Finite-time and fixed-time stabilization of inertial memristive Cohen-Grossberg neural networks via non-reduced order method

1 School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China 2 School of Mathematics, Southeast University, Nanjing 210096, China 3 Yonsei Frontier Lab, Yonsei University, Seoul 03722, South Korea 4 The Computer Science and Engineering Department, Yunnan University, Qunming 210096, China 5 School of Mathematics and Physics, Yancheng Institute of Technology, Yancheng 224051, China 6 School of Arts and Sciences, Shaanxi University of Science and Technology, Xi’an, China


Introduction
Recently, neural networks (NNs) has received considerable attention from various fields of engineering and science, including secure communication, brain science and pattern recognition. As we know, Cohen-Grossberg NNs is a well-known network model, many famous networks like Hopfield NNs and Cellular NNs can be regarded as its special case. Since it was first proposed in 1983 [1], Cohen-Grossberg NNs has received increasing attention from different areas, such as neurobiology, pattern recognition and parallel computation. To better understand the application of Cohen-Grossberg NNs, the dynamical behavior of this network need to be further studied [2][3][4][5].
The second-order term in NNs is called inertial term. Physically, it represents the inductance in a neural circuit system [6,7]. It is known that, the inertial NNs have strong backgrounds from both biology and engineering [8,9]. The inertial term can arouse more complicated dynamical behavior to the systems, like chaos and bifurcations. Recently, the dynamical behavior of inertial NNs has become a hot topic both in theory and applications.
To cope with the effect caused by inertial (second-order) term, most of the previous literature use the variable translation to reduce the second-order system to first-order system. However, this approach may increase the system's dimension and add the complexity of analysis process. Due to this great shortage, it is necessary to propose a non-reduced order method to deal with the second-order system directly and some results have come out recently [10][11][12][13]. In [10], the asymptotic stabilization of inertial memristive NNs with mixed time delays is explored by non-reduced order method. In [12], by proposing a direct analysis method, the asymptotic stabilization of Cohen-Grossberg inertial NNs is investigated. In [13], a novel criteria is derived for global stabilization of delayed inertial MNNs, both feedback control and adaptive control are considered. However, these investigations only focus on the asymptotic stabilization, the corresponding results on finite-time or fixed-time stabilization have not been reported yet.
Memristor is considered as the fourth fundamental circuit element, which can memorize the passed amount of electronics [14,15]. Due to the function of memory, the memristor can be used to emulate the neural synapses with better performance than normal resistors [16][17][18][19][20]. Thus, by introducing the memristor into NNs, the memristive NNs (MNNs) is formulated. Recently, the research of dynamical behavior of MNNs has attracted various attention [21][22][23][24][25][26]. As we concern, the combination of inertial term and memristive connections can widen the application scope of our model.
In recent decades, a great many literatures have been published on asymptotic and exponential stabilization [24][25][26][27][28]. However, the convergence time of asymptotic stability tend to infinity, owing to the limited life span of engineering machines, the asymptotic stability is invalid for some applications. Hence, to overcome this shortage, the issue of finite-time stability is introduced, in which the system state is required to converge to zero in finite time. Also, the finite-time control shows strong robustness and fast convergence rate [29][30][31][32][33][34]. However, the settling time of finite-time control is dependent on initial information of system, which may be difficult to obtain some times. To overcome such shortage, the concept of fixed-time control is proposed later [36], in which the settling time can be computed without the knowledge of initial condition. Hence, fixed-time control shows great advantage among various areas [37][38][39][40][41][42]. As we considered, there are still no results on finite-time (fixed-time) stabilization of inertial Cohen-Grossberg NNs by non-reduced order approach. Thus, our research is novel and has a good application prospect.
Owing to the finite switching rate of amplifier, time delay is an unavoidable phenomenon in signal transmission and information processing. Due to the parallel pathways in NNs, there exists another type of time delay named distributed time delay. It is known that, mixed time delays can cause oscillation and instability to systems [25,26,31]. Thus, the stabilization of NNs with mixed time delays is a meaningful topic and deserves further discussion.
The main contribution of this work is as follows.
1) Different from the reduced-order method in [37,38], this paper uses a non-reduced order method to cope with the second-order Cohen-Grossberg neural networks. Novel criteria on finite-time (fixedtime) stabilization of inertial Cohen-Grossberg NNs are derived.
2) Two types of time delays are considered in this work: the time-varying delays and mixed time delays. To cope with such time delays, different types of feedback controllers and Lyapunov functionals are designed to solve the problem. Moreover, the memristive connections are taken into the network model, which gives a better application prospect.
3) The non-reduced order approach has been used to solve the asymptotic stabilization of inertial NNs in many previous literature. However, the finite-time or fixed-time stabilization problem has not been investigated by this approach yet. In this work, we extend the asymptotic stabilization to fixedtime stabilization by designing novel control schemes.
The contents of our work is as follows. In part 2, the preliminaries are introduced. The main results are given in part 3. Simulations are given to demonstrate our theorem in part 4. At last, the conclusion is achieved in part 5.

Preliminaries
The model of inertial memristive Cohen-Grossberg NNs is as follows: where i = 1, · · · , n; x i (t) ∈ R is the state value of ith neuron. f j (x j (·)), g j (x j (·)) ∈ R denote the activation function satisfying that f j (0) = g j (0) = 0. c i (x i (t)) is the amplification function. k i (x i (t)) is the behaved function of the ith node satisfying k i (0) = 0. a i j (x i (t)) and b i j (x i (t)) represent the memristor connection weights. u i (t) is the control input to achieve the stabilization goal. The time delay τ(t) satisfyτ(t) ≤ ρ < 1 and 0 ≤ τ(t) < τ, ρ, τ are positive numbers. The initial condition of system (2.1) is taken as According to the Kirchhoff's current law, the memristive connections are defined as: where the switching jump T i > 0,â i j ,ǎ i j ,b i j ,b i j are known constants.
Remark 1. From the characteristic of memristors, the connection weights a i j (x i (t)), b i j (x i (t)) are statedependent switched according to the state x i (t). Obviously, the righthand of Eq (2.1) is discontinuous. Hence, the solution is in the sense of Filippov.
x), x ∈ R n , with discontinuous right-hand sides, a set-valued map is defined as where co[E] is the closure of the convex hull of set E, B(x, δ) = {y : y − x ≤ δ}, and µ(N) is the Lebesgue measure of set N. A solution in Filippov's sense of the Cauchy problem for this system with initial condition x(0) = x 0 is an absolutely continuous function

Definition 3. [29]
If there exists positive constant T * that depends (or not depend) on initial condition of system (2.1), such that Lemma 1. [29] Assume that the function V(t) is continuous and nonnegative, if there is positive constant q 1 such thatV for some 0 < α < 1. Then, V(t) = 0 for all t ≥ T 0 , and the settling time is [36] Assume that function V(t) is continuous and nonnegative, if there are positive constants q 1 and q 2 such thatV for some 0 < α < 1, β > 1. Then, V(t) = 0 for all t ≥ T max , and the settling time is

Main results
Assumption 1. Suppose there exists positive constants l i , h i such that for any x, y ∈ R Suppose there exists constants 0 < c i ≤c i and 0 < k i ≤k i such that for any x ∈ R, To achieve the finite-time(or fixed-time) stabilization of system (2.1), the following feedback controller is designed.
Remark 3. The finite-time or fixed-time stabilization of second-order system has been investigated by reduced-order method in some previous papers [32][33][34]37,38]. For example, in [32][33][34], the finite-time synchronization of delayed inertial neural networks is investigated by integral inequality method and maximum value approach. However, the above results are achieved by reduced order method, there are still no results on finite-time (fixed-time) stabilization by non-reduced order method yet. Thus, the research gap is filled in this work.
Next, we consider the inertial memristive Cohen-Grossberg NNs with mixed time delays: the memristive weights are defined as: where the switching jump T i > 0,â i j ,ǎ i j ,b i j ,b i j ,d i j ,ď i j are known constants. The initial condition of system (3.15) is taken as Assumption 3. Suppose there exists positive constants σ,σ such that To achieve our control goal, the following feedback controller is proposed: Then, (1) When 0 < α < 1 and 0 < β < 1, the system (2.1) can reach finite-time stabilization under controller (3.17), the settling time T 1 is estimated as (2) When 0 < α < 1 and β > 1, the system (2.1) can reach fixed-time stabilization under controller (3.17), the settling time T 2 is estimated as Proof. Constructing the Lyapunov functional |x j (s)|ds Computing the derivative along trajectories (3.15), we havė sgn(ẋ i (t))ẍ i (t) It is obvious that Then, we haveV Case 1: When 0 < α < 1 and 0 < β < 1, Case 2: When 0 < α < 1 and β > 1,V Following the same discussion in Theorem 1, the conclusion can be derived.
Remark 4. In this work, two types of time delays are considered. It is known that, mixed time delays may cause oscillation and instability of NNs. This may arouse more complex dynamical behavior in this network model. Moreover, the memristive connections is taken into our network, which gives a better application prospect. Compared with the previous literature [12], the model is more general and the results can be applied to wider application.

Numerical examples
To verify the validity of the theoretical results, some simulations are provided in this part. First, we give the numerical example of Theorem 1.

Conclusion
By designing novel control scheme and constructing Lyapunov functional, this work proposed a non-reduced order method to deal with the finite-time (fixed-time) stabilization problem of secondorder memristive Cohen-Grossberg neural networks. Compared with the traditional reduced-order approach, the method adopted in this paper can simplify the analysis process and solve the control problem of second-order system directly. Two kinds of time delays: time-varying delays and mixed time delays are taken into consideration. Then, two criteria are derived to ensure the finite-time (fixedtime) stabilization of inertial Cohen-Grossberg NNs. Lastly, simulations are presented to verify the validity of our results.
In our future study, we will try to extend our results to complex-valued and quaternion-valued field.