Input-to-state stability of delayed reaction-diffusion neural networks with multiple impulses

Abstract: This paper concerns the input-to-state stability problem of delayed reaction-diffusion neural networks with multiple impulses. After reformulating the neural-network model in term of an abstract impulsive functional differential equation, the criteria of input-to-state stability are established by the direct estimate of mild solution and an integral inequality with infinite distributed delay. It shows that the input-to-state stability of the continuous dynamics can be retained under certain multiple impulsive disturbance and the unstable continuous dynamics can be stabilised by the multiple impulsive control, if the intervals between the multiple impulses are bounded. The numerical simulation of two examples is given to show the effectiveness of theoretical results.


Introduction
Over the past few decades, neural networks have been widely applied in the field of smart grid, secure communication, machine learning, among many others [1][2][3][4][5][6]. Such applications heavily depend on their dynamical behaviors such as stability, synchronization, periodicity, passivity, to name just a few. Among them, the input-to-state stability (ISS), which is originally developed by E. D. Sontag in the late 1980s [7] and measures the influence of external input to stability, is of comparable significance in the dynamical analysis of nonlinear systems including neural networks. Extensions of ISS have also been proposed for various kinds of nonlinear systems and ignited plenty of valuable works [8][9][10][11][12][13][14][15].
In the ISS analysis of neural networks, the time delays are usually involved in neural networks, since the finite speed of signal transmission and amplifier switching inevitably causes the hysteresis of neural networks. There are usually two kinds of time delays considered in neural networks: the time-varying delays and the infinite distributed delays [16][17][18][19][20][21]. For instance, the exponential ISS of recurrent neural networks with multiple time-varying delays was studied by the Lyapunov method in [18], whose results were further extended to the stochastic case with both time-varying delays and infinite distributed delays [19]. In [20], the pth moment exponential ISS of stochastic recurrent neural networks with time-varying delay was investigated by the vector Lyapunov function to reduce the conservatism caused by the scalar Lyapunov function used in previous literature.
Recently, the reaction-diffusion is introduced in neural-network models because the electrons sometimes have the diffusive shift trajectory in nonuniform electromagnetic field. Different from the delayed neural networks (DNNs) without reaction-diffusion, the delayed reaction-diffusion neural networks (DRDNNs) are described by PDEs because their dynamics depends on both the spatial derivative and the time derivative. Therefore, the dynamical analysis of DRDNNs has attracted the interest of plenty of researchers and the extension of ISS has also been carried out from DNNs to DRDNNs [22][23][24][25][26][27][28][29][30]. In addition, the ISS is used not only for DRDNNs, but is in fact a key concept in robust control of infinite-dimensional systems, with the expectation that ISS will enable similar advances in the control theory of infinite-dimensional systems as it has for finite-dimensional systems. See [31,32] and the references therein.
On the other hand, the impulsive effects may occur in the hardware implementation of neural networks since the nodes may be shocked by defective connections, sudden attacks, and abrupt changes [24], so the impulsive delayed neural networks (IDNNs) have been massively studied [33][34][35][36][37] where the impulses are classified into two kinds: the stabilising impulses which force the trajectory of neural network into desirable pattern, and the destabilising impulses which bring fluctuation to neural networks. However, the multiple impulses containing both stabilising impulses and destabilising impulses are more elegant to model the instantaneous shocks of the neural networks. Even though some ISS properties of impulsive nonlinear system with multiple impulses are unveiled in recent literature [38,39], the ISS of DNNs with multiple impulses, is rarely investigated, not to mention the ISS of DRDNNs with multiple impulses, because the multiple impulses are difficult to handle with the infinite distributed delays included in neural networks. Motivated by the above discussion, the aim of this paper is to establish the ISS criteria of DRDNNs with multiple impulses. The contributions lie in the following aspects: (1) The multiple impulses, infinite distributed delays, and reaction-diffusion are considered simultaneously in neural-network model; (2) The ISS conditions of the DRDNNs with multiple impulses are obtained by the direct estimate of mild solution and an integral inequality; (3) It show that the ISS property of continuous dynamics can be retained under certain multiple impulsive disturbance and the unstable continuous dynamics can be stabilised by multiple impulsive control, if the intervals between the multiple impulses are bounded.
The remainder of this paper is organized as follows. Section 2 introduces the neural-network model and preliminaries. Section 3 gives the sufficient conditions for ISS of the DRDNNs with multiple impulses. Section 4 presents the numerical simulation of two examples. Finally, the conclusions are drawn in Section 5.

Model description and preliminaries
In this paper, unless otherwise specified, the following notations are used.n = {1, 2, · · · , n} and N = {1, 2, 3, · · · }. For a, b ∈ R, a ∧ b denotes the minimum of a and b. L (L 2 (O)) n and L 2 (O) is a Hilbert space with inner product z 1 , z 2 = O z 1 (x)z 2 (x)dx and norm z 2 = z, z , where O = {x|x = (x 1 , · · · , x w ) T , |x j | ≤ ρ j , ρ j ∈ R + , j ∈w}. Here, we also use the same symbol · to denote the usual norm of linear bounded operators from L to L.
, · · · } be the sequences of impulse times satisfying 0 = t 0 < t 1 < t 2 < · · · < t k < · · · to prevent the occurrence of accumulation points. Forή > 0 andὴ > 0, F + (ὴ), F − (ή), and F (ή,ὴ) denote the sets of admissible sequences of impulse times in F 0 satisfying 0 < t k − t k−1 ≤ὴ,ή ≤ t k − t k−1 < ∞, andή ≤ t k − t k−1 ≤ὴ for any k ∈ N, respectively. PC(R, J) represents the space of functions f : R → J which are continuous on (t k−1 , t k ) for k ∈ N and f (t + ) = f (t) for ∀t ∈ R where R = R or R = R + and J is an Euclidean space or a Hilbert space. U PC(R + , H ∩ L). PC represents the space of functions f : (−∞, 0] → L ∩ H which have at most a finite number of jump discontinuities on (−∞, 0] and f (t x) is said to be piecewise continuous if z(t, x) is piecewise continuous for all x ∈ O. K represents the class of continuous strictly increasing function κ : R + → R + with κ(0) = 0. K ∞ is the subset of K functions that are unbounded. A function β is said to belong to the class of KL, if β(·, t) is of class K for each fixed t > 0 and β(s, t) decreases to 0 as t → +∞ for each fixed s ≥ 0.
Consider the following DRDNNs with multiple impulses is the state variable of the ith neuron at time t and space x, d i represents the positive transmission diffusion coefficient of the ith neuron, w j=1 represents the reactiondiffusion term, a i > 0 stands for the recovery rate, b i j > 0, p i j > 0, and q i j > 0 are the connection weight strengths of the jth neuron on the ith neuron,f j stands for the activation function,û i is the external input, i, j ∈n. The delay kernel k : [0, +∞) → R + is a nonnegative continuous function satisfying that there exists a positive constant λ * such that k(s) ≤ e −λ * s for s ≥ 0. Then, the neuralnetwork model (2.1) can be rewritten in terms of the following vector form Dirichlet boundary condition and initial condition, associated with (2.1) or (2.2), are given bŷ As standard hypotheses, we assume that (H1) there exist positive constants l i such that, for ∀ẑ 1 ,ẑ 2 ∈ R, i ∈n, (H2) there exist constants N ∈ N and σ k > 0, k ∈ N such that σ k+N = σ k and |1 + c k | ≤ σ k .
In this paper, we always assume that (H1) and (H2) are satisfied. The sets of stabilising strengths and destabilising strengths are denoted by , respectively. Define a linear operator D from H to L by Dẑ = D∆ẑ − Aẑ, then D is an infinitesimal generator of a strongly continuous C 0 -semigroup S (t) [40]. Furthermore, the neural networks (2.2)-(2.4) can be reformulated in terms of the following abstract impulsive functional differential equation is said to be a mild solution of (2.5), if z(t) satisfies the following equation The DRDNNs with multiple impulses (2.5) are called uniformly input-to-state stable (UISS) over a given set F of admissible sequences of impulse times if it is input-to-state stable for every sequence in F with β and γ independent of the choice of the sequence from the class F .

Lemma 1 ( [23]).
Let O be a cube |x j | ≤ ρ j ( j ∈w) and let h(x) be a real-valued function belonging to Lemma 2. Assume that (H1) holds. Then, the mild solution of (2.5) can be represented by Proof. The proof is analogous to the proof of Lemma 2.2 in [43] so as to be omitted. Lemma 3. Consider the following abstract Cauchy problem where ψ ∈ H. Then, the strongly continuous semigroup S (t) generated by D is contractive and satisfies Recalling that the solution of (2.9) is z(t) = S (t)ψ. Combining the Gaussian theorem, the homogeneous Dirichlet boundary condition, and Lemma 1, we obtain Therefore, z(t) 2 ≤ e −2ϑt ψ 2 for all ψ ∈ H. By the density of H in L [Theorem 1.2, 42], the result holds for all ψ ∈ L so as to complete the proof.
where c > 0 and t ≥ t 1 .

Input-to-state stability of DRDNNs with multiple impulses
In this section, the ISS of the DRDNNs with multiple impulses will be investigated by the direct estimate of the mild solution. First, let us consider the following integral inequality with infinite distributed delay:
Remark 2. In [28], the ISS property of stochastic delayed neural networks are investigated to show that the ISS of continuous dynamics can be retained under certain destabilising impulses. Then, the ISS criteria of DRDNNs with impulses were established by an impulsive delay inequality in [27], where two scenarios are considered: stabilising continuous dynamics with destabilising impulses and destabilising continuous dynamics with stabilising impulses. One can notice that these results in [27,28] focused on the single impulse effect (stabilising or destabilising impulses), and ignored the hybrid effect of multiple impulses. In comparison, the results established here indicate that the ISS property of continuous dynamics can be retained under certain multiple impulsive disturbance and the unstable continuous dynamics can be stabilised by multiple impulsive control, if the intervals between the multiple impulses are bounded.
Remark 3. In most of the existing works on ISS of neural networks [20,27,28], the ISS criteria are usually established by the Lyapunov method and extended Halanay-type inequalities. In comparison, the ISS criteria in this paper are established by direct estimate of mild solution and an integral inequality to handle the multiple impulses and infinite distributed delays.
If the multiple impulses degenerate into single impulses, that is, stabilising impulses or destabilising impulses, we have the following corollaries from Theorem 1.
Remark 4. The corollaries with stabilising or destabilising impulses accord with the results in [27,43]. But the infinite distributed delays are additionally included in neural-network model, so our results are more general.
Remark 5. Because of the multiple impulses and the infinite distributed delays, the existing results in [27,28,43] are invalid for these two numerical examples.

Conclusions
This work addresses the ISS issues of DRDNNs with multiple impulses after reformulating the neural-network model in term of an abstract impulsive functional differential equation. The ISS property is studied by the direct estimate of mild solution and an integral inequality with infinite distributed delay. The obtained results show that the ISS property can be ensured if the intervals between the multiple impulses are bounded. Note that the impulsive sequences considered here have a fixed dwell-time. A more general class of impulsive sequences satisfying the average dwell-time condition is considered in ISS literature [8,9]. Thus, the further work will focus on ISS analysis of impulsive DRDNNs under the average dwell-time condition.