Passivity analysis for Markovian jumping neutral type neural networks with leakage and mode-dependent delay

Natarajan Mala; Arumugam Vinodkumar; Jehad Alzabut; Natarajan Mala; Arumugam Vinodkumar; Jehad Alzabut

doi:10.3934/biophy.2023012

AIMS Biophysics

2023, Volume 10, Issue 2: 184-204. doi: 10.3934/biophy.2023012

Previous Article Next Article

Research article Special Issues

Passivity analysis for Markovian jumping neutral type neural networks with leakage and mode-dependent delay

1.
Department of Mathematics, Kovai Kalaimagal College of Arts and Science, Coimbatore-641 109, Tamil Nadu, India
2.
Department of Mathematics, Amrita School of Physical Sciences, Coimbatore, Amrita Vishwa Vidyapeetham, India
3.
Department of Mathematics and Sciences, Prince Sultan University, 11586 Riyadh, Saudi Arabia
4.
Department of Industrial Engineering, OSTİM Technical University, Ankara 06374, Turkey

Received: 09 January 2023 Revised: 05 April 2023 Accepted: 16 April 2023 Published: 28 April 2023

In this study, we discuss the passivity analysis for Markovian jumping Neural Networks of neural-type. The results are demonstrated using phases of linear matrix inequalities as well as an improved Lyapunov-Krasovskii functional (LKF) of the triple integral terms and quadruple integrals. The information of the mode-dependent of all delays have been taken into account in the constructed Lyapunov–Krasovskii functional and novel stability criterion is derived. The value of selecting as many Lyapunov matrices that are mode-dependent as possible is demonstrated. The effectiveness and decreased conservatism of the aforementioned theoretical results are eventually demonstrated by a numerical example.

Keywords:

Citation: Natarajan Mala, Arumugam Vinodkumar, Jehad Alzabut. Passivity analysis for Markovian jumping neutral type neural networks with leakage and mode-dependent delay[J]. AIMS Biophysics, 2023, 10(2): 184-204. doi: 10.3934/biophy.2023012

Related Papers:

[1]	Muhammad Asif Zahoor Raja, Adeeba Haider, Kottakkaran Sooppy Nisar, Muhammad Shoaib . Intelligent computing knacks for infected media and time delay impacts on dynamical behaviors and control measures of rumor-spreading model. AIMS Biophysics, 2024, 11(1): 1-17. doi: 10.3934/biophy.2024001
[2]	Julia C. Arciero, Paola Causin, Francesca Malgaroli . Mathematical methods for modeling the microcirculation. AIMS Biophysics, 2017, 4(3): 362-399. doi: 10.3934/biophy.2017.3.362
[3]	Mario Lefebvre . A Wiener process with jumps to model the logarithm of new epidemic cases. AIMS Biophysics, 2022, 9(3): 271-281. doi: 10.3934/biophy.2022023
[4]	Ramesh Rijal, Hari Prasad Lamichhane, Kiran Pudasainee . Molecular structure, homo-lumo analysis and vibrational spectroscopy of the cancer healing pro-drug temozolomide based on dft calculations. AIMS Biophysics, 2022, 9(3): 208-220. doi: 10.3934/biophy.2022018
[5]	Dafina Xhako, Niko Hyka, Elda Spahiu, Suela Hoxhaj . Medical image analysis using deep learning algorithms (DLA). AIMS Biophysics, 2025, 12(2): 121-143. doi: 10.3934/biophy.2025008
[6]	T. M. Medvedeva, A. K. Lüttjohann, M. V. Sysoeva, G. van Luijtelaar, I. V. Sysoev . Estimating complexity of spike-wave discharges with largest Lyapunov exponent in computational models and experimental data. AIMS Biophysics, 2020, 7(2): 65-75. doi: 10.3934/biophy.2020006
[7]	Kieran Greer . New ideas for brain modelling 6. AIMS Biophysics, 2020, 7(4): 308-322. doi: 10.3934/biophy.2020022
[8]	Dina Falah Noori Al-Sabak, Leila Sadeghi, Gholamreza Dehghan . Predicting the correlation between neurological abnormalities and thyroid dysfunction using artificial neural networks. AIMS Biophysics, 2024, 11(4): 403-426. doi: 10.3934/biophy.2024022
[9]	Soumyajit Podder, Abhishek Mallick, Sudipta Das, Kartik Sau, Arijit Roy . Accurate diagnosis of liver diseases through the application of deep convolutional neural network on biopsy images. AIMS Biophysics, 2023, 10(4): 453-481. doi: 10.3934/biophy.2023026
[10]	Arjun Acharya, Madan Khanal, Rajesh Maharjan, Kalpana Gyawali, Bhoj Raj Luitel, Rameshwar Adhikari, Deependra Das Mulmi, Tika Ram Lamichhane, Hari Prasad Lamichhane . Quantum chemical calculations on calcium oxalate and dolichin A and their binding efficacy to lactoferrin: An in silico study using DFT, molecular docking, and molecular dynamics simulations. AIMS Biophysics, 2024, 11(2): 142-165. doi: 10.3934/biophy.2024010

Abstract

1. Introduction

Numerous studies on recurrent neural networks (RNNs), including those on bidirectional associative memory neural networks, cellular neural networks, and Hopfield neural networks, among others, have been proposed. In the meantime, neural networks (NNs) have been employed as a tool to address issues that have arisen in associative memory, signal processing, image processing, static image treatment, pattern recognition, and optimization. Additionally, it seems that time delays are crucial to networked control systems, chemical reactions, communication systems, etc. The authors of [1], [2] investigated time-delayed artificial neural network electronic implementations. Several studies have looked into different sorts of delays inside NNs in this context; see the papers [38]–[40] and references therein. The passivity theory is widely applied in many engineering problems. Indeed, it is intimately related to the circuit analysis which is a useful and significant tool to analyze the stability of nonlinear systems, signal processing and chaos control. Thus, it has been employed in various fields of science and engineering [7], [8], [44]–[48]. In [9], the authors proposed neural adaptive output feedback control based on passivity with adaptive set-point regulation of nonlinear uncertain non-negative and compartmental systems. In the last decade, great attention has been paid to the passivity analysis of RNNs with delay-independent [10], [49] and delay-dependent [11]–[14], [36]. In [13], the author studied the passivity analysis of Markovian jump neural network with leakage time varying delay, discrete and distributed time varying delays. We can see the discussion of the extended dissipativity of discrete-time neural networks NNs with time-varying delay in [55]. However, comparatively less interest has been shown towards the passivity analysis of mode dependent delay on neutral type neural networks (NNNs) with Markovian jumping (Mj). On the other hand, a Markov jump system is a special classes of a hybrid system. Indeed, they have great ability to model the dynamical systems and their application can be found in manufacturing systems, economic systems, network control systems, modeling production system, communication systems and so on. In the recent years, several results are reported on the stability analysis for neural networks with Markovian jump parameters, see the references [16]–[19], [41]–[43]. In [20], the authors studied global exponential estimates of delayed stochastic NNs with Markovian switching by constructing with positive definite matrices in stochastic Lyapunov functional which are dependent on the system mode and a triple-integral term. The ℋ_∞ synchronization issue for singularly perturbed coupled neural networks (SPCNNs) affected by both nonlinear constraints and gain uncertainties was explored in [57] using a novel double-layer switching regulation containing Markov chain and persistent dwell-time switching regulation (PDTSR). Convolutional Neural Networks (CNNs) are efficient tools for pattern recognition applications. More on this topic can be seen in [58], [60]. An exponential synchronization problem for the multi-weighted complex dynamical network (MCDN) with hybrid delays on a time scale is investigated in [61]. We can see the establishment of fixed-point and coincidence-point consequences in generalized metric spaces in [59]. The nonfragile ℋ_∞ synchronization issue for a class of discrete-time Takagi–Sugeno (T–S) fuzzy Markov jump systems was investigated in [56]. In [21], the authors have studied stochastic NNNs with mixed time-delays under adaptive synchronization. Additionally, the problem of state estimation of RNNs with Mj parameters and mixed delays based on mode-dependent approach was investigated in [22].

However, some of the researchers discussed the robust passive filtering for NNNs with delays in [25]. In [26], the authors investigated the global asymptotic stability of NNNs with delays by utilizing the Lyapunov-Krasovskii functional (LKF) and the linear matrix inequality approach. While employing the method of Lyapunov–Krasovskii functional, we necessarily need these three steps for the derivation of a global asymptotic stability criterion: constructing a Lyapunov–Krasovskii functional, estimating the derivative of the Lyapunov–Krasovskii functional, and formulating a global asymptotic stability criterion. You will get an overview of recent developments in each of the above steps if you refer to [54].The author in [52] studied passivity and exponential passivity for NNNs with various delays. In [51], the authors investigated the robust passivity analysis of mixed delayed NNs with distributed time-varying delays. The exponential passivity of discrete-time switched NNs with transmission delay was studied in [53]. Recently, the authors in [5] studied passivity analysis for NNNs with Mj parameters and time delay in the leakage term. New delay-dependent passivity conditions are derived in terms of LMIs with a proper construction of LKF, and it can be checked easily via standard numerical packages. However, triple and quadruple integrals have not been taken into account to derive the passivity conditions and, moreover, the mode-dependent time delays have not been included in [5]. Recently, a novel Lyapunov functional with some terms involving triple or quadruple integrals are taken into account to study the state estimation problem with mode-dependent approach in [22]. Motivated by the above discussion, the main purpose of this paper is to study the global passivity of Mj for NNNs with leakage and mode-dependent delay terms. By construction of a new LKF involving mode-dependent Lyapunov matrices, some sufficient conditions are derived in terms of LMIs. For the sake of illustration, a numerical example is given to demonstrate the usefulness and effectiveness of the presented results. Unlike previous results, we will introduce an improved Lyapunov–Krasovskii functional with triple and quadruple integrals for deriving the reported stability results in this paper. Based on this discussion, our technique not only provides different approach but also gives less conservative conditions than those studied in [5], [22]. The rest of this paper is organized as follows. The problem and some preliminaries are introduced in Section 2. In Section 3, the main results are stated and proved. Some sufficient conditions for global passivity results are developed here. In Section 4, an illustrative example is provided to demonstrate the effectiveness of the proposed criteria. We conclude the results of this paper in Section 5.

Notations: Throughout this paper the following notations are used: $ℝ^{n}$ - n-dimensional Euclidean space; $ℝ^{n \times n}$ - the set of all $n \times n$ real matrices; $d i a g (\dots)$ - a block diagonal matrix; I -the identity matrix with compatible dimensions; $C^{T}$ - the transpose of C; X and Y are symmetric matrices, where $X \geq Y$ (similarly $X > Y$ )- $X - Y$ is a positive semi-definite (similarly positive definite); $(Ω, ℱ, 𝒫)$ - a complete probability space with a natural filtration ${ℱ_{t}}_{t \geq 0}$ ; $E [\cdot]$ - expectation operator with respect to the given probability measure $𝒫$ ; $𝒞 ([- d,0]; ℝ^{n})$ - the family of continuously differentiable function; $‖ φ ‖ = m a x {m a x_{- τ \leq θ \leq 0} | φ (θ) |, m a x_{- d \leq θ \leq 0} | φ^{'} (θ) |}$ ; $𝒞_{ℱ_{0}}^{2} ([- d,0]; ℝ^{n})$ - the family of bounded $ℱ_{0}$ -measurable; $𝒞 ([- d,0]; ℝ^{n})$ -valued stochastic variables $ξ = {ξ (θ) : - d \leq θ \leq 0}$ such that $\int_{- d}^{0} E | ξ (θ {) |}^{2} d s < \infty$ ; *-the symmetric block in one symmetric matrix.

2. Problem formulation and preliminaries

Let ${\nabla (t), t \in ℤ^{+}}$ be right-continuous on $(Ω, ℱ, 𝒫)$ . Here $(Ω, ℱ, 𝒫)$ is a complete probability space with Markov chain, $\nabla (t)$ takes the values from a finite state space $𝒮 = {1,2, \dots, N}$ with generator $Γ = (π_{i j})_{N \times N}$ given by

$P {\nabla (t + Δ t) = j | \nabla (t) = i} = {\begin{array}{l} π_{i j} Δ t + o (Δ t), i \neq j, \\ 1 + π_{i i} Δ t + o (Δ t), i = j, \end{array}$

where $Δ t > 0$ and ${l i m}_{Δ t \to 0} \frac{o (Δ t)}{Δ t} = 0$ , $π_{i j} \geq 0$ $(i \neq j)$ is the transition rate from i to j, and $π_{i i} = - \sum_{j = 1, j \neq i}^{N} π_{i j}$ .

We consider Mode-dependent Markov jump $N N N s$ with mixed time-delays:

${\begin{array}{l} \dot{x} (t) & = & - 𝒜 (\nabla (t)) x (t - σ (\nabla (t))) + ℬ (\nabla (t)) ℋ (x (t)) + 𝒞 (\nabla (t)) ℋ (x (t - τ (t, \nabla (t)))) \\ + 𝒟 (\nabla (t)) \dot{x} (t - h (t, \nabla (t))) + ℰ (\nabla (t)) \int_{t - d (t, \nabla (t))}^{t} ℋ (x (s)) d s + u (t) \\ y (t) & = & ℋ (x (t)) \end{array}$ (2.1)

where $x (t) = [x_{1} (t), x_{2} (t), \dots, x_{n} (t {)]}^{T} \in ℝ^{n}$ is the state vector linked with n neurons. The diagonal matrix $𝒜 (\nabla (t)) = d i a g (A_{1} (\nabla (t)), A_{2} (\nabla (t)), \dots, A_{n} (\nabla (t)))$ has positive entries $A_{i} (\nabla (t)) > 0$ $(i = 1,2, \dots, n)$ . $ℬ (\nabla (t))$ , $𝒞 (\nabla (t))$ , $𝒟 (\nabla (t))$ , $ℰ (\nabla (t))$ are known appropriate dimensional constant matrices. Here the neuron activation function is $ℋ (x (t)) = [ℋ_{1} (x_{1} (t)), ℋ_{2} (x_{2} (t)), \dots, ℋ_{n} (x_{n} (t {))]}^{T}$ . $u (t)$ denotes a constant input. $τ (t, \nabla (t))$ , $h (t, \nabla (t))$ , $d (t, \nabla (t))$ are mode dependent discrete, neutral and distributed delays, respectively and $σ (\nabla (t))$ is the mode dependent leakage delay.

Throughout this paper, we assume the following.

Assumption 1. For any $j = 1,2, \dots, n$ , $ℋ_{j} (0) = 0$ and there exist constants ${\hat{l}}_{j}^{-}$ and ${\hat{l}}_{j}^{+}$ such that

${\hat{l}}_{j}^{-} \leq \frac{ℋ_{j} (γ_{1}) - ℋ_{j} (γ_{2})}{γ_{1} - γ_{2}} \leq {\hat{l}}_{j}^{+},$ (2.2)

where $γ_{1}, γ_{2} \in ℝ$ , and $γ_{1} \neq γ_{2}$ .

For the sake of convenience, we denote $𝒜 (\nabla (t) = i) = 𝒜_{i}, ℬ (\nabla (t) = i) = ℬ_{i}, 𝒞 (\nabla (t) = i) = 𝒞_{i}$ , $𝒟 (\nabla (t) = i) = 𝒟_{i}$ , $ℰ (\nabla (t) = i) = ℰ_{i}$ , respectively.

System (2.1) can be rewritten as

${\begin{array}{l} \dot{x} (t) & = & - 𝒜_{i} x (t - σ_{i}) + ℬ_{i} ℋ (x (t)) + 𝒞_{i} ℋ (x (t - τ_{i} (t))) + 𝒟_{i} \dot{x} (t - h_{i} (t)) \\ + ℰ_{i} \int_{t - d_{i} (t)}^{t} ℋ (x (s)) d s + u (t) \\ y (t) & = & ℋ (x (t)) \end{array}$ (2.3)

and the parameters associated with time delays are assumed to satisfy following:

$0 \leq τ_{i} (t) \leq τ_{i}, {\dot{τ}}_{i} (t) \leq τ_{µ_{i}}, 0 \leq h_{i} (t) \leq h_{i}, {\dot{h}}_{i} (t) \leq h_{µ_{i}}, 0 \leq d_{i} (t) \leq d_{i}, {\dot{d}}_{i} (t) \leq d_{µ_{i}}, σ_{i} > 0$ (2.4)

where $τ_{i}, h_{i}, d_{i}, τ_{µ_{i}}, h_{µ_{i}}$ and $d_{µ_{i}}$ are some real constants and $τ = \underset{i \in S}{m a x} {τ_{i}}$ , $h = \underset{i \in S}{m a x} {h_{i}}$ , $d = \underset{i \in S}{m a x} {d_{i}}$ , $σ = \underset{i \in S}{m a x} {σ_{i}}$ .

Now we can see a few necessary lemmas and a definition.

$(b - a) [\int_{a}^{b} y {(s)}^{T} M y (s) d s] \geq {[\int_{a}^{b} y (s) d s]}^{T} M [\int_{a}^{b} y (s) d s] .$

holds.

Proof. By Schur complement,

$[\begin{matrix} y {(s)}^{T} M y (s) & y {(s)}^{T} \\ y (s) & M^{- 1} \end{matrix}] \geq 0, s \in [a, b] .$

On integration from a to b yields, $[\begin{matrix} \int_{a}^{b} y {(s)}^{T} M y (s) d s & \int_{a}^{b} y {(s)}^{T} d s \\ \int_{a}^{b} y (s) d s & (b - a) M^{- 1} \end{matrix}]$ $\geq 0, s \in [a, b] .$ Now using the Schur complement on this inequality, we obtain our desired result. □

$\pm 2 x^{T} y \leq x^{T} ℳ x + y^{T} ℳ^{- 1} y .$

Lemma 3. [31] (Schur complement) Given $Ω_{1}, Ω_{2}$ and $Ω_{3}$ are constant matrices with appropriate dimensions, where $Ω_{1}, Ω_{2} > 0$ are symmetric matrices, then

$Ω_{1} + Ω_{3}^{T} Ω_{2}^{- 1} Ω_{3} < 0 \Leftrightarrow [\begin{matrix} Ω_{1} & Ω_{3}^{T} \\ * & - Ω_{2} \end{matrix}] < 0, o r [\begin{matrix} - Ω_{2} & Ω_{3} \\ * & Ω_{1} \end{matrix}] < 0.$

Definition 1. [32] If there exists a scalar $ν \geq 0$ such that $\forall t_{p} \geq 0$ and for all solutions of (2.1), the following inequality holds under zero initial conditions,

$2 \int_{0}^{t_{p}} E {y {(s)}^{T} u (s)} d s \geq - γ \int_{0}^{t_{p}} E {u {(s)}^{T} u (s)} d s,$ (2.5)

then the system (2.3) is said to be passive.

3. Main results

Now, we denote

${\hat{L}}_{1} = d i a g {{\hat{l}}_{1}^{-} {\hat{l}}_{1}^{+}, {\hat{l}}_{2}^{-} {\hat{l}}_{2}^{+}, \dots, {\hat{l}}_{m}^{-} {\hat{l}}_{m}^{+}}, a n d {\hat{L}}_{2} = d i a g {\frac{{\hat{l}}_{1}^{-} + {\hat{l}}_{1}^{+}}{2}, \frac{{\hat{l}}_{2}^{-} + {\hat{l}}_{2}^{+}}{2}, \dots, \frac{{\hat{l}}_{m}^{-} + {\hat{l}}_{m}^{+}}{2}} .$

$[\begin{matrix} X_{i} & N_{i} \\ X_{i} \end{matrix}] \geq 0,$ (3.1)

$\sum_{j = 1}^{N} π_{i j} V_{j} - V \leq 0,$ (3.2)

$\sum_{j = 1, j \neq i}^{N} π_{i j} 𝒢_{j} - 𝒢 \leq 0,$ (3.3)

$Φ = [\begin{matrix} Ω & Γ^{T} \\ - \frac{1}{π_{i j}} K_{j} \end{matrix}] < 0,$ (3.4)

where $𝒢_{j}$ in (3.3) respectively represents $Q_{j}$ , $W_{j}$ , $R_{j}$ , $S_{j}$ , $U_{j}$ , $X_{j}$ , $Y_{j}$ , $T_{j}$ , $Z_{j}$ , $L_{j}$ and correspondingly 𝒢 represents $Q$ , $W$ , $R$ , $S$ , $U$ , $X$ , $Y$ , $T$ , $Z$ , (e.q.,when 𝒢_j is $Q_{j}$ , $𝒢$ is $Q$ ) and

$Ω = (ϑ_{i, j})_{15 \times 15},$

$ϑ_{1,1} = - P_{i} 𝒜_{i} - 𝒜_{i}^{T} P_{i} + π_{i i} P_{i} + \sum_{j \neq i} π_{i j} P_{j} + \sum_{j \neq i} π_{i j} K_{j} + Q_{1 i} + τ Q_{1} + W_{i} + τ W + R_{i} + σ R$

$+ τ_{i} U_{i} + \frac{τ^{2}}{2} U - \frac{1}{τ_{i}} X_{i} - \frac{1}{h_{i}} Y_{i} - 2 T_{i} - 2 Z_{i} - 2 L_{i} - {\hat{L}}_{1} H_{1 i} - {\hat{L}}_{1} H_{3 i} - {\hat{L}}_{1} H_{4 i}$

$- {\hat{L}}_{1} H_{5 i} - {\hat{L}}_{1} H_{6 i} - 2 σ^{2} M, ϑ_{1,2} = - \frac{1}{τ_{i}} N_{i}^{T} + \frac{1}{τ_{i}} X_{i}, ϑ_{1,3} = \frac{1}{τ_{i}} N_{i}^{T} + {\hat{L}}_{2} H_{3 i},$

$ϑ_{1,4} = \frac{2}{τ_{i}} T_{i}, ϑ_{1,5} = P_{i} ℬ_{i} + Q_{2 i} + Q_{2} + {\hat{L}}_{2} H_{1 i} + J_{1} ℬ_{i}, ϑ_{1,6} = P_{i} 𝒞_{i} + J_{1} 𝒞_{i},$

$ϑ_{1,7} = - J_{1} 𝒜_{i} - 𝒜_{i}^{T} J_{1}^{T} + {\hat{L}}_{2} H_{5 i}, ϑ_{1,8} = 𝒜_{i}^{T} P_{i} 𝒜_{i} - π_{i i} P_{i} 𝒜_{i} - \frac{2}{σ_{i}} L_{i}, ϑ_{1,9} = 2 σ M,$

$ϑ_{1,10} = P_{i}, ϑ_{1,11} = P_{i} 𝒟_{i} + J_{1} 𝒟_{i}, ϑ_{1,12} = \frac{1}{h_{i}} Y_{i} + {\hat{L}}_{2} H_{4 i}, ϑ_{1,13} = \frac{2}{h_{i}} Z_{i},$

$ϑ_{1,14} = J_{1} ℰ_{i}, ϑ_{1,15} = - J_{1} + {\hat{L}}_{2} H_{6 i}, ϑ_{2,2} = - (1 - τ_{µ_{i}}) Q_{1 i} - {\hat{L}}_{1} H_{2 i} - \frac{2}{τ_{i}} X_{i} + \frac{1}{τ_{i}} N_{i}^{T},$

$ϑ_{2,3} = \frac{1}{τ_{i}} X_{i} - \frac{1}{τ_{i}} N_{i}^{T}, ϑ_{2,6} = - (1 - τ_{µ_{i}}) Q_{1 i} + {\hat{L}}_{2} H_{2 i}, ϑ_{3,3} = - W_{i} - \frac{1}{τ_{i}} X_{i} - H_{3 i},$

$ϑ_{4,4} = - \frac{1}{τ_{i}} U_{i} - \frac{2}{τ_{i}^{2}} T_{i}, ϑ_{5,5} = Q_{3 i} + Q_{3} + d_{i} V_{i} + \frac{d^{2}}{2} V - H_{1 i}, ϑ_{5,8} = - ℬ_{i}^{T} P_{i} 𝒜_{i},$

$ϑ_{5,15} = ℬ_{i}^{T} J_{2}^{T}, ϑ_{6,6} = - (1 - τ_{µ_{i}}) Q_{3 i} - H_{2 i}, ϑ_{6,8} = - 𝒞_{i}^{T} P_{i} 𝒜_{i}, ϑ_{6,15} = 𝒞_{i}^{T} J_{2}^{T},$

$ϑ_{7,7} = - R_{i} - H_{5 i}, ϑ_{7,15} = - 𝒜_{i}^{T} J_{2}^{T}, ϑ_{8,8} = π_{i i} 𝒜_{i}^{T} P_{i} 𝒜_{i} - \frac{2}{σ_{i}^{2}} L_{i}, ϑ_{8,10} = - 𝒜_{i}^{T} P_{i},$

$ϑ_{8,11} = - 𝒜_{i}^{T} P_{i} 𝒟_{i}, ϑ_{8,14} = - 𝒜_{i}^{T} P_{i} ℰ_{i}, ϑ_{9,9} = \sum_{j \neq i} π_{i j} 𝒜_{j}^{T} P_{j} 𝒜_{j} - 2 M, ϑ_{10,10} = - γ I,$

$ϑ_{10,15} = J_{2}^{T}, ϑ_{11,11} = - S_{i} (1 - h_{µ_{i}}), ϑ_{11,15} = 𝒟_{i}^{T} J_{2}^{T}, ϑ_{12,12} = - \frac{1}{h_{i}} Y_{i} - H_{4 i},$

$ϑ_{13,13} = - \frac{2}{h_{i}^{2}} Z_{i}, ϑ_{14,14} = - \frac{(1 - d_{µ_{i}})}{d_{i}} V_{i}, ϑ_{14,15} = ℰ_{i}^{T} J_{2}^{T}, ϑ_{15,15} = S_{i} + h S + τ_{i} X_{i} + \frac{τ^{2}}{2} X$

$+ h_{i} Y_{i} + \frac{h^{2}}{2} Y + \frac{τ_{i}^{2}}{2} T_{i} + \frac{τ^{3}}{6} T + \frac{h_{i}^{2}}{2} Z_{i} + \frac{h^{3}}{6} Z + \frac{σ_{i}^{2}}{2} L_{i} + \frac{σ^{3}}{6} L + \frac{σ^{4}}{2} Z - (J_{2} + J_{2}^{T}) - H_{6 i},$

$Γ^{T} = [{\underset{}{\underset{︸}{0.....0}}}_{8 e l e m e n t s} \sum_{j \neq i} π_{i j} {(𝒜_{j}^{T} P_{j})}^{T} {\underset{}{\underset{︸}{0.....0}}}_{15 e l e m e n t s}]^{T},$

and the other coefficients are zero.

Proof. Here, we consider LKF candidate:

$V (x_{t}, i, t) = \sum_{κ = 1}^{13} V_{κ} (x_{t}, i, t),$ (3.5)

where

$V_{1} (x_{t}, i, t) = {[x (t) - 𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s]}^{T} P_{i} [x (t) - 𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s],$

$V_{2} (x_{t}, i, t) = \int_{t - τ_{i} (t)}^{t} ζ^{T} (s) Q_{i} ζ (s) d s + \int_{- τ}^{0} \int_{t + θ}^{t} ζ^{T} (s) Q ζ (s) d s d θ,$

$V_{3} (x_{t}, i, t) = \int_{t - τ_{i}}^{t} x^{T} (s) W_{i} x (s) d s + \int_{- τ}^{0} \int_{t + θ}^{t} x^{T} (s) W x (s) d s d θ,$

$V_{4} (x_{t}, i, t) = \int_{t - σ_{i}}^{t} x^{T} (s) R_{i} x (s) d s + \int_{- σ}^{0} \int_{t + θ}^{t} x^{T} (s) R x (s) d s d θ,$

$V_{5} (x_{t}, i, t) = \int_{t - h_{i}}^{t} x^{T} (s) S_{i} x (s) d s + \int_{- h}^{0} \int_{t + θ}^{t} x^{T} (s) S x (s) d s d θ,$

$V_{6} (x_{t}, i, t) = \int_{- d_{i} (t)}^{0} \int_{t + θ}^{t} ℋ^{T} (x (s)) V_{i} ℋ (x (s)) d s d θ + \int_{- d}^{0} \int_{θ}^{0} \int_{t + β}^{t} ℋ^{T} (x (s)) V ℋ (x (s)) d s d β d θ,$

$V_{7} (x_{t}, i, t) = \int_{- τ_{i}}^{0} \int_{t + θ}^{t} x^{T} (s) U_{i} x (s) d s d θ + \int_{- τ}^{0} \int_{θ}^{0} \int_{t + β}^{t} x^{T} (s) U x (s) d s d β d θ,$

$V_{8} (x_{t}, i, t) = \int_{- τ_{i}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) X_{i} \dot{x} (s) d s d θ + \int_{- τ}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) X \dot{x} (s) d s d β d θ,$

$V_{9} (x_{t}, i, t) = \int_{- h_{i}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) Y_{i} \dot{x} (s) d s d θ + \int_{- h}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) Y \dot{x} (s) d s d β d θ,$

$V_{10} (x_{t}, i, t) = \int_{- τ_{i}}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) T_{i} \dot{x} (s) d s d β d θ + \int_{- τ}^{0} \int_{θ}^{0} \int_{β}^{0} \int_{t + α}^{t} {\dot{x}}^{T} (s) T \dot{x} (s) d s d α d β d θ,$

$V_{11} (x_{t}, i, t) = \int_{- h_{i}}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) Z_{i} \dot{x} (s) d s d β d θ + \int_{- h}^{0} \int_{θ}^{0} \int_{β}^{0} \int_{t + α}^{t} {\dot{x}}^{T} (s) Z \dot{x} (s) d s d α d β d θ,$

$V_{12} (x_{t}, i, t) = \int_{- σ_{i}}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) L_{i} \dot{x} (s) d s d β d θ + \int_{- σ}^{0} \int_{θ}^{0} \int_{β}^{0} \int_{t + α}^{t} {\dot{x}}^{T} (s) L \dot{x} (s) d s d α d β d θ,$

$V_{13} (x_{t}, i, t) = σ^{2} \int_{- σ}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) M \dot{x} (s) d s d β d θ,$

where $ζ^{T} (t) = [x^{T} (t), ℋ^{T} (x (t {))]}^{T} .$

From (2.1), we get

$ℒ V (x_{t}, i, t) = \sum_{κ = 1}^{13} ℒ V_{κ} (x_{t}, i, t),$ (3.6)

where

$ℒ V_{1} (x_{t}, i, t) = 2 {[x (t) - 𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s]}^{T} P_{i} \frac{d}{d t} [x (t) - 𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s]$

$+ \sum_{j = 1}^{N} π_{i j} {[x (t) - 𝒜_{j} \int_{t - σ_{j}}^{t} x (s) d s]}^{T} P_{j} [x (t) - 𝒜_{j} \int_{t - σ_{j}}^{t} x (s) d s],$

$\leq 2 {[x (t) - 𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s]}^{T} P_{i} [- 𝒜_{i} x (t) + ℬ_{i} ℋ (x (t)) + 𝒞_{i} ℋ (x (t - τ_{i} (t)))$

$+ 𝒟_{i} \dot{x} (t - h_{i} (t)) + ℰ_{i} \int_{t - d_{i} (t)}^{t} ℋ (x (s)) d s + u (t)]$

$+ π_{i i} {[x (t) - 𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s]}^{T} P_{i} [x (t) - 𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s]$

$+ \sum_{j \neq i} π_{i j} [x^{T} (t) P_{j} x (t) + \int_{t - σ_{j}}^{t} x^{T} (s) d s 𝒜_{j} P_{j} K_{j}^{- 1} P_{j} 𝒜_{j} \int_{t - σ_{j}}^{t} x (s) d s$

$+ \int_{t - σ}^{t} x^{T} (s) d s 𝒜_{j}^{T} P_{j} 𝒜_{j} \int_{t - σ}^{t} x (s) d s],$

$ℒ V_{2} (x_{t}, i, t) \leq ζ^{T} (t) Q_{i} ζ (t) - ζ^{T} (t - τ_{i} (t)) Q_{i} ζ (t - τ_{i} (t)) (1 - τ_{µ_{i}}) + \sum_{j = 1}^{N} π_{i j} \int_{t - τ_{j} (t)}^{t} ζ^{T} (s) Q_{j} ζ (s) d s$

$+ τ ζ^{T} (t) Q ζ (t) - \int_{t - τ}^{t} ζ^{T} (s) Q ζ (s) d s,$

$ℒ V_{3} (x_{t}, i, t) = x^{T} (t) W_{i} x (t) - x^{T} (t - τ_{i}) W_{i} x (t - τ_{i}) + \sum_{j = 1}^{N} π_{i j} \int_{t - τ_{j}}^{t} x^{T} (s) W_{j} x (s) d s$

$+ τ x^{T} (t) W x (t) - \int_{t - τ}^{t} x^{T} (s) W x (s) d s,$

$ℒ V_{4} (x_{t}, i, t) = x^{T} (t) R_{i} x (t) - x^{T} (t - σ_{i}) R_{i} x (t - σ_{i}) + \sum_{j = 1}^{N} π_{i j} \int_{t - σ_{j}}^{t} x^{T} (s) R_{j} x (s) d s$

$+ σ x^{T} (t) R x (t) - \int_{t - σ}^{t} x^{T} (s) R x (s) d s,$

$ℒ V_{5} (x_{t}, i, t) \leq {\dot{x}}^{T} (t) S_{i} \dot{x} (t) - {\dot{x}}^{T} (t - h_{i} (t)) S_{i} \dot{x} (t - h_{i} (t)) (1 - h_{µ_{i}}) + \sum_{j = 1}^{N} π_{i j} \int_{t - h_{j} (t)}^{t} {\dot{x}}^{T} (s) S_{j} \dot{x} (s) d s$

$+ h {\dot{x}}^{T} (t) S \dot{x} (t) - \int_{t - h}^{t} {\dot{x}}^{T} (s) S \dot{x} (s) d s,$

$ℒ V_{6} (x_{t}, i, t) = d_{i} (t) ℋ^{T} (x (t)) V_{i} ℋ (x (t)) - (1 - d_{µ_{i}}) \int_{t - d_{i} (t)}^{t} ℋ^{T} (x (s)) V_{i} ℋ (x (s)) d s$

$+ \sum_{j = 1}^{N} π_{i j} \int_{- d_{j} (t)}^{0} \int_{t + θ}^{t} ℋ^{T} (x (s)) V_{j} ℋ (x (s)) d s d θ$

$+ \frac{d^{2}}{2} ℋ^{T} (x (t)) V ℋ (x (t)) - \int_{- d}^{0} \int_{t + θ}^{t} ℋ^{T} (x (s)) V ℋ (x (s)) d s d θ,$

$ℒ V_{7} (x_{t}, i, t) = τ_{i} x^{T} (t) U_{i} x (t) - \int_{t - τ_{i}}^{t} x^{T} (s) U_{i} x (s) d s + \sum_{j = 1}^{N} π_{i j} \int_{- τ_{j}}^{0} \int_{t + θ}^{t} x^{T} (s) U_{j} x (s) d s d θ$

$+ \frac{τ^{2}}{2} x^{T} (t) U x (t) - \int_{- τ}^{0} \int_{t + θ}^{t} x^{T} (s) U x (s) d s d θ,$

$ℒ V_{8} (x_{t}, i, t) = τ_{i} {\dot{x}}^{T} (t) X_{i} \dot{x} (t) - \int_{t - τ_{i}}^{t} {\dot{x}}^{T} (s) X_{i} \dot{x} (s) d s + \sum_{j = 1}^{N} π_{i j} \int_{- τ_{j}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) X_{j} \dot{x} (s) d s d θ$

$+ \frac{τ^{2}}{2} {\dot{x}}^{T} (t) X \dot{x} (t) - \int_{- τ}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) X \dot{x} (s) d s d θ,$

$ℒ V_{9} (x_{t}, i, t) = h_{i} {\dot{x}}^{T} (t) Y_{i} \dot{x} (t) - \int_{t - h_{i}}^{t} {\dot{x}}^{T} (s) Y_{i} \dot{x} (s) d s + \sum_{j = 1}^{N} π_{i j} \int_{- h_{j}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) Y_{j} \dot{x} (s) d s d θ$

$+ \frac{h^{2}}{2} {\dot{x}}^{T} (t) Y \dot{x} (t) - \int_{- h}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) Y \dot{x} (s) d s d θ,$

$ℒ V_{10} (x_{t}, i, t) = \frac{τ_{i}^{2}}{2} {\dot{x}}^{T} (t) T_{i} \dot{x} (t) - \int_{- τ_{i}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) T_{i} \dot{x} (s) d s d θ + \frac{τ^{3}}{6} {\dot{x}}^{T} (t) T \dot{x} (t)$

$+ \sum_{j = 1}^{N} π_{i j} \int_{- τ_{j}}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) T_{j} \dot{x} (s) d s d β d θ - \int_{- τ}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) T \dot{x} (s) d s d β d θ,$

$ℒ V_{11} (x_{t}, i, t) = \frac{h_{i}^{2}}{2} {\dot{x}}^{T} (t) Z_{i} \dot{x} (t) - \int_{- h_{i}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) Z_{i} \dot{x} (s) d s d θ + \frac{h^{3}}{6} {\dot{x}}^{T} (t) Z \dot{x} (t)$

$+ \sum_{j = 1}^{N} π_{i j} \int_{- h_{j}}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) Z_{j} \dot{x} (s) d s d β d θ - \int_{- h}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) Z \dot{x} (s) d s d β d θ,$

$ℒ V_{12} (x_{t}, i, t) = \frac{σ_{i}^{2}}{2} {\dot{x}}^{T} (t) L_{i} \dot{x} (t) - \int_{- σ_{i}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) L_{i} \dot{x} (s) d s d θ + \frac{σ^{3}}{6} {\dot{x}}^{T} (t) L \dot{x} (t)$

$+ \sum_{j = 1}^{N} π_{i j} \int_{- σ_{j}}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) L_{j} \dot{x} (s) d s d β d θ - \int_{- σ}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) L \dot{x} (s) d s d β d θ,$

$ℒ V_{13} (x_{t}, i, t) = \frac{σ^{4}}{2} {\dot{x}}^{T} (t) M \dot{x} (t) - σ^{2} \int_{- σ}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) M \dot{x} (s) d s d θ .$

Using the upper bounds of discrete, neutral, distributed time-varying delays, leakage delays, Lemma 2 and with $π_{i i} < 0,$ the following relationship is obtained

$\begin{array}{l} \sum_{j \neq i} (- 2 π_{i j} x^{T} (t) P_{j} 𝒜_{j} \int_{t - σ_{j}}^{t} x (s) d s) \\ \leq \sum_{j \neq i} π_{i j} (x^{T} (t) K_{j} x (t) + \int_{t - σ_{j}}^{t} x^{T} (s) d s 𝒜_{j}^{T} P_{j} K_{j}^{- 1} P_{j} 𝒜_{j} \int_{t - σ_{j}}^{t} x (s) d s) \\ \leq \sum_{j \neq i} π_{i j} (x^{T} (t) K_{j} x (t) + \int_{t - σ}^{t} x^{T} (s) d s 𝒜_{j}^{T} P_{j} K_{j}^{- 1} P_{j} 𝒜_{j} \int_{t - σ}^{t} x (s) d s), \end{array}$ (3.7)

$\begin{array}{l} \sum_{j \neq i} π_{i j} (\int_{t - σ_{j}}^{t} x^{T} (s) d s) 𝒜_{j}^{T} P_{j} 𝒜_{j} (\int_{t - σ_{j}}^{t} x (s) d s) \\ \leq \sum_{j \neq i} π_{i j} (\int_{t - σ}^{t} x^{T} (s) d s) 𝒜_{j}^{T} P_{j} 𝒜_{j} (\int_{t - σ}^{t} x (s) d s) . \end{array}$ (3.8)

Similarly,

$\begin{array}{l} \sum_{j = 1}^{N} π_{i j} \int_{t - τ_{j} (t)}^{t} ζ^{T} (s) Q_{j} ζ (s) d s \leq \sum_{j = 1, j \neq i}^{N} π_{i j} \int_{t - τ_{j} (t)}^{t} ζ^{T} (s) Q_{j} ζ (s) d s \\ \leq \sum_{j = 1, j \neq i}^{N} π_{i j} \int_{t - τ}^{t} ζ^{T} (s) Q_{j} ζ (s) d s \\ \leq \int_{t - τ}^{t} ζ^{T} (s) Q ζ (s) d s, \end{array}$ (3.9)

$\sum_{j = 1}^{N} π_{i j} \int_{t - τ_{j}}^{t} x^{T} (s) W_{j} x (s) d s \leq \int_{t - τ}^{t} x^{T} (s) W x (s) d s,$ (3.10)

$\sum_{j = 1}^{N} π_{i j} \int_{t - σ_{j}}^{t} x^{T} (s) R_{j} x (s) d s \leq \int_{t - σ}^{t} x^{T} (s) R x (s) d s,$ (3.11)

$\sum_{j = 1}^{N} π_{i j} \int_{t - h_{j} (t)}^{t} {\dot{x}}^{T} (s) S_{j} \dot{x} (s) d s \leq \int_{t - h}^{t} {\dot{x}}^{T} (s) S \dot{x} (s) d s,$ (3.12)

$\sum_{j = 1}^{N} π_{i j} \int_{- d_{j} (t)}^{0} \int_{t + θ}^{t} ℋ^{T} (x (s)) V_{j} ℋ (x (s)) d s d θ \leq - \int_{- d}^{0} \int_{t + θ}^{t} ℋ^{T} (x (s)) V ℋ (x (s)) d s d θ,$ (3.13)

$\sum_{j = 1}^{N} π_{i j} \int_{- τ_{j}}^{0} \int_{t + θ}^{t} x^{T} (s) U_{j} x (s) d s d θ \leq \int_{- τ}^{0} \int_{t + θ}^{t} x^{T} (s) U x (s) d s d θ,$ (3.14)

$\sum_{j = 1}^{N} π_{i j} \int_{- τ_{j}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) X_{j} \dot{x} (s) d s d θ \leq \int_{- τ}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) X \dot{x} (s) d s d θ,$ (3.15)

$\sum_{j = 1}^{N} π_{i j} \int_{- h_{j}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) Y_{j} \dot{x} (s) d s d θ \leq \int_{- h}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) Y \dot{x} (s) d s d θ,$ (3.16)

$\sum_{j = 1}^{N} π_{i j} \int_{- τ_{j}}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) T_{j} \dot{x} (s) d s d β d θ \leq \int_{- τ}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) T \dot{x} (s) d s d β d θ,$ (3.17)

$\sum_{j = 1}^{N} π_{i j} \int_{- h_{j}}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) Z_{j} \dot{x} (s) d s d β d θ \leq \int_{- h}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) Z \dot{x} (s) d s d β d θ,$ (3.18)

$\sum_{j = 1}^{N} π_{i j} \int_{- σ_{j}}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) L_{j} \dot{x} (s) d s d β d θ \leq \int_{- σ}^{0} \int_{θ}^{0} \int_{t + β}^{t} {\dot{x}}^{T} (s) L \dot{x} (s) d s d β d θ .$ (3.19)

By using Lemma 1, we get

$- (1 - d_{µ_{i}}) \int_{t - d_{i} (t)}^{t} ℋ^{T} (x (s)) V_{i} ℋ (x (s)) d s \leq - \frac{(1 - d_{µ_{i}})}{d_{i}} \int_{t - d_{i} (t)}^{t} ℋ^{T} (x (s)) d s V_{i} \int_{t - d_{i} (t)}^{t} ℋ (x (s)) d s,$

$- \int_{t - τ_{i}}^{t} x^{T} (s) U_{i} x (s) d s \leq - \frac{1}{τ_{i}} \int_{t - τ_{i}}^{t} x^{T} (s) d s U_{i} \int_{t - τ_{i}}^{t} x (s) d s,$

$- \int_{t - h_{i}}^{t} {\dot{x}}^{T} (s) Y_{i} \dot{x} (s) d s \leq - \frac{1}{h_{i}} \int_{t - h_{i}}^{t} {\dot{x}}^{T} (s) d s Y_{i} \int_{t - h_{i}}^{t} \dot{x} (s) d s,$

$- \int_{- τ_{i}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) T_{i} \dot{x} (s) d s d θ \leq - \frac{2}{τ_{i}^{2}} \int_{- τ_{i}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) d s d θ T_{i} \int_{- τ_{i}}^{0} \int_{t + θ}^{t} \dot{x} (s) d s d θ,$

$- \int_{- h_{i}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) Z_{i} \dot{x} (s) d s d θ \leq - \frac{2}{h_{i}^{2}} \int_{- h_{i}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) d s d θ Z_{i} \int_{- h_{i}}^{0} \int_{t + θ}^{t} \dot{x} (s) d s d θ,$

$- \int_{- σ_{i}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) Z_{i} \dot{x} (s) d s d θ \leq - \frac{2}{σ_{i}^{2}} \int_{- σ_{i}}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) d s d θ Z_{i} \int_{- σ_{i}}^{0} \int_{t + θ}^{t} \dot{x} (s) d s d θ,$

$\int_{- σ}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) M \dot{x} (s) d s d θ \leq - \frac{2}{σ^{2}} \int_{- σ}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) d s d θ Z_{i} \int_{- σ}^{0} \int_{t + θ}^{t} \dot{x} (s) d s d θ .$

Note that from (3.1) and using the reciprocally convex technique in [34], we obtain

$- \int_{t - τ_{i}}^{t} {\dot{x}}^{T} (s) X_{i} \dot{x} (s) d s \leq - \int_{t - τ_{i}}^{t - τ_{i} (t)} {\dot{x}}^{T} (s) X_{i} \dot{x} (s) d s - \int_{t - τ_{i} (t)}^{t} {\dot{x}}^{T} (s) X_{i} \dot{x} (s) d s$

$\leq - \frac{1}{τ_{i}} ϖ^{T} (t) [\begin{matrix} R_{i} & X_{i} \\ R_{i} \end{matrix}] ϖ (t),$

where $ϖ (t) = [x^{T} (t - τ_{i} (t)) - x^{T} (t - τ_{i}), x^{T} (t) - x^{T} (t - τ_{i} (t))]$ . For positive diagonal matrices $H_{1 i}, H_{2 i}, H_{3 i}, H_{4 i}, H_{5 i}, H_{6 i}$ , by Assumption 1, we get

${[\begin{matrix} x (t) \\ ℋ (x (t)) \end{matrix}]}^{T} [\begin{matrix} {\hat{L}}_{1} H_{1 i} & - {\hat{L}}_{2} H_{1 i} \\ * & H_{1 i} \end{matrix}] [\begin{matrix} x (t) \\ ℋ (x (t)) \end{matrix}] \leq 0,$ (3.20)

${[\begin{matrix} x (t - τ_{i} (t)) \\ ℋ (x (t - τ_{i} (t))) \end{matrix}]}^{T} [\begin{matrix} {\hat{L}}_{1} H_{2 i} & - {\hat{L}}_{2} H_{2 i} \\ * & H_{2 i} \end{matrix}] [\begin{matrix} x (t - τ_{i} (t)) \\ ℋ (x (t - τ_{i} (t))) \end{matrix}] \leq 0,$ (3.21)

${[\begin{matrix} x (t) \\ x (t - τ_{i}) \end{matrix}]}^{T} [\begin{matrix} {\hat{L}}_{1} H_{3 i} & - {\hat{L}}_{2} H_{3 i} \\ * & H_{3 i} \end{matrix}] [\begin{matrix} x (t) \\ x (t - τ_{i}) \end{matrix}] \leq 0,$ (3.22)

${[\begin{matrix} x (t) \\ x (t - h_{i}) \end{matrix}]}^{T} [\begin{matrix} {\hat{L}}_{1} H_{4 i} & - {\hat{L}}_{2} H_{4 i} \\ * & H_{4 i} \end{matrix}] [\begin{matrix} x (t) \\ x (t - h_{i}) \end{matrix}] \leq 0,$ (3.23)

${[\begin{matrix} x (t) \\ x (t - σ_{i}) \end{matrix}]}^{T} [\begin{matrix} {\hat{L}}_{1} H_{5 i} & - {\hat{L}}_{2} H_{5 i} \\ * & H_{5 i} \end{matrix}] [\begin{matrix} x (t) \\ x (t - σ_{i}) \end{matrix}] \leq 0,$ (3.24)

${[\begin{matrix} x (t) \\ \dot{x} (t) \end{matrix}]}^{T} [\begin{matrix} {\hat{L}}_{1} H_{6 i} & - {\hat{L}}_{2} H_{6 i} \\ * & H_{6 i} \end{matrix}] [\begin{matrix} x (t) \\ \dot{x} (t) \end{matrix}] \leq 0.$ (3.25)

Hence, for any matrices J₁, J₂ of appropriate dimensions, we get

$\begin{array}{l} 0 = 2 [x^{T} (t) J_{1} + {\dot{x}}^{T} (t) J_{2}] [- 𝒜_{i} x (t - σ_{i}) + ℬ_{i} ℋ (x (t)) + 𝒞_{i} ℋ (x (t - τ_{i} (t))) \\ + 𝒟_{i} \dot{x} (t - h_{i} (t)) + ℰ_{i} \int_{t - d_{i} (t)}^{t} ℋ (x (s)) d s + u (t) - \dot{x} (t)] . \end{array}$ (3.26)

Using (3.6) and adding (3.20)-(3.26), we have

$ℒ V (x_{t}, i, t) - 2 y^{T} (t) u (t) - γ u^{T} (t) u (t) \leq ℒ {η^{T} (t) Φ η (t)},$ (3.27)

where

$η^{T} (t) = [x^{T} (t) x^{T} (t - τ_{i} (t)) x^{T} (t - τ_{i}) \int_{t - τ_{i}}^{t} x^{T} (s) d s ℋ^{T} (x (t)) ℋ^{T} (x (t - τ_{i} (t)))$

$x^{T} (t - σ_{i}) \int_{t - σ_{i}}^{t} x^{T} (s) d s \int_{t - σ}^{t} x^{T} (s) d s u^{T} (t) {\dot{x}}^{T} (t - h_{i} (t)) x^{T} (t - h_{i})$

$\int_{t - h_{i}}^{t} x^{T} (s) d s \int_{t - d_{i} (t)}^{t} ℋ^{T} (x (s)) d s {\dot{x}}^{T} (t)] .$

Hence from equation (3.4) we have,

$ℒ V (x_{t}, i, t) - 2 y {(t)}^{T} u (t) - γ u {(t)}^{T} u (t) \leq 0.$

Now, to show the passivity of the delayed NNs in (2.3), we take

$J (t_{p}) = 𝔼 {\int_{0}^{t_{p}} [- γ u {(t)}^{T} u (t) - 2 y {(t)}^{T} u (t)] d t}$ (3.28)

where $t_{p} \geq 0$ .

From Dynkin's formula, we get

$𝔼 [\int_{0}^{t_{p}} ℒ V (x_{t}, i, t) d t] = 𝔼 [V (x_{t_{p}}, i, t_{p})] - 𝔼 [V (x_{0}, \nabla (0),0)] .$

Therefore,

$\begin{array}{l} J (t_{p}) = 𝔼 {\int_{0}^{t_{p}} [- γ u {(t)}^{T} u (t) - 2 y {(t)}^{T} u (t) + ℒ V (x_{t}, i, t)] d t} - 𝔼 [\int_{0}^{t_{p}} ℒ V (x_{t}, i, t) d t] \\ = 𝔼 {\int_{0}^{t_{p}} [- γ u {(t)}^{T} u (t) - 2 y {(t)}^{T} u (t) + ℒ V (x_{t}, i, t)] d t} \\ - 𝔼 [V (x_{t}, i, t)] + 𝔼 [V (x_{0}, \nabla (0),0)] . \end{array}$ (3.29)

By applying lemma 3 to (3.4), we have

$Φ < 0.$ (3.30)

Thus, if (3.30) holds, then $𝔼 [V (x_{t_{p}}, i, t_{p})] \geq 0$ and $V (x_{0}, \nabla (0),0) = 0$ holds with zero initial conditions. From (3.30), it follows that $J (t_{p}) \leq 0$ for any $t_{p} \geq 0$ , which implies (2.5) is satisfied and hence the delayed NNs (2.3) is locally passive.

Now, we prove the global passivity of the system.

By taking expectation of (3.27) and then integration from 0 to t we get,

$\int_{0}^{t} 𝔼 [ℒ V (x_{s}, r (s), s)] d s - 2 \int_{0}^{t} 𝔼 [y^{T} (s) u (s)] d s - γ \int_{0}^{t} 𝔼 [u^{T} (s) u (s)] d s \leq \int_{0}^{t} 𝔼 [η^{T} (s) Φ η (s)] d s .$

Then by Dynkin's formula,

$𝔼 [ℒ V (x_{t}, i, t)] - 𝔼 [ℒ V (x_{0}, \nabla (0),0)] - 2 \int_{0}^{t} 𝔼 [y^{T} (s) u (s)] d s - γ \int_{0}^{t} 𝔼 [u^{T} (s) u (s)] d s$

$\leq \int_{0}^{t} 𝔼 [η^{T} (s) Φ η (s)] d s .$

Hence,

$\begin{matrix} 𝔼 [ℒ V (x_{t}, i, t)] - \int_{0}^{t} 𝔼 [η^{T} (s) Φ η (s)] d s \leq 𝔼 [ℒ V (x_{0}, \nabla (0),0)] + 2 \int_{0}^{t} 𝔼 [y^{T} (s) u (s)] d s \\ + γ \int_{0}^{t} 𝔼 [u^{T} (s) u (s)] d s \\ < \infty, t \geq 0. \end{matrix}$ (3.31)

By Jenson's inequality and (3.6), we get

$\begin{array}{l} E {∥ 𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s ∥}^{2} = E {[𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s]}^{T} [𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s] \\ \leq λ_{m a x} (𝒜_{i}^{2}) E {[\int_{t - σ_{i}}^{t} x (s) d s]}^{T} [\int_{t - σ_{i}}^{t} x (s) d s] \\ \leq \frac{λ_{m a x} (𝒜_{i}^{2})}{λ_{m i n} (R_{i})} {[\int_{t - σ_{i}}^{t} E x (s) d s]}^{T} R_{i} [\int_{t - σ_{i}}^{t} E x (s) d s] \\ \leq σ_{i} \frac{λ_{m a x} (𝒜_{i}^{2})}{λ_{m i n} (R_{i})} {\int_{t - σ_{i}}^{t} E x^{T} (s) R_{i} x (s) d s} \\ \leq σ_{i} \frac{λ_{m a x} (𝒜_{i}^{2})}{λ_{m i n} (R_{i})} E V_{4} (x_{t}, i, t) \\ \leq σ \frac{λ_{m a x} (𝒜_{i}^{2})}{λ_{m i n} (R_{i})} E V (x_{t}, i, t) \\ \leq σ \frac{λ_{m a x} (𝒜_{i}^{2})}{λ_{m i n} (R_{i})} E V (x_{0}, 𝒜 (0),0), t \geq 0. \end{array}$ (3.32)

Similarly, it follows from the definition of $V_{1} (x_{t}, i, t)$ that

$E {∥ x (t) - 𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s ∥}^{2} = E {[𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s]}^{T} [𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s]$

$\leq \frac{E V_{1} (x_{t}, i, t)}{λ_{m i n} (P_{i})}$

$\leq \frac{E V (x_{t}, i, t)}{λ_{m i n} (P_{i})}$

$\leq \frac{E V (x_{0}, \nabla (0),0)}{λ_{m i n} (P_{i})}, t \geq 0.$

Hence, it can be obtained that

$\begin{array}{l} E {∥ x (t) ∥}^{2} = E {∥ x (t) - 𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s + 𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s ∥}^{2} \\ \leq 2 E {∥ A_{i} \int_{t - σ_{i}}^{t} x (s) d s ∥}^{2} + 2 E {∥ x (t) - 𝒜_{i} \int_{t - σ_{i}}^{t} x (s) d s ∥}^{2} \\ \leq 2 σ \frac{λ_{m a x} (𝒜_{i}^{2})}{λ_{m i n} (R_{i})} E V (x_{0}, \nabla (0),0) + 2 \frac{E V (x_{0}, \nabla (0),0)}{λ_{m i n} (P_{i})} < \infty, t \geq 0, \end{array}$ (3.33)

$\begin{array}{l} E V (x_{0}, \nabla (0),0) \\ \leq {2 λ_{\underset{i \in S}{m a x}} (P_{i}) (1 + σ_{i}^{2} \underset{i \in S}{m a x} 𝒜_{i}) + τ \underset{i \in S}{m a x} {λ_{m a x} (Q_{i})} + τ^{2} λ_{m a x} (Q) + τ \underset{i \in S}{m a x} {λ_{m a x} (W_{i})} \\ + τ^{2} λ_{m a x} (W) + σ \underset{i \in S}{m a x} {λ_{m a x} (R_{i})} + σ^{2} λ_{m a x} (R) + h \underset{i \in S}{m a x} {λ_{m a x} (S_{i})} + h^{2} λ_{m a x} (S) \\ + d^{2} \underset{i \in S}{m a x} {λ_{m a x} (V_{i})} + d^{3} λ_{m a x} (V) + τ^{2} \underset{i \in S}{m a x} {λ_{m a x} (U_{i})} + τ^{3} λ_{m a x} (U) + τ^{2} \underset{i \in S}{m a x} {λ_{m a x} (X_{i})} \\ + τ^{3} λ_{m a x} (X) + h^{2} \underset{i \in S}{m a x} {λ_{m a x} (Y_{i})} + h^{3} λ_{m a x} (Y) + τ^{3} \underset{i \in S}{m a x} {λ_{m a x} (T_{i})} + τ^{4} λ_{m a x} (T) \\ + h^{3} \underset{i \in S}{m a x} {λ_{m a x} (Z_{i})} + h^{4} λ_{m a x} (Z) + σ^{3} \underset{i \in S}{m a x} {λ_{m a x} (L_{i})} + σ^{4} λ_{m a x} (L) + σ^{5} λ_{m a x} (M)} < \infty . \end{array}$ (3.34)

From (3.33) and (3.34), we get that the solution of the system (2.3) is locally passive. Then the solutions $x (t) = x (t,0, φ)$ of system (2.3) is bounded on $[0, \infty)$ . The solution $x (t)$ on $[0, \infty)$ is uniformly continuous because $\dot{x} (t)$ is bounded on $[0, \infty)$ . Further, from the equation (3.31), the following holds:

$λ_{m i n} (Φ) \int_{0}^{t} E [x^{T} (s) x (s)] d s \leq 𝔼 [ℒ V (x_{t}, i, t)] - \int_{0}^{t} 𝔼 [η^{T} (s) Φ η (s)] d s$

$\leq 𝔼 [ℒ V (x_{0}, \nabla (0),0)] + 2 \int_{0}^{t} 𝔼 [y^{T} (s) u (s)] d s$

$+ γ \int_{0}^{t} 𝔼 [u^{T} (s) u (s)] d s$

$< \infty, t \geq 0.$

From Barbalats lemma [30], $𝔼 [{‖ x (t) ‖}^{2}] \to 0$ as $t \to \infty$ holds. Hence the proof is complete.

Remark 1. Recently, studies on passivity analysis for neural networks of neutral type with Markovian jumping parameters and time delay in the leakage term were conducted in [5]. By constructing proper Lyapunov–Krasovskii functional, new delay-dependent passivity conditions are derived in terms of LMIs and it can easily be checked using standard numerical packages. Moreover, it is well known that the passivity behaviour of neural networks is very sensitive to the time delay in the leakage term. Triple and quadruple integrals have not been taken into account to derive the passivity conditions in [5]. Mode-dependent time delays were not included in [5]. Very recently, a mode-dependent approach is proposed by constructing a novel Lyapunov functional, where some terms involving triple or quadruple integrals are taken into account to study the state estimation problem in [22]. Motivated by this reason, we have introduced improved Lyapunov–Krasovskii functional with triple and quadruple integrals for deriving the reported stability results in this paper. Based on this discussion, our results will give less conservative results than those studied in [5], [22].

4. Numerical example

In this section, a numerical example is provided to demonstrate the validity of the proposed theorems.

Example 1. Consider a 2-D Mode-dependent Markov jump NNNs with mixed time-delays (2.3) with the following parameters

$𝒜_{1} = [\begin{matrix} 8.4 & 0 \\ 0 & 9 \end{matrix}], 𝒜_{2} = [\begin{matrix} 7.8 & 0 \\ 0 & 8.5 \end{matrix}], ℬ_{1} = [\begin{matrix} - 0.21 & - 0.19 \\ - 0.24 & 0.1 \end{matrix}], ℬ_{2} = [\begin{matrix} 0.9 & - 0.9 \\ 0.5 & - 0.8 \end{matrix}], 𝒞_{1} = [\begin{matrix} - 0.09 & - 0.2 \\ 0.2 & 0.1 \end{matrix}], 𝒞_{2} = [\begin{matrix} 0.1 & 0.1 \\ 0.2 & 0.3 \end{matrix}], 𝒟_{1} = [\begin{matrix} - 0.2 & 0 \\ 0.2 & - 0.09 \end{matrix}], 𝒟_{2} = [\begin{matrix} 0.1 & 0 \\ 0.5 & - 0.1 \end{matrix}], ℰ_{1} = [\begin{matrix} - 0.5 & 0 \\ 0 & - 0.5 \end{matrix}], ℰ_{2} = [\begin{matrix} 0.1 & - 0.02 \\ - 0.2 & 0.07 \end{matrix}], {\hat{L}}_{1} = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}], {\hat{L}}_{2} = [\begin{matrix} 0.25 & 0 \\ 0 & 0.25 \end{matrix}] .$

Take $ℋ_{1} (s) = ℋ_{2} (s) = t a n h (s)$ , $τ_{1} (t) = τ_{2} (t) = h_{1} (t) = h_{2} (t) = d_{1} (t) = d_{2} (t) = 0.1 c o s t + 0.4$ , $σ_{1} = σ_{2} = 0.1$ , $τ_{µ_{1}} = τ_{µ_{2}} = h_{µ_{1}} = h_{µ_{2}} = d_{µ_{1}} = d_{µ_{2}} = 0.1$ . $Γ = [\begin{matrix} - 7 & 7 \\ 6 & - 6 \end{matrix}] .$

By using the MATLAB LMI toolbox, we can obtain the following feasible solution for the LMIs (3.1)–(3.4):

$P_{1} = [\begin{matrix} 0.0157 & 0.0008 \\ 0.0008 & 0.0143 \end{matrix}], P_{2} = [\begin{matrix} 0.0068 & 0.0004 \\ 0.0004 & 0.0060 \end{matrix}], Q_{11} = [\begin{matrix} 0.2401 & 0.0007 \\ 0.0007 & 0.2592 \end{matrix}], Q_{12} = [\begin{matrix} 0.3770 & - 0.0192 \\ - 0.0192 & 0.3992 \end{matrix}], Q_{21} = [\begin{matrix} - 0.4101 & 0.0043 \\ 0.0201 & - 0.4457 \end{matrix}], Q_{22} = [\begin{matrix} - 0.6428 & 0.0773 \\ 0.0231 & - 0.6959 \end{matrix}], Q_{31} = [\begin{matrix} 1.3786 & 0.0133 \\ 0.0133 & 1.4063 \end{matrix}], Q_{32} = [\begin{matrix} 1.9101 & - 0.0284 \\ - 0.0284 & 1.9903 \end{matrix}], W_{1} = [\begin{matrix} 0.1039 & 0.0088 \\ 0.0088 & 0.1018 \end{matrix}], W_{2} = [\begin{matrix} 0.1517 & 0.0121 \\ 0.0121 & 0.1487 \end{matrix}], R_{1} = [\begin{matrix} 0.1758 & 0.0160 \\ 0.0160 & 0.1721 \end{matrix}], R_{2} = [\begin{matrix} 0.5675 & 0.0333 \\ 0.0333 & 0.5563 \end{matrix}], S_{1} = [\begin{matrix} 1.0817 & - 0.0075 \\ - 0.0075 & 0.7204 \end{matrix}], S_{2} = [\begin{matrix} 1.2231 & 0.0093 \\ 0.0093 & 0.9256 \end{matrix}], V_{1} = [\begin{matrix} 10.7113 & 0.5692 \\ 0.5692 & 9.9973 \end{matrix}], V_{2} = [\begin{matrix} 10.8483 & 0.5953 \\ 0.5953 & 10.2619 \end{matrix}], U_{1} = [\begin{matrix} 0.5120 & 0.0419 \\ 0.0419 & 0.5056 \end{matrix}], U_{2} = [\begin{matrix} 0.6238 & 0.0492 \\ 0.0492 & 0.6133 \end{matrix}], X_{1} = [\begin{matrix} 21.0470 & - 0.1153 \\ - 0.1153 & 21.2070 \end{matrix}], X_{2} = [\begin{matrix} 18.5642 & - 0.3139 \\ - 0.3139 & 18.6620 \end{matrix}], Y_{1} = [\begin{matrix} 27.5946 & - 0.0019 \\ - 0.0019 & 26.6557 \end{matrix}], Y_{2} = [\begin{matrix} 28.4115 & 0.1096 \\ 0.1096 & 27.4131 \end{matrix}], T_{1} = [\begin{matrix} 3.0928 & 0.0943 \\ 0.0943 & 3.0505 \end{matrix}], T_{2} = [\begin{matrix} 3.8182 & 0.0431 \\ 0.0431 & 3.7941 \end{matrix}], Z_{1} = [\begin{matrix} 3.0867 & 0.1080 \\ 0.1080 & 3.0591 \end{matrix}], Z_{2} = [\begin{matrix} 3.8647 & 0.0504 \\ 0.0504 & 3.8446 \end{matrix}], L_{1} = [\begin{matrix} 0.0068 & 0.0001 \\ 0.0001 & 0.0068 \end{matrix}], L_{2} = [\begin{matrix} 0.0059 & 0.0004 \\ 0.0004 & 0.0062 \end{matrix}], K_{1} = [\begin{matrix} 1.3461 & - 0.1410 \\ - 0.1410 & 1.3280 \end{matrix}], K_{2} = [\begin{matrix} 0.0919 & 0.0135 \\ 0.0135 & 0.0907 \end{matrix}], Q_{1} = [\begin{matrix} 4.0913 & - 0.1850 \\ - 0.1850 & 4.4495 \end{matrix}], Q_{2} = [\begin{matrix} - 6.6332 & 0.7686 \\ 0.3444 & - 7.3948 \end{matrix}], γ = 0.2272.$

Continuing in this way, the remaining feasible matrices are obtained. This shows that the given system (2.3) is globally passive in the mean square.

5. Conclusion

In this paper, passivity analysis of Markovian jumping NNNs with time delays in the leakage term is considered. Delay-mode-dependent passivity conditions are derived by taking the inherent characteristic of such kinds of NNs into account. An improved LKF, with the triple integral terms and quadruple integrals, is constructed and the results are derived in terms of linear matrix inequalities. The information of the mode-dependent of all delays have been taken into account in the constructed LKF and derived novel stability criterion. Theoretical results are validated through a numerical example.

Acknowledgments

J. Alzabut is thankful to Prince Sultan University and OSTİM Technical University for their endless support during the achievement of this paper.

Conflicts of interest

Availability of data and materials: Not applicable. On behalf of all authors, the corresponding author declares that they have no conflict of interest.

References

[1]	Faydasicok O, Arik S (2012) Robust stability analysis of a class of neural networks with discrete time delays. Neural Net 29: 52-59. https://doi.org/10.1016/j.neunet.2012.02.001
[2]	Wu A, Zeng Z (2012) Dynamic behaviors of memristor-based recurrent neural networks with time-varying delays. Neural Net 36: 1-10. https://doi.org/10.1016/j.neunet.2012.08.009
[3]	Zhang H, Liu Z, Huang GB, et al. (2009) Novel weighting-delay-based stability criteria for recurrent neural networks with time-varying delay. IEEE T Neur Net 21: 91-106. https://doi.org/10.1109/TNN.2009.2034742
[4]	Huang H, Feng G, Cao J (2010) State estimation for static neural networks with time-varying delay. Neural Net 23: 1202-1207. https://doi.org/10.1016/j.neunet.2010.07.001
[5]	Balasubramaniam P, Nagamani G, Rakkiyappan R (2011) Passivity analysis for neural networks of neutral type with Markovian jumping parameters and time delay in the leakage term. Commun Nonlinear Sci Numer Simul 16: 4422-4437. https://doi.org/10.1016/j.cnsns.2011.03.028
[6]	Ahn CK (2012) Switched exponential state estimation of neural networks based on passivity theory. Nonlinear Dynam 67: 573-586. https://doi.org/10.1007/s11071-011-0010-x
[7]	Yu W, Li X (2007) Passivity analysis of dynamic neural networks with different time-scales. Neural Process Lett 25: 143-155. https://doi.org/10.1007/s11063-007-9034-0
[8]	Zhu S, Shen Y (2011) Passivity analysis of stochastic delayed neural networks with Markovian switching. Neurocomputing 74: 1754-1761. https://doi.org/10.1016/j.neucom.2011.02.010
[9]	Haddad WM, Bailey JM, Hovakimyan N (2005) Passivity-based neural network adaptive output feedback control for nonlinear nonnegative dynamical systems. IEEE T Neur Net 16: 387-398. https://doi.org/10.1109/TNN.2004.841782
[10]	Mahmoud M, Xia Y (2011) Improved exponential stability analysis for delayed recurrent neural networks. J Franklin Inst 348: 201-211. https://doi.org/10.1016/j.jfranklin.2010.11.002
[11]	Xu S, Zheng WX, Zou Y (2009) Passivity analysis of neural networks with time-varying delays. IEEE T Circuits II 56: 325-329. https://doi.org/10.1109/TCSII.2009.2015399
[12]	Zhang Z, Mou S, Lam J, et al. (2010) New passivity criteria for neural networks with time-varying delay. Neural Net 22: 864-868. https://doi.org/10.1016/j.neunet.2009.05.012
[13]	Mala N, Sudamani Ramaswamy AR (2013) Passivity analysis of Markovian jumping neural networks with leakage time varying delays. J Comput Methods Phys 2013: 172906. https://doi.org/10.1155/2013/172906
[14]	Wu ZG, Park JH, Su H, et al. (2012) New results on exponential passivity of neural networks with time-varying delays. Nonlinear Anal Real World Appl 13: 1593-1599. https://doi.org/10.1016/j.nonrwa.2011.11.017
[15]	Balasubramaniam P, Lakshmanan S (2009) Delay-range dependent stability criteria for neural networks with Markovian jumping parameters. Nonlinear Anal Hybri 3: 749-756. https://doi.org/10.1016/j.nahs.2009.06.012
[16]	Zhu Q, Cao J (2010) Robust exponential stability of Markovian jump impulsive stochastic Cohen-Grossberg neural networks with mixed time delays. IEEE T Neur Net 21: 1314-1325. https://doi.org/10.1109/TNN.2010.2054108
[17]	Zhu Q, Cao J (2010) Stability analysis for stochastic neural networks of neutral type with both Markovian jumping parameters and mixed time delays. Neurocomputing 73: 2671-2680. https://doi.org/10.1016/j.neucom.2010.05.002
[18]	Ma Q, Xu S, Zou Y, et al. (2011) Stability of stochastic Markovian jumping neural networks with mode-dependent delays. Neurocomputing 74: 21570-2163. https://doi.org/10.1016/j.neucom.2011.01.016
[19]	Tian J, Li Y, Zhao J, et al. (2012) Delay-dependent stochastic stability criteria for Markovian jumping neural networks with mode-dependent time-varying delays and partially known transition rates. Appl Math Comput 218: 5769-5781. https://doi.org/10.1016/j.amc.2011.11.087
[20]	Huang H, Huang T, Chen X (2012) Global exponential estimates of delayed stochastic neural networks with Markovian switching. Neural Net 36: 136-145. https://doi.org/10.1016/j.neunet.2012.10.002
[21]	Zhu Q, Zhou W, Tong D (2013) Adaptive synchronization for stochastic neural networks of neutral-type with mixed time-delays. Neurocomputing 99: 477-485. https://doi.org/10.1016/j.neucom.2012.07.013
[22]	Huang H, Huang T, Chen X (2013) A mode-dependent approach to state estimation of recurrent neural networks with Markovian jumping parameters and mixed delays. Neural Net 46: 50-61. https://doi.org/10.1016/j.neunet.2013.04.014
[23]	Niu Y, Lam J, Wang X (2004) Sliding-mode control for uncertain neutral delay systems. IEE Proc Part D: Control Theory Appl 151: 38-44. https://doi.org/10.1049/ip-cta:20040009
[24]	Cheng CJ, Liao TL, Yan JJ, et al. (2006) Globally asymptotic stability of a class of neutral-type neural networks with delays. IEEE T Syst. Man Cybern. Part B 36: 1191-1195. https://doi.org/10.1109/TSMCB.2006.874677
[25]	Lin X, Zhang X, Wang Y (2013) Robust passive filtering for neutral-type neural networks with time-varying discrete and unboundeddistributed delays. J Frank Inst 350: 966-989. https://doi.org/10.1016/j.jfranklin.2013.01.021
[26]	Rakkiyappan R, Balasubramaniam P (2008) LMI conditions for global asymptotic stability results for neutral-type neural networks with distributed time delays. Appl Math Comput 204: 317-324. https://doi.org/10.1016/j.amc.2008.06.049
[27]	Park JH, Kwon OM, Lee SM (2008) LMI optimization approach on stability for delayed neural network of neutral-type. J Comput Appl Math 196: 224-236. https://doi.org/10.1016/j.amc.2007.05.047
[28]	Mahmoud MS, Ismail A (2010) Improved results on robust exponential stability criteria for neutral type delayed neural networks. Appl Math Comput 217: 3011-3019. https://doi.org/10.1016/j.amc.2010.08.034
[29]	Zhang H, Dong M, Wang Y, et al. (2010) Stochastic stability analysis of neutral-type impulsive neural networks with mixed time-varying delays and Markovian jumping. Neurocomputing 73: 2689-2695. https://doi.org/10.1016/j.neucom.2010.04.016
[30]	Gopalsamy K (1992) Stability and Oscillations in Delay Differential Equations of Population Dynamics. Dordrecht: Kluwer Academic publishers.
[31]	Li C, Huang T (2009) On the stability of nonlinear systems with leakage delay. J Frank Inst 346: 366-377. https://doi.org/10.1016/j.jfranklin.2008.12.001
[32]	Fu J, Zhang H, Ma T, et al. (2010) On passivity analysis for stochastic neural networks with interval time-varying delay. Neurocomputing 73: 795-801. https://doi.org/10.1016/j.neucom.2009.10.010
[33]	Gu K, Chen J, Kharitonov VL (2003) Stability of Time-delay Systems. Massachusetts: Springer Science and Business Media.
[34]	Park P, Ko JW, Jeong C (2011) Reciprocally convex approach to stability of systems with time-varying delays. Automatica 47: 235-238. https://doi.org/10.1016/j.automatica.2010.10.014
[35]	Wang J, Jiang H, Hu C, et al. (2021) Exponential passivity of discrete-time switched neural networks with transmission delays via an event-triggered sliding mode control. Neural Net 143: 271-282. https://doi.org/10.1016/j.neunet.2021.06.014
[36]	Padmaja N, Balasubramaniam P (2021) New delay and order-dependent passivity criteria for impulsive fractional-order neural networks with switching parameters and proportional delays. Neurocomputing 454: 113-123. https://doi.org/10.1016/j.neucom.2021.04.099
[37]	Yan M, Jian J, Zheng S (2021) Passivity analysis for uncertain BAM inertial neural networks with time-varying delays. Neurocomputing 435: 114-125. https://doi.org/10.1016/j.neucom.2020.12.073
[38]	Dong Y, Wang H (2020) Robust output feedback stabilization for uncertain discrete-time stochastic neural networks with time-varying delay. Neural Process Lett 51: 83-103. https://doi.org/10.1007/s11063-019-10077-x
[39]	Sun B, Cao Y, Guo Z, et al. (2020) Synchronization of discrete-time recurrent neural networks with time-varying delays via quantized sliding mode control. Appl Math Comput 375: 125093. https://doi.org/10.1016/j.amc.2020.125093
[40]	Miaadi F, Li X (2021) Impulse-dependent settling-time for finite time stabilization of uncertain impulsive static neural networks with leakage delay and distributed delays. Math Comput Simulat 182: 259-276. https://doi.org/10.1016/j.matcom.2020.11.003
[41]	Yang H, Wang Z, Shen Y, et al. (2021) Event-triggered state estimation for Markovian jumping neural networks: On mode-dependent delays and uncertain transition probabilities. Neurocomputing 424: 226-235. https://doi.org/10.1016/j.neucom.2020.10.050
[42]	Xu G, Bao H (2020) Further results on mean-square exponential input-to-state stability of time-varying delayed BAM neural networks with Markovian switching. Neurocomputing 376: 191-201. https://doi.org/10.1016/j.neucom.2019.09.033
[43]	Lin WJ, He Y, Zhang CK, et al. (2020) Stochastic finite-time H_∞ state estimation for discrete-time semi-Markovian jump neural networks with time-varying delays. IEEE T Neural Net Learn Syst 31: 5456-5467. https://doi.org/10.1109/TNNLS.2020.2968074
[44]	Xiao J, Zeng Z (2020) Finite-time passivity of neural networks with time varying delay. J Frank Inst 357: 2437-2456. https://doi.org/10.1016/j.jfranklin.2020.01.023
[45]	Liao Y, Wang X, Blaabjerg F (2020) Passivity-based analysis and design of linear voltage controllers for voltage-source converters. IEEE Open J Ind Electron 1: 114-126. https://doi.org/10.1109/OJIES.2020.3001406
[46]	Wang Y, Cao Y, Guo Z, et al. (2020) Passivity and passification of memristive recurrent neural networks with multi-proportional delays and impulse. Appl Math Comput 369: 124838. https://doi.org/10.1016/j.amc.2019.124838
[47]	Grienggrai R, Sriraman R (2021) Robust passivity and stability analysis of uncertain complex-valued impulsive neural networks with time-varying delays. Neural Process Lett 53: 581-606. https://doi.org/10.1007/s11063-020-10401-w
[48]	Yan M, Jian J, Zheng S (2021) Passivity analysis for uncertain BAM inertial neural networks with time-varying delays. Neurocomputing 435: 114-125. https://doi.org/10.1016/j.neucom.2020.12.073
[49]	Gunasekaran N, Ali MS (2021) Design of stochastic passivity and passification for delayed BAM neural networks with Markov jump parameters via non-uniform sampled-data control. Neural Process Lett 53: 391-404. https://doi.org/10.1007/s11063-020-10394-6
[50]	Hu X, Nie L (2018) Exponential stability of nonlinear systems with impulsive effects and disturbance input. Adv Differ Equ 2018: 354. https://doi.org/10.1186/s13662-018-1798-1
[51]	Botmart T, Noun S, Mukdasai K, et al. (2021) Robust passivity analysis of mixed delayed neural networks with interval nondifferentiable time-varying delay based on multiple integral approach. AIMS Math 6: 2778-2795. https://doi.org/10.3934/math.2021170
[52]	Maharajan C, Raja R, Cao J, et al. (2018) Novel results on passivity and exponential passivity for multiple discrete delayed neutral-type neural networks with leakage and distributed time-delays. Chaos Soliton Fract 115: 268-282. https://doi.org/10.1016/j.chaos.2018.07.008
[53]	Wang J, Jiang H, Hu C, et al. (2021) Exponential passivity of discrete-time switched neural networks with transmission delays via an event-triggered sliding mode control. Neural Net 143: 271-282. https://doi.org/10.1016/j.neunet.2021.06.014
[54]	Zhang XM, Han QL, Ge X, et al. (2018) An overview of recent developments in Lyapunov–Krasovskii functionals and stability criteria for recurrent neural networks with time-varying delays. Neurocomputing 313: 392-401. https://doi.org/10.1016/j.neucom.2018.06.038
[55]	Zhang XM, Han QL, Ge X, et al. (2021) Delay-variation-dependent criteria on extended dissipativity for discrete-time neural networks with time-varying delay. IEEE Trans Neural Netw Learn Syst 34: 1578-1587. https://doi.org/10.1109/TNNLS.2021.3105591
[56]	Wang J, Xia J, Shen H, et al. (2020) ℋ_∞ Synchronization for fuzzy Markov jump chaotic systems With piecewise-constant transition probabilities subject to PDT switching rule. IEEE T Fuzzy Syst 29: 3082-3092. https://doi.org/10.1109/TFUZZ.2020.3012761
[57]	Shen H, Hu X, Wang J, et al. (2021) Non-Fragile ℋ_∞ Synchronization for Markov jump singularly perturbed coupled neural networks subject to double-layer switching regulation. IEEE T Neur Net Lear . https://doi.org/10.1109/tnnls.2021.3107607
[58]	Ghanem HS, Al-makhlasawy RM, El-shafai W, et al. (2022) Wireless modulation classification based on Radon transform and convolutional neural networks. J Amb Intel Hub Comp : 922. https://doi.org/10.1007/s12652-021-03650-7
[59]	Hammad HA, Abdeljawad T (2022) Quadruple fixed-point techniques for solving integral equations involved with matrices and the Markov process in generalized metric spaces. J Inequal Appl 2022: 44. https://doi.org/10.1186/s13660-022-02780-6
[60]	Atitallah SB, Driss M, Almomani I (2022) A novel detection and multi-classification approach for IoT-malware using random forest voting of fine-tuning convolutional neural networks. Sensors 22: 4302. https://doi.org/10.3390/s22114302
[61]	Aadhithiyan S, Raja R, Zhu Q, et al. (2021) Exponential synchronization of nonlinear multi-weighted complex dynamic networks with hybrid time varying delays. Neural Process Lett 53: 1035-1063. https://doi.org/10.1007/s11063-021-10428-7

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Biophysics

1.1 2.4

Metrics

Article views(1108) PDF downloads(110) Cited by(0)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

AIMS Biophysics

Passivity analysis for Markovian jumping neutral type neural networks with leakage and mode-dependent delay

Related Papers:

Abstract

1. Introduction

2. Problem formulation and preliminaries

3. Main results

4. Numerical example

5. Conclusion

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Catalog

AIMS Biophysics

Passivity analysis for Markovian jumping neutral type neural networks with leakage and mode-dependent delay

Related Papers:

Abstract

1. Introduction

2. Problem formulation and preliminaries

3. Main results

4. Numerical example

5. Conclusion

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog