Stability Analysis of Bidirectional Associative Memory Neural Networks with Time-Varying Delays via Second-Order Reciprocally Convex Approach

Chandran, Kalaivani; Kuppusamy, Renuga; Vaitheeswaran, Vembarasan

doi:10.3390/sym17111852

Open AccessArticle

Stability Analysis of Bidirectional Associative Memory Neural Networks with Time-Varying Delays via Second-Order Reciprocally Convex Approach

by

Kalaivani Chandran

^1,*

,

Renuga Kuppusamy

²

and

Vembarasan Vaitheeswaran

³

¹

Department of Mathematics, Sri Sivasubramaniya Nadar College of Engineering, Kalavakkam 603110, Tamil Nadu, India

²

Department of Mathematics, St. Joseph’s Institute of Technology, OMR, Chennai 600119, Tamil Nadu, India

³

School of Sciences and Humanities, Shiv Nadar University Chennai, Kalavakkam 603110, Tamil Nadu, India

^*

Author to whom correspondence should be addressed.

Symmetry 2025, 17(11), 1852; https://doi.org/10.3390/sym17111852

Submission received: 3 September 2025 / Revised: 7 October 2025 / Accepted: 16 October 2025 / Published: 3 November 2025

(This article belongs to the Section Mathematics)

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Abstract

This research examines the Lyapunov-based criterion for global asymptotic stability of Bidirectional Associative Memory (BAM) neural networks that have mixed-interval time-varying delays. Using a second-order reciprocally convex approach, this paper introduces a novel stability criterion for BAM neural networks with time delays. The literature has recently incorporated a few triple integral expressions in the Lyapunov–Krasovskii functional to lessen conservatism in the analysis of system stability with interval time-varying delays using a second-order reciprocally convex combination strategy. This research work establishes the negative definiteness of the Lyapunov–Krasovskii functional derivative and is formulated using Linear Matrix Inequalities (LMIs). The effectiveness of the proposed result is demonstrated through numerical examples.

Keywords:

Bidirectional Associative Memory; neural networks; time-varying delays; Lyapunov-Krasovskii functional; Linear Matrix Inequality; second-order reciprocally convex combination

1. Introduction

The field of neural networks (NNs) has vast applications in the domains of pattern recognition, signal processing, associative memory, optimization, etc., Furthermore, several NN architectures, including Hopfield Networks [1], Fuzzy Neural Networks with Markovian jumps [2,3] and Bidirectional Associative Memory (BAM) models [4,5,6], have undergone extensive analysis. By extending the single-layer auto-associative Hebbian network into a two-layer, pattern-based hetero-associative network, the BAM framework, which was first presented by Kosko [7,8], allows reconstructing whole unambiguous patterns from incomplete or noisy inputs [9]. In this design, every neuron in each layer is completely connected to all neurons in the opposing layer. A substantial body of work has explored the stability of BAM networks, deriving suitable conditions for both exponential and asymptotic stability [10,11,12]. Consequently, BAM architectures demonstrate superior performance in memory storage and pattern association.

Time delays play a crucial role in synchronization control, sliding mode control, industrial systems, ecological modeling, and networked environments, such as the Internet, among others [13,14,15]. These delays often degrade system performance and can potentially lead to instability. One of the key objectives in this area is to identify the maximum allowable time in which the given system continues to maintain asymptotic stability [13].

Time-varying delays will inevitably be imposed during the deployment of artificial NNs due to the intrinsic communication time between neurons and the limiting slew rate of amplifiers [16,17,18]. These signal transmission delays can trigger unwanted behaviors, including oscillations and instability. Consequently, the study of stability of neural network under time-varying delays has become a central research focus, with many significant contributions reported in the literature [19]. Therefore, it is essential to establish a proper delay-dependent stability criterion to accurately assess how delays impact NN stability. Consequently, delayed neural networks’ (DNNs) [20] delay-dependent stability has been a major focus of research in recent decades, leading to significant progress in the field.

The main objective of studying time delay systems for system stability is to develop less conservative stability conditions. One key factor in this is the upper limit of the delay, which helps to measure the conservativeness of the stability condition [21]. For systems where delay affects the outcome, the Lyapunov–Krasovskii functionals (LKFs) [22,23,24] method is widely considered one of the most effective tools for analyzing stability. The primary challenge in this approach is how to estimate the integral terms accurately so that the final stability condition can be written with respect to LMIs [25,26] which are easier to work with mathematically. To address this, a new and improved LKF has been introduced, which includes a triple integral term for better accuracy.

Before 2013, Jensen’s inequality was one of the most widely used tools for analyzing system stability involving time delays [27]. New techniques, such as inequality employing an auxiliary function [28,29], and an integral bound derived using slack matrices [20], have been introduced to enhance results and reduce conservative estimates. Later, a more general form called the Bessel–Legendre inequality was proposed [30,31], which helped to reduce the error in estimating single integral terms. However, it has become increasingly difficult to make further improvements in reducing conservatism by just refining these single-integral-based methods.

Another way to improve the stability conditions is by designing more advanced Lyapunov–Krasovskii functionals (LKFs) [23] than the basic ones used in earlier studies. Some of the improved methods include delay-partitioning LKFs [32], multiple-integral LKFs [33,34,35], and augmented LKFs. These methods often use free-weighting matrices [36] to include more detailed information about the system’s state and the time delay. These techniques have helped reduce conservatism to some extent, but as the complexity of LKFs increases, the improvements become smaller, while the computational cost becomes significantly higher. Still, among all, Jensen’s inequality is considered very effective and widely used.

Furthermore, systems with time-varying delays have been studied for stability using the reciprocally convex approach [37]. Combining this technique with Jensen’s inequality [27] helps reduce conservatism even further. In the field of stability analysis, a new technique called the second-order reciprocally convex combination method [38,39] was developed to improve upon the limitations of Jensen’s method [40,41]. This new approach provides a more accurate and less conservative way to estimate the upper bounds of quadratic terms, which is crucial for determining the stability of a system.

From the above discussion, we will propose a novel stability criterion for BAM NNs along with time delays in this research paper by combining the reciprocally convex combination approach [37,38,42] and second-order reciprocally convex combination technique [39] for the newly constructed LKF. It should be noted that our method does not require the monotonicity, differentiability, and boundedness of the activation function. Our findings show that this technique can give a less conservative condition for ensuring stability. Several examples are also provided to demonstrate how well the method works in practice.

This paper is organized into Section 1, Section 2, Section 3, Section 4 and Section 5. Section 1 outlines the problem statement and presents the necessary background information. Section 2 gives conditions, using LMIs, that ensure the BAM NN’s global asymptotic stability behaviour. Section 3 presents further LMI conditions for the globally asymptotic stability of these networks under time-delay conditions. Section 4 shows practical examples that support our theoretical work. Section 5 gives the conclusion.

Notations: In this paper, any positive definite matrix

X

is represented as

X > 0

.

M^{T}

is the transpose of

M

;

I

is the identity matrix with suitable dimension;

R^{n}

represents the n-D Cartesian space,

R^{n \times n}

represents the set of all

n \times n

real matrices, and ∗ indicates the symmetric block in one symmetric matrix.

2. Problem Overview and Basic Notions

Let us examine the following BAM NNs for stability, incorporating the time-varying delays:

\begin{matrix} {\dot{X}}_{r} (t) & = & - {a_{1}}_{r} X_{r} (t) + \sum_{s = 1}^{n} {b_{1}}_{s r} f_{s} (У_{s} (t)) + \sum_{s = 1}^{n} {c_{1}}_{s r} f_{s} (У_{s} (t - ς_{s r} (t))) + \sum_{s = 1}^{n} {d_{1}}_{s r} \int_{{t - η_{s r}}_{2} (t)}^{{t - η_{s r}}_{1} (t)} f_{s} (У_{s} (ϰ)) d ϰ, \\ {\dot{У}}_{s} (t) & = & - {a_{2}}_{s} У_{s} (t) + \sum_{r = 1}^{n} {b_{2}}_{r s} g_{r} (X_{r} (t)) + \sum_{r = 1}^{n} {c_{2}}_{r s} g_{r} (X_{r} (t - ϱ_{r s} (t))) + \sum_{r = 1}^{n} {d_{2}}_{r s} \int_{{t - γ r s}_{2} (t)}^{{t - γ_{r s}}_{1} (t)} g_{r} (X_{r} (ϰ)) d ϰ, \end{matrix}

for

t = 1, 2, \dots

, where

X_{r} (t)

represents the state of the

r^{t h}

neuron at any time

t

, and similarly

У_{s} (t)

represents the state of the

s^{t h}

neuron at any time

t

;

{a_{1}}_{r}, {a_{2}}_{r}

describes the stability of internal neuron processes.

{b_{1}}_{s r}, {b_{2}}_{r s}

are connection weights, and

{c_{1}}_{s r}, {c_{2}}_{r s}, {d_{1}}_{s r}, {d_{2}}_{r s}

are delayed connection weights.

g_{r}, f_{s}

indicates the activation functions of the

r^{t h}

neuron and

s^{t h}

neuron, respectively.

The vector form of the above system is

\begin{matrix} \dot{X} (t) & = & - A_{1} X (t) + B_{1} f (У (t)) + C_{1} f (У (t - ς (t))) + D_{1} \int_{t - η_{2} (t)}^{t - η_{1} (t)} f (У (ϰ)) d ϰ \\ \dot{У} (t) & = & - A_{2} У (t) + B_{2} g (X (t)) + C_{2} g (X (t - ϱ (t))) + D_{2} \int_{t - γ_{2} (t)}^{t - γ_{1} (t)} g (X (ϰ)) d ϰ, \end{matrix}

(1)

The delays

ς (t), ϱ (t), η (t), γ (t)

satisfy

ς (t) \in [ς_{1}, ς_{2}], 0 \leq ς_{1} \leq ς_{2}; \dot{ς} (t) \leq υ_{1}; ϱ (t) \in [ϱ_{1}, ϱ_{2}], 0 \leq ϱ_{1} \leq ϱ_{2}; \dot{ϱ} (t) \leq υ_{2}; 0 \leq η_{1} \leq η_{1} (t) \leq η_{2} (t) \leq η_{2}; 0 \leq γ_{1} \leq γ_{1} (t) \leq γ_{2} (t) \leq γ_{2}; ϱ_{12} = ϱ_{2} - ϱ_{1}; ϱ_{12}^{2} = {(ϱ_{2} - ϱ_{1})}^{2}; ς_{12} = ς_{2} - ς_{1}; ς_{12}^{2} = {(ς_{2} - ς_{1})}^{2};

where

ς_{1}, ς_{2}, η_{1}, η_{2}, ϱ_{1}, ϱ_{2}, γ_{1}, γ_{2}

are non-negative real constants with initial parameters

X (s) = ϖ (s), s \in [- K_{1}, 0]; У (s) = φ (s), s \in [- K_{2}, 0]

where

K_{1} = m a x {ϱ_{2}, γ_{2}}, K_{2} = m a x {ς_{2}, η_{2}}

and

A_{1} = [{a_{1}}_{i}]

and

A_{2} = [{a_{2}}_{j}]

are diagonal matrices with

{a_{1}}_{i} > 0, i = 1 t o n, {a_{2}}_{j} > 0, j = 1 t o n,

B_{1}, B_{2}

are the connection weight matrices,

C_{1},

and

C_{2}, D_{1}, D_{2}

are delayed connection weight matrices.

By making the following assumptions, we achieve the stability conditions for the above system.

Assumption A1.

The activation function

g (X (t)) = {g_{1} (X_{1} (t)), g_{2} (X_{2} (t)), \dots, g_{n} (X_{n} (t))}^{T} \in R^{n}

is assumed to satisfy the following condition:

\begin{matrix} 0 \leq \frac{g_{i} (ζ_{1}) - g_{i} (ζ_{2})}{ζ_{1} - ζ_{2}} \leq l_{i}, g (0) = 0, ζ_{1}, ζ_{2} \in R, ζ_{1} \neq ζ_{2}, i = 1, 2, \dots, n \end{matrix}

where

l_{i}, i = 1, 2, \dots, n

are positive real constants. Denote

L = [l_{i}], i = 1, 2, \dots, n

as a diagonal matrix. The following Lemmas are extensively used in the analysis of our time delay systems.

Lemma 1

(Lower bound theorem [42]). If an open subset D

\in R^{n}

has positive values for

f_{1}, f_{2}, \dots, f_{N} \in R^{n} \to R

, the reciprocal convex combination of

f_{r}

over D satisfies

\begin{matrix} min_{{α_{r} | α_{r} > 0, \sum α_{r} = 1}} \sum_{r} \frac{1}{α_{r}} f_{r} (t) = \sum_{r} f_{r} (t) + max_{g_{r, s} (t)} \sum_{r \neq s} g_{(r, s)} (t) \end{matrix}

subject to

\begin{matrix} \{g_{(r, s)} (t) : R^{n} \to R, g_{r, s} (t) = g_{s, r} (t), [\begin{matrix} f_{r} (t) & g_{(r, s)} (t) \\ g_{s, r} (t) & f_{s} (t) \end{matrix}]\} \geq 0 . \end{matrix}

Lemma 2

([25] ). For the symmetric matrices

R > 0

, Ω and matrix

Γ,

the following are equivalent.

1.: $Ω - Γ^{T} R Γ < 0$ .
2.: There exists an appropriate dimensional matrix Π such that

$\begin{matrix} [\begin{matrix} Ω + Γ^{T} Π + Π^{T} Γ & Π^{T} \\ * & - R \end{matrix}] < 0 . \end{matrix}$

Lemma 3

(Schur Complement [25] ). For the given constant matrices

Ψ_{1}

,

Ψ_{2}

,

Ψ_{3}

with appropriate dimensions, and

Ψ_{1}^{T} = Ψ_{1}

,

Ψ_{2}^{T} = Ψ_{2} > 0

, then

Ψ_{1} + Ψ_{3}^{T} Ψ_{2}^{- 1} Ψ_{3} < 0,

if and only if

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

3. Main Results

This work explores a novel delay-dependent stability condition for the system (1) as given below:

Theorem 1.

For any scalars

0 \leq ς_{1} \leq ς_{2}

,

0 \leq ϱ_{1} \leq ϱ_{2}

, our BAM NNs (1) is globally asymptotically stable provided there are some symmetric matrices

P = {[P_{i j}]}_{4 \times 4} > 0

,

Q = {[Q_{i j}]}_{4 \times 4} > 0

,

R_{k} > 0

,

S_{k} > 0

,

T_{k} > 0

,

k = 1, 2, 3, 4,

N_{1}, N_{2} > 0

,

U_{j} > 0

,

j = 1, 2, \dots, 8,

[\begin{matrix} R_{11} & R_{12} \\ * & R_{22} \end{matrix}] > 0, [\begin{matrix} R_{31} & R_{32} \\ * & R_{33} \end{matrix}] > 0,

J_{1}, J_{2} > 0

, and any matrices

M_{1}, M_{2}, \dots, M_{10}

and

Π_{1}, Π_{2}, Π_{3}, Π_{4}

with appropriate dimensions and

a_{m} > 0, b_{m} > 0, m = 1, 2

exists subject to the following constraints holding for l = 1 to 4,

\begin{matrix} [\begin{matrix} ϕ_{l} & Π_{1}^{T} & Π_{2}^{T} & Π_{3}^{T} & Π_{4}^{T} \\ * & - \bar{U_{3}} & 0 & 0 & 0 \\ * & * & - \bar{U_{4}} & 0 & 0 \\ * & * & * & - \bar{U_{7}} & 0 \\ * & * & * & * & - \bar{U_{8}} \end{matrix}] < 0, \end{matrix}

(2)

\begin{matrix} [\begin{matrix} 2 U_{3} & 0 & M_{1} & 0 \\ * & U_{3} & 0 & M_{3} \\ * & * & 2 U_{3} & 0 \\ * & * & * & U_{3} \end{matrix}] > 0, [\begin{matrix} 2 U_{4} & 0 & M_{5} & 0 \\ * & U_{4} & 0 & M_{6} \\ * & * & 2 U_{4} & 0 \\ * & * & * & U_{4} \end{matrix}] > 0, \end{matrix}

\begin{matrix} [\begin{matrix} 2 U_{7} & 0 & M_{7} & 0 \\ * & U_{7} & 0 & M_{8} \\ * & * & 2 U_{7} & 0 \\ * & * & * & U_{7} \end{matrix}] > 0, [\begin{matrix} 2 U_{8} & 0 & M_{9} & 0 \\ U_{8} & 0 & M_{10} \\ * & * & 2 U_{8} & 0 \\ * & * & * & U_{8} \end{matrix}] > 0, \end{matrix}

(3)

\begin{matrix} [\begin{matrix} S_{2} + \frac{ϱ_{12}^{2}}{2} U_{3} & M_{2} \\ * & S_{2} + \frac{ϱ_{12}^{2}}{2} U_{4} \end{matrix}] \geq 0, [\begin{matrix} S_{4} + \frac{ς_{12}^{2}}{2} U_{7} & M_{4} \\ * & S_{4} + \frac{ς_{12}^{2}}{2} U_{8} \end{matrix}] \geq 0, \end{matrix}

\begin{matrix} [\begin{matrix} T_{2} & N_{1} \\ * & T_{2} \end{matrix}] \geq 0, [\begin{matrix} T_{4} & N_{2} \\ * & T_{4} \end{matrix}] \geq 0, \end{matrix}

(4)

where

\begin{matrix} ϕ_{l} = [\begin{matrix} Ω & 2 Γ_{1 l}^{T} \hat{U_{3}} & 2 Γ_{2 l}^{T} \hat{U_{4}} & 2 Γ_{3 l}^{T} \hat{U_{7}} & 2 Γ_{4 l}^{T} \hat{U_{8}} \\ * & - 2 \hat{U_{3}} & 0 & 0 & 0 \\ * & * & - 2 \hat{U_{4}} & 0 & 0 \\ * & * & * & - 2 \hat{U_{7}} & 0 \\ * & * & * & * & - 2 \hat{U_{8}} \end{matrix}] + [\begin{matrix} Γ_{1 l}^{T} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \end{matrix}] Π_{1} + Π_{1}^{T} {[\begin{matrix} Γ_{1 l}^{T} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \end{matrix}]}^{T} + [\begin{matrix} Γ_{2 l}^{T} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \end{matrix}] Π_{2} \\ + Π_{2}^{T} {[\begin{matrix} Γ_{2 l}^{T} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \end{matrix}]}^{T} + [\begin{matrix} Γ_{3 l}^{T} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \end{matrix}] Π_{3} + Π_{3}^{T} {[\begin{matrix} Γ_{3 l}^{T} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \end{matrix}]}^{T} + [\begin{matrix} Γ_{4 l}^{T} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \end{matrix}] Π_{4} + Π_{4}^{T} {[\begin{matrix} Γ_{4 l}^{T} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \end{matrix}]}^{T}, \end{matrix}

\begin{matrix} \bar{U_{3}} & = & [\begin{matrix} 3 U_{3} & M_{1} + M_{3} \\ * & 3 U_{3} \end{matrix}], \bar{U_{4}} = [\begin{matrix} 3 U_{4} & M_{5} + M_{6} \\ * & 3 U_{4} \end{matrix}], \\ \bar{U_{7}} & = & [\begin{matrix} 3 U_{7} & M_{7} + M_{8} \\ * & 3 U_{7} \end{matrix}], \bar{U_{8}} = [\begin{matrix} 3 U_{8} & M_{9} + M_{10} \\ * & 3 U_{8} \end{matrix}], \end{matrix}

where

Ω = {[Ω_{r, s}]}_{20 \times 20}, Ω_{r, s} = Ω_{s, r}^{T}, r, s = 1, 2, \dots, 20, Γ_{1 l}, Γ_{2 l}, Γ_{3 l}, Γ_{4 l}, \hat{U_{3}}, \hat{U_{4}}, \hat{U_{7}}, and \hat{U_{8}}

are defined as

\begin{matrix} Ω_{1, 1} & = & - P_{11} A_{1} - A_{1}^{T} P_{11}^{T} + P_{12} + P_{12}^{T} + R_{1} + R_{11} + A_{1}^{T} (ϱ_{1}^{2} S_{1} + ϱ_{12}^{2} S_{2}) A_{1} - S_{1} \\ + ϱ_{1}^{2} T_{1} + ϱ_{12}^{2} T_{2} + A_{1}^{T} U_{α} A_{1} - ϱ_{1}^{2} U_{1} + a_{1} W_{1}^{T} W_{1}, \\ Ω_{1, 2} & = & (1 - υ_{2}) P_{13} - (1 - υ_{2}) P_{14}, Ω_{1, 3} = - P_{12} + P_{14} + S_{1}, \\ Ω_{1, 4} & = & - P_{13}, Ω_{1, 5} = P_{22}^{T} + ϱ_{1} U_{1}^{T}, Ω_{1, 6} = R_{12}, \\ Ω_{1, 18} & = & P_{11} B_{1} - A_{1}^{T} (ϱ_{1}^{2} S_{1} + ϱ_{12}^{2} S_{2}) B_{1} - A_{1}^{T} U_{α} B_{1}, \\ Ω_{1, 19} & = & P_{11} C_{1} - A_{1}^{T} (ϱ_{1}^{2} S_{1} + ϱ_{12}^{2} S_{2}) C_{1} - A_{1}^{T} U_{α} C_{1}, \\ Ω_{1, 20} & = & P_{11} D_{1} - A_{1}^{T} (ϱ_{1}^{2} S_{1} + ϱ_{12}^{2} S_{2}) D_{1} - A_{1}^{T} U_{α} D_{1}, \\ Ω_{2, 2} & = & - (1 - υ_{2}) R_{11} - 2 S_{2} + M_{2} + M_{2}^{T} + a_{2} W_{1}^{T} W_{1}, Ω_{2, 3} = - M^{T} + S_{2}, \\ Ω_{2, 4} & = & - M_{2} + S_{2}, Ω_{2, 5} = (1 - υ_{2}) P_{23}^{T} - (1 - υ_{2}) P_{24}^{T}, Ω_{2, 6} = - (1 - υ_{2}) P_{44}^{T}, \\ Ω_{2, 7} & = & (1 - υ_{2}) P_{33}^{T} - (1 - υ_{2}) P_{34}^{T}, Ω_{2, 9} = - (1 - υ_{2}) R_{12}, \\ Ω_{3, 3} & = & R_{2} - R_{1} - S_{1} - S_{2} - ϱ_{1}^{2} U_{2}, Ω_{3, 5} = - P_{22}^{T} + P_{24}^{T} + ϱ_{1} U_{2}^{T}, Ω_{3, 6} = P_{44}^{T}, \\ Ω_{3, 7} & = & P_{34}^{T}, Ω_{4, 4} = - R_{2}, Ω_{4, 5} = - P_{23}^{T}, Ω_{4, 7} = - P_{33}^{T}, Ω_{5, 5} = - T_{1} - U_{1} - U_{2}, \\ Ω_{6, 6} & = & - T_{2}, Ω_{6, 7} = - N_{1}, Ω_{7, 7} = - T_{2}, \\ Ω_{8, 8} & = & R_{22} + B_{2}^{T} (ς_{1}^{2} S_{3} + ς_{12}^{2} S_{4}) B_{2} + B_{2}^{T} U_{β} B_{2} + γ_{12}^{2} J_{1} - a_{1} I, \\ Ω_{8, 9} & = & B_{2}^{T} (ς_{1}^{2} S_{3} + ς_{12}^{2} S_{4}) C_{2} + B_{2}^{T} U_{β} C_{2}, Ω_{8, 10} = B_{2}^{T} (ς_{1}^{2} S_{3} + ς_{12}^{2} S_{4}) D_{2} + B_{2}^{T} U_{β} D_{2}, \\ Ω_{8, 11} & = & B_{2}^{T} Q_{11}^{T} - B_{2}^{T} (ς_{1}^{2} S_{3} + ς_{12}^{2} S_{4}) A_{2} - B_{2}^{T} U_{β} A_{2}, \\ Ω_{9, 9} & = & - (1 - υ_{2}) R_{22} + C_{2}^{T} (ς_{1}^{2} S_{3} + ς_{12}^{2} S_{4}) C_{2} + C_{2}^{T} U_{β} C_{2} - a_{2} I, \\ Ω_{9, 10} & = & C_{2}^{T} (ς_{1}^{2} S_{3} + ς_{12}^{2} S_{4}) D_{2} + C_{2}^{T} U_{β} D_{2}, \\ Ω_{9, 11} & = & C_{2}^{T} Q_{11}^{T} - C_{2}^{T} (ς_{1}^{2} S_{3} + ς_{12}^{2} S_{4}) A_{2} - C_{2}^{T} U_{β} A_{2}, \\ Ω_{10, 10} & = & D_{2}^{T} (ς_{1}^{2} S_{3} + ς_{12}^{2} S_{4}) D_{2} + D_{2}^{T} U_{β} D_{2} - J_{1}, \\ Ω_{10, 11} & = & D_{2}^{T} Q_{11}^{T} - D_{2}^{T} (ς_{1}^{2} S_{3} + ς_{12}^{2} S_{4}) A_{2} - D_{2}^{T} U_{β} A_{2}, \\ Ω_{11, 11} & = & - Q_{11} A_{2} - A_{2}^{T} Q_{11}^{T} + Q_{12} + Q_{12}^{T} + R_{3} + R_{31} + A_{2}^{T} (ς_{1}^{2} S_{3} + ς_{12}^{2} S_{4}) A_{2} \\ - S_{3} + ς_{1}^{2} T_{3} + ς_{12}^{2} T_{4} + A_{2}^{T} U_{β} A_{2} - ς_{1}^{2} U_{5} + b_{1} W_{2}^{T} W_{2} \\ Ω_{11, 12} & = & (1 - υ_{1}) Q_{13} - (1 - υ_{1}) Q_{14}, Ω_{11, 13} = - Q_{12} + Q_{14} + S_{3}, Ω_{11, 14} = - Q_{13}, \\ Ω_{11, 15} & = & Q_{22}^{T} + ς_{1} U_{5}^{T}, Ω_{11, 18} = R_{32}, Ω_{12, 12} = - (1 - υ_{1}) R_{31} - 2 S_{4} + M_{4} + M_{4}^{T} + b_{2} W_{2}^{T} W_{2}, \\ Ω_{12, 13} & = & - M_{4}^{T} + S_{4}, Ω_{12, 14} = - M_{4} + S_{4}, \\ Ω_{12, 15} & = & (1 - υ_{1}) Q_{23}^{T} - (1 - υ_{1}) Q_{24}^{T}, Ω_{12, 16} = - (1 - υ_{1}) Q_{44}^{T}, \\ Ω_{12, 17} & = & (1 - υ_{1}) Q_{33}^{T} - (1 - υ_{1}) Q_{34}^{T}, Ω_{12, 19} = - (1 - υ_{1}) R_{32} \\ Ω_{13, 13} & = & R_{4} - R_{3} - S_{3} - S_{4} - ς_{1}^{2} U_{6}, Ω_{13, 14} = - M_{4}, \\ Ω_{13, 15} & = & - Q_{22}^{T} + Q_{24}^{T} + ς_{1} U_{6}^{T}, Ω_{13, 16} = Q_{44}^{T}, Ω_{13, 17} = Q_{34}^{T}, \\ Ω_{14, 14} & = & - R_{4} - S_{4}, Ω_{14, 15} = - Q_{23}^{T}, Ω_{14, 17} = - Q_{33}^{T}, \\ Ω_{15, 15} & = & - T_{3} - U_{5} - U_{6}, \\ Ω_{16, 16} & = & - T_{4}, Ω_{16, 17} = - N_{2}, \\ Ω_{17, 17} & = & - T_{4}, \\ Ω_{18, 18} & = & R_{33} + B_{1}^{T} (ϱ_{1}^{2} S_{1} + ϱ_{12}^{2} S_{2}) B_{1} + B_{1}^{T} U_{α} B_{1} + η_{12}^{2} J_{2} - b_{1} I, \\ Ω_{18, 19} & = & B_{1}^{T} (ϱ_{1}^{2} S_{1} + ϱ_{12}^{2} S_{2}) C_{1} + B_{1}^{T} U_{α} C_{1}, Ω_{18, 20} = B_{1}^{T} (ϱ_{1}^{2} S_{1} + ϱ_{12}^{2} S_{2}) D_{1} + B_{1}^{T} U_{α} D_{1}, \\ Ω_{19, 19} & = & - (1 - υ_{1}) R_{33} + C_{1}^{T} (ϱ_{1}^{2} S_{1} + ϱ_{12}^{2} S_{2}) C_{1} + C_{1}^{T} U_{α} C_{1} - b_{1} I, \\ Ω_{19, 20} & = & C_{1}^{T} (ϱ_{1}^{2} S_{1} + ϱ_{12}^{2} S_{2}) D_{1} + C_{1}^{T} U_{α} D_{1}, Ω_{20, 20} = D_{1}^{T} (ϱ_{1}^{2} S_{1} + ϱ_{12}^{2} S_{2}) D_{1} + D_{1}^{T} U_{α} D_{1} - J_{2}, \\ U_{α} & = & \frac{ϱ_{1}^{4} U_{1} + ϱ_{1}^{4} U_{2} + ϱ_{12}^{4} U_{3} + ϱ_{12}^{4} U_{4}}{4}, U_{β} = \frac{ς_{1}^{4} U_{5} + ς_{1}^{4} U_{6} + ς_{12}^{4} U_{7} + ς_{12}^{4} U_{8}}{4}; \end{matrix}

the remaining terms are all zeros and

\begin{matrix} Γ_{11} & = & [\begin{matrix} 0_{n} & 0_{n} & 3 t e r m s (0_{n}) & - I_{n} & 0_{n} & 13 t e r m s (0_{n}) \\ 0_{n} & ϱ_{12} I_{n} & 3 t e r m s (0_{n}) & 0_{n} & - I_{n} & 13 t e r m s (0_{n}) \end{matrix}], \\ Γ_{12} & = & [\begin{matrix} 2 t e r m s (0_{n}) & ϱ_{12} I_{n} & 2 t e r m s (0_{n}) & - I_{n} & 0_{n} & 13 t e r m s (0_{n}) \\ 2 t e r m s (0_{n}) & 0_{n} & 2 t e r m s (0_{n}) & 0_{n} & - I_{n} & 13 t e r m s (0_{n}) \end{matrix}], \\ Γ_{21} & = & [\begin{matrix} 3 t e r m s (0_{n}) & 0_{n} & 0_{n} & I_{n} & 0_{n} & 13 t e r m s (0_{n}) \\ 3 t e r m s (0_{n}) & - ϱ_{12} I_{n} & 0_{n} & 0_{n} & I_{n} & 13 t e r m s (0_{n}) \end{matrix}], \\ Γ_{22} & = & [\begin{matrix} 0_{n} & - ϱ_{12} I_{n} & 3 t e r m s (0_{n}) & I_{n} & 0_{n} & 13 t e r m s (0_{n}) \\ 0_{n} & 0_{n} & 3 t e r m s (0_{n}) & 0_{n} & I_{n} & 13 t e r m s (0_{n}) \end{matrix}], \\ Γ_{31} & = & [\begin{matrix} 11 t e r m s (0_{n}) & 0_{n} & 3 t e r m s (0_{n}) & - I_{n} & 0_{n} & 3 t e r m s (0_{n}) \\ 11 t e r m s (0_{n}) & ς_{12} I_{n} & 3 t e r m s (0_{n}) & 0_{n} & - I_{n} & 3 t e r m s (0_{n}) \end{matrix}], \\ Γ_{32} & = & [\begin{matrix} 12 t e r m s (0_{n}) & ς_{12} I_{n} & 2 t e r m s (0_{n}) & - I_{n} & 0_{n} & 3 t e r m s (0_{n}) \\ 12 t e r m s (0_{n}) & 0_{n} & 2 t e r m s (0_{n}) & 0_{n} & - I_{n} & 3 t e r m s (0_{n}) \end{matrix}], \\ Γ_{41} & = & [\begin{matrix} 13 t e r m s (0_{n}) & 0_{n} & 0_{n} & I_{n} & 0_{n} & 3 t e r m s (0_{n}) \\ 13 t e r m s (0_{n}) & - ς_{12} I_{n} & 0_{n} & 0_{n} & I_{n} & 3 t e r m s (0_{n}) \end{matrix}], \\ Γ_{42} & = & [\begin{matrix} 11 t e r m s (0_{n}) & - ς_{12} I_{n} & 3 t e r m s (0_{n}) & I_{n} & 0_{n} & 3 t e r m s (0_{n}) \\ 11 t e r m s (0_{n}) & 0_{n} & 3 t e r m s (0_{n}) & 0_{n} & I_{n} & 3 t e r m s (0_{n}) \end{matrix}], \\ \hat{U_{3}} & = & [\begin{matrix} U_{3} & 0 \\ 0 & U_{3} \end{matrix}], \hat{U_{4}} = [\begin{matrix} U_{4} & 0 \\ 0 & U_{4} \end{matrix}], \hat{U_{7}} = [\begin{matrix} U_{7} & 0 \\ 0 & U_{7} \end{matrix}], \hat{U_{8}} = [\begin{matrix} U_{8} & 0 \\ 0 & U_{8} \end{matrix}] . \end{matrix}

Proof.

Consider the newly constructed LKF

\begin{matrix} V (t) & = & \sum_{r = 1}^{6} V_{r} (t), \end{matrix}

(5)

where

\begin{matrix} V_{1} (t) & = & ζ_{1}^{T} (t) P ζ_{1} (t) + ζ_{2}^{T} (t) Q ζ_{2} (t), in which \\ ζ_{1} (t) & = & c o l {X (t), \int_{t - ϱ_{1}}^{t} X (ϰ) d ϰ, \int_{t - ϱ_{2}}^{t - ϱ (t)} X (ϰ) d ϰ, \int_{t - ϱ (t)}^{t - ϱ_{1}} X (ϰ) d ϰ}, \\ ζ_{2} (t) & = & c o l {У (t), \int_{t - ς_{1}}^{t} У (ϰ) d ϰ, \int_{t - ς_{2}}^{t - ς (t)} У (ϰ) d ϰ, \int_{t - ς (t)}^{t - ς_{1}} У (ϰ) d ϰ}, \\ P & = & [\begin{matrix} P_{11} & P_{12} & P_{13} & P_{14} \\ * & P_{22} & P_{23} & P_{24} \\ * & * & P_{33} & P_{34} \\ * & * & * & P_{44} \end{matrix}] > 0, Q = [\begin{matrix} Q_{11} & Q_{12} & Q_{13} & Q_{14} \\ * & Q_{22} & Q_{23} & Q_{24} \\ * & * & Q_{33} & Q_{34} \\ * & * & * & Q_{44} \end{matrix}] > 0, \end{matrix}

\begin{matrix} V_{2} (t) & = & \int_{t - ϱ_{1}}^{t} X^{T} (ϰ) R_{1} X (ϰ) d ϰ + \int_{t - ϱ_{2}}^{t - ϱ_{1}} X^{T} (ϰ) R_{2} X (ϰ) d ϰ \\ + \int_{t - ϱ (t)}^{t} [\begin{matrix} X^{T} (ϰ) & g^{T} (X (ϰ)) \end{matrix}] [\begin{matrix} R_{11} & R_{12} \\ * & R_{22} \end{matrix}] [\begin{matrix} X (ϰ) \\ g (X (ϰ)) \end{matrix}] d ϰ \\ + \int_{t - ς_{1}}^{t} У^{T} (ϰ) R_{3} У (ϰ) d ϰ + \int_{t - ς_{2}}^{t - ς_{1}} У^{T} (ϰ) R_{4} У (ϰ) d ϰ \\ + \int_{t - ς (t)}^{t} [\begin{matrix} У^{T} (ϰ) & f^{T} (У (ϰ)) \end{matrix}] [\begin{matrix} R_{31} & R_{32} \\ * & R_{33} \end{matrix}] [\begin{matrix} У (ϰ) \\ f (У (ϰ)) \end{matrix}] d ϰ, \\ V_{3} (t) & = & ϱ_{1} \int_{- ϱ_{1}}^{0} \int_{t + λ}^{t} {\dot{X}}^{T} (ϰ) S_{1} \dot{X} (ϰ) d ϰ d λ + ϱ_{12} \int_{- ϱ_{2}}^{- ϱ_{1}} \int_{t + λ}^{t} {\dot{X}}^{T} (ϰ) S_{2} \dot{X} (ϰ) d ϰ d λ \\ + ς_{1} \int_{- ς_{1}}^{0} \int_{t + λ}^{t} {\dot{У}}^{T} (ϰ) S_{3} \dot{У} (ϰ) d ϰ d λ + ς_{12} \int_{- ς_{2}}^{- ς_{1}} \int_{t + λ}^{t} {\dot{У}}^{T} (ϰ) S_{4} \dot{У} (ϰ) d ϰ d λ, \\ V_{4} (t) & = & ϱ_{1} \int_{- ϱ_{1}}^{0} \int_{t + λ}^{t} X^{T} (ϰ) T_{1} X (ϰ) d ϰ d λ + ϱ_{12} \int_{- ϱ_{2}}^{- ϱ_{1}} \int_{t + λ}^{t} X^{T} (ϰ) T_{2} X (ϰ) d ϰ d λ λ \\ + ς_{1} \int_{- ς_{1}}^{0} \int_{t + λ}^{t} У^{T} (ϰ) T_{3} У (ϰ) d ϰ d λ + ς_{12} \int_{- ς_{2}}^{- ς_{1}} \int_{t + λ}^{t} У^{T} (ϰ) T_{4} У (ϰ) d ϰ d λ, \end{matrix}

\begin{matrix} V_{5} (t) & = & \frac{ϱ_{1}^{2}}{2} \int_{- ϱ_{1}}^{0} \int_{η}^{0} \int_{t + θ}^{t} {\dot{X}}^{T} (ϰ) U_{1} \dot{X} (ϰ) d ϰ d θ d η + \frac{ϱ_{1}^{2}}{2} \int_{- ϱ_{1}}^{0} \int_{- ϱ_{1}}^{η} \int_{t + θ}^{t} {\dot{X}}^{T} (ϰ) U_{2} \dot{X} (ϰ) d ϰ d θ d η \\ + \frac{ϱ_{12}^{2}}{2} \int_{- ϱ_{2}}^{- ϱ_{1}} \int_{η}^{- ϱ_{1}} \int_{t + θ}^{t} {\dot{X}}^{T} (ϰ) U_{3} \dot{X} (ϰ) d ϰ d θ d η + \frac{ϱ_{12}^{2}}{2} \int_{- ϱ_{2}}^{- ϱ_{1}} \int_{- ϱ_{2}}^{η} \int_{t + θ}^{t} {\dot{X}}^{T} (ϰ) U_{4} \dot{X} (ϰ) d ϰ d θ d η \\ + \frac{ς_{1}^{2}}{2} \int_{- ς_{1}}^{0} \int_{η}^{0} \int_{t + θ}^{t} {\dot{У}}^{T} (ϰ) U_{5} \dot{У} (ϰ) d ϰ d θ d η + \frac{ς_{1}^{2}}{2} \int_{- ς_{1}}^{0} \int_{- ς_{1}}^{η} \int_{t + θ}^{t} {\dot{У}}^{T} (ϰ) U_{6} \dot{У} (ϰ) d ϰ d θ d η \\ + \frac{ς_{12}^{2}}{2} \int_{- ς_{2}}^{- ς_{1}} \int_{η}^{- ς_{1}} \int_{t + θ}^{t} {\dot{Y}}^{T} (ϰ) U_{7} \dot{У} (ϰ) d ϰ d θ d η + \frac{ς_{12}^{2}}{2} \int_{- ς_{2}}^{- ς_{1}} \int_{- ς_{2}}^{η} \int_{t + θ}^{t} {\dot{У}}^{T} (ϰ) U_{8} \dot{У} (ϰ) d ϰ d θ d η, \\ V_{6} (t) & = & γ_{12} \int_{- γ_{2}}^{- γ_{1}} \int_{t + θ}^{t} g^{T} (X (ϰ)) J_{1} g (X (ϰ)) d ϰ d θ + η_{12} \int_{- η_{2}}^{- η_{1}} \int_{t + θ}^{t} f^{T} (У (ϰ)) J_{2} f (У (ϰ)) d ϰ d θ, \end{matrix}

where

\begin{matrix} ϱ_{12} & = & ϱ_{2} - ϱ_{1}, ς_{12} = ς_{2} - ς_{1}, γ_{12} = γ_{2} - γ_{1}, η_{12} = η_{2} - η_{1}, \\ α_{1} = \frac{ϱ (t) - ϱ_{1}}{ϱ_{12}}, β_{1} = \frac{ϱ_{2} - ϱ (t)}{ϱ_{12}}, α_{2} = \frac{ς (t) - ς_{1}}{ς_{12}}, β_{2} = \frac{ς_{2} - ς (t)}{ς_{12}} . \end{matrix}

Calculating the time derivatives of Equation (5), we get

\begin{matrix} {\dot{V}}_{1} (t) & = & 2 ζ_{1}^{T} (t) P \dot{ζ_{1}} (t) + 2 ζ_{2}^{T} (t) Q \dot{ζ_{2}} (t) . \end{matrix}

(6)

Calculating

{\dot{V}}_{2} (t)

and substituting

\dot{ς} (t) \leq υ_{1}

and

\dot{ϱ} (t) \leq υ_{2}

leads to

\begin{matrix} {\dot{V}}_{2} (t) & \leq & X^{T} (t) (R_{1}) X (t) + X^{T} (t - ϱ_{1}) (R_{2} - R_{1}) X (t - ϱ_{1}) - X^{T} (t - ϱ_{2}) (R_{2}) X (t - ϱ_{2}) \\ + X^{T} (t) (R_{11}) X (t) + X^{T} (t) (R_{12}) g (X (t)) + g^{T} (X (t)) (R_{12}^{T}) X (t) \\ + g^{T} (X (t)) (R_{22}) g (X (t)) - (1 - υ_{2}) X^{T} (t - ϱ (t)) (R_{11}) X (t - ϱ (t)) \\ - (1 - υ_{2}) X^{T} (t - ϱ (t)) (R_{12}) g (X (t - ϱ (t))) - (1 - υ_{2}) g^{T} (X (t - ϱ (t))) (R_{22}) g (X (t - ϱ (t))) \\ - (1 - υ_{2}) g^{T} (X (t - ϱ (t))) (R_{12}^{T}) X (t - ϱ (t)) + У^{T} (t) (R_{3}) У (t) \\ + У^{T} (t - ς_{1}) (R_{4} - R_{3}) У (t - ς_{1}) - У^{T} (t - ς_{2}) (R_{4}) У (t - ς_{2}) + У^{T} (t) (R_{31}) У (t) \\ + У^{T} (t) (R_{32}) f (У (t)) + f^{T} (У (t)) (R_{32}^{T}) У (t) + f^{T} (У (t)) (R_{33}) f (У (t)) \\ - (1 - υ_{1}) У^{T} (t - ς (t)) (R_{31}) У (t - ς (t)) - (1 - υ_{1}) У^{T} (t - ς (t)) (R_{32}) f (У (t - ς (t))) \\ - (1 - υ_{1}) f^{T} (У (t - ς (t))) (R_{33}) f (У (t - ς (t))) \\ - (1 - υ_{1}) f^{T} (У (t - ς (t))) (R_{32}^{T}) У (t - ς (t)) . \end{matrix}

(7)

Calculating the time derivative of

V_{3} (t)

and applying Jensen’s inequality [27], we obtain

\begin{matrix} l {\dot{V}}_{3} (t) & \leq & ϱ_{1}^{2} {\dot{X}}^{T} (t) S_{1} \dot{X} (t) - \int_{t - ϱ_{1}}^{t} {\dot{X}}^{T} (ϰ) d ϰ S_{1} \int_{t - ϱ_{1}}^{t} \dot{X} (ϰ) d ϰ \\ + ς_{1}^{2} {\dot{У}}^{T} (t) S_{3} \dot{У} (t) - \int_{t - ς_{1}}^{t} {\dot{У}}^{T} (ϰ) d ϰ S_{3} \int_{t - ς_{1}}^{t} \dot{У} (ϰ) d ϰ \\ + ϱ_{12}^{2} {\dot{X}}^{T} (t) S_{2} \dot{X} (t) - \frac{1}{α_{1}} \int_{t - ϱ (t)}^{t - ϱ_{1}} {\dot{X}}^{T} (ϰ) d ϰ S_{2} \int_{t - ϱ (t)}^{t - ϱ_{1}} \dot{X} (ϰ) d ϰ \\ - \frac{1}{β_{1}} \int_{t - ϱ_{2}}^{t - ϱ (t)} {\dot{X}}^{T} (ϰ) d ϰ S_{2} \int_{t - ϱ_{2}}^{t - ϱ (t)} \dot{X} (ϰ) d ϰ \\ + ς_{12}^{2} {\dot{У}}^{T} (t) S_{4} \dot{У} (t) - \frac{1}{α_{2}} \int_{t - ς (t)}^{t - ς_{1}} {\dot{У}}^{T} (ϰ) d ϰ S_{4} \int_{t - ς (t)}^{t - ς_{1}} \dot{У} (ϰ) d ϰ \\ - \frac{1}{β_{2}} \int_{t - ς_{2}}^{t - ς (t)} {\dot{У}}^{T} (ϰ) d ϰ S_{4} \int_{t - ς_{2}}^{t - ς (t)} \dot{У} (ϰ) d ϰ, \end{matrix}

By applying Lemma 1 to

{\dot{V}}_{3} (t)

, we have

\begin{matrix} {\dot{V}}_{3} (t) & \leq & ϱ_{1}^{2} {\dot{X}}^{T} (t) S_{1} \dot{X} (t) + ϱ_{12}^{2} {\dot{X}}^{T} (t) S_{2} \dot{X} (t) - \int_{t - ϱ_{1}}^{t} {\dot{X}}^{T} (ϰ) d ϰ S_{1} \int_{t - ϱ_{1}}^{t} \dot{X} (ϰ) d ϰ \\ + ς_{1}^{2} {\dot{У}}^{T} (t) S_{3} \dot{У} (t) + ς_{12}^{2} {\dot{У}}^{T} (t) S_{4} \dot{У} (t) - \int_{t - ς_{1}}^{t} {\dot{У}}^{T} (ϰ) d ϰ S_{3} \int_{t - ς_{1}}^{t} \dot{У} (ϰ) d ϰ \\ - [\begin{matrix} \int_{t - ϱ (t)}^{t - ϱ_{1}} {\dot{X}}^{T} (ϰ) d ϰ & \int_{t - ϱ_{2}}^{t - ϱ (t)} {\dot{X}}^{T} (ϰ) d ϰ \end{matrix}] [\begin{matrix} S_{2} & M_{2} \\ * & S_{2} \end{matrix}] [\begin{matrix} \int_{t - ϱ (t)}^{t - ϱ_{1}} \dot{X} (ϰ) d ϰ \\ \int_{t - ϱ_{2}}^{t - ϱ (t)} \dot{X} (ϰ) d ϰ \end{matrix}] \\ - [\begin{matrix} \int_{t - ς (t)}^{t - ς_{1}} {\dot{У}}^{T} (ϰ) d ϰ & \int_{t - ϱ_{2}}^{t - ϱ (t)} {\dot{У}}^{T} (ϰ) d ϰ \end{matrix}] [\begin{matrix} S_{4} & M_{4} \\ * & S_{4} \end{matrix}] [\begin{matrix} \int_{t - ς (t)}^{t - ς_{1}} \dot{У} (ϰ) d ϰ \\ \int_{t - ϱ_{2}}^{t - ϱ (t)} \dot{У} (ϰ) d ϰ \end{matrix}], \end{matrix}

(8)

\begin{matrix} {\dot{V}}_{4} (t) & = & ϱ_{1}^{2} X^{T} (t) T_{1} X (t) - ϱ_{1} \int_{t - ϱ_{1}}^{t} X^{T} (ϰ) T_{1} X (ϰ) d ϰ + ς_{1}^{2} У^{T} (t) T_{3} У (t) - ς_{1} \int_{t - ς_{1}}^{t} У^{T} (ϰ) T_{3} У (ϰ) d ϰ \\ + ϱ_{12}^{2} X^{T} (t) T_{2} X (t) - ϱ_{12} \int_{t - ϱ_{2}}^{t - ϱ_{1}} X^{T} (ϰ) T_{2} X (ϰ) d ϰ + ς_{12}^{2} У^{T} (t) T_{4} У (t) - ς_{12} \int_{t - ς_{2}}^{t - ς_{1}} У^{T} (ϰ) T_{4} У (ϰ) d ϰ \end{matrix}

\begin{matrix} \leq & ϱ_{1}^{2} X^{T} (t) T_{1} X (t) + ς_{1}^{2} У^{T} (t) T_{3} У (t) + ϱ_{12}^{2} X^{T} (t) T_{2} X (t) + ς_{12}^{2} У^{T} (t) T_{4} У (t) \\ - \int_{t - ϱ_{1}}^{t} X^{T} (ϰ) d ϰ T_{1} \int_{t - ϱ_{1}}^{t} X (ϰ) d ϰ - \int_{t - ς_{1}}^{t} У^{T} (ϰ) d ϰ T_{3} \int_{t - ς_{1}}^{t} У (ϰ) d ϰ \\ - \frac{1}{α_{1}} \int_{t - ϱ (t)}^{t - ϱ_{1}} X^{T} (ϰ) d ϰ T_{2} \int_{t - ϱ (t)}^{t - ϱ_{1}} X (ϰ) d ϰ - \frac{1}{β_{1}} \int_{t - ϱ_{2}}^{t - ϱ (t)} X^{T} (ϰ) d ϰ T_{2} \int_{t - ϱ_{2}}^{t - ϱ (t)} X (ϰ) d ϰ \\ - \frac{1}{α_{2}} \int_{t - ς (t)}^{t - ς_{1}} У^{T} (ϰ) d ϰ T_{4} \int_{t - ς (t)}^{t - ς_{1}} У (ϰ) d ϰ - \frac{1}{β_{2}} \int_{t - ς_{2}}^{t - ς (t)} У^{T} (ϰ) d ϰ T_{4} \int_{t - ς_{2}}^{t - ς (t)} У (ϰ) d ϰ, \end{matrix}

By applying Jensen’s inequality [27] and Lemma 1 to

{\dot{V}}_{4} (t)

, we get

\begin{matrix} {\dot{V}}_{4} (t) & \leq & X^{T} (t) {ϱ_{1}^{2} T_{1} + ϱ_{12}^{2} T_{2}} X (t) + У^{T} (t) {ς_{1}^{2} T_{3} + ς_{12}^{2} T_{4}} У (t) \\ - \int_{t - ϱ_{1}}^{t} X^{T} (ϰ) d ϰ T_{1} \int_{t - ϱ_{1}}^{t} X (ϰ) d ϰ - \int_{t - ς_{1}}^{t} У^{T} (ϰ) d ϰ T_{3} \int_{t - ς_{1}}^{t} У (ϰ) d ϰ \\ - [\begin{matrix} \int_{t - ϱ (t)}^{t - ϱ_{1}} X^{T} (ϰ) d ϰ & \int_{t - ϱ_{2}}^{t - ϱ (t)} X^{T} (ϰ) d ϰ \end{matrix}] [\begin{matrix} T_{2} & N_{1} \\ * & T_{2} \end{matrix}] [\begin{matrix} \int_{t - ϱ (t)}^{t - ϱ_{1}} X (ϰ) d ϰ \\ \int_{t - ϱ_{2}}^{t - ϱ (t)} X (ϰ) d ϰ \end{matrix}] \\ - [\begin{matrix} \int_{t - ς (t)}^{t - ς_{1}} У^{T} (ϰ) d ϰ & \int_{t - ϱ_{2}}^{t - ϱ (t)} У^{T} (ϰ) d ϰ \end{matrix}] [\begin{matrix} T_{4} & N_{2} \\ * & T_{4} \end{matrix}] [\begin{matrix} \int_{t - ς (t)}^{t - ς_{1}} У (ϰ) d ϰ \\ \int_{t - ϱ_{2}}^{t - ϱ (t)} У (ϰ) d ϰ \end{matrix}] . \end{matrix}

(9)

When applying Jensen’s inequality [27] and second-order reciprocal convex combination Lemma 1 to

{\dot{V}}_{5} (t)

,

\begin{matrix} {\dot{V}}_{5} (t) & = & \frac{ϱ_{1}^{2}}{2} \int_{- ϱ_{1}}^{0} \int_{η}^{0} [{\dot{X}}^{T} (t) U_{1} \dot{X} (t) - {\dot{X}}^{T} (t + ϰ) U_{1} \dot{X} (t + ϰ)] d ϰ d η \\ + \frac{ϱ_{1}^{2}}{2} \int_{- ϱ_{1}}^{0} \int_{ϱ_{1}}^{η} [{\dot{X}}^{T} (t) U_{2} \dot{X} (t) - {\dot{X}}^{T} (t + ϰ) U_{2} \dot{X} (t + ϰ)] d ϰ d η \\ + \frac{ϱ_{12}^{2}}{2} \int_{- ϱ_{2}}^{ϱ_{1}} \int_{η}^{- ϱ_{1}} [{\dot{X}}^{T} (t) U_{3} \dot{X} (t) - {\dot{X}}^{T} (t + ϰ) U_{3} \dot{X} (t + ϰ)] d ϰ d η \\ + \frac{ϱ_{12}^{2}}{2} \int_{- ϱ_{2}}^{- ϱ_{1}} \int_{- ϱ_{2}}^{η} [{\dot{X}}^{T} (t) U_{4} \dot{X} (t) - {\dot{X}}^{T} (t + ϰ) U_{4} \dot{X} (t + ϰ)] d ϰ d η \\ + \frac{ς_{1}^{2}}{2} \int_{- ς_{1}}^{0} \int_{η}^{0} [{\dot{У}}^{T} (t) U_{5} \dot{У} (t) - {\dot{У}}^{T} (t + ϰ) U_{5} \dot{У} (t + ϰ)] d ϰ d η \\ + \frac{ς_{1}^{2}}{2} \int_{- ς_{1}}^{0} \int_{ς_{1}}^{η} [{\dot{У}}^{T} (t) U_{6} \dot{У} (t) - {\dot{У}}^{T} (t + ϰ) U_{6} \dot{У} (t + ϰ)] d ϰ d η \\ + \frac{ς_{12}^{2}}{2} \int_{- ς_{2}}^{ς_{1}} \int_{η}^{- ς_{1}} [{\dot{У}}^{T} (t) U_{7} \dot{У} (t) - {\dot{У}}^{T} (t + ϰ) U_{7} \dot{У} (t + ϰ)] d ϰ d η \\ + \frac{ς_{12}^{2}}{2} \int_{- ς_{2}}^{- ς_{1}} \int_{- ς_{2}}^{η} [{\dot{У}}^{T} (t) U_{8} \dot{У} (t) - {\dot{У}}^{T} (t + ϰ) U_{8} \dot{У} (t + ϰ)] d ϰ d η \end{matrix}

\begin{matrix} = & \frac{ϱ_{1}^{4}}{4} {\dot{X}}^{T} (t) U_{1} \dot{X} (t) + \frac{ϱ_{1}^{4}}{4} {\dot{X}}^{T} (t) U_{2} \dot{X} (t) - \frac{ϱ_{1}^{2}}{2} \int_{- ϱ_{1}}^{0} \int_{t + η}^{t} {\dot{X}}^{T} (ϰ) U_{1} \dot{X} (ϰ) d ϰ d η \\ - \frac{ϱ_{1}^{2}}{2} \int_{- ϱ_{1}}^{0} \int_{t - ϱ_{1}}^{t + η} {\dot{X}}^{T} (ϰ) U_{2} \dot{X} (ϰ) d ϰ d η + \frac{ϱ_{12}^{4}}{4} {\dot{X}}^{T} (t) U_{3} \dot{X} (t) - \frac{ϱ_{12}^{2}}{2} \int_{- ϱ_{2}}^{- ϱ_{1}} \int_{t + η}^{t - ϱ_{1}} {\dot{X}}^{T} (ϰ) U_{3} \dot{X} (ϰ) d ϰ d η \\ + \frac{ϱ_{12}^{4}}{4} {\dot{X}}^{T} (t) U_{4} \dot{X} (t) - \frac{ϱ_{12}^{2}}{2} \int_{- ϱ_{2}}^{- ϱ_{1}} \int_{t - ϱ_{2}}^{t + η} {\dot{X}}^{T} (ϰ) U_{4} \dot{X} (ϰ) d ϰ d η + \frac{ς_{1}^{4}}{4} {\dot{У}}^{T} (t) U_{5} \dot{У} (t) \\ - \frac{ς_{1}^{2}}{2} \int_{- ς_{1}}^{0} \int_{t + η}^{t} {\dot{У}}^{T} (ϰ) U_{5} \dot{У} (ϰ) d ϰ d η + \frac{ς_{1}^{4}}{4} {\dot{У}}^{T} (t) U_{6} \dot{У} (t) - \frac{ς_{1}^{2}}{2} \int_{- ς_{1}}^{0} \int_{t - ς_{1}}^{t + η} {\dot{У}}^{T} (ϰ) U_{6} \dot{У} (ϰ) d ϰ d η \\ + \frac{ς_{12}^{4}}{4} {\dot{У}}^{T} (t) U_{7} \dot{У} (t) - \frac{ς_{12}^{2}}{2} \int_{- ς_{2}}^{- ς_{1}} \int_{t + η}^{t - ς_{1}} {\dot{У}}^{T} (ϰ) U_{7} \dot{У} (ϰ) d ϰ d η + \frac{ς_{12}^{4}}{4} {\dot{У}}^{T} (t) U_{8} \dot{У} (t) \\ - \frac{ς_{12}^{2}}{2} \int_{- ς_{2}}^{- ς_{1}} \int_{t - ς_{2}}^{t + η} {\dot{У}}^{T} (ϰ) U_{8} \dot{У} (ϰ) d ϰ d η \end{matrix}

\begin{matrix} \leq & {\dot{X}}^{T} (t) U_{α} \dot{X} (t) + {\dot{У}}^{T} (t) U_{β} \dot{У} (t) - [ϱ_{1} X^{T} (t) - \int_{t - ϱ_{1}}^{t} X^{T} (ϰ) d ϰ] U_{1} [ϱ_{1} X (t) - \int_{t - ϱ_{1}}^{t} X (ϰ) d ϰ] \\ - [\int_{t - ϱ_{1}}^{t} X^{T} (ϰ) d ϰ - ϱ_{1} X^{T} (t - ϱ_{1})] U_{2} [\int_{t - ϱ_{1}}^{t} X (ϰ) d ϰ - ϱ_{1} X (t - ϱ_{1})] \\ - \frac{ϱ_{12}^{2}}{2} . \frac{β_{1}}{α_{1}} (ϱ (t) - ϱ_{1}) \int_{t - ϱ (t)}^{t - ϱ_{1}} {\dot{X}}^{T} (ϰ) U_{3} \dot{X} (ϰ) d ϰ - \frac{1}{α_{1}^{2}} . \frac{{(ϱ (t) - ϱ_{1})}^{2}}{2} \int_{- ϱ (t)}^{- ϱ_{1}} \int_{t + η}^{t - ϱ_{1}} \dot{X} (ϰ) U_{3} \dot{X} (ϰ) d ϰ d η \\ - \frac{1}{β_{1}^{2}} . \frac{{(ϱ_{2} - ϱ (t))}^{2}}{2} \int_{- ϱ_{2}}^{- ϱ (t)} \int_{t + η}^{t - ϱ_{1}} \dot{X} (ϰ) U_{3} \dot{X} (ϰ) d ϰ d η - \frac{ϱ_{12}^{2}}{2} . \frac{α_{1}}{β_{1}} (ϱ_{2} - ϱ (t)) \int_{t - ϱ_{2}}^{t - ϱ (t)} {\dot{X}}^{T} (ϰ) U_{4} \dot{X} (ϰ) d ϰ \\ - \frac{1}{α_{1}^{2}} . \frac{{(ϱ (t) - ϱ_{1})}^{2}}{2} \int_{- ϱ (t)}^{- ϱ_{1}} \int_{t - ϱ (t)}^{t + η} \dot{X} (ϰ) U_{4} \dot{X} (ϰ) d ϰ d η - \frac{1}{β_{1}^{2}} . \frac{{(ϱ_{2} - ϱ (t))}^{2}}{2} \int_{- ϱ_{2}}^{- ϱ (t)} \int_{t - ϱ_{2}}^{t + η} \dot{X} (ϰ) U_{4} \dot{X} (ϰ) d ϰ d η \\ - [ς_{1} У^{T} (t) - \int_{t - ς_{1}}^{t} У^{T} (ϰ) d ϰ] U_{5} [ς_{1} У (t) - \int_{t - ς_{1}}^{t} У (ϰ) d ϰ] \\ - [\int_{t - ς_{1}}^{t} У^{T} (ϰ) d ϰ - ς_{1} У^{T} (t - ς_{1})] U_{6} [\int_{t - ς_{1}}^{t} У (ϰ) d ϰ - ς_{1} Y (t - ς_{1})] \\ - \frac{ς_{12}^{2}}{2} . \frac{β_{2}}{α_{2}} (ς (t) - ς_{1}) \int_{t - ς (t)}^{t - ς_{1}} {\dot{У}}^{T} (ϰ) U_{7} \dot{У} (ϰ) d ϰ - \frac{1}{α_{2}^{2}} . \frac{{(ς (t) - ς_{1})}^{2}}{2} \int_{- ς (t)}^{- ς_{1}} \int_{t + η}^{t - ς_{1}} \dot{У} (ϰ) U_{7} \dot{У} (ϰ) d ϰ d η \\ - \frac{1}{β_{2}^{2}} . \frac{{(ς_{2} - ς (t))}^{2}}{2} \int_{- ς_{2}}^{- ς (t)} \int_{t + η}^{t - ς_{1}} \dot{У} (ϰ) U_{7} \dot{У} (ϰ) d ϰ d η - \frac{ς_{12}^{2}}{2} . \frac{α_{2}}{β_{2}} (ς_{2} - ς (t)) \int_{t - ς_{2}}^{t - ς (t)} {\dot{У}}^{T} (ϰ) U_{8} \dot{У} (ϰ) d ϰ \\ - \frac{1}{α_{2}^{2}} . \frac{{(ς (t) - ς_{1})}^{2}}{2} \int_{- ς (t)}^{- ς_{1}} \int_{t - ς (t)}^{t + η} \dot{У} (ϰ) U_{8} \dot{У} (ϰ) d ϰ d η \\ - \frac{1}{β_{2}^{2}} . \frac{{(ς_{2} - ς (t))}^{2}}{2} \int_{- ς_{2}}^{- ς (t)} \int_{t - ς_{2}}^{t + η} \dot{У} (ϰ) U_{8} \dot{У} (ϰ) d ϰ d η, \end{matrix}

(10)

where

\begin{matrix} U_{α} & = & \frac{ϱ_{1}^{4} U_{1} + ϱ_{1}^{4} U_{2} + ϱ_{12}^{4} U_{3} + ϱ_{12}^{4} U_{4}}{4}, U_{β} = \frac{ς_{1}^{4} U_{5} + ς_{1}^{4} U_{6} + ς_{12}^{4} U_{7} + ς_{12}^{4} U_{8}}{4}; \end{matrix}

from lower bound lemma 1, and if

ϱ (t) = ϱ_{1}, ϱ (t) = ϱ_{2}, ς (t) = ς_{1}, ς (t) = ς_{2}, w e h a v e

\begin{matrix} \int_{t - ϱ (t)}^{t - ϱ_{1}} \dot{X} (ϰ) d ϰ = 0, \int_{t - ϱ (t)}^{t - ϱ_{1}} X (ϰ) d ϰ = 0, \int_{t - ς (t)}^{t - ς_{1}} \dot{У} (ϰ) d ϰ = 0, \int_{t - ς (t)}^{t - ς_{1}} У (ϰ) d ϰ = 0, \\ \int_{t - ϱ_{2}}^{t - ϱ (t)} \dot{X} (ϰ) d ϰ = 0, \int_{t - ϱ_{2}}^{t - ϱ (t)} X (ϰ) d ϰ = 0, \int_{t - ς_{2}}^{t - ς (t)} \dot{У} (ϰ) d ϰ = 0, \int_{t - ς_{2}}^{t - ς (t)} У (ϰ) d ϰ = 0 . \end{matrix}

The second-order convex combination’s upper bounds can be obtained in

{\dot{V}}_{5} (t)

from the lower bound Lemma 1 if there exist matrices

M_{1}, M_{2}, \dots, M_{10}

such that (3) and (4) hold

\begin{matrix} - \frac{1}{α_{1}^{2}} \int_{- ϱ (t)}^{- ϱ_{1}} \int_{t + η}^{t - ϱ_{1}} {\dot{X}}^{T} (ϰ) d ϰ d η U_{3} \int_{- ϱ (t)}^{- ϱ_{1}} \int_{t + η}^{t - ϱ_{1}} \dot{X} (ϰ) d ϰ d η \\ - \frac{1}{β_{1}^{2}} \int_{- ϱ_{2}}^{- ϱ (t)} \int_{t + η}^{t - ϱ (t)} {\dot{X}}^{T} (ϰ) d ϰ d η U_{3} \int_{- ϱ_{2}}^{- ϱ (t)} \int_{t + η}^{t - ϱ (t)} \dot{X} (ϰ) d ϰ d η \\ - \frac{1}{α_{1}^{2}} \int_{- ϱ (t)}^{- ϱ_{1}} \int_{t - ϱ (t)}^{t + η} {\dot{X}}^{T} (ϰ) d ϰ d η U_{4} \int_{- ϱ (t)}^{- ϱ_{1}} \int_{t - ϱ (t)}^{t + η} \dot{X} (ϰ) d ϰ d η \\ - \frac{1}{β_{1}^{2}} \int_{- ϱ_{2}}^{- ϱ (t)} \int_{t - ϱ_{2}}^{t + η)} {\dot{X}}^{T} (ϰ) d ϰ d η U_{4} \int_{- ϱ_{2}}^{- ϱ (t)} \int_{t - ϱ_{2}}^{t + η} \dot{X} (ϰ) d ϰ d η \\ - \frac{1}{α_{2}^{2}} \int_{- ς (t)}^{- ς_{1}} \int_{t + η}^{t - ς_{1}} {\dot{У}}^{T} (ϰ) d ϰ d η U_{7} \int_{- ς (t)}^{- ς_{1}} \int_{t + η}^{t - ς_{1}} \dot{У} (ϰ) d ϰ d η \\ - \frac{1}{β_{2}^{2}} \int_{- ς_{2}}^{- ς (t)} \int_{t + η}^{t - ς (t)} {\dot{У}}^{T} (ϰ) d ϰ d η U_{7} \int_{- ς_{2}}^{- ς (t)} \int_{t + η}^{t - ς (t)} \dot{У} (ϰ) d ϰ d η \\ - \frac{1}{α_{2}^{2}} \int_{- ς (t)}^{- ς_{1}} \int_{t - ς (t)}^{t + η} {\dot{У}}^{T} (ϰ) d ϰ d η U_{8} \int_{- ς (t)}^{- ς_{1}} \int_{t - ς (t)}^{t + η} \dot{У} (ϰ) d ϰ d η \\ - \frac{1}{β_{2}^{2}} \int_{- ς_{2}}^{- ς (t)} \int_{t - ς_{2}}^{t + η)} {\dot{У}}^{T} (ϰ) d ϰ d η U_{8} \int_{- ς_{2}}^{- ς (t)} \int_{t - ς_{2}}^{t + η} \dot{У} (ϰ) d ϰ d η \end{matrix}

\begin{matrix} \leq & - {[\begin{matrix} \int_{- ϱ (t)}^{- ϱ_{1}} \int_{t + η}^{t - ϱ_{1}} \dot{X} (ϰ) d ϰ d η \\ \int_{- ϱ_{2}}^{- ϱ (t)} \int_{t + η}^{t - ϱ (t)} \dot{X} (ϰ) d ϰ d η \end{matrix}]}^{T} [\begin{matrix} U_{3} & M_{1} + M_{3} \\ * & U_{3} \end{matrix}] [\begin{matrix} \int_{- ϱ (t)}^{- ϱ_{1}} \int_{t + η}^{t - ϱ_{1}} \dot{X} (ϰ) d ϰ d η \\ \int_{- ϱ_{2}}^{- ϱ (t)} \int_{t + η}^{t - ϱ (t)} \dot{X} (ϰ) d ϰ d η \end{matrix}] \\ - {[\begin{matrix} \int_{- ϱ (t)}^{- ϱ_{1}} \int_{t - ϱ (t)}^{t + η} \dot{X} (ϰ) d ϰ d η \\ \int_{- ϱ_{2}}^{- ϱ (t)} \int_{t - ϱ_{2}}^{t + η} \dot{X} (ϰ) d ϰ d η \end{matrix}]}^{T} [\begin{matrix} U_{4} & M_{5} + M_{6} \\ * & U_{4} \end{matrix}] [\begin{matrix} \int_{- ϱ (t)}^{- ϱ_{1}} \int_{t - ϱ (t)}^{t + η} \dot{X} (ϰ) d ϰ d η \\ \int_{- ϱ_{2}}^{- ϱ (t)} \int_{t - ϱ_{2}}^{t + η} \dot{X} (ϰ) d ϰ d η \end{matrix}] \\ - {[\begin{matrix} \int_{- ς (t)}^{- ς_{1}} \int_{t + η}^{t - ς_{1}} \dot{У} (ϰ) d ϰ d η \\ \int_{- ς_{2}}^{- ς (t)} \int_{t + η}^{t - ς (t)} \dot{У} (ϰ) d ϰ d η \end{matrix}]}^{T} [\begin{matrix} U_{7} & M_{7} + M_{8} \\ * & U_{7} \end{matrix}] [\begin{matrix} \int_{- ς (t)}^{- ς_{1}} \int_{t + η}^{t - ς_{1}} \dot{У} (ϰ) d ϰ d η \\ \int_{- ς_{2}}^{- ς (t)} \int_{t + η}^{t - ς (t)} \dot{У} (ϰ) d ϰ d η \end{matrix}] \\ - {[\begin{matrix} \int_{- ς (t)}^{- ς_{1}} \int_{t - ς (t)}^{t + η} \dot{У} (ϰ) d ϰ d η \\ \int_{- ς_{2}}^{- ς (t)} \int_{t - ς_{2}}^{t + η} \dot{У} (ϰ) d ϰ d η \end{matrix}]}^{T} [\begin{matrix} U_{8} & M_{9} + M_{10} \\ * & U_{8} \end{matrix}] [\begin{matrix} \int_{- ς (t)}^{- ς_{1}} \int_{t - ς (t)}^{t + η} \dot{У} (ϰ) d ϰ d η \\ \int_{- ς_{2}}^{- ς (t)} \int_{t - ς_{2}}^{t + η} \dot{У} (ϰ) d ϰ d η \end{matrix}], \end{matrix}

(11)

\begin{matrix} - \frac{1}{α_{1}} \int_{t - ϱ (t)}^{t - ϱ_{1}} {\dot{X}}^{T} (ϰ) d ϰ S_{2} \int_{t - ϱ (t)}^{t - ϱ_{1}} \dot{X} (ϰ) d ϰ - \frac{1}{β_{1}} \int_{t - ϱ_{2}}^{t - ϱ (t)} {\dot{X}}^{T} (ϰ) d ϰ S_{2} \int_{t - ϱ_{2}}^{t - ϱ (t)} \dot{X} (ϰ) d ϰ \\ - \frac{ϱ_{12}^{2}}{2} . \frac{β_{1}}{α_{1}} \int_{t - ϱ (t)}^{t - ϱ_{1}} {\dot{X}}^{T} (ϰ) d ϰ U_{3} \int_{t - ϱ (t)}^{t - ϱ_{1}} \dot{X} (ϰ) d ϰ - \frac{ϱ_{12}^{2}}{2} . \frac{α_{1}}{β_{1}} \int_{t - ϱ_{2}}^{t - ϱ (t)} {\dot{X}}^{T} (ϰ) d ϰ U_{4} \int_{t - ϱ_{2}}^{t - ϱ (t)} \dot{X} (ϰ) d ϰ \\ \leq - {[\begin{matrix} \int_{t - ϱ (t)}^{t - ϱ_{1}} \dot{X} (ϰ) d ϰ \\ \int_{t - ϱ_{2}}^{t - ϱ (t)} \dot{X} (ϰ) d ϰ \end{matrix}]}^{T} [\begin{matrix} S_{2} + \frac{ϱ_{12}^{2}}{2} U_{3} & M_{2} \\ * & S_{2} + \frac{ϱ_{12}^{2}}{2} U_{4} \end{matrix}] [\begin{matrix} \int_{t - ϱ (t)}^{t - ϱ_{1}} \dot{X} (ϰ) d ϰ \\ \int_{t - ϱ_{2}}^{t - ϱ (t)} \dot{X} (ϰ) d ϰ \end{matrix}], \end{matrix}

(12)

\begin{matrix} - \frac{1}{α_{2}} \int_{t - ς (t)}^{t - ς_{1}} {\dot{У}}^{T} (ϰ) d ϰ S_{4} \int_{t - ς (t)}^{t - ς_{1}} \dot{У} (ϰ) d ϰ - \frac{1}{β_{2}} \int_{t - ς_{2}}^{t - ς (t)} {\dot{У}}^{T} (ϰ) d ϰ S_{4} \int_{t - ς_{2}}^{t - ς (t)} \dot{У} (ϰ) d ϰ \\ - \frac{ς_{12}^{2}}{2} . \frac{β_{2}}{α_{2}} \int_{t - ς (t)}^{t - ς_{1}} {\dot{У}}^{T} (ϰ) d ϰ U_{7} \int_{t - ς (t)}^{t - ς_{1}} \dot{У} (ϰ) d ϰ - \frac{ς_{12}^{2}}{2} . \frac{α_{2}}{β_{2}} \int_{t - ς_{2}}^{t - ς (t)} {\dot{У}}^{T} (ϰ) d ϰ U_{8} \int_{t - ς_{2}}^{t - ς (t)} \dot{У} (ϰ) d ϰ \\ \leq - {[\begin{matrix} \int_{t - ς (t)}^{t - ς_{1}} \dot{У} (ϰ) d ϰ \\ \int_{t - ς_{2}}^{t - ς (t)} \dot{У} (ϰ) d ϰ \end{matrix}]}^{T} [\begin{matrix} S_{4} + \frac{ς_{12}^{2}}{2} U_{7} & M_{4} \\ * & S_{4} + \frac{ς_{12}^{2}}{2} U_{8} \end{matrix}] [\begin{matrix} \int_{t - ς (t)}^{t - ς_{1}} \dot{У} (ϰ) d ϰ \\ \int_{t - ς_{2}}^{t - ς (t)} \dot{У} (ϰ) d ϰ \end{matrix}], \end{matrix}

(13)

\begin{matrix} - \frac{1}{α_{1}} \int_{t - ϱ (t)}^{t - ϱ_{1}} X^{T} (ϰ) d ϰ T_{2} \int_{t - ϱ (t)}^{t - ϱ_{1}} X (ϰ) d ϰ - \frac{1}{β_{1}} \int_{t - ϱ_{2}}^{t - ϱ (t)} X^{T} (ϰ) d ϰ T_{2} \int_{t - ϱ_{2}}^{t - ϱ (t)} X (ϰ) d ϰ \\ \leq - {[\begin{matrix} \int_{t - ϱ (t)}^{t - ϱ_{1}} X (ϰ) d ϰ \\ \int_{t - ϱ_{2}}^{t - ϱ (t)} X (ϰ) d ϰ \end{matrix}]}^{T} [\begin{matrix} T_{2} & N_{1} \\ * & T_{2} \end{matrix}] [\begin{matrix} \int_{t - ϱ (t)}^{t - ϱ_{1}} X (ϰ) d ϰ \\ \int_{t - ϱ_{2}}^{t - ϱ (t)} X (ϰ) d ϰ \end{matrix}], \end{matrix}

(14)

\begin{matrix} - \frac{1}{α_{2}} \int_{t - ς (t)}^{t - ς_{1}} У^{T} (ϰ) d ϰ T_{4} \int_{t - ς (t)}^{t - ς_{1}} У (ϰ) d ϰ - \frac{1}{β_{2}} \int_{t - ς_{2}}^{t - ς (t)} У^{T} (ϰ) d ϰ T_{4} \int_{t - ς_{2}}^{t - ς (t)} У (ϰ) d ϰ \\ \leq - {[\begin{matrix} \int_{t - ς (t)}^{t - ς_{1}} У (ϰ) d ϰ \\ \int_{t - ς_{2}}^{t - ς (t)} У (ϰ) d ϰ \end{matrix}]}^{T} [\begin{matrix} T_{4} & N_{2} \\ * & T_{4} \end{matrix}] [\begin{matrix} \int_{t - ς (t)}^{t - ς_{1}} У (ϰ) d ϰ \\ \int_{t - ς_{2}}^{t - ς (t)} У (ϰ) d ϰ \end{matrix}], \end{matrix}

(15)

Substituting (11)–(15) in (10), we have

\begin{matrix} {\dot{V}}_{5} (t) & \leq & {\dot{X}}^{T} (t) U_{α} \dot{X} (t) + {\dot{У}}^{T} (t) U_{β} \dot{У} (t) - [ϱ_{1} X^{T} (t) - \int_{t - ϱ_{1}}^{t} X^{T} (ϰ) d ϰ] U_{1} [ϱ_{1} X (t) - \int_{t - ϱ_{1}}^{t} X (ϰ) d ϰ] \\ - [\int_{t - ϱ_{1}}^{t} X^{T} (ϰ) d ϰ - ϱ_{1} X^{T} (t - ϱ_{1})] U_{2} [\int_{t - ϱ_{1}}^{t} X (ϰ) d ϰ - ϱ_{1} X (t - ϱ_{1})] \\ - [ς_{1} У^{T} (t) - \int_{t - ς_{1}}^{t} У^{T} (ϰ) d ϰ] U_{5} [ς_{1} У (t) - \int_{t - ς_{1}}^{t} У (ϰ) d ϰ] \\ - [\int_{t - ς_{1}}^{t} У^{T} (ϰ) d ϰ - ς_{1} У^{T} (t - ς_{1})] U_{6} [\int_{t - ς_{1}}^{t} У (ϰ) d ϰ - ς_{1} У (t - ς_{1})] \\ - {[\begin{matrix} \int_{- ϱ (t)}^{- ϱ_{1}} \int_{t + η}^{t - ϱ_{1}} \dot{X} (ϰ) d ϰ d η \\ \int_{- ϱ_{2}}^{- ϱ (t)} \int_{t + η}^{t - ϱ (t)} \dot{X} (ϰ) d ϰ d η \end{matrix}]}^{T} [\begin{matrix} U_{3} & M_{1} + M_{3} \\ * & U_{3} \end{matrix}] [\begin{matrix} \int_{- ϱ (t)}^{- ϱ_{1}} \int_{t + η}^{t - ϱ_{1}} \dot{X} (ϰ) d ϰ d η \\ \int_{- ϱ_{2}}^{- ϱ (t)} \int_{t + η}^{t - ϱ (t)} \dot{X} (ϰ) d ϰ d η \end{matrix}] \\ - {[\begin{matrix} \int_{- ϱ (t)}^{- ϱ_{1}} \int_{t - ϱ (t)}^{t + η} \dot{X} (ϰ) d ϰ d η \\ \int_{- ϱ_{2}}^{- ϱ (t)} \int_{t - ϱ_{2}}^{t + η} \dot{X} (ϰ) d ϰ d η \end{matrix}]}^{T} [\begin{matrix} U_{4} & M_{5} + M_{6} \\ * & U_{4} \end{matrix}] [\begin{matrix} \int_{- ϱ (t)}^{- ϱ_{1}} \int_{t - ϱ (t)}^{t + η} \dot{X} (ϰ) d ϰ d η \\ \int_{- ϱ_{2}}^{- ϱ (t)} \int_{t - ϱ_{2}}^{t + η} \dot{X} (ϰ) d ϰ d η \end{matrix}] \\ - {[\begin{matrix} \int_{- ς (t)}^{- ς_{1}} \int_{t + η}^{t - ς_{1}} \dot{У} (ϰ) d ϰ d η \\ \int_{- ς_{2}}^{- ς (t)} \int_{t + η}^{t - ς (t)} \dot{У} (ϰ) d ϰ d η \end{matrix}]}^{T} [\begin{matrix} U_{7} & M_{7} + M_{8} \\ * & U_{7} \end{matrix}] [\begin{matrix} \int_{- ς (t)}^{- ς_{t}} \int_{t + η}^{t - ς_{1}} \dot{У} (ϰ) d ϰ d η \\ \int_{- ς_{2}}^{- ς (t)} \int_{t + η}^{t - ς (t)} \dot{У} (ϰ) d ϰ d η \end{matrix}] \\ - {[\begin{matrix} \int_{- ς (t)}^{- ς_{1}} \int_{t - ς (t)}^{t + η} \dot{У} (ϰ) d ϰ d η \\ \int_{- ς_{2}}^{- ς (t)} \int_{t - ς_{2}}^{t + η} \dot{У} (ϰ) d ϰ d η \end{matrix}]}^{T} [\begin{matrix} U_{8} & M_{9} + M_{10} \\ * & U_{8} \end{matrix}] [\begin{matrix} \int_{- ς (t)}^{- ς_{1}} \int_{t - ς (t)}^{t + η} \dot{У} (ϰ) d ϰ d η \\ \int_{- ς_{2}}^{- ς (t)} \int_{t - ς_{2}}^{t + η} \dot{У} (ϰ) d ϰ d η \end{matrix}] \end{matrix}

\begin{matrix} \leq & {\dot{X}}^{T} (t) U_{α} \dot{X} (t) + {\dot{У}}^{T} (t) U_{β} \dot{У} (t) - [ϱ_{1} X^{T} (t) - \int_{t - ϱ_{1}}^{t} X^{T} (ϰ) d ϰ] U_{1} [ϱ_{1} X (t) - \int_{t - ϱ_{1}}^{t} X (ϰ) d ϰ] \\ - [\int_{t - ϱ_{1}}^{t} X^{T} (ϰ) d ϰ - ϱ_{1} X^{T} (t - ϱ_{1})] U_{2} [\int_{t - ϱ_{1}}^{t} X (ϰ) d ϰ - ϱ_{1} X (t - ϱ_{1})] \\ - [ς_{1} У^{T} (t) - \int_{t - ς_{1}}^{t} У^{T} (ϰ) d ϰ] U_{5} [ς_{1} У (t) - \int_{t - ς_{1}}^{t} У (ϰ) d ϰ] \\ - [\int_{t - ς_{1}}^{t} У^{T} (ϰ) d ϰ - ς_{1} У^{T} (t - ς_{1})] U_{6} [\int_{t - ς_{1}}^{t} У (ϰ) d ϰ - ς_{1} У (t - ς_{1})] \\ - Ξ_{1}^{T} (t) {Γ_{1}^{T} (t) [\begin{matrix} U_{3} & M_{1} + M_{3} \\ * & U_{3} \end{matrix}] Γ_{1} (t) - Γ_{2}^{T} (t) [\begin{matrix} U_{4} & M_{5} + M_{6} \\ * & U_{4} \end{matrix}] Γ_{2} (t) \\ - Γ_{3}^{T} (t) [\begin{matrix} U_{7} & M_{7} + M_{8} \\ * & U_{7} \end{matrix}] Γ_{3} (t) - Γ_{4}^{T} (t) [\begin{matrix} U_{8} & M_{9} + M_{10} \\ * & U_{8} \end{matrix}] Γ_{4} (t)} Ξ_{1} (t), \end{matrix}

(16)

where

\begin{matrix} Γ_{1} (t) & = & [\begin{matrix} 0_{n} & 0_{n} & (ϱ (t) - ϱ_{1}) I_{n} & 2 t e r m s (0_{n}) & - (I_{n}) & 0_{n} & 13 t e r m s (0_{n}) \\ 0_{n} & (ϱ_{2} - ϱ (t)) I_{n} & 0_{n} & 2 t e r m s (0_{n}) & 0_{n} & - (I_{n}) & 13 t e r m s (0_{n}) \end{matrix}], \end{matrix}

\begin{matrix} Γ_{2} (t) & = & [\begin{matrix} 0_{n} & (ϱ (t) - ϱ_{1}) (- I_{n}) & 0_{n} & 0_{n} & 0_{n} & I_{n} & 0_{n} & 13 t e r m s (0_{n}) \\ 0_{n} & 0_{n} & 0_{n} & (ϱ_{2} - ϱ (t)) (- I_{n}) & 0_{n} & 0_{n} & I_{n} & 13 t e r m s (0_{n}) \end{matrix}], \end{matrix}

\begin{matrix} Γ_{3} (t) & = & [\begin{matrix} 11 t e r m s (0_{n}) & 0_{n} & (ς (t) - ς_{1}) I_{n} & 2 t e r m s (0_{n}) & - (I_{n}) & 0_{n} & 3 t e r m s (0_{n}) \\ 11 t e r m s (0_{n}) & (ς_{2} - ς (t)) I_{n} & 0_{n} & 2 t e r m s (0_{n}) & 0_{n} & - (I_{n}) & 3 t e r m s (0_{n}) \end{matrix}], \end{matrix}

\begin{matrix} Γ_{4} (t) & = & [\begin{matrix} 12 t e r m s (0_{n}) & (ς (t) - ς_{1}) (- I_{n}) & 0_{n} & 0_{n} & I_{n} & 0_{n} & 3 t e r m s (0_{n}) \\ 12 t e r m s (0_{n}) & 0_{n} & (ς_{2} - ς (t)) (- I_{n}) & 0_{n} & 0_{n} & I_{n} & 3 t e r m s (0_{n}) \end{matrix}], \end{matrix}

\begin{matrix} Ξ_{1} (t) & = & c o l {X (t), X (t - ϱ (t), X (t - ϱ_{1}), X (t - ϱ_{2}), \int_{t - ϱ_{1}}^{t} X (ϰ) d ϰ, \int_{t - ϱ (t)}^{t - ϱ_{1}} X (ϰ) d ϰ, \int_{t - ϱ_{2}}^{t - ϱ (t)} X (ϰ) d ϰ, g (X (t)), \\ g (X (t - ϱ (t), \int_{t - γ_{2} (t)}^{t - γ_{1} (t)} g (X (ϰ)) d ϰ, У (t), У (t - ς (t), У (t - ς_{1}), У (t - ς_{2}), \int_{t - ς_{1}}^{t} У (ϰ) d ϰ, \\ \int_{t - ς (t)}^{t - ς_{1}} У (ϰ) d ϰ, \int_{t - ς_{2}}^{t - ς (t)} У (ϰ) d ϰ, f (У (t)), f (У (t - ς (t), \int_{t - η_{2} (t)}^{t - η_{1} (t)} f (У (ϰ)) d ϰ} . \end{matrix}

Calculating the time derivative of

V_{6} (t)

, we obtain

\begin{matrix} {\dot{V}}_{6} (t) & = & γ_{12}^{2} g^{T} (X (t)) J_{1} g (X (t)) - γ_{12} \int_{t - γ_{2}}^{t - γ_{1}} g^{T} (X (ϰ)) J_{1} g (X (ϰ)) d ϰ + η_{12}^{2} f^{T} (У (t)) J_{2} f (У (t)) \\ - η_{12} \int_{t - η_{2}}^{t - η_{1}} f^{T} (У (ϰ)) J_{2} f (У (ϰ)) d ϰ, \\ \leq γ_{12}^{2} g^{T} (X (t)) J_{1} g (X (t)) - \int_{t - γ_{2} (t)}^{t - γ_{1} (t)} g^{T} (X (ϰ)) d ϰ J_{1} \int_{t - γ_{2} (t)}^{t - γ_{1} (t)} g (X (ϰ)) d ϰ + η_{12}^{2} f^{T} (У (t)) J_{2} f (У (t)) \\ - \int_{t - η_{2} (t)}^{t - η_{1} (t)} f^{T} (У (ϰ)) d ϰ J_{2} \int_{t - η_{2} (t)}^{t - η_{1} (t)} f (У (ϰ)) d ϰ . \end{matrix}

(17)

In addition, from Assumption 1, we have

\begin{matrix} g^{T} (X (t - ϱ (t))) g (X (t - ϱ (t))) \leq X^{T} (t - ϱ (t)) W_{1}^{T} W_{1} X (t - ϱ (t)), \end{matrix}

(18)

\begin{matrix} f^{T} (У (t - ς (t))) f (У (t - ς (t))) \leq У^{T} (t - ς (t)) W_{2}^{T} W_{2} У (t - ς (t)), \end{matrix}

(19)

and from (18) and (19), it is easily verified that

\begin{matrix} a_{1} X^{T} (t) W_{1}^{T} W_{1} X (t) - a_{1} g^{T} (X (t)) g (X (t)) \geq 0, \end{matrix}

(20)

\begin{matrix} a_{2} X^{T} (t - ϱ (t)) W_{1}^{T} W_{1} X (t - ϱ (t)) - a_{2} g^{T} (X (t - ϱ (t))) g (X (t - ϱ (t))) \geq 0, \end{matrix}

(21)

\begin{matrix} b_{1} У^{T} (t) W_{2}^{T} W_{2} Y (t) - b_{1} f^{T} (У (t)) f (У (t)) \geq 0, \end{matrix}

(22)

\begin{matrix} b_{2} У^{T} (t - ς (t)) W_{2}^{T} W_{2} У (t - ς (t)) - b_{2} f^{T} (У (t - ς (t))) f (У (t - ς (t))) \geq 0 . \end{matrix}

(23)

From the estimation of the time derivatives (6)–(9), (16) and (17) of

V (t)

along with the activation functions (20)–(23), it is evident that

\begin{matrix} \dot{V} (t) \leq Ξ_{1}^{T} (t) {Ω - Γ_{1}^{T} (t) [\begin{matrix} U_{3} & M_{1} + M_{3} \\ * & U_{3} \end{matrix}] Γ_{1} (t) - Γ_{2}^{T} (t) [\begin{matrix} U_{4} & M_{5} + M_{6} \\ * & U_{4} \end{matrix}] Γ_{2} (t) \\ - Γ_{3}^{T} (t) [\begin{matrix} U_{7} & M_{7} + M_{8} \\ * & U_{7} \end{matrix}] Γ_{3} (t) - Γ_{4}^{T} (t) [\begin{matrix} U_{8} & M_{9} + M_{10} \\ * & U_{8} \end{matrix}] Γ_{4} (t)} Ξ_{1} (t) . \end{matrix}

(24)

Furthermore,

\begin{matrix} Ω - Γ_{1}^{T} (t) [\begin{matrix} U_{3} & M_{1} + M_{3} \\ * & U_{3} \end{matrix}] Γ_{1} (t) - Γ_{2}^{T} (t) [\begin{matrix} U_{4} & M_{5} + M_{6} \\ * & U_{4} \end{matrix}] Γ_{2} (t) \\ - Γ_{3}^{T} (t) [\begin{matrix} U_{7} & M_{7} + M_{8} \\ * & U_{7} \end{matrix}] Γ_{3} (t) - Γ_{4}^{T} (t) [\begin{matrix} U_{8} & M_{9} + M_{10} \\ * & U_{8} \end{matrix}] Γ_{4} (t) < 0, \end{matrix}

(25)

is intrinsically linear in

ϱ (t) & ς (t)

and from Lemma 2 and Schur Complement Lemma 3, we can see that

\begin{matrix} [\begin{matrix} ϕ_{l} & Π_{1}^{T} & Π_{2}^{T} & Π_{3}^{T} & Π_{4}^{T} \\ * & - \bar{U_{3}} & 0 & 0 & 0 \\ * & * & - \bar{U_{4}} & 0 & 0 \\ * & * & * & - \bar{U_{7}} & 0 \\ * & * & * & * & - \bar{U_{8}} \end{matrix}] < 0, \end{matrix}

(26)

where

ϕ_{l} = [\begin{matrix} Ω & 2 Γ_{1 l}^{T} \hat{U_{3}} & 2 Γ_{2 l}^{T} \hat{U_{4}} & 2 Γ_{3 l}^{T} \hat{U_{7}} & 2 Γ_{4 l}^{T} \hat{U_{8}} \\ * & - 2 \hat{U_{3}} & 0 & 0 & 0 \\ * & * & - 2 \hat{U_{4}} & 0 & 0 \\ * & * & * & - 2 \hat{U_{7}} & 0 \\ * & * & * & * & - 2 \hat{U_{8}} \end{matrix}] + [\begin{matrix} Γ_{1 l}^{T} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \end{matrix}] Π_{1} + Π_{1}^{T} {[\begin{matrix} Γ_{1 l}^{T} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \end{matrix}]}^{T} + [\begin{matrix} Γ_{2 l}^{T} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \end{matrix}] Π_{2}

+ Π_{2}^{T} {[\begin{matrix} Γ_{2 l}^{T} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \end{matrix}]}^{T} + [\begin{matrix} Γ_{3 l}^{T} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \end{matrix}] Π_{3} + Π_{3}^{T} {[\begin{matrix} Γ_{3 l}^{T} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \end{matrix}]}^{T} + [\begin{matrix} Γ_{4 l}^{T} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \end{matrix}] Π_{4} + Π_{4}^{T} {[\begin{matrix} Γ_{4 l}^{T} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \\ 0_{2 n} \end{matrix}]}^{T},

Therefore, four corresponding boundary LMIs can be treated non-conservatively in (26): two for

ϱ (t) = ϱ_{1}, ς (t) = ς_{1}

and the other two for

ϱ (t) = ϱ_{2}, ς (t) = ς_{2}

, which imply

\dot{V} (t) < 0 .

This completes the proof. □

Theorem 2.

A given scalar

0 \leq ς_{1} \leq ς_{2}

,

0 \leq ϱ_{1} \leq ϱ_{2}

system (1) is globally asymptotically stable if there exist symmetric matrices

P = {[P_{i j}]}_{4 \times 4} > 0

,

Q = {[Q_{i j}]}_{4 \times 4} > 0

,

R_{k} > 0

,

S_{k} > 0

,

T_{k} > 0

,

k = 1, 2, 3, 4,

U_{j} > 0

,

j = 1, 2, \dots, 8,

[\begin{matrix} R_{11} & R_{12} \\ * & R_{22} \end{matrix}] > 0, [\begin{matrix} R_{31} & R_{32} \\ * & R_{33} \end{matrix}] > 0,

J_{1}, J_{2} > 0

,

a_{k} > 0, b_{k} > 0, k = 1, 2

such that the LMI condition

Λ < 0

holds, where

Λ = {[Λ_{i j}]}_{32 \times 32}, Λ_{r, s} = Λ_{s, r}^{T}, r, s = 1, 2, \dots, 32 .

\begin{matrix} Λ_{1, 1} & = & - P_{11} A_{1} - A_{1}^{T} P_{11}^{T} + P_{12} + P_{12}^{T} + R_{1} + R_{11} + A_{1}^{T} (ϱ_{1}^{2} S_{1} + ϱ_{12}^{2} S_{2}) A_{1} - S_{1} + ϱ_{1}^{2} T_{1} + ϱ_{12}^{2} T_{2} \\ + ϱ_{12}^{2} T_{2} + A_{1}^{T} U_{α} A_{1} + a_{1} W_{1}^{T} W_{1}, \\ Λ_{1, 2} & = & (1 - υ_{2}) (P_{13} - P_{14}), Λ_{1, 3} = - P_{12} + P_{14} + S_{1}, Λ_{1, 4} = - P_{13}, \\ Λ_{1, 5} & = & P_{22}^{T}, Λ_{1, 14} = R_{12}, Λ_{1, 30} = P_{11} B_{1} - A_{1}^{T} (ϱ_{1}^{2} S_{1} + ϱ_{12}^{2} S_{2}) B_{1} - A_{1}^{T} U_{α} B_{1}, \\ Λ_{1, 31} & = & P_{11} C_{1} - A_{1}^{T} (ϱ_{1}^{2} S_{1} + ϱ_{12}^{2} S_{2}) C_{1} - A_{1}^{T} U_{α} C_{1}, Λ_{1, 32} = P_{11} D_{1} - A_{1}^{T} (ϱ_{1}^{2} S_{1} + ϱ_{12}^{2} S_{2}) D_{1} - A_{1}^{T} U_{α} D_{1}, \\ Λ_{2, 2} & = & - (1 - υ_{2}) R_{11} + a_{2} W_{1}^{T} W_{1}, Λ_{2, 5} = (1 - υ_{2}) P_{23}^{T} - (1 - υ_{2}) P_{24}^{T}, Λ_{2, 6} = - (1 - υ_{2}) P_{44}^{T}, \\ Λ_{2, 7} & = & (1 - υ_{2}) P_{33}^{T} - (1 - υ_{2}) P_{34}^{T}, Λ_{2, 15} = - (1 - υ_{2}) R_{12}, \\ Λ_{3, 3} & = & R_{2} - R_{1} - S_{1} - S_{2}, Λ_{3, 4} = S_{2}, Λ_{3, 5} = - P_{22}^{T} + P_{24}^{T}, Λ_{3, 6} = P_{44}^{T}, Λ_{3, 7} = P_{34}^{T}, \\ Λ_{4, 4} & = & - R_{2} - S_{2}, Λ_{4, 5} = - P_{23}^{T}, Λ_{4, 7} = - P_{33}^{T}, Λ_{5, 5} = - T_{1}, Λ_{6, 6} = - T_{2}, Λ_{6, 7} = - {\bar{T_{2}}}^{T}, \\ Λ_{7, 7} & = & - T_{2}, Λ_{8, 8} = - U_{1}, Λ_{9, 9} = - U_{3}, Λ_{9, 10} = - {\bar{U_{3}}}^{T}, \\ Λ_{10, 10} & = & - U_{3}, Λ_{11, 11} = - U_{2}, Λ_{12, 12} = - U_{4}, Λ_{12, 13} = - {\bar{U_{4}}}^{T}, Λ_{13, 13} = - U_{4} \\ Λ_{14, 14} & = & R_{22} + B_{2}^{T} (ς_{1}^{2} S_{3} + ς_{12}^{2} S_{4}) B_{2} + γ_{12}^{2} J_{1} - a_{1} I + B_{2}^{T} U_{β} B_{2}, \\ Λ_{14, 15} & = & B_{2}^{T} (ς_{1}^{2} S_{3} + ς_{12}^{2} S_{4}) C_{2} + B_{2}^{T} U_{β} C_{2}, \\ Λ_{14, 16} & = & B_{2}^{T} (ς_{1}^{2} S_{3} + ς_{12}^{2} S_{4}) D_{2} + B_{2}^{T} U_{β} D_{2}, Λ_{14, 17} = B_{2}^{T} Q_{11}^{T} - B_{2}^{T} (ς_{1}^{2} S_{3} + ς_{12}^{2} S_{4}) A_{2} - B_{2}^{T} U_{β} A_{2}, \\ Λ_{15, 15} & = & - (1 - υ_{2}) R_{22} + C_{2}^{T} (ς_{1}^{2} S_{3} + ς_{12}^{2} S_{4}) C_{2} + C_{2}^{T} U_{β} C_{2} - a_{2} I, \\ Λ_{15, 16} & = & C_{2}^{T} (ς_{1}^{2} S_{3} + ς_{12}^{2} S_{4}) D_{2} + C_{2}^{T} U_{β} D_{2}, \\ Λ_{15, 17} & = & C_{2}^{T} Q_{11}^{T} - C_{2}^{T} (ς_{1}^{2} S_{3} + ς_{12}^{2} S_{4}) A_{2} - C_{2}^{T} U_{β} A_{2}, \\ Λ_{16, 16} & = & D_{2}^{T} (ς_{1}^{2} S_{3} + ς_{12}^{2} S_{4}) D_{2} + D_{2}^{T} U_{β} D_{2} - J_{1}, \\ Λ_{16, 17} & = & D_{2}^{T} Q_{11}^{T} - D_{2}^{T} (ς_{1}^{2} S_{3} + ς_{12}^{2} S_{4}) A_{2} - D_{2}^{T} U_{β} A_{2}, \\ Λ_{17, 17} & = & - Q_{11} A_{2} - A_{2}^{T} Q_{11}^{T} + Q_{12} + Q_{12}^{T} + R_{3} + R_{31} + A_{2}^{T} (ς_{1}^{2} S_{3} + ς_{12}^{2} S_{4}) A_{2} - S_{3} + ς_{1}^{2} T_{3} + ς_{12}^{2} T_{4} \\ + A_{2}^{T} U_{β} A_{2} + b_{1} W_{2}^{T} W_{2}, \\ Λ_{17, 18} & = & (1 - υ_{1}) (Q_{13} - Q_{14}), Λ_{17, 19} = - Q_{12} + Q_{14} + S_{3}, Λ_{17, 20} = - Q_{13}, Λ_{17, 21} = Q_{22}^{T}, \\ Λ_{17, 30} & = & R_{32}, Λ_{18, 18} = - (1 - υ_{1}) R_{31} + b_{2} W_{2}^{T} W_{2}, \\ Λ_{18, 21} & = & (1 - υ_{1}) (Q_{23}^{T} - Q_{24}^{T}), Λ_{18, 22} = - (1 - υ_{1}) Q_{44}^{T}, \\ Λ_{18, 23} & = & (1 - υ_{1}) (Q_{33}^{T} - Q_{34}^{T}), Λ_{18, 31} = - (1 - υ_{1}) R_{32}, Λ_{19, 19} = - R_{3} + R_{4} - S_{3} - S_{4}, Λ_{19, 20} = S_{4}, \\ Λ_{19, 21} & = & - Q_{22}^{T} + Q_{24}^{T}, Λ_{19, 22} = Q_{44}^{T}, Λ_{19, 23} = Q_{34}^{T}, Λ_{20, 20} = - R_{4} - S_{4}, Λ_{20, 21} = - Q_{23}^{T}, Λ_{20, 23} = - Q_{33}^{T}, \\ Λ_{21, 21} & = & - T_{3}, Λ_{22, 22} = - T_{4}, Λ_{22, 23} = - {\bar{T_{4}}}^{T}, Λ_{23, 23} = - T_{4}, Λ_{24, 24} = - U_{5}, Λ_{25, 25} = - U_{7}, Λ_{25, 26} = - {\bar{U_{7}}}^{T}, \\ Λ_{26, 26} & = & - U_{7}, Λ_{27, 27} = - U_{6}, Λ_{28, 28} = - U_{8}, Λ_{28, 29} = - {\bar{U_{8}}}^{T}, Λ_{29, 29} = - U_{8}, \\ Λ_{30, 30} & = & R_{33} + B_{1}^{T} (ϱ_{1}^{2} S_{1} + ϱ_{12}^{2} S_{2}) B_{1} + B_{1}^{T} U_{α} B_{1} + η_{12}^{2} J_{2} - b_{1} I, Λ_{30, 31} = B_{1}^{T} (ϱ_{1}^{2} S_{1} + ϱ_{12}^{2} S_{2}) C_{1} + B_{1}^{T} U_{α} C_{1}, \\ Λ_{30, 32} & = & B_{1}^{T} (ϱ_{1}^{2} S_{1} + ϱ_{12}^{2} S_{2}) D_{1} + B_{1}^{T} U_{α} D_{1}, Λ_{31, 31} = - (1 - u p s i l o n_{1}) R_{33} + C_{1}^{T} (ϱ_{1}^{2} S_{1} + ϱ_{12}^{2} S_{2}) C_{1} + C_{1}^{T} U_{α} C_{1} - b_{2} I, \\ Λ_{31, 32} & = & C_{1}^{T} (ϱ_{1}^{2} S_{1} + ϱ_{12}^{2} S_{2}) D_{1} + C_{1}^{T} U_{α} D_{1}, Λ_{32, 32} = D_{1}^{T} (ϱ_{1}^{2} S_{1} + ϱ_{12}^{2} S_{2}) D_{1} + D_{1}^{T} U_{α} D_{1} - J_{2}, \\ w h e r e U_{α} & = & \frac{ϱ_{1}^{4} U_{1} + ϱ_{1}^{4} U_{2} + ϱ_{12}^{4} U_{3} + ϱ_{12}^{4} U_{4}}{4}, U_{β} = \frac{ς_{1}^{4} U_{5} + ς_{1}^{4} U_{6} + ς_{12}^{4} U_{7} + ς_{12}^{4} U_{8}}{4}; \end{matrix}

Proof.

Consider the LKF candidates as in Equation (5)

\begin{matrix} V (t) & = & \sum_{r = 1}^{6} V_{r} (t) . \end{matrix}

Calculating the time derivatives, we have

\begin{matrix} {\dot{V}}_{1} (t) & = & 2 ζ_{1}^{T} (t) P \dot{ζ_{1}} (t) + 2 ζ_{2}^{T} (t) Q \dot{ζ_{2}} (t), \end{matrix}

(27)

\begin{matrix} {\dot{V}}_{2} (t) & \leq & X^{T} (t) (R_{1}) X (t) + X^{T} (t - ϱ_{1}) (R_{2} - R_{1}) X (t - ϱ_{1}) - X^{T} (t - ϱ_{2}) (R_{2}) X (t - ϱ_{2}) \\ + X^{T} (t) (R_{11}) X (t) + X^{T} (t) (R_{12}) g (X (t)) + g^{T} (X (t)) (R_{12}^{T}) X (t) \\ + g^{T} (X (t)) (R_{22}) g (X (t)) - (1 - υ_{2}) X^{T} (t - ϱ (t)) (R_{11}) X (t - ϱ (t)) \\ - (1 - υ_{2}) X^{T} (t - ϱ (t)) (R_{12}) g (X (t - ϱ (t))) - (1 - υ_{2}) g^{T} (X (t - ϱ (t))) (R_{22}) g (X (t - ϱ (t))) \\ - (1 - υ_{2}) g^{T} (X (t - ϱ (t))) (R_{12}^{T}) X (t - ϱ (t)) + У^{T} (t) (R_{3}) У (t) \\ + У^{T} (t - ς_{1}) (R_{4} - R_{3}) У (t - ς_{1}) - У^{T} (t - ς_{2}) (R_{4}) У (t - ς_{2}) + У^{T} (t) (R_{31}) У (t) \\ + У^{T} (t) (R_{32}) f (У (t)) + f^{T} (У (t)) (R_{32}^{T}) У (t) + f^{T} (У (t)) (R_{33}) f (У (t)) \\ - (1 - υ_{1}) У^{T} (t - ς (t)) (R_{31}) У (t - ς (t)) - (1 - υ_{1}) У^{T} (t - ς (t)) (R_{32}) f (У (t - ς (t))) \\ - (1 - υ_{1}) f^{T} (У (t - ς (t))) (R_{33}) f (У (t - ς (t))) \\ - (1 - υ_{1}) f^{T} (У (t - ς (t))) (R_{32}^{T}) У (t - ς (t)), \end{matrix}

(28)

By applying Jensen’s inequality [27], we have

\begin{matrix} {\dot{V}}_{3} (t) & \leq & ϱ_{1}^{2} {\dot{X}}^{T} (t) S_{1} \dot{X} (t) - \int_{t - ϱ_{1}}^{t} {\dot{X}}^{T} (ϰ) d ϰ S_{1} \int_{t - ϱ_{1}}^{t} \dot{X} (ϰ) d ϰ + ς_{1}^{2} {\dot{У}}^{T} (t) S_{3} \dot{У} (t) \\ - \int_{t - ς_{1}}^{t} {\dot{У}}^{T} (ϰ) d ϰ S_{3} \int_{t - ς_{1}}^{t} \dot{У} (ϰ) d ϰ + ϱ_{12}^{2} {\dot{X}}^{T} (t) S_{2} \dot{X} (t) \\ - \int_{t - ϱ (t)}^{t - ϱ_{1}} {\dot{X}}^{T} (ϰ) d ϰ S_{2} \int_{t - ϱ (t)}^{t - ϱ_{1}} \dot{X} (ϰ) d ϰ - \int_{t - ϱ_{2}}^{t - ϱ_{1}} {\dot{X}}^{T} (ϰ) d ϰ S_{2} \int_{t - ϱ_{2}}^{t - ϱ_{1}} \dot{X} (ϰ) d ϰ \\ + ς_{12}^{2} {\dot{У}}^{T} (t) S_{4} \dot{У} (t) - \int_{t - ς_{2}}^{t - ς_{1}} {\dot{У}}^{T} (ϰ) d ϰ S_{4} \int_{t - ς_{2}}^{t - ς_{1}} \dot{У} (ϰ) d ϰ, \end{matrix}

(29)

\begin{matrix} {\dot{V}}_{4} (t) & = & ϱ_{1}^{2} X^{T} (t) T_{1} X (t) - ϱ_{1} \int_{t - ϱ_{1}}^{t} X^{T} (ϰ) T_{1} X (ϰ) d ϰ + ς_{1}^{2} У^{T} (t) T_{3} У (t) - ς_{1} \int_{t - ς_{1}}^{t} У^{T} (ϰ) T_{3} У (ϰ) d ϰ \\ + ϱ_{12}^{2} X^{T} (t) T_{2} X (t) - ϱ_{12} \int_{t - ϱ_{2}}^{t - ϱ_{1}} X^{T} (ϰ) T_{2} X (ϰ) d ϰ \\ + ς_{12}^{2} У^{T} (t) T_{4} У (t) - ς_{12} \int_{t - ς_{2}}^{t - ς_{1}} У^{T} (ϰ) T_{4} У (ϰ) d ϰ, \end{matrix}

(30)

By applying Jensen’s inequality [27] and the lower bound Lemma 1 to

{\dot{V}}_{4} (t)

, we obtain

\begin{matrix} - ϱ_{1} \int_{t - ϱ_{1}}^{t} X^{T} (ϰ) T_{1} X (ϰ) d ϰ & \leq & - {\{\int_{t - ϱ_{1}}^{t} X (ϰ) d ϰ\}}^{T} T_{1} \{\int_{t - ϱ_{1}}^{t} X (ϰ) d ϰ\}, \end{matrix}

(31)

\begin{matrix} - ς_{1} \int_{t - ς_{1}}^{t} У^{T} (ϰ) T_{3} У (ϰ) d ϰ & \leq & - {\{\int_{t - ς_{1}}^{t} У (ϰ) d ϰ\}}^{T} T_{3} \{\int_{t - ς_{1}}^{t} У (ϰ) d ϰ\}, \end{matrix}

(32)

\begin{matrix} - ϱ_{12} \int_{t - ϱ_{2}}^{t - ϱ_{1}} X^{T} (ϰ) T_{2} X (ϰ) d ϰ & = & - ϱ_{12} \int_{t - ϱ_{2}}^{t - ϱ (t)} X^{T} (ϰ) T_{2} X (ϰ) d ϰ - ϱ_{12} \int_{t - ϱ (t)}^{t - ϱ_{1}} X^{T} (ϰ) T_{2} X (ϰ) d ϰ \\ \leq & - \frac{ϱ_{12}}{ϱ_{2} - ϱ (t)} {\{\int_{t - ϱ_{2}}^{t - ϱ (t)} X (ϰ) d ϰ\}}^{T} T_{2} \{\int_{t - ϱ_{2}}^{t - ϱ (t)} X (ϰ) d ϰ\} \\ - \frac{ϱ_{12}}{ϱ (t) - ϱ_{1}} {\{\int_{t - ϱ (t)}^{t - ϱ_{1}} X (ϰ) d ϰ\}}^{T} T_{2} \{\int_{t - ϱ (t)}^{t - ϱ_{1}} X (ϰ) d ϰ\} \\ \leq & {[\begin{matrix} \int_{t - ϱ_{2}}^{t - ϱ (t)} X (ϰ) d ϰ \\ \int_{t - ϱ (t)}^{t - ϱ_{1}} X (ϰ) d ϰ \end{matrix}]}^{T} [\begin{matrix} T_{2} & \bar{T_{2}} \\ * & T_{2} \end{matrix}] [\begin{matrix} \int_{t - ϱ_{2}}^{t - ϱ (t)} X (ϰ) d ϰ \\ \int_{t - ϱ (t)}^{t - ϱ_{1}} X (ϰ) d ϰ \end{matrix}], \end{matrix}

(33)

Similarly,

\begin{matrix} - ς_{12} \int_{t - ς_{2}}^{t - ς_{1}} У^{T} (ϰ) T_{4} У (ϰ) d ϰ & \leq & {[\begin{matrix} \int_{t - ς_{2}}^{t - ς (t)} У (ϰ) d ϰ \\ \int_{t - ς (t)}^{t - ς_{1}} У (ϰ) d ϰ \end{matrix}]}^{T} [\begin{matrix} T_{4} & \bar{T_{4}} \\ * & T_{4} \end{matrix}] [\begin{matrix} \int_{t - ς_{2}}^{t - ς (t)} У (ϰ) d ϰ \\ \int_{t - ς (t)}^{t - ς_{1}} У (ϰ) d ϰ \end{matrix}], \end{matrix}

(34)

By substituting (31)–(34) in (30), we get

\begin{matrix} {\dot{V}}_{4} (t) & \leq & ϱ_{1}^{2} X^{T} (t) T_{1} X (t) + ς_{1}^{2} У^{T} (t) T_{3} У (t) + ϱ_{12}^{2} X^{T} (t) T_{2} X (t) + ς_{12}^{2} У^{T} (t) T_{4} У (t) \\ - {\{\int_{t - ϱ_{1}}^{t} X (ϰ) d ϰ\}}^{T} T_{1} \{\int_{t - ϱ_{1}}^{t} X (ϰ) d ϰ\} - {\{\int_{t - ς_{1}}^{t} У (ϰ) d ϰ\}}^{T} T_{3} \{\int_{t - ς_{1}}^{t} У (ϰ) d ϰ\} \\ - {[\begin{matrix} \int_{t - ϱ_{2}}^{t - ϱ (t)} X (ϰ) d ϰ \\ \int_{t - ϱ (t)}^{t - ϱ_{1}} X (ϰ) d ϰ \end{matrix}]}^{T} [\begin{matrix} T_{2} & \bar{T_{2}} \\ * & T_{2} \end{matrix}] [\begin{matrix} \int_{t - ϱ_{2}}^{t - ϱ (t)} X (ϰ) d ϰ \\ \int_{t - ϱ (t)}^{t - ϱ_{1}} X (ϰ) d ϰ \end{matrix}] \\ - {[\begin{matrix} \int_{t - ς_{2}}^{t - ς (t)} У (ϰ) d ϰ \\ \int_{t - ς (t)}^{t - ς_{1}} У (ϰ) d ϰ \end{matrix}]}^{T} [\begin{matrix} T_{4} & \bar{T_{4}} \\ * & T_{4} \end{matrix}] [\begin{matrix} \int_{t - ς_{2}}^{t - ς (t)} У (ϰ) d ϰ \\ \int_{t - ς (t)}^{t - ς_{1}} У (ϰ) d ϰ \end{matrix}], \end{matrix}

(35)

\begin{matrix} {\dot{V}}_{5} (t) & = & \frac{ϱ_{1}^{2}}{2} \int_{- ϱ_{1}}^{0} \int_{η}^{0} [{\dot{X}}^{T} (t) U_{1} \dot{X} (t) - {\dot{X}}^{T} (t + ϰ) U_{1} \dot{X} (t + ϰ)] d ϰ d η \\ + \frac{ϱ_{1}^{2}}{2} \int_{- ϱ_{1}}^{0} \int_{ϱ_{1}}^{η} [{\dot{X}}^{T} (t) U_{2} \dot{X} (t) - {\dot{X}}^{T} (t + ϰ) U_{2} \dot{X} (t + ϰ)] d ϰ d η \\ + \frac{ϱ_{12}^{2}}{2} \int_{- ϱ_{2}}^{ϱ_{1}} \int_{η}^{- ϱ_{1}} [{\dot{X}}^{T} (t) U_{3} \dot{X} (t) - {\dot{X}}^{T} (t + ϰ) U_{3} \dot{X} (t + ϰ)] d ϰ d η \\ + \frac{ϱ_{12}^{2}}{2} \int_{- ϱ_{2}}^{- ϱ_{1}} \int_{- ϱ_{2}}^{η} [{\dot{X}}^{T} (t) U_{4} \dot{X} (t) - {\dot{X}}^{T} (t + ϰ) U_{4} \dot{X} (t + ϰ)] d ϰ d η \\ + \frac{ς_{1}^{2}}{2} \int_{- ς_{1}}^{0} \int_{η}^{0} [{\dot{У}}^{T} (t) U_{5} \dot{У} (t) - {\dot{У}}^{T} (t + ϰ) U_{5} \dot{У} (t + ϰ)] d ϰ d η \\ + \frac{ς_{1}^{2}}{2} \int_{- ς_{1}}^{0} \int_{ς_{1}}^{η} [{\dot{У}}^{T} (t) U_{6} \dot{У} (t) - {\dot{У}}^{T} (t + ϰ) U_{6} \dot{У} (t + ϰ)] d ϰ d η \\ + \frac{ς_{12}^{2}}{2} \int_{- ς_{2}}^{ς_{1}} \int_{η}^{- ς_{1}} [{\dot{У}}^{T} (t) U_{7} \dot{У} (t) - {\dot{У}}^{T} (t + ϰ) U_{7} \dot{У} (t + ϰ)] d ϰ d η \\ + \frac{ς_{12}^{2}}{2} \int_{- ς_{2}}^{- ς_{1}} \int_{- ς_{2}}^{η} [{\dot{У}}^{T} (t) U_{8} \dot{У} (t) - {\dot{У}}^{T} (t + ϰ) U_{8} \dot{У} (t + ϰ)] d ϰ d η, \end{matrix}

\begin{matrix} = & \frac{ϱ_{1}^{4}}{4} {\dot{X}}^{T} (t) U_{1} \dot{X} (t) + \frac{ϱ_{1}^{4}}{4} {\dot{X}}^{T} (t) U_{2} \dot{X} (t) - \frac{ϱ_{1}^{2}}{2} \int_{- ϱ_{1}}^{0} \int_{t + η}^{t} {\dot{X}}^{T} (ϰ) U_{1} \dot{X} (ϰ) d ϰ d η \\ - \frac{ϱ_{1}^{2}}{2} \int_{- ϱ_{1}}^{0} \int_{t - ϱ_{1}}^{t + η} {\dot{X}}^{T} (ϰ) U_{2} \dot{X} (ϰ) d ϰ d η + \frac{ϱ_{12}^{4}}{4} {\dot{X}}^{T} (t) U_{3} \dot{X} (t) - \frac{ϱ_{12}^{2}}{2} \int_{- ϱ_{2}}^{- ϱ_{1}} \int_{t + η}^{t - ϱ_{1}} {\dot{X}}^{T} (ϰ) U_{3} \dot{X} (ϰ) d ϰ d η \\ + \frac{ϱ_{12}^{4}}{4} {\dot{X}}^{T} (t) U_{4} \dot{X} (t) - \frac{ϱ_{12}^{2}}{2} \int_{- ϱ_{2}}^{- ϱ_{1}} \int_{t - ϱ_{2}}^{t + η} {\dot{X}}^{T} (ϰ) U_{4} \dot{X} (ϰ) d ϰ d η + \frac{ς_{1}^{4}}{4} {\dot{У}}^{T} (t) U_{5} \dot{У} (t) \\ - \frac{ς_{1}^{2}}{2} \int_{- ς_{1}}^{0} \int_{t + η}^{t} {\dot{У}}^{T} (ϰ) U_{5} \dot{У} (ϰ) d ϰ d η + \frac{ς_{1}^{4}}{4} {\dot{У}}^{T} (t) U_{6} \dot{У} (t) - \frac{ς_{1}^{2}}{2} \int_{- ς_{1}}^{0} \int_{t - ς_{1}}^{t + η} {\dot{У}}^{T} (ϰ) U_{6} \dot{У} (ϰ) d ϰ d η \\ + \frac{ς_{12}^{4}}{4} {\dot{У}}^{T} (t) U_{7} \dot{У} (t) - \frac{ς_{12}^{2}}{2} \int_{- ς_{2}}^{- ς_{1}} \int_{t + η}^{t - ς_{1}} {\dot{У}}^{T} (ϰ) U_{7} \dot{У} (ϰ) d ϰ d η + \frac{ς_{12}^{4}}{4} {\dot{У}}^{T} (t) U_{8} \dot{У} (t) \\ - \frac{ς_{12}^{2}}{2} \int_{- ς_{2}}^{- ς_{1}} \int_{t - ς_{2}}^{t + η} {\dot{У}}^{T} (ϰ) U_{8} \dot{У} (ϰ) d ϰ d η, \end{matrix}

(36)

Further,

\begin{matrix} - \frac{ϱ_{1}^{2}}{2} \int_{- ϱ_{1}}^{0} \int_{t + η}^{t} {\dot{X}}^{T} (ϰ) U_{1} \dot{X} (ϰ) d ϰ d η - \frac{ϱ_{1}^{2}}{2} \int_{- ϱ_{1}}^{0} \int_{t - ϱ_{1}}^{t + η} {\dot{X}}^{T} (ϰ) U_{2} \dot{X} (ϰ) d ϰ d η \\ - \frac{ς_{1}^{2}}{2} \int_{- ς_{1}}^{0} \int_{t + η}^{t} {\dot{У}}^{T} (ϰ) U_{5} \dot{У} (ϰ) d ϰ d η - \frac{ς_{1}^{2}}{2} \int_{- ς_{1}}^{0} \int_{t - ς_{1}}^{t + η} {\dot{У}}^{T} (ϰ) U_{6} \dot{У} (ϰ) d ϰ d η \\ \leq - {\{\int_{- ϱ_{1}}^{0} \int_{t + η}^{t} \dot{X} (ϰ) d ϰ d η\}}^{T} U_{1} \{\int_{- ϱ_{1}}^{0} \int_{t + η}^{t} \dot{X} (ϰ) d ϰ d η\} \\ - {\{\int_{- ϱ_{1}}^{0} \int_{t - ϱ_{1}}^{t + η} \dot{X} (ϰ) d ϰ d η\}}^{T} U_{2} \{\int_{- ϱ_{1}}^{0} \int_{t - ϱ_{1}}^{t + η} \dot{X} (ϰ) d ϰ d η\} \\ - {\{\int_{- ς_{1}}^{0} \int_{t + η}^{t} \dot{У} (ϰ) d ϰ d η\}}^{T} U_{5} \{\int_{- ς_{1}}^{0} \int_{t + η}^{t} \dot{У} (ϰ) d ϰ d η\} \\ - {\{\int_{- ς_{1}}^{0} \int_{t - ς_{1}}^{t + η} \dot{У} (ϰ) d ϰ d η\}}^{T} U_{6} \{\int_{- ς_{1}}^{0} \int_{t - ς_{1}}^{t + η} \dot{У} (ϰ) d ϰ d η\}, \end{matrix}

(37)

and

\begin{matrix} - \frac{ϱ_{12}^{2}}{2} \int_{- ϱ_{2}}^{- ϱ (t)} \int_{t + η}^{t - ϱ_{1}} {\dot{X}}^{T} (ϰ) U_{3} \dot{X} (ϰ) d ϰ d η - \frac{ϱ_{12}^{2}}{2} \int_{- ϱ (t)}^{- ϱ_{1}} \int_{t + η}^{t - ϱ_{1}} {\dot{X}}^{T} (ϰ) U_{3} \dot{X} (ϰ) d ϰ d η \\ - \frac{ϱ_{12}^{2}}{2} \int_{- ϱ_{2}}^{- ϱ (t)} \int_{t - ϱ_{2}}^{t + η} {\dot{X}}^{T} (ϰ) U_{4} \dot{X} (ϰ) d ϰ d η - \frac{ϱ_{12}^{2}}{2} \int_{- ϱ (t)}^{- ϱ_{1}} \int_{t - ϱ_{2}}^{t + η} {\dot{X}}^{T} (ϰ) U_{4} \dot{X} (ϰ) d ϰ d η \\ - \frac{ς_{12}^{2}}{2} \int_{- ς_{2}}^{- ς (t)} \int_{t + η}^{t - ς_{1}} {\dot{У}}^{T} (ϰ) U_{7} \dot{У} (ϰ) d ϰ d η - \frac{ς_{12}^{2}}{2} \int_{- ς (t)}^{- ς_{1}} \int_{t + η}^{t - ς_{1}} {\dot{У}}^{T} (ϰ) U_{7} \dot{У} (ϰ) d ϰ d η \\ - \frac{ς_{12}^{2}}{2} \int_{- ς_{2}}^{- ς (t)} \int_{t - ς_{2}}^{t + η} {\dot{У}}^{T} (ϰ) U_{8} \dot{У} (ϰ) d ϰ d η - \frac{ς_{12}^{2}}{2} \int_{- ς (t)}^{- ς_{1}} \int_{t - ς_{2}}^{t + η} {\dot{У}}^{T} (ϰ) U_{8} \dot{У} (ϰ) d ϰ d η \\ \leq - {[\begin{matrix} \int_{- ϱ_{2}}^{- ϱ (t)} \int_{t + η}^{t - ϱ_{1}} \dot{X} (ϰ) d ϰ d η \\ \int_{- ϱ (t)}^{- ϱ_{1}} \int_{t + η}^{t - ϱ_{1}} \dot{X} (ϰ) d ϰ d η \end{matrix}]}^{T} [\begin{matrix} U_{3} & \bar{U_{3}} \\ * & U_{3} \end{matrix}] [\begin{matrix} \int_{- ϱ_{2}}^{- ϱ (t)} \int_{t + η}^{t - ϱ_{1}} \dot{X} (ϰ) d ϰ d η \\ \int_{- ϱ (t)}^{- ϱ_{1}} \int_{t + η}^{t - ϱ_{1}} \dot{X} (ϰ) d ϰ d η \end{matrix}] \\ - {[\begin{matrix} \int_{- ϱ_{2}}^{- ϱ (t)} \int_{t - ϱ_{2}}^{t + η} \dot{X} (ϰ) d ϰ d η \\ \int_{- ϱ (t)}^{- ϱ_{1}} \int_{t - ϱ_{2}}^{t + η} \dot{X} (ϰ) d ϰ d η \end{matrix}]}^{T} [\begin{matrix} U_{4} & \bar{U_{4}} \\ * & U_{4} \end{matrix}] [\begin{matrix} \int_{- ϱ_{2}}^{- ϱ (t)} \int_{t - ϱ_{2}}^{t + η} \dot{X} (ϰ) d ϰ d η \\ \int_{- ϱ (t)}^{- ϱ_{1}} \int_{t - ϱ_{2}}^{t + η} \dot{X} (ϰ) d ϰ d η \end{matrix}] \\ - {[\begin{matrix} \int_{- ς_{2}}^{- ς (t)} \int_{t + η}^{t - ς_{1}} \dot{У} (ϰ) d ϰ d η \\ \int_{- ς (t)}^{- ς_{1}} \int_{t + η}^{t - ς_{1}} \dot{У} (ϰ) d ϰ d η \end{matrix}]}^{T} [\begin{matrix} U_{7} & \bar{U_{7}} \\ * & U_{7} \end{matrix}] [\begin{matrix} \int_{- ς_{2}}^{- ς (t)} \int_{t + η}^{t - ς_{1}} \dot{У} (ϰ) d ϰ d η \\ \int_{- ς (t)}^{- ς_{1}} \int_{t + η}^{t - ς_{1}} \dot{У} (ϰ) d ϰ d η \end{matrix}] \\ - {[\begin{matrix} \int_{- ς_{2}}^{- ς (t)} \int_{t - ς_{2}}^{t + η} \dot{У} (ϰ) d ϰ d η \\ \int_{- ς (t)}^{- ς_{1}} \int_{t - ς_{2}}^{t + η} \dot{У} (ϰ) d ϰ d η \end{matrix}]}^{T} [\begin{matrix} U_{8} & \bar{U_{8}} \\ * & U_{8} \end{matrix}] [\begin{matrix} \int_{- ς_{2}}^{- ς (t)} \int_{t - ς_{2}}^{t + η} \dot{У} (ϰ) d ϰ d η \\ \int_{- ς (t)}^{- ς_{1}} \int_{t - ς_{2}}^{t + η} \dot{У} (ϰ) d ϰ d η \end{matrix}], \end{matrix}

(38)

By substituting (37) and (38) in (36), we get

\begin{matrix} \dot{V_{5}} (t) & \leq & {\dot{X}}^{T} (t) U_{α} \dot{X} (t) + {\dot{У}}^{T} (t) U_{β} \dot{У} (t) \\ - {\{\int_{- ϱ_{1}}^{0} \int_{t + η}^{t} \dot{X} (ϰ)\}}^{T} U_{1} \{\int_{- ϱ_{1}}^{0} \int_{t + η}^{t} \dot{X} (ϰ)\} - {\{\int_{- ϱ_{1}}^{0} \int_{t - ϱ_{1}}^{t + η} \dot{X} (ϰ)\}}^{T} U_{2} \{\int_{- ϱ_{1}}^{0} \int_{t - ϱ_{1}}^{t + η} \dot{X} (ϰ)\} \\ - {\{\int_{- ς_{1}}^{0} \int_{t + η}^{t} \dot{У} (ϰ)\}}^{T} U_{5} \{\int_{- ς_{1}}^{0} \int_{t + η}^{t} \dot{У} (ϰ)\} - {\{\int_{- ς_{1}}^{0} \int_{t - ς_{1}}^{t + η} \dot{У} (ϰ)\}}^{T} U_{6} \{\int_{- ς_{1}}^{0} \int_{t - ς_{1}}^{t + η} \dot{У} (ϰ)\} \\ - {[\begin{matrix} \int_{- ϱ_{2}}^{- ϱ (t)} \int_{t + η}^{t - ϱ_{1}} \dot{X} (ϰ) d ϰ d η \\ \int_{- ϱ (t)}^{- ϱ_{1}} \int_{t + η}^{t - ϱ_{1}} \dot{X} (ϰ) d ϰ d η \end{matrix}]}^{T} [\begin{matrix} U_{3} & \bar{U_{3}} \\ * & U_{3} \end{matrix}] [\begin{matrix} \int_{- ϱ_{2}}^{- ϱ (t)} \int_{t + η}^{t - ϱ_{1}} \dot{X} (ϰ) d ϰ d η \\ \int_{- ϱ (t)}^{- ϱ_{1}} \int_{t + η}^{t - ϱ_{1}} \dot{X} (ϰ) d ϰ d η \end{matrix}] \\ - {[\begin{matrix} \int_{- ϱ_{2}}^{- ϱ (t)} \int_{t - ϱ_{2}}^{t + η} \dot{X} (ϰ) d ϰ d η \\ \int_{- ϱ (t)}^{- ϱ_{1}} \int_{t - ϱ_{2}}^{t + η} \dot{X} (ϰ) d ϰ d η \end{matrix}]}^{T} [\begin{matrix} U_{4} & \bar{U_{4}} \\ * & U_{4} \end{matrix}] [\begin{matrix} \int_{- ϱ_{2}}^{- ϱ (t)} \int_{t - ϱ_{2}}^{t + η} \dot{X} (ϰ) d ϰ d η \\ \int_{- ϱ (t)}^{- ϱ_{1}} \int_{t - ϱ_{2}}^{t + η} \dot{X} (ϰ) d ϰ d η \end{matrix}] \\ - {[\begin{matrix} \int_{- ς_{2}}^{- ς (t)} \int_{t + η}^{t - ς_{1}} \dot{У} (ϰ) d ϰ d η \\ \int_{- ς (t)}^{- ς_{1}} \int_{t + η}^{t - ς_{1}} \dot{У} (ϰ) d ϰ d η \end{matrix}]}^{T} [\begin{matrix} U_{7} & \bar{U_{7}} \\ * & U_{7} \end{matrix}] [\begin{matrix} \int_{- ς_{2}}^{- ς (t)} \int_{t + η}^{t - ς_{1}} \dot{У} (ϰ) d ϰ d η \\ \int_{- ς (t)}^{- ς_{1}} \int_{t + η}^{t - ς_{1}} \dot{У} (ϰ) d ϰ d η \end{matrix}] \\ - {[\begin{matrix} \int_{- ς_{2}}^{- ς (t)} \int_{t - ς_{2}}^{t + η} \dot{У} (ϰ) d ϰ d η \\ \int_{- ς (t)}^{- ς_{1}} \int_{t - ς_{2}}^{t + η} \dot{У} (ϰ) d ϰ d η \end{matrix}]}^{T} [\begin{matrix} U_{8} & \bar{U_{8}} \\ * & U_{8} \end{matrix}] [\begin{matrix} \int_{- ς_{2}}^{- ς (t)} \int_{t - ς_{2}}^{t + η} \dot{У} (ϰ) d ϰ d η \\ \int_{- ς (t)}^{- ς_{1}} \int_{t - ς_{2}}^{t + η} \dot{У} (ϰ) d ϰ d η \end{matrix}], \end{matrix}

(39)

\begin{matrix} w h e r e U_{α} & = & \frac{ϱ_{1}^{4} U_{1} + ϱ_{1}^{4} U_{2} + ϱ_{12}^{4} U_{3} + ϱ_{12}^{4} U_{4}}{4}, U_{β} = \frac{ς_{1}^{4} U_{5} + ς_{1}^{4} U_{6} + ς_{12}^{4} U_{7} + ς_{12}^{4} U_{8}}{4} . \end{matrix}

\begin{matrix} {\dot{V}}_{6} (t) & \leq & γ_{12}^{2} g^{T} (X (t)) J_{1} g (X (t)) - \int_{t - γ_{2} (t)}^{t - γ_{1} (t)} g^{T} (X (ϰ)) d ϰ J_{1} \int_{t - γ_{2} (t)}^{t - γ_{1} (t)} g (X (ϰ)) d ϰ + η_{12}^{2} f^{T} (У (t)) J_{2} f (У (t)) \\ - \int_{t - η_{2} (t)}^{t - η_{1} (t)} f^{T} (У (ϰ)) d ϰ J_{2} \int_{t - η_{2} (t)}^{t - η_{1} (t)} f (У (ϰ)) d ϰ . \end{matrix}

(40)

Adding the activation functions (20) to (23) along with the calculation of the time derivatives (27)–(29), (35), (39) and (40) of

V (t)

, we can see that

\begin{matrix} \dot{V} (t) \leq Ξ_{2}^{T} (t) Λ Ξ_{2} (t), \end{matrix}

(41)

where

\begin{matrix} Ξ_{2} (t) & = & c o l {X (t), X (t - ϱ (t)), X (t - ϱ_{1}), X (t - ϱ_{2}), \int_{t - ϱ_{1}}^{t} X (ϰ) d ϰ, \int_{t - ϱ (t)}^{t - ϱ_{1}} X (ϰ) d ϰ, \int_{t - ϱ_{2}}^{t - ϱ (t)} X (ϰ) d ϰ, \\ \int_{- ϱ_{1}}^{0} \int_{t + η}^{t} X (ϰ) d ϰ d η, \int_{- ϱ (t)}^{- ϱ_{1}} \int_{t + η}^{t - ϱ_{1}} X (ϰ) d ϰ d η, \int_{- ϱ_{2}}^{- ϱ (t)} \int_{t + η}^{t - ϱ_{1}} X (ϰ) d ϰ d η, \int_{- ϱ_{1}}^{0} \int_{t - ϱ_{1}}^{t + η} X (ϰ) d ϰ d η, \\ \int_{- ϱ (t)}^{- ϱ_{1}} \int_{t - ϱ_{2}}^{t + η} X (ϰ) d ϰ d η, \int_{- ϱ_{2}}^{- ϱ (t)} \int_{t - ϱ_{2}}^{t + η} X (ϰ) d ϰ d η, g (X (t)), g (X (t - ϱ (t))), \int_{t - γ_{2} (t)}^{t - γ_{1} (t)} g (X (ϰ)) d ϰ, \\ У (t), У (t - ς (t)), У (t - ς_{1}), У (t - ς_{2}), \int_{t - ς_{1}}^{t} У (ϰ) d ϰ, \int_{t - ς (t)}^{t - ς_{1}} У (ϰ) d ϰ, \int_{t - ς_{2}}^{t - ς (t)} У (ϰ) d ϰ, \\ \int_{- ς_{1}}^{0} \int_{t + η}^{t} У (ϰ) d ϰ d η, \int_{- ς (t)}^{- ς_{1}} \int_{t + η}^{t - ς_{1}} У (ϰ) d ϰ d η, \int_{- ς_{2}}^{- ς (t)} \int_{t + η}^{t - ς_{1}} У (ϰ) d ϰ d η, \int_{- ς_{1}}^{0} \int_{t - ς_{1}}^{t + η} У (ϰ) d ϰ d η, \\ \int_{- ς (t)}^{- ς_{1}} \int_{t - ς_{2}}^{t + η} У (ϰ) d ϰ d η, \int_{- ς_{2}}^{- ς (t)} \int_{t - ς_{2}}^{t + η} У (ϰ) d ϰ d η, f (У (t)), f (У (t - ς (t))), \int_{t - η_{2} (t)}^{t - η_{1} (t)} f (У (ϰ)) d ϰ}, \end{matrix}

Furthermore,

\begin{matrix} Λ < 0 \end{matrix}

is intrinsically linear in

ϱ (t) & ς (t)

, which implies

\dot{V} (t) < 0 .

□

4. Numerical Examples

Example 1.

Consider the two-neuron BAM NN model with mixed time-varying delays (1) with the following parameters:

\begin{matrix} A_{1} & = & [\begin{matrix} 4 & 0 \\ 0 & 4 \end{matrix}], B_{1} = [\begin{matrix} 0.55 & 0.32 \\ 0.42 & 0.76 \end{matrix}], C_{1} = [\begin{matrix} 0.9 & 0.6 \\ 0.3 & 0.1 \end{matrix}], D_{1} = [\begin{matrix} - 0.4 & 0.1 \\ 0.1 & 0.2 \end{matrix}], \\ A_{2} & = & [\begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}], B_{2} = [\begin{matrix} 0.3 & 0.6 \\ 0.5 & 0.4 \end{matrix}], C_{2} = [\begin{matrix} 0.7 & 0.5 \\ 0.6 & 0.2 \end{matrix}], D_{2} = [\begin{matrix} - 0.2 & 0.3 \\ 0.2 & 0.1 \end{matrix}], \\ W_{1} & = & [\begin{matrix} 0.1 & 0 \\ 0 & 0.1 \end{matrix}], W_{2} = [\begin{matrix} 0.1 & 0 \\ 0 & 0.1 \end{matrix}], I = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}], \\ υ_{1} & = & 0.5, υ_{2} = 0.5, η_{1} = 0.25, η_{2} = 0.4756, γ_{1} = 0.25, γ_{2} = 0.4756, \\ ς (t) & = & 1.5 + 0.5 s i n t, ϱ (t) = 1.5 + 0.5 s i n t . C l e a r l y ς_{1} = 1, ς_{2} = 2, ϱ_{1} = 1, ϱ_{2} = 2 . \end{matrix}

(42)

By Theorem 1, using the MATLAB R2025a LMI solver toolbox, we can find that the system (1) is globally asymptotically stable and a part of the feasible solution is as follows:

\begin{matrix} P_{11} & = & [\begin{matrix} 49.1139 & - 5.4309 \\ - 5.4309 & 55.3414 \end{matrix}], P_{12} = [\begin{matrix} 13.1157 & - 1.1130 \\ - 1.1130 & 14.3562 \end{matrix}], P_{13} = [\begin{matrix} 7.4812 & - 0.0846 \\ - 0.0846 & 7.5586 \end{matrix}], \\ P_{14} & = & [\begin{matrix} 9.7913 & - 0.0041 \\ - 0.0041 & 9.8517 \end{matrix}], P_{22} = [\begin{matrix} 13.2004 & - 0.4010 \\ - 0.4010 & 13.6412 \end{matrix}], P_{23} = [\begin{matrix} 5.8365 & - 0.0464 \\ - 0.0464 & 5.8844 \end{matrix}], \\ P_{24} & = & [\begin{matrix} 6.7410 & 0.0640 \\ 0.0640 & 6.6803 \end{matrix}], P_{33} = [\begin{matrix} 14.7311 & - 0.0933 \\ - 0.0933 & 14.8188 \end{matrix}], P_{34} = [\begin{matrix} 7.7115 & 0.0106 \\ 0.0106 & 7.7015 \end{matrix}], \\ P_{44} & = & [\begin{matrix} 12.7271 & 0.0027 \\ 0.0027 & 12.7502 \end{matrix}], Q_{11} = [\begin{matrix} 53.0313 & - 6.4562 \\ - 6.4562 & 56.5498 \end{matrix}], Q_{12} = [\begin{matrix} 11.9544 & - 1.8408 \\ - 1.8408 & 12.9746 \end{matrix}], \\ Q_{13} & = & [\begin{matrix} 7.1708 & - 0.2469 \\ - 0.2469 & 7.2948 \end{matrix}], Q_{14} = [\begin{matrix} 9.8058 & - 0.0548 \\ - 0.0548 & 9.8537 \end{matrix}], Q_{22} = [\begin{matrix} 12.7087 & - 0.7036 \\ - 0.7036 & 13.0921 \end{matrix}], \\ Q_{23} & = & [\begin{matrix} 5.5854 & - 0.1589 \\ - 0.1589 & 5.6705 \end{matrix}], Q_{24} = [\begin{matrix} 6.5708 & - 0.0641 \\ - 0.0641 & 6.6120 \end{matrix}], Q_{33} = [\begin{matrix} 13.9142 & - 0.1976 \\ - 0.1976 & 14.0179 \end{matrix}], \\ Q_{34} & = & [\begin{matrix} 6.9318 & - 0.0065 \\ - 0.0065 & 6.9338 \end{matrix}], Q_{44} = [\begin{matrix} 14.7622 & - 0.2441 \\ - 0.2441 & 14.8952 \end{matrix}], R_{1} = [\begin{matrix} 45.1445 & 2.4128 \\ 2.4128 & 42.4111 \end{matrix}], \\ R_{2} & = & [\begin{matrix} 24.1601 & - 0.3687 \\ - 0.3687 & 24.5842 \end{matrix}], R_{3} = [\begin{matrix} 27.8471 & 2.9449 \\ 2.9449 & 26.2354 \end{matrix}], R_{4} = [\begin{matrix} 19.6855 & - 0.0687 \\ - 0.0687 & 19.6998 \end{matrix}], e t c ., \end{matrix}

The follwing Table 1 shows the Maximum Allowable Upper Bounds (MAUBs) obatined for different values of υ by Theorem 1.

Example 2.

Consider the 2-neuron BAM NN model with mixed time-varying delays (1) with the same parameters as in (42):

By Theorem 2, using the MATLAB LMI solver toolbox, we can find that the system (1) is globally asymptotically stable and a part of the feasible solution is as follows:

\begin{matrix} P_{11} & = & [\begin{matrix} 91.8077 & - 12.8486 \\ - 12.8486 & 106.4923 \end{matrix}], P_{12} = [\begin{matrix} 23.1396 & - 0.0511 \\ - 0.0511 & 23.1529 \end{matrix}], P_{13} = [\begin{matrix} 16.0093 & 0.2931 \\ 0.2931 & 15.7299 \end{matrix}], \\ P_{14} & = & [\begin{matrix} 22.6700 & - 0.1689 \\ - 0.1689 & 22.6655 \end{matrix}], P_{22} = [\begin{matrix} 25.7340 & 0.6961 \\ 0.6961 & 24.8843 \end{matrix}], P_{23} = [\begin{matrix} 11.5107 & 0.2773 \\ 0.2773 & 11.2136 \end{matrix}], \\ P_{24} & = & [\begin{matrix} 15.2282 & 0.3415 \\ 0.3415 & 14.7799 \end{matrix}], P_{33} = [\begin{matrix} 2.6639 & - 0.0157 \\ - 0.0157 & 2.6828 \end{matrix}], P_{34} = [\begin{matrix} 12.9641 & 0.0595 \\ 0.0595 & 12.9106 \end{matrix}], \\ P_{44} & = & [\begin{matrix} 25.2266 & 0.3279 \\ 0.3279 & 24.8217 \end{matrix}], Q_{11} = [\begin{matrix} 100.1672 & - 14.9924 \\ - 14.9924 & 108.2465 \end{matrix}], Q_{12} = [\begin{matrix} 20.9750 & - 0.8581 \\ - 0.8581 & 21.5058 \end{matrix}], \\ Q_{13} & = & [\begin{matrix} 15.1543 & - 0.2330 \\ - 0.2330 & 15.2414 \end{matrix}], Q_{14} = [\begin{matrix} 20.9160 & - 0.0844 \\ - 0.0844 & 21.0900 \end{matrix}], Q_{22} = [\begin{matrix} 23.7231 & 0.1029 \\ 0.1029 & 23.7113 \end{matrix}], \\ Q_{23} & = & [\begin{matrix} 10.8709 & 0.0140 \\ 0.0140 & 10.8634 \end{matrix}], Q_{24} = [\begin{matrix} 14.1974 & 0.0961 \\ 0.0961 & 14.1894 \end{matrix}], Q_{33} = [\begin{matrix} 24.3363 & - 0.4064 \\ - 0.4064 & 24.5526 \end{matrix}], \\ Q_{34} & = & [\begin{matrix} 12.7398 & - 0.1551 \\ - 0.1551 & 12.8310 \end{matrix}], Q_{44} = [\begin{matrix} 24.2265 & 0.1200 \\ 0.1200 & 24.2192 \end{matrix}], R_{1} = [\begin{matrix} 76.1489 & 5.3839 \\ 5.3839 & 70.0132 \end{matrix}], \\ R_{2} & = & [\begin{matrix} 41.1025 & 1.0392 \\ 1.0392 & 39.9201 \end{matrix}], R_{3} = [\begin{matrix} 64.4080 & 6.2536 \\ 6.2536 & 61.1712 \end{matrix}], R_{4} = [\begin{matrix} 38.4461 & 0.9853 \\ 0.9853 & 37.8757 \end{matrix}], e t c ., \end{matrix}

Table 2 shows the MAUBs obtained for different values of υ by Theorem 2.

Remark 1.

From Example 1, it is observed that Theorem 1 gives less conservatism compared to the conservatism obtained by Example 2 of Theorem 2. Even though the number of variables is higher in Theorem 1 as compared to Theorem 2, we obtain less conservatism while using Theorem 1. So the second-order reciprocally convex combination using Theorem 1 gives less conservatism to the system (1). Table 3 gives the comparison of Example 1 and Example 2.

Remark 2.

In [42], the stability criterion for linear time-delay systems is obtained by using the reciprocally convex approach and in [38], the stability criterion of the delayed neural network system is obtained by using the reciprocally convex approach. In [39], the second-order convex combination technique is applied to the genetic regulatory neural network system, whereas we have taken BAM NNs to show the effectiveness of the second-order convex combination technique. However, in this paper, we have applied second-order convex combination to obtain reduced conservatism of BAM NNs with mixed time-varying delays.

Remark 3.

In this study, we investigated the stability issues of delayed BAM NNs using Jensen’s inequality and second-order reciprocally convex combination approach, and we came up with a less conservative criterion. These methods can also be used to investigate the stability of a variety of systems, including linear time-delay systems, neutral type delay systems, and other NNs [41].

5. Conclusions

This paper examined BAM NNs that had both discrete and distributed delays, as well as mixed-interval time-varying delays. We focused on interval time-varying delays, where the lower bounds could be either zero or positive. Based on a reciprocally convex approach, this research work suggests a new approach for dealing linear combinations of positive functions scaled by the reciprocal-squared values of convex weight parameters. By introducing modifications of the lower bound lemma to handle different types of function combinations emerging from triple integral terms in the formulation of LMI requirements, a novel delay-dependent stability criterion is constructed. Furthermore, this second-order convex approach can be extended to Cohen–Grossberg Neural Networks for the superiority of the lemma and the third-order convex combination approach can also be applied to our BAMNN’s system to obtain a novel delay-dependent stability criterion. This criterion enhances the robustness of the system by allowing greater flexibility in managing delays. Consequently, it opens new avenues for improving performance in various applications.

Author Contributions

Conceptualization, R.K. and V.V.; methodology, R.K.; software, R.K. and V.V.; validation, K.C., R.K. and V.V.; formal analysis, K.C.; investigation, R.K.; resources, V.V.; data curation, V.V.; writing—original draft preparation, R.K.; writing—review and editing, K.C. and V.V.; visualization, K.C.; supervision, K.C.; project administration, R.K. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Acknowledgments

Kalaivani Chandran thanks the management of Sri Sivasubramaniya Nadar College of Engineering for supporting her research work. Renuga Kuppusamy is thankful to the management of Sri Sivasubramaniya Nadar College of Engineering and St. Joseph’s Institute of Technology, OMR, Chennai, Tamil Nadu 600119, for supporting her research work. Vembarasan Vaitheeswaran is thankful to the management of Shiv Nadar University Chennai for supporting his research work.

Conflicts of Interest

The authors declare no conflicts of interest.

References

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Table 1. The MAUB for delays

ς_{2} = ϱ_{2} = \bar{r} = \bar{γ}

with different

υ (υ_{1} = υ_{2})

.

Table 1. The MAUB for delays

ς_{2} = ϱ_{2} = \bar{r} = \bar{γ}

with different

υ (υ_{1} = υ_{2})

.

		Values of $υ$
	Theorem 1	0.5	1.0	1.5	2.0	2.5
values of $ς_{1} = ϱ_{1}$	0.0	145.1342	145.3240	145.4740	145.4943	145.5407
	0.5	146.2320	146.4451	146.5670	146.6116	146.6683
	1.0	147.1399	147.3313	147.4210	147.5522	147.5958
	1.5	148.2490	148.4354	148.5136	148.5840	148.6148
	2.0	149.1221	149.2633	149.4471	149.5325	149.6182

Table 2. The MAUB for delays

ς_{2} = ϱ_{2} = \bar{r} = \bar{γ}

with different

υ (υ_{1} = υ_{2})

.

Table 2. The MAUB for delays

ς_{2} = ϱ_{2} = \bar{r} = \bar{γ}

with different

υ (υ_{1} = υ_{2})

.

		Values of $υ$
	Theorem 2	0.25	0.5	0.75	0.9	0.99
values of $ς_{1} = ϱ_{1}$	0.0	120.1468	120.2357	120.4543	120.5462	120.6128
	0.5	121.2548	121.3829	121.4517	121.5015	121.6882
	1.0	122.2789	122.3982	122.4028	122.5234	122.6789
	1.5	123.2167	123.3454	123.4675	123.5678	123.6248
	2.0	124.3925	124.5689	124.6895	124.7052	124.8962

Table 3. Comparison of theorems.

Comparison of Theorems	Theorem 1	Theorem 2
MAUB obtained	nearly 150	nearly 125
Value of $υ$	$0 \leq υ \leq a n y v a l u e$	$0 \leq υ < 1$
Symmetric Matrices	50	54
Any Matrices	10	Nil
Unknown Constants	228	4
No. of Decision Variables	$35 n^{2} + 25 n + 228$	$27 n^{2} + 27 n + 4$
when n = 2	418	166

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Chandran, K.; Kuppusamy, R.; Vaitheeswaran, V. Stability Analysis of Bidirectional Associative Memory Neural Networks with Time-Varying Delays via Second-Order Reciprocally Convex Approach. Symmetry 2025, 17, 1852. https://doi.org/10.3390/sym17111852

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Chandran K, Kuppusamy R, Vaitheeswaran V. Stability Analysis of Bidirectional Associative Memory Neural Networks with Time-Varying Delays via Second-Order Reciprocally Convex Approach. Symmetry. 2025; 17(11):1852. https://doi.org/10.3390/sym17111852

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Chandran, Kalaivani, Renuga Kuppusamy, and Vembarasan Vaitheeswaran. 2025. "Stability Analysis of Bidirectional Associative Memory Neural Networks with Time-Varying Delays via Second-Order Reciprocally Convex Approach" Symmetry 17, no. 11: 1852. https://doi.org/10.3390/sym17111852

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Chandran, K., Kuppusamy, R., & Vaitheeswaran, V. (2025). Stability Analysis of Bidirectional Associative Memory Neural Networks with Time-Varying Delays via Second-Order Reciprocally Convex Approach. Symmetry, 17(11), 1852. https://doi.org/10.3390/sym17111852

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Stability Analysis of Bidirectional Associative Memory Neural Networks with Time-Varying Delays via Second-Order Reciprocally Convex Approach

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1. Introduction

2. Problem Overview and Basic Notions

3. Main Results

4. Numerical Examples

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