent years [8][9][10][11][12].The earlier HNNs model proposed by Hopfield [13,14] was based on the theory of analog circuit consisting of capacitors, resistors, and amplifiers and can be formulated as a system of ordinary equations.Time delays are inevitable in the interactions of neurons in biological and artificial neural networks.The existence of delays is frequently a source of instability for neural networks [9,10,[15][16][17][18][19].
Over the past few decades, the stability of HNNs with time delays has attracted considerable attention in the literature [20][21][22][23].One of the most investigated problems in the study of HNNs is global exponential stability of the equilibrium point.An equilibrium point of HNNs is globally exponentially stable, if the domain of attraction of the equilibrium point is the whole space and the convergence is in real time.
It is worth noting that although the signal propagation is sometimes instantaneous and can be modeled with discrete delays, it may also be distributed during a certain time period so that the distributed delays should be incorporated in the model [24].Discrete delays and distributed delays attract the attention of many scholars and have been widely studied [17,19,25].To the best of our knowledge, the stability problem for system with both discrete and distributed delays has been a challenging issue, mainly due to the mathematical difficulties in dealing with discrete and distributed delays simultaneously.In 2002, Wang and Xu [26] presented a new neural network model with S-type distributed time delays and demonstrated that S-type distributed time delays include discrete or continuously distributed time delays, but it is not true in the opposite way.In the following years, S-type distributed time delayed neural network models have raised great interest [12,[27][28][29].
Compared with traditional Hopfield neural networks, the high-order Hopfield type neural networks (HOHNNs) [11,12,[30][31][32][33][34] have the advantages of stronger approximation properties, faster convergence rate, greater storage capacity, and higher fault tolerance.Therefore, it is of considerable interest to explore the theoretical foundations and practical applications of HOHNNs.
Motivated by the aforementioned discussion, we studied the problem of global exponential robust stability of HOHNNs with S-type distributed time delays.By employing differential inequality technique and a new Lyapunov functional method, some criteria for the global exponential
It is obvious that the solutions to (11) are the equilibrium points of system (1).
Let us define homotopic mapping as follows: where By homotopy invariance theorem (see [35]), topological degree theory (see [36]), ( 2 ), and the proof, which is similar to Theorem 1 in [28], we can conclude that ( 13) has at least one solution.
That is, system (1) has at least an equilibrium point.
Let us choose a positive constant  such that So which leads by contradiction to (18).Hence, (17) holds.That is, the solutions to system (1) are bounded.So the solutions to system (1) are of global existence.(1).From ( 2 ), we know that there exists constant   > 0, such that

Part 3: Global Exponential Stability of System
So, from ( 3 ), we can choose a constant  > 0 sufficiently small, such that Then, we will show that there exists Π > 0 such that       −   Therefore, it follows from Theorem 2 that the null solution to system (37) is globally exponentially robustly stable.

Conclusion
We have investigated the global exponential robust stability of high-order Hopfield neural networks with S-type distributed time delays, which is of theoretical as well as practical importance for the development of neural networks with time delays.The system (1) considered here is more general compared to the systems in literatures [10,12,26,31].By employing differential inequality technique and Lyapunov functional method, some criteria of global exponential robust stability for the high-order neural networks with S-type distributed time delays are established, which are easily verifiable and have a wider applicable range.The linear matrix inequality (LMI) approach is also widely used to establish the desired sufficient conditions for stability analysis of delayed neural networks [11,37].Wen et al. [17] have done some great work in control and filtering problems for neural systems.In future extension, we will do some research in stability of highorder Hopfield neural networks with S-type distributed time delays using LMI method.