Remarks on the Schrödinger-Lohe model

Hyungjin Huh; Hyungjin Huh

doi:10.3934/nhm.2019030

Networks and Heterogeneous Media

2019, Volume 14, Issue 4: 759-769. doi: 10.3934/nhm.2019030

Previous Article Next Article

Remarks on the Schrödinger-Lohe model

Hyungjin Huh ^,

Department of Mathematics, Chung-Ang University, Seoul 156-756, Republic of Korea

Received: 01 October 2018 Revised: 01 July 2019
82C10, 34E10, 35C05

We study the Schrödinger-Lohe model. Making use of the principal fundamental matrix $ Y $ of linear ODEs with variable coefficients, the coupled nonlinear Schrödinger-Lohe system is transformed into the decoupled linear Schrödinger equations. The boundedness of $ Y $ is shown for the case of complete synchronization. We also study the cases where the principal fundamental matrices can be derived explicitly.

Keywords:

Citation: Hyungjin Huh. Remarks on the Schrödinger-Lohe model[J]. Networks and Heterogeneous Media, 2019, 14(4): 759-769. doi: 10.3934/nhm.2019030

Related Papers:

[1]	Yang Liu . Global attractors for a nonlinear plate equation modeling the oscillations of suspension bridges. Communications in Analysis and Mechanics, 2023, 15(3): 436-456. doi: 10.3934/cam.2023021
[2]	Chunyou Sun, Junyan Tan . Attractors for a Navier–Stokes–Allen–Cahn system with unmatched densities. Communications in Analysis and Mechanics, 2025, 17(1): 237-262. doi: 10.3934/cam.2025010
[3]	Panyu Deng, Jun Zheng, Guchuan Zhu . Well-posedness and stability for a nonlinear Euler-Bernoulli beam equation. Communications in Analysis and Mechanics, 2024, 16(1): 193-216. doi: 10.3934/cam.2024009
[4]	Hongxia Lin, Sabana, Qing Sun, Ruiqi You, Xiaochuan Guo . The stability and decay of 2D incompressible Boussinesq equation with partial vertical dissipation. Communications in Analysis and Mechanics, 2025, 17(1): 100-127. doi: 10.3934/cam.2025005
[5]	Hilal Essaouini, Pierre Capodanno . Analysis of small oscillations of a pendulum partially filled with a viscoelastic fluid. Communications in Analysis and Mechanics, 2023, 15(3): 388-409. doi: 10.3934/cam.2023019
[6]	Tingfu Feng, Yan Dong, Kelei Zhang, Yan Zhu . Global existence and blow-up to coupled fourth-order parabolic systems arising from modeling epitaxial thin film growth. Communications in Analysis and Mechanics, 2025, 17(1): 263-289. doi: 10.3934/cam.2025011
[7]	Yachun Li, Chenmu Wang . On a partially synchronizable system for a coupled system of wave equations in one dimension. Communications in Analysis and Mechanics, 2023, 15(3): 470-493. doi: 10.3934/cam.2023023
[8]	Senhao Duan, Yue MA, Weidong Zhang . Conformal-type energy estimates on hyperboloids and the wave-Klein-Gordon model of self-gravitating massive fields. Communications in Analysis and Mechanics, 2023, 15(2): 111-131. doi: 10.3934/cam.2023007
[9]	Anthony Bloch, Marta Farré Puiggalí, David Martín de Diego . Metriplectic Euler-Poincaré equations: smooth and discrete dynamics. Communications in Analysis and Mechanics, 2024, 16(4): 910-927. doi: 10.3934/cam.2024040
[10]	Yao Sun, Pan Wang, Xinru Lu, Bo Chen . A boundary integral equation method for the fluid-solid interaction problem. Communications in Analysis and Mechanics, 2023, 15(4): 716-742. doi: 10.3934/cam.2023035

Abstract

1. Introduction

Time-delay systems exist in many practical situations as industry process, biological, ecological groups, telecommunication, economy, mechanical engineering, and so on. A time-delay in a system often induces oscillation and instability, which motivated a huge number of researchers to study the stability analysis with various criteria ^[1,2,3]. Evaluation of system stability with a constant delay has been studied extensively and lots of theoretical tools have been presented like characteristic equation and eigenvalues analysis ^[4,5]. Those methods have been well established currently which can derive effective criteria smoothly with numerical efficiency. However, this type of criteria cannot be applied to a time-varying delay system and some other methodologies have been employed.

Generally, two different methodologies have been employed: the first one is so called input-output method that treats a delay as an uncertain operator, and transforms the original time-varying delay system into a closed loop between a nominal LTI system and a perturbation depending on the delay. The stability criteria of which have been well developed by using conventional robustness tools like Small Gain Theorem ^[6,7], Integral Quadratic Constraint or Quadratic Separation ^[8,9]. The conservativeness is small for a slowly varying delay, but large for a quickly one because it depending on the upper bound on the derivative of the delay. Another technique is based on the proper construction of Lyapunov-Krasovskii functions. The conservativeness of this method comes from two aspects: the choice of functional and the bound on its derivative. It is not easy to find an appropriate Lyapunov-Krasovskii functional (LFK) to obtain less conservative criteria since it contains both the delay and its bounds.

In earlier research, only a single integral term was employed as a part of LFK to analysis and handle the time delay in systems ^[10,11,12]. Up to now, double, triple, even quadruple integral terms has been developed which usually bring more effective stability criteria ^[13,14,15]. And also an augmented and a delay-partitioning LKF method were proposed to reduce the conservativeness, and the difficulty now lies in the bounds of the integrals that appear in the derivative of the functional for a stability condition ^[16,17].

Previously, The Jensen inequality and Wirtinger-based integral inequality were reported as the integral inequality method that yields less conservative stability criteria ^[2,18]. Delay-dependent strategy and delay-independent approach under time-varying delays, uncertainties and disturbance are employed to stability analysis. Delay-dependent strategy has been received many attentions as a result of its less conservatism than delay-independent ^{[19,20,21,22,23,24,25,26,27]}. Later, the first- and second-order reciprocally convex approach were proposed based on a new kind of linear combination of positive functions weighted by the inverses of squared convex parameters emerges when the Jensen inequality was applied to partitioned double integral terms in the derivation of LMI conditions ^[28,29]. And the optimal divided method and the secondary partitioning method were provided for stability criteria in double integral terms in LPF ^[30,31].

Recently, the integral term with higher order approximation has been proposed, such as Wirtinger-based double integral inequality ^[32], free-matrix-based integral inequality ^[33], auxiliary function-based integral inequality ^[34]. These inequalities provided less conservation of stability criteria that those of the Jensen or Wirtinger-based single integral inequities. Especially, a novel integral inequality which called Bessel-Legendre (B-L) inequality has only been applied to the system with constant delays ^{[35,36,37,38]}. And also multiple-integral inequalities were newly developed to give high-order approximation to the original integral, the associated integral terms in LPF are also increased ^[39,40].

In this study, a new single integral inequality is proposed through using shifted Legendre polynomials, and then the double integral inequality is developed with the utilization of Cholesky decomposition. Both single and double integral inequalities are with arbitrary approximation order, which encompasses the well-known Jensen and Wirtinger-based inequalities, auxiliary function-based integral inequalities, and even the B-L inequality. The proposed two inequalities yield improved stability criteria with less conservativeness.

This paper is organized as follows. Section 2 introduces the relevant theories of shifted Legendre polynomials-based single and double integral inequalities, and section 3 and 4 provide application of proposed methods to systems with constant and time-varying delays, including numerical examples.

2. Shifted Legendre polynomials-based single and double integral inequalities

2.1. Shifted Legendre polynomials for single integral

The classical shifted Legendre polynomials are a set of functions analogous to the Legendre polynomials, but defined on the interval $ [0, 1] $ as follows

$\begin{equation} p_i^{}(s) = \sum\limits_{j = 0}^i {{w_{i, j}}{s^j}} {, } \quad j = 0{, _{}}1{, _{}} \cdots {, _{}}i \end{equation}$ $

(2.1)

where $ p_i(s) $ denotes the $ i $-order shifted Legendre polynomial, $ w_{i, j} $ denotes the $ j $th coefficient of $ p_i(s) $.

We here call classical shifted Legendre polynomials as the shifted Legendre polynomials for single integral with the following coefficient

$\begin{equation} {w_{i, j}} = {( - 1)^i}C_{i + j}^iC_i^j \end{equation}$ $

(2.2)

where $ C_i^j $ denotes the combination which can be written using factorials as

$\begin{equation} C_i^j = \frac{{i!}}{{j!(i - j)!}} \end{equation}$ $

(2.3)

Shifted Legendre polynomials obey the orthogonality relationship, i. e.

$\begin{equation} \begin{aligned} \int_0^1 {{p_l}(s){p_m}(s)ds} & = \sum\limits_{i = 0}^l {\sum\limits_{j = 0}^m {{{( - 1)}^{i + j}}C_{l + i}^lC_l^iC_{m + j}^mC_m^j} \int_0^1 {{s^{i + j}}ds} } \\ & = \sum\limits_{i = 0}^l {\sum\limits_{j = 0}^m {{{( - 1)}^{i + j}}C_{l + i}^lC_l^iC_{m + j}^mC_m^j} \frac{1}{{i + j + 1}}} \\ & = \frac{1}{{2m + 1}}{\delta _{lm}} \\ \end{aligned} \end{equation}$ $

(2.4)

where $ {\delta _{nm}} $ denotes the Kronecker delta.

Also we can represent shifted Legendre polynomials for single integral in the matrix form as follows

$\begin{equation} {U_m}(s) = \left[ {\begin{array}{*{20}{c}} 1 \\ s \\ \vdots \\ {{s^m}} \\ \end{array}} \right]{, _{}} \quad {L_m}(s) = \left[ {\begin{array}{*{20}{c}} {{p_0}(s)} \\ {{p_1}(s)} \\ \vdots \\ {{p_m}(s)} \\ \end{array}} \right] \end{equation}$ $

(2.5)

The relationship between $ {L_m}(s) $ and $ {U_m}(s) $ is obtained

$\begin{equation} {L_m}(s) = {W_m}{U_m}(s) \end{equation}$ $

(2.6)

where $ W_m $ is the coefficient matrix with the following form

$\begin{equation} {W_m} = \underbrace {[{{( - 1)}^j}C_{i + j}^iC_i^j]}_{i \ge j} = \left[ {\begin{array}{*{20}{c}} 1 & {} & {} & \cdots & {} \\ 1 & { - 2} & {} & \cdots & {} \\ 1 & { - 6} & 6 & \cdots & {} \\ \vdots & \vdots & \vdots & \ddots & {} \\ 1 & { - m(m + 1)} & {C_{m + 2}^mC_m^2} & \cdots & {{{( - 1)}^m}C_{2m}^m} \\ \end{array}} \right] \end{equation}$ $

(2.7)

It's obvious that $ W_n $ is a lower triangular matrix.

With similar formulation, (2.4) can be rewritten as

$\begin{equation} {G_m} = \int_0^1 {{L_m}(s)L_m^{\mathop{\rm T}\nolimits} (s)du} = \left[ {{g_{ij}}} \right] = \left[ {\begin{array}{*{20}{c}} 1 & {} & {} & \cdots & {} \\ {} & {\frac{1}{3}} & {} & \cdots & {} \\ {} & {} & {\frac{1}{5}} & \cdots & {} \\ \vdots & \vdots & \vdots & \ddots & {} \\ 0 & 0 & 0 & \cdots & {\frac{1}{{2m + 1}}} \\ \end{array}} \right] \end{equation}$ $

(2.8)

2.2. Shifted Legendre polynomials for double integral

The interest of shifted Legendre polynomials for double integral is that the orthogonality relationship exists if we use double integral instead of single integral.

The double integral of the product of two classical shifted Legendre polynomials can be obtained as follows

$\begin{equation} \begin{aligned} {h_{lm}} & = \int_0^1 {\int_s^1 {{p_l}(u){p_m}(u)du} {\mathop{\rm d}\nolimits} s} \\ & = \sum\limits_{i = 0}^l {\sum\limits_{j = 0}^m {{{( - 1)}^{i + j}}C_{l + i}^lC_l^iC_{m + j}^mC_m^j} \int_0^1 {\int_s^1 {{u^{i + j}}du{\mathop{\rm d}\nolimits} s} } } \\ & = \sum\limits_{i = 0}^n {\sum\limits_{j = 0}^m {{{( - 1)}^{i + j}}C_{l + i}^lC_l^iC_{m + j}^mC_m^j} \frac{1}{{i + j + 2}}} \\ & = \left\{ {\begin{array}{*{20}{c}} {\frac{1}{{2(2m + 1)}}, } \hfill & {l = m} \hfill \\ { - \frac{m}{{2(2m - 1)(2m + 1)}}, } \hfill & {l = m - 1} \hfill \\ { - \frac{l}{{2(2l - 1)(2l + 1)}}, } \hfill & {l = m + 1} \hfill \\ {0, } \hfill & {{\mathop{\rm otherwise}\nolimits} } \hfill \\ \end{array}} \right. \\ \end{aligned} \end{equation}$ $

(2.9)

which can also be extended using the form of matrix

$\begin{equation} \begin{aligned} {H_m} & = \int_0^1 {\int_s^1 {{L_m}(u)L_m^{\mathop{\rm T}\nolimits} (u)duds} } \\ & = \frac{1}{2}\left[ {\begin{array}{*{20}{c}} {\frac{1}{1}} & { - \frac{1}{{1 \times 3}}} & {} & {} & {} & {} \\ { - \frac{1}{{1 \times 3}}} & {\frac{1}{3}} & { - \frac{2}{{3 \times 5}}} & {} & {} & {} \\ {} & { - \frac{2}{{3 \times 5}}} & {\frac{1}{5}} & { - \frac{3}{{5 \times 7}}} & {} & {} \\ {} & {} & { - \frac{3}{{5 \times 7}}} & \ddots & \ddots & {} \\ {} & {} & {} & \ddots & {\frac{1}{{2m - 1}}} & { - \frac{m}{{(2m - 1)(2m + 1)}}} \\ {} & {} & {} & {} & { - \frac{m}{{(2m - 1)(2m + 1)}}} & {\frac{1}{{2m + 1}}} \\ \end{array}} \right] \\ \end{aligned} \end{equation}$ $

(2.10)

Considering that $ H_m $ is a real-valued symmetric positive semi-definite matrix, we can gain the associated lower triangular matrix using Cholesky decomposition

$\begin{equation} {H_m} = {B_m}B_m^{\mathop{\rm T}\nolimits} \end{equation}$ $

(2.11)

where

$\begin{equation} {B_m} = \frac{{\sqrt 2 }}{2}\left[ {\begin{array}{*{20}{c}} 1 & {} & {} & {} & {} \\ { - \frac{1}{3}} & {\frac{{\sqrt 2 }}{3}} & {} & {} & {} \\ {} & { - \frac{{\sqrt 2 }}{5}} & {\frac{{\sqrt 3 }}{5}} & {} & {} \\ {} & {} & \ddots & \ddots & {} \\ {} & {} & {} & { - \frac{{\sqrt m }}{{2m + 1}}} & {\frac{{\sqrt {m + 1} }}{{2m + 1}}} \\ \end{array}} \right] \end{equation}$ $

(2.12)

Since $ {B_m} > 0 $, $ H_m $ has the unique Cholesky decomposition. Unfortunately, (2.10) shows that $ {L_m}(u) $ is not a proper set of basic functions when the double integral is employed instead of single integral. Thus, we need to find new ones. We introduce the linear combination of $ \{ {p_j}(s)\} $ as follows

$\begin{equation} \bar p_i^{}(s) = \sum\limits_{j = 0}^i {{d_{i, j}}{p_j}(s)} \end{equation}$ $

(2.13)

i.e.

$\begin{equation} {\bar L_m}(u) = \left[ {\begin{array}{*{20}{c}} {{{\bar p}_0}(u)} \\ {{{\bar p}_1}(u)} \\ \vdots \\ {{{\bar p}_m}(u)} \\ \end{array}} \right] = {D_m}L_m^{}(u) \end{equation}$ $

(2.14)

where $ D_m $ denotes the transition matrix from $ L_m^{}(u) $ to $ {\bar L_m}(u) $ with the form

$\begin{equation} {D_m} = \underbrace {[{d_{ij}}]}_{i \ge j} = \left[ {\begin{array}{*{20}{c}} {{d_{00}}} & {} & \cdots & {} \\ {{d_{10}}} & {{d_{11}}} & \cdots & {} \\ \vdots & \vdots & \ddots & \vdots \\ {{d_{m0}}} & {{d_{m1}}} & \cdots & {{d_{mm}}} \\ \end{array}} \right] \end{equation}$ $

(2.15)

In order to obtain the proper shifted Legendre polynomials for double integral, the following equation should be solved.

$\begin{equation} {\bar H_m} = \int_0^1 {\int_s^1 {{{\bar L}_m}(u)\bar L_m^{\mathop{\rm T}\nolimits} (u)duds} } = {D_m}{H_m}D_m^{\mathop{\rm T}\nolimits} = \left[ {\begin{array}{*{20}{c}} {{{\bar h}_{11}}} & {} & {} & {} \\ {} & {{{\bar h}_{22}}} & {} & {} \\ {} & {} & \ddots & {} \\ {} & {} & {} & {{{\bar h}_{mm}}} \\ \end{array}} \right] \end{equation}$ $

(2.16)

where $ \{ {d_{ij}}\} $ and $ \{ {h_{ij}}\} $ are coefficients to be determined.

Substituting (2.11) into (2.16) yields

$\begin{equation} {D_m}{V_m} = \sqrt {{{\bar H}_m}} = \left[ {\begin{array}{*{20}{c}} {\sqrt {{{\bar h}_{00}}} } & {} & {} & {} \\ {} & {\sqrt {{{\bar h}_{11}}} } & {} & {} \\ {} & {} & \ddots & {} \\ {} & {} & {} & {\sqrt {{{\bar h}_{mm}}} } \\ \end{array}} \right] \end{equation}$ $

(2.17)

By solving a serial of linear equations of (2.17), the matrices $ D_m $ and $ {\bar H_m} $ are achieved as following

$\begin{equation} {D_m} = \underbrace {[{d_{ij}} = \frac{{2j + 1}}{{i + 1}}]}_{i \ge j} = \left[ {\begin{array}{*{20}{c}} 1 & {} & {} & {} \\ {\frac{1}{2}} & {\frac{3}{2}} & {} & {} \\ \vdots & \vdots & \ddots & {} \\ {\frac{1}{{m + 1}}} & {\frac{3}{{m + 1}}} & \cdots & {\frac{{2m + 1}}{{m + 1}}} \\ \end{array}} \right] \end{equation}$ $

(2.18)

$\begin{equation} {\bar H_m} = \underbrace {[{{\bar h}_{ii}} = \frac{1}{{2i + 2}}]}_{i = j} = {\mathop{\rm diag}\nolimits} \{ \frac{1}{2}{, _{}}\frac{1}{4}{, _{}} \cdots {, _{}}\frac{1}{{2m + 2}}\} \end{equation}$ $

(2.19)

Thus the vector of shifted Legendre polynomials are achieved

$\begin{equation} {\bar L_m}(u) = {D_m}{L_m}(u) = {D_m}{W_m}{U_m}(u) = {\bar W_m}{U_m}(u) \end{equation}$ $

(2.20)

where, by (2.6),

$\begin{equation} \begin{aligned} {{\bar W}_m} & = \underbrace {\left[ {{{( - 1)}^j}\sum\limits_{k = j}^i {\frac{{2k + 1}}{{i + 1}}C_{k + j}^kC_k^j} } \right]}_{i \ge j} \\ & = \left[ {\begin{array}{*{20}{c}} 1 & {} & {} & {} & {} \\ 2 & { - 3} & {} & {} & {} \\ 3 & { - 12} & {10} & {} & {} \\ \vdots & \vdots & \vdots & \ddots & {} \\ m & {\sum\limits_{k = 1}^m {\frac{{2k + 1}}{{m + 1}}k(k + 1)} } & {\sum\limits_{k = 2}^m {\frac{{2k + 1}}{{m + 1}}C_{k + 2}^kC_k^2} } & \cdots & {{{( - 1)}^m}\frac{{2m + 1}}{{m + 1}}C_{2m}^m} \\ \end{array}} \right] \\ \end{aligned} \end{equation}$ $

(2.21)

2.3. Shifted Legendre polynomials-based single integral inequality

For continuously vector function $ \dot x(\tau):[a{, _{}}b] \to {\textbf{R}^n} $, the associated function $ \dot{ \tilde x}(s):[0{, _{}}1] \to {\textbf{R}^n} $ is defined as follows

$\begin{equation} \dot {\tilde x}(s) = \dot x(\tau ) = \dot x((b - a)s + a) \end{equation}$ $

(2.22)

where $ \tau = (b - a)s + a $.

We can develop the relationships between the single integrals of $ \dot x(\tau) $ and $ \dot {\tilde x}(s) $

$\begin{equation} (b - a)\int_0^1 {{s^k}\dot {\tilde x}(s)ds} = \frac{1}{{{{(b - a)}^k}}}\int_a^b {{{(\tau - a)}^k}\dot x(\tau )d\tau } {, _{}} \quad k = 0{, _{}}1{, _{}}2{, _{}} \cdots \end{equation}$ $

(2.23)

The best weighted square approximation can be obtained with minimizing the following cost function

$\begin{equation} \begin{aligned} {J_s} & = \int_a^b {{{(f(\tau ) - \dot x(\tau ))}^{\mathop{\rm T}\nolimits} }R(f(\tau ) - \dot x(\tau ))d\tau } \\ & = (b - a)\int_0^1 {{{(\tilde f(s) - \dot {\tilde x}(s))}^{\mathop{\rm T}\nolimits} }R(\tilde f(s) - \dot {\tilde x}(s))ds} \\ \end{aligned} \end{equation}$ $

(2.24)

where $ R > 0 $ denotes a symmetric positive-defined matrix with proper dimensions, $ \tilde f(s) $ denotes the approximation function defined as follows

$\begin{equation} \tilde f(s) = \sum\limits_{i = 0}^m {{\beta _i}{p_i}(s)} \end{equation}$ $

(2.25)

where $ {\beta _i} \in {\textbf{R}^n} $ denotes the weight corresponding to the shifted Legendre polynomial $ {p_i}(s) $ for single integral.

Substituting (2.25) into (2.24) yields

$\begin{equation} \begin{aligned} {J_s} & = (b - a)\int_0^1 {{{(\sum\limits_{i = 0}^m {{\beta _i}{p_i}(s)} - \dot {\tilde x}(s))}^{\mathop{\rm T}\nolimits} }R(\sum\limits_{i = 0}^m {{\beta _i}{p_i}(s)} - \dot {\tilde x}(s))ds} \\ & = (b - a)\left[ \begin{array}{l} \sum\limits_{i = 0}^m {\sum\limits_{j = 0}^m {\beta _i^{\mathop{\rm T}\nolimits} R{\beta _j}} } \int_0^1 {{p_i}(s){p_j}(s){\mathop{\rm d}\nolimits} s} \\ - {\mathop{\rm sym}\nolimits} \left( {\sum\limits_{j = 0}^m {\beta _i^{\mathop{\rm T}\nolimits} } R\int_0^1 {\dot {\tilde x}(s){p_i}(s){\mathop{\rm d}\nolimits} s} } \right) \\ \end{array} \right] + \int_a^b {{{\dot x}^{\mathop{\rm T}\nolimits} }(\tau )R\dot x(\tau ){\mathop{\rm d}\nolimits} \tau } \\ & = (b - a)\sum\limits_{i = 0}^m {\frac{1}{{2i + 1}}\beta _i^{\mathop{\rm T}\nolimits} R{\beta _i}} - \sum\limits_{i = 0}^m {{\mathop{\rm sym}\nolimits} (\beta _i^{\mathop{\rm T}\nolimits} } R{\omega _i}) + \int_a^b {{{\dot x}^{\mathop{\rm T}\nolimits} }(\tau )R\dot x(\tau ){\mathop{\rm d}\nolimits} } \tau \\ \end{aligned} \end{equation}$ $

(2.26)

where $ {\omega _i} $ denotes the integral of the product of $ \dot {\tilde x}(s) $ and the i-th shifted Legendre polynomial $ {p_i}(s) $ for single integral. sym() is defined as the sum of vector/matrix with its own transpose sym$ (x) = x+x^{\rm T} $.

$\begin{equation} {\omega _i} = (b - a)\int_0^1 {\dot {\tilde x}(s){p_i}(s){\mathop{\rm d}\nolimits} s} \end{equation}$ $

(2.27)

i.e.

$\begin{equation} {\varpi _m} = \left[ {\begin{array}{*{20}{c}} {{\omega _0}} \\ {{\omega _1}} \\ \vdots \\ {{\omega _m}} \\ \end{array}} \right] = (b - a)\left[ {\begin{array}{*{20}{c}} {\int_0^1 {\dot {\tilde x}(s){p_0}(s){\mathop{\rm d}\nolimits} s} } \\ {\int_0^1 {\dot {\tilde x}(s){p_1}(s){\mathop{\rm d}\nolimits} s} } \\ \vdots \\ {\int_0^1 {\dot {\tilde x}(s){p_m}(s){\mathop{\rm d}\nolimits} s} } \\ \end{array}} \right] = (b - a){\widehat W_m}\left[ {\begin{array}{*{20}{c}} {\int_0^1 {\dot {\tilde x}(s){\mathop{\rm d}\nolimits} s} } \\ {\int_0^1 {\dot {\tilde x}(s)s{\mathop{\rm d}\nolimits} s} } \\ \vdots \\ {\int_0^1 {\dot {\tilde x}(s){s^m}{\mathop{\rm d}\nolimits} s} } \\ \end{array}} \right] \end{equation}$ $

(2.28)

where $ {\widehat W_m} $ denotes the extension matrix associated to $ {W_m} $

$\begin{equation} {\widehat W_m} = \underbrace {[{{( - 1)}^j}C_{i + j}^iC_i^jI]}_{i \ge j} = \left[ {\begin{array}{*{20}{c}} I & {} & {} & \cdots & {} \\ I & { - 2I} & {} & \cdots & {} \\ I & { - 6I} & {6I} & \cdots & {} \\ \vdots & \vdots & \vdots & \ddots & {} \\ I & { - m(m + 1)I} & {C_{m + 2}^mC_m^2I} & \cdots & {{{( - 1)}^m}C_{2m}^mI} \\ \end{array}} \right] \end{equation}$ $

(2.29)

where $ I $ denotes the identity matrix with proper dimensions.

Substituting (2.23) into (2.28) yields

$\begin{equation} {\varpi _m} = \left[ {\begin{array}{*{20}{c}} {{\omega _0}} \\ {{\omega _1}} \\ \vdots \\ {{\omega _m}} \\ \end{array}} \right] = {\widehat W_m}\left[ {\begin{array}{*{20}{c}} {\int_a^b {\dot x(\tau ){\mathop{\rm d}\nolimits} \tau } } \\ {\frac{1}{{b - a}}\int_a^b {(\tau - a)\dot x(\tau ){\mathop{\rm d}\nolimits} \tau } } \\ \vdots \\ {\frac{1}{{{{(b - a)}^m}}}\int_a^b {{{(\tau - a)}^m}\dot x(\tau ){\mathop{\rm d}\nolimits} \tau } } \\ \end{array}} \right] \end{equation}$ $

(2.30)

According to the static condition of (2.26), we obtain

$\begin{equation} \frac{{\partial {J_s}}}{{\partial {\beta _i}}} = (R + {R^{\mathop{\rm T}\nolimits} })\left( {\frac{{b - a}}{{2i + 1}}{\beta _i} - {\omega _i}} \right) = 0 \end{equation}$ $

(2.31)

The second condition of (2.26)

$\begin{equation} [\frac{{{\partial ^2}{J_s}}}{{\partial {\beta _i}\partial {\beta _j}}}] = \frac{{b - a}}{{2i + 1}}(R + {R^{\mathop{\rm T}\nolimits} }){\delta _{ij}} \gt 0 \end{equation}$ $

(2.32)

It means that the optimal $ \beta _i^* = (2i + 1){\omega _i}/(b-a) $ leads to the only minimum cost value

$\begin{equation} {L_s} \ge L_s^ * = \int_a^b {{{\dot x}^{\mathop{\rm T}\nolimits} }(s)R\dot x(s){\mathop{\rm d}\nolimits} s} - \frac{1}{{b - a}}\sum\limits_{i = 0}^m {\omega _i^{\mathop{\rm T}\nolimits} [(2i + 1)R]{\omega _i}} \gt 0 \end{equation}$ $

(2.33)

Lemma 1 (shifted Legendre polynomials-based single integral inequality): For any symmetric positive-defined constant matrix $ R \in {\textbf{R}^{n \times n}} $, $ R > 0 $, and vector function $ \dot x(t):[a{, _{}}b] \to {\textbf{R}^n} $ such that the integrations concerned are well defined, then the following inequality exists

$\begin{equation} \begin{aligned} \int_a^b {{{\dot x}^{\rm T}}(\tau )R\dot x(\tau ){\mathop{\rm d}\nolimits} \tau } &\ge \frac{1}{{b - a}}\varpi _m^{\rm T}{\Omega _m(R)}{\varpi _m} \\ & = \frac{1}{{b - a}}{{\left[ {\begin{array}{*{20}{c}} {{\omega _0}} \\ {{\omega _1}} \\ {{\omega _2}} \\ \vdots \\ {{\omega _m}} \\ \end{array}} \right]}^{\mathop{\rm T}\nolimits} }\left[ {\begin{array}{*{20}{c}} R & 0 & 0 & \cdots & 0 \\ 0 & {3R} & 0 & \cdots & 0 \\ 0 & 0 & {5R} & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & {(2m + 1)R} \\ \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\omega _0}} \\ {{\omega _1}} \\ {{\omega _2}} \\ \vdots \\ {{\omega _m}} \\ \end{array}} \right] \\ \end{aligned} \end{equation}$ $

(2.34)

Proof: It can be obtained from (2.33) observably.

Remark 1: The right term of the proposed single integral inequality (2.34) is approximation with arbitrary order to the left term, i.e., when $ \dot x(t) = {c_0} + {c_1}t + \cdots + {c_m}{t^m} $, $ {c_i} \in {\textbf{R}^n} $, $ i = 0{, _{}}1{, _{}} \cdots {, _{}}m $, the left term is exactly equal to the right term.

Proof: The function $ \dot x(t) = {c_0} + {c_1}t + \cdots + {c_m}{t^m} $ can be rewritten as

$\begin{equation} \begin{aligned} \dot x((b - a)s + a) & = {c_0} + {c_1}[(b - a)s + a] + \cdots + {c_m}{{[(b - a)s + a]}^m} \\ & = {{\tilde c}_0} + {{\tilde c}_1}s + \cdots + {{\tilde c}_m}{s^m} \\ & = \dot {\tilde x}(s) \\ \end{aligned} \end{equation}$ $

(2.35)

where

$\begin{equation} {\tilde c_k} = {(b - a)^k}\sum\limits_{i = k}^m {{a^{k - i}}C_k^i} \end{equation}$ $

(2.36)

$ \dot {\tilde x}(s) $ can also be expressed by serial of shifted Legendre polynomials $ \{ {p_k}(s)\} $ as follows

$\begin{equation} \dot {\tilde x}(s) = {\lambda _0}{p_0}(s) + {\lambda _1}{p_1}(s) + \cdots + {\lambda _m}{p_m}(s) \end{equation}$ $

(2.37)

where

$\begin{equation} {\lambda _i} = \frac{{\int_0^1 {\dot {\tilde x}(s){p_i}(s){\mathop{\rm d}\nolimits} s} }}{{\int_0^1 {{p_i}(s){p_i}(s){\mathop{\rm d}\nolimits} s} }} = \frac{{2i + 1}}{{b - a}}{\omega _i} \end{equation}$ $

(2.38)

Thus the left term of (2.34) becomes

$\begin{equation} \begin{aligned} \int_a^b {{{\dot x}^T}(\tau )R\dot x(\tau ){\mathop{\rm d}\nolimits} \tau } & = (b - a)\int_0^1 {\left( {\sum\limits_{i = 0}^m {\lambda _i^{}{p_i}(s)} } \right)R\left( {\sum\limits_{i = 0}^m {\lambda _i^{}{p_i}(s)} } \right){\mathop{\rm d}\nolimits} s} \\ & = (b - a)\sum\limits_{i = 0}^m {\sum\limits_{j = 0}^m {\lambda _i^{\rm T} R{\lambda _j}} } \int_0^1 {{p_i}(s){p_j}(s){\mathop{\rm d}\nolimits} s} \\ & = (b - a)\sum\limits_{i = 0}^m {\frac{1}{{2i + 1}}\lambda _i^{\rm T} R{\lambda _i}} \\ & = (b - a)\sum\limits_{i = 0}^m {\frac{1}{{2i + 1}}{{\left( {\frac{{2i + 1}}{{b - a}}{\omega _i}} \right)}^{\rm T}}R{{\left( {\frac{{2i + 1}}{{b - a}}{\omega _i}} \right)}^{\rm T}}} \\ & = \frac{1}{{b - a}}\sum\limits_{i = 0}^m {(2i + 1)\omega _i^{\rm T} R{\omega _i}} \\ \end{aligned} \end{equation}$ $

(2.39)

This complete the proof.

Remark 2: The integral inequality (2.34) degenerates to Jensen inequality when $ m = 0 $^[2].

Proof: Substituting $ m = 0 $ into (2.34) yields

$\begin{equation} \begin{aligned} \int_a^b {{{\dot x}^{\rm T}}(\tau )R\dot x(\tau ){\mathop{\rm d}\nolimits} \tau } &\ge \frac{1}{{b - a}}{\varpi ^{\rm T}}\Omega \varpi = \frac{1}{{b - a}}\omega _0^{\mathop{\rm T}\nolimits} R{\omega _0} \\ & = \frac{1}{{b - a}}{\left( {\int_a^b {\dot x(\tau ){\mathop{\rm d}\nolimits} \tau } } \right)^{\mathop{\rm T}\nolimits} }R\left( {\int_a^b {\dot x(\tau ){\mathop{\rm d}\nolimits} \tau } } \right) \\ & = \frac{1}{{b - a}}{(x(b) - x(a))^{\mathop{\rm T}\nolimits} }R(x(b) - x(a)) \\ \end{aligned} \end{equation}$ $

(2.40)

This complete the proof.

Remark 3: The integral inequality (2.34) degenerates to Wirtinger-based inequality when $ m = 1 $^[18].

Proof: According to (2.30) we have

$\begin{equation} {\omega _0} = \int_a^b {\dot x(\tau ){\mathop{\rm d}\nolimits} \tau } = x(b) - x(a) = {\omega _{{\mathop{\rm Wirtinger}\nolimits} , 0}} \end{equation}$ $

(2.41)

$\begin{equation} \begin{aligned} {\omega _1} & = \int_a^b {\dot x(\tau ){\mathop{\rm d}\nolimits} \tau } - \frac{2}{{b - a}}\int_a^b {(\tau - a)\dot x(\tau ){\mathop{\rm d}\nolimits} \tau } \\ & = x(b) - x(a) - \frac{2}{{b - a}}\left[ {(b - a)x(b) - \int_a^b {x(\tau ){\mathop{\rm d}\nolimits} \tau } } \right] \\ & = - \left[ {x(a) + x(b) - \frac{2}{{b - a}}\int_a^b {x(\tau ){\mathop{\rm d}\nolimits} \tau } } \right] \\ & = - {\omega _{{\mathop{\rm Wirtinger}\nolimits} , 1}} \\ \end{aligned} \end{equation}$ $

(2.42)

Substituting (2.41) and (2.42) into (2.34) yields

$\begin{equation} \begin{aligned} \int_a^b {{{\dot x}^{\rm T}}(\tau )R\dot x(\tau ){\mathop{\rm d}\nolimits} \tau } &\ge \frac{1}{{b - a}}{{\left[ {\begin{array}{*{20}{c}} {{\omega _0}} \\ {{\omega _1}} \\ \end{array}} \right]}^{\mathop{\rm T}\nolimits} }\left[ {\begin{array}{*{20}{c}} R & {} \\ {} & {3R} \\ \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\omega _0}} \\ {{\omega _1}} \\ \end{array}} \right] \\ & = \frac{1}{{b - a}}{{\left[ {\begin{array}{*{20}{c}} {{\omega _{{\mathop{\rm Wirtinger}\nolimits} , 0}}} \\ {{\omega _{{\mathop{\rm Wirtinger}\nolimits} , 1}}} \\ \end{array}} \right]}^{\mathop{\rm T}\nolimits} }\left[ {\begin{array}{*{20}{c}} R & {} \\ {} & {3R} \\ \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\omega _{{\mathop{\rm Wirtinger}\nolimits} , 0}}} \\ {{\omega _{{\mathop{\rm Wirtinger}\nolimits} , 1}}} \\ \end{array}} \right] \\ \end{aligned} \end{equation}$ $

(2.43)

This complete the proof.

2.4. Shifted Legendre polynomials-based double integral inequality

For continuously vector function $ \dot x(\tau):[a{, _{}}b] \to {\textbf{R}^n} $, and it's associated function $ \dot {\tilde x}(s):[0{, _{}}1] \to {\textbf{R}^n} $ defined in (2.22), we can develop the relationships between the double integrals of $ \dot x(\tau) $ and $ \dot {\tilde x}(s) $ as follows

$\begin{equation} \begin{array}{l} {(b - a)^2}\int_0^1 {\int_s^1 {{u^k}\dot {\tilde x}(u)duds} } = \frac{1}{{{{(b - a)}^k}}}\int_a^b {\int_\theta ^b {{{(\tau - a)}^k}\dot x(\tau )d\tau {\mathop{\rm d}\nolimits} \theta } } \\ \quad k = 0{, _{}}1{, _{}}2{, _{}} \cdots \\ \end{array} \end{equation}$ $

(2.44)

where

$ u = \frac{{\tau - a}}{{b - a}}{, _{}} \quad s = \frac{{\theta - a}}{{b - a}} $

The best weighted square approximation with double integral can be obtained with minimizing the following cost function

$\begin{equation} \begin{aligned} {J_d} & = \int_a^b {\int_\theta ^b {{{(g(\tau ) - \dot x(\tau ))}^{\mathop{\rm T}\nolimits} }R(g(\tau ) - \dot x(\tau ))d\tau {\mathop{\rm d}\nolimits} \theta } } \\ & = {{(b - a)}^2}\int_0^1 {\int_s^1 {{{(\tilde g(u) - \dot {\tilde x}(u))}^{\mathop{\rm T}\nolimits} }R(\tilde g(u) - \dot {\tilde x}(u))duds} } \\ \end{aligned} \end{equation}$ $

(2.45)

where $ R > 0 $ denotes a positive-defined matrix with proper dimensions, $ \tilde g(u) $ denotes the approximation function defined as follows

$\begin{equation} \tilde g(u) = \sum\limits_{i = 0}^m {{\beta _i}{{\bar p}_i}(s)} \end{equation}$ $

(2.46)

where $ {\beta _i} \in {\textbf{R}^n} $ denotes the weight corresponding to the shifted Legendre polynomial $ {\bar p_i}(s) $ for double integral.

Substituting (2.46) into (2.45) yields

$\begin{equation} \begin{aligned} {J_d} & = {{(b - a)}^2}\int_0^1 {\int_s^1 {{{\left( {\sum\limits_{i = 0}^m {{\beta _i}{{\bar p}_i}(u)} - \dot {\tilde x}(u)} \right)}^{\mathop{\rm T}\nolimits} }R\left( {\sum\limits_{i = 0}^m {{\beta _i}{{\bar p}_i}(u)} - \dot {\tilde x}(u)} \right)duds} } \\ & = {(b - a)^2}\left[ \begin{array}{l} \sum\limits_{i = 0}^m {\sum\limits_{j = 0}^m {\beta _i^{\mathop{\rm T}\nolimits} R{\beta _j}} } \int_0^1 {\int_s^1 {{{\bar p}_i}(s){{\bar p}_j}(s)du{\mathop{\rm d}\nolimits} s} } \\ - {\mathop{\rm sym}\nolimits} \left( {\sum\limits_{j = 0}^m {\beta _i^{\mathop{\rm T}\nolimits} } R\int_0^1 {\int_s^1 {\dot {\tilde x}(s){{\bar p}_j}(s)du{\mathop{\rm d}\nolimits} s} } } \right) \\ \end{array} \right] + \int_a^b {\int_\theta ^b {{{\dot x}^{\mathop{\rm T}\nolimits} }(\tau )R\dot x(\tau ){\mathop{\rm d}\nolimits} \tau {\mathop{\rm d}\nolimits} \theta } } \\ & = {(b - a)^2}\sum\limits_{i = 0}^m {\frac{1}{{2i + 2}}\beta _i^{\mathop{\rm T}\nolimits} R{\beta _i}} - (b - a)\sum\limits_{i = 0}^m {{\mathop{\rm sym}\nolimits} (\beta _i^{\mathop{\rm T}\nolimits} } R{\nu _i}) + \int_a^b {\int_\theta ^b {{{\dot x}^{\mathop{\rm T}\nolimits} }(\tau )R\dot x(\tau ){\mathop{\rm d}\nolimits} \tau {\mathop{\rm d}\nolimits} \theta } } \\ \end{aligned} \end{equation}$ $

(2.47)

where $ {\nu _i} $ denotes the integral of the product of $ \dot {\tilde x}(s) $ and the i-th shifted Legendre polynomial $ {p_i}(s) $ for single integral

$\begin{equation} {\nu _i} = (b - a)\int_0^1 {\int_s^1 {\dot {\tilde x}(s){{\bar p}_i}(u)du{\mathop{\rm d}\nolimits} s} } \end{equation}$ $

(2.48)

i.e.

$\begin{equation} {\bar \nu _m} = \left[ {\begin{array}{*{20}{c}} {{\nu _0}} \\ {{\nu _1}} \\ \vdots \\ {{\nu _m}} \\ \end{array}} \right] = (b - a)\left[ {\begin{array}{*{20}{c}} {\int_0^1 {\int_s^1 {\dot {\tilde x}(s){{\bar p}_0}(u)du{\mathop{\rm d}\nolimits} s} } } \\ {\int_0^1 {\int_s^1 {\dot {\tilde x}(s){{\bar p}_1}(u)du{\mathop{\rm d}\nolimits} s} } } \\ \vdots \\ {\int_0^1 {\int_s^1 {\dot {\tilde x}(s){{\bar p}_m}(u)du{\mathop{\rm d}\nolimits} s} } } \\ \end{array}} \right] = (b - a){\widehat{\bar W}_m}\left[ {\begin{array}{*{20}{c}} {\int_0^1 {\int_s^1 {\dot {\tilde x}(s)du{\mathop{\rm d}\nolimits} s} } } \\ {\int_0^1 {\int_s^1 {\dot {\tilde x}(s)du{\mathop{\rm d}\nolimits} s} } } \\ \vdots \\ {\int_0^1 {\int_s^1 {\dot {\tilde x}(s){u^m}du{\mathop{\rm d}\nolimits} s} } } \\ \end{array}} \right] \end{equation}$ $

(2.49)

where $ {\widehat{\bar W}_m} $ denotes the extension matrix associated to $ {\bar W_m} $

$\begin{equation} \begin{aligned} {{\widehat{\bar W}}_m} & = \underbrace {[{{( - 1)}^j}\sum\limits_{k = j}^i {\frac{{2k + 1}}{{i + 1}}C_{k + j}^kC_k^j} I]}_{i \ge j} \\ & = \left[ {\begin{array}{*{20}{c}} I & {} & {} & {} & {} \\ {2I} & { - 3I} & {} & {} & {} \\ {3I} & { - 12I} & {10} & {} & {} \\ \vdots & \vdots & \vdots & \ddots & {} \\ {mI} & {\sum\limits_{k = 1}^m {\frac{{2k + 1}}{{m + 1}}k(k + 1)} I} & {\sum\limits_{k = 2}^m {\frac{{2k + 1}}{{m + 1}}C_{k + 2}^kC_k^2} I} & \cdots & {{{( - 1)}^m}\frac{{2m + 1}}{{m + 1}}C_{2m}^mI} \\ \end{array}} \right] \\ \end{aligned} \end{equation}$ $

(2.50)

Substituting (2.44) into (2.49) yields

$\begin{equation} {\bar \nu _m} = \left[ {\begin{array}{*{20}{c}} {{\nu _0}} \\ {{\nu _1}} \\ \vdots \\ {{\nu _m}} \\ \end{array}} \right] = {\widehat{\bar W}_m}\left[ {\begin{array}{*{20}{c}} {\frac{1}{{b - a}}\int_a^b {\int_\theta ^b {\dot x(\tau )d\tau {\mathop{\rm d}\nolimits} \theta } } } \\ {\frac{1}{{{{(b - a)}^2}}}\int_a^b {\int_\theta ^b {(\tau - a)\dot x(\tau )d\tau {\mathop{\rm d}\nolimits} \theta } } } \\ \vdots \\ {\frac{1}{{{{(b - a)}^{m + 1}}}}\int_a^b {\int_\theta ^b {{{(\tau - a)}^m}\dot x(\tau )d\tau {\mathop{\rm d}\nolimits} \theta } } } \\ \end{array}} \right] = {\widehat{\bar W}_m}\left[ {\begin{array}{*{20}{c}} {\frac{1}{{b - a}}\int_a^b {(\tau - a)\dot x(\tau )d\tau } } \\ {\frac{1}{{{{(b - a)}^2}}}\int_a^b {{{(\tau - a)}^2}\dot x(\tau )d\tau } } \\ \vdots \\ {\frac{1}{{{{(b - a)}^{m + 1}}}}\int_a^b {{{(\tau - a)}^{m + 1}}\dot x(\tau )d\tau } } \\ \end{array}} \right] \end{equation}$ $

(2.51)

According to the static condition of (2.47), we obtain

$\begin{equation} \frac{{\partial {J_d}}}{{\partial {\beta _i}}} = (R + {R^{\mathop{\rm T}\nolimits} })\left[ {\frac{{{{(b - a)}^2}}}{{2i + 2}}{\beta _i} - (b - a){\nu _i}} \right] = 0 \end{equation}$ $

(2.52)

The second condition of (2.47)

$\begin{equation} [\frac{{{\partial ^2}{J_d}}}{{\partial {\beta _i}\partial {\beta _j}}}] = \frac{{{{(b - a)}^2}}}{{2i + 2}}(R + {R^{\mathop{\rm T}\nolimits} }){\delta _{ij}} \gt 0 \end{equation}$ $

(2.53)

It means that the optimal $ \beta _i^* = \frac{{2i + 2}}{{b - a}}{\nu _i} $ leads to the only minimum cost value

$\begin{equation} {L_d} \ge L_d^ * = \int_a^b {\int_\theta ^b {{{\dot x}^{\mathop{\rm T}\nolimits} }(\tau )R\dot x(\tau ){\mathop{\rm d}\nolimits} \tau {\mathop{\rm d}\nolimits} \theta } } - \sum\limits_{i = 0}^m {\nu _i^{\mathop{\rm T}\nolimits} [(2i + 2)R]{\nu _i}} \gt 0 \end{equation}$ $

(2.54)

Lemma 2 (shifted Legendre polynomials-based double integral inequality): For any positive-defined constant matrix $ R \in {\textbf{R}^{n \times n}} $, $ R > 0 $, and vector function $ \dot x(t):[a{, _{}}b] \to {\textbf{R}^n} $ such that the integrations concerned are well defined, then the following inequality exists

$\begin{equation} \int_a^b {\int_\theta ^b {{{\dot x}^{\mathop{\rm T}\nolimits} }(\tau )R\dot x(\tau ){\mathop{\rm d}\nolimits} \tau {\mathop{\rm d}\nolimits} \theta } } \ge \bar \nu {_m^T}{\bar \Omega _m(R)}\bar \nu _m^{} = {\left[ {\begin{array}{*{20}{c}} {{\nu _0}} \\ {{\nu _1}} \\ {{\nu _2}} \\ \vdots \\ {{\nu _m}} \\ \end{array}} \right]^{\mathop{\rm T}\nolimits} }\left[ {\begin{array}{*{20}{c}} {2R} & 0& 0& \cdots & 0 \\ 0& {4R}& 0& \cdots& 0 \\ 0& 0& {6R}& \cdots& 0 \\ \vdots & \vdots& \vdots& \ddots & \vdots \\ 0& 0& 0& \cdots & {(2m + 2)R} \\ \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\nu _0}} \\ {{\nu _1}} \\ {{\nu _2}} \\ \vdots \\ {{\nu _m}} \\ \end{array}} \right] \end{equation}$ $

(2.55)

Proof: It can be obtained from (2.54) observably.

Proof: The function $ \dot x(t) = {c_0} + {c_1}t + \cdots + {c_m}{t^m} $ can be rewritten as

(2.56)

where

$\begin{equation} {\tilde c_k} = {(b - a)^k}\sum\limits_{i = k}^m {{a^{k - i}}C_k^i} \end{equation}$ $

(2.57)

$ \dot {\tilde x}(s) $ can also be expressed by serial of shifted Legendre polynomials $ \{ {\bar p_k}(s)\} $ as follows

$\begin{equation} \dot {\tilde x}(s) = {\lambda _0}{\bar p_0}(s) + {\lambda _1}{\bar p_1}(s) + \cdots + {\lambda _m}{\bar p_m}(s) \end{equation}$ $

(2.58)

where

$\begin{equation} {\lambda _i} = \frac{{\int_0^1 {\int_s^1 {\dot {\tilde x}(s){{\bar p}_i}(u)du{\mathop{\rm d}\nolimits} s} } }}{{\int_0^1 {\int_s^1 {{{\bar p}_i}{{\bar p}_i}(u)du{\mathop{\rm d}\nolimits} s} } }} = \frac{{2i + 2}}{{b - a}}{\nu _i} \end{equation}$ $

(2.59)

Thus the left term of (2.34) becomes

$\begin{equation} \begin{aligned} {{{_{}}_{}}_{}\int_a^b {\int_\theta ^b {{{\dot x}^{\mathop{\rm T}\nolimits} }(\tau )R\dot x(\tau ){\mathop{\rm d}\nolimits} \tau {\mathop{\rm d}\nolimits} \theta } } } & = {{(b - a)}^2}\int_0^1 {\int_s^1 {\dot {\tilde x}{{(u)}^{\mathop{\rm T}\nolimits} }R\dot {\tilde x}(s)duds} } \\ & = {(b - a)^2}\sum\limits_{i = 0}^m {\sum\limits_{j = 0}^m {{{\left( {\frac{{2i + 2}}{{b - a}}{\nu _i}} \right)}^{\mathop{\rm T}\nolimits} }R\left( {\frac{{2j + 2}}{{b - a}}{\nu _j}} \right)} } \int_0^1 {\int_s^1 {{{\bar p}_i}(s){{\bar p}_j}(s)du{\mathop{\rm d}\nolimits} s} } \\ & = \sum\limits_{i = 0}^m {\nu _i^{\rm T}[(2i + 2)R]{\nu _i}} \\ \end{aligned} \end{equation}$ $

(2.60)

This complete the proof.

Remark 2: The integral inequality (2.34) degenerates to auxiliary function-based integral inequality when $ m = 1 $^[34].

Proof: According to (2.51) we have

$\begin{equation} {\nu _0} = \frac{1}{{b - a}}\int_a^b {\int_\theta ^b {\dot x(\tau )d\tau {\mathop{\rm d}\nolimits} \theta } } = x(b) - \frac{1}{{b - a}}\int_a^b {x(\tau ){\mathop{\rm d}\nolimits} \tau } \end{equation}$ $

(2.61)

$\begin{equation} \begin{aligned} {\nu _1} & = \frac{2}{{b - a}}\int_a^b {\int_\theta ^b {\dot x(\tau )d\tau {\mathop{\rm d}\nolimits} \theta } } - \frac{2}{{{{(b - a)}^2}}}\int_a^b {\int_\theta ^b {(\tau - a)\dot x(\tau )d\tau {\mathop{\rm d}\nolimits} \theta } } \\ & = 2\left[ {x(b) - \frac{1}{{b - a}}\int_a^b {x(\tau ){\mathop{\rm d}\nolimits} \tau } } \right] - 3\left[ {x(b) - \frac{2}{{{{(b - a)}^2}}}\int_a^b {\int_\theta ^b {x(\tau )d\tau } {\mathop{\rm d}\nolimits} \theta } } \right] \\ & = - x(b) - \frac{2}{{b - a}}\int_a^b {x(\tau ){\mathop{\rm d}\nolimits} \tau } + \frac{6}{{{{(b - a)}^2}}}\int_a^b {\int_\theta ^b {x(\tau )d\tau } {\mathop{\rm d}\nolimits} \theta } \\ \end{aligned} \end{equation}$ $

(2.62)

Note that $ {\nu _0} $ and $ {\nu _1} $ are just the coefficients of auxiliary function-based integral inequality. This complete the proof.

3. Applications to systems with constant delays

3.1. Systems with constant delays

Let us consider the following linear system with constant delay interval

$\begin{equation} \begin{array}{l} \dot x(t) = Ax(t) + {A_h}x(t - h) \\ x(t) = \varphi (t){, _{}} \quad t \in [ - h{, _{}}0] \\ \end{array} \end{equation}$ $

(3.1)

where $ x(t) \in {\textbf{R}^n} $ denotes the state vector of the system with $ n $ dimensions, $ A $ and $ A_h $ are real known constant matrices with appropriate dimensions, the continuously differentiable functions $ \varphi (t) $ denote the initial condition, $ h \ge 0 $ denotes the system's constant delay.

Theorem 1: The system (3.1) is asymptotically stable if there exist matrices $ P > 0 $, $ Q > 0 $, $ R > 0 $ and $ S > 0 $ such that the following conditions hold ^[41]:

$\begin{equation} \left[ \begin{array}{l} B_{}^{\mathop{\rm T}\nolimits} PC + C_{}^{\mathop{\rm T}\nolimits} PB + e_1^{\mathop{\rm T}\nolimits} Q{e_1} - e_2^{\mathop{\rm T}\nolimits} Q{e_2} + {h^2}A_e^{\mathop{\rm T}\nolimits} R{A_e} + \frac{1}{2}{h^2}A_e^{\mathop{\rm T}\nolimits} S{A_e} \\ - \Psi _{}^{\rm T}\widehat W_{m}^{\rm T}\Omega_m(R) \widehat W_m\Psi - \bar \Psi _{}^{\rm T}\widehat{\bar W}_{m}^{\rm T}\bar \Omega_m(S) \widehat{\bar W}_m\bar \Psi \\ \end{array} \right] \lt 0 \end{equation}$ $

(3.2)

where the notations in (3.2) are intermediate variables that defined properly in previous and in the process of proof, which can be found as $ B $ in (3.10), $ C $ in (3.12), $ e_1, e_2 $ in (3.10), $ h $ in (3.1), $ A_e $ in (3.11), $ \Psi $ in (3.13), $ \widehat W_m $ in (2.29), $ \Omega_m $ in (2.34), $ \bar \Psi $ in (3.14), $ \widehat{\bar W}_m $ in (3.7), $ \bar \Omega_m $ in (3.18).

Proof: We define a set of functions $ \{ {y_k}(t)\} $ as follows

$\begin{equation} \begin{aligned} {y_k}(t) &\buildrel \Delta \over = h\int_0^1 {\dot {\tilde x}(s){u^k}{\mathop{\rm d}\nolimits} u} = \frac{1}{{{h^k}}}\int_{t - h}^t {\dot x(\tau ){{(\tau - t + h)}^k}{\mathop{\rm d}\nolimits} \tau } \\ k & = 0{, _{}}1{, _{}}2{, _{}} \cdots \\ \end{aligned} \end{equation}$ $

(3.3)

The time derivatives of $ {y_k}(t) $ can be obtained as follows

$\begin{equation} \begin{aligned} {{\dot y}_k}(t) & = \frac{{\mathop{\rm d}\nolimits} }{{{\mathop{\rm d}\nolimits} t}}\left[ {\frac{1}{{{h^k}}}\int_{t - h}^t {\dot x(\tau ){{(\tau - t + h)}^k}{\mathop{\rm d}\nolimits} \tau } } \right] \\ & = \dot x(t) - \frac{k}{{{h^k}}}\int_{t - h}^t {\dot x(\tau ){{(\tau - t + h)}^{k - 1}}{\mathop{\rm d}\nolimits} \tau } \\ & = \dot x(t) - \frac{k}{h}{y_{k - 1}}(t) \\ & = Ax(t) + {A_h}x(t - h) - \frac{k}{h}{y_{k - 1}}(t) \\ (k \ge 1) \\ \end{aligned} \end{equation}$ $

(3.4)

And the initial we have

$\begin{equation} \begin{aligned} {y_0}(t) & = \int_{t - h}^t {\dot x(\tau ){\mathop{\rm d}\nolimits} \tau } = x(t) - x(t - h) \\ {{\dot y}_1}(t) & = \dot x(t) - \frac{1}{h}{y_0}(t) = (A - \frac{1}{h}I)x(t) + ({A_h} + \frac{1}{h}I)x(t-h) \\ \end{aligned} \end{equation}$ $

(3.5)

Let $ a = t - h $, $ b = t $, we can obtain $ \{ {\omega _k}\} $ and $ \{ {\nu _k}\} $ for shifted Legendre polynomials-based single and double integral inequalities, respectively

$\begin{equation} \left[ {\begin{array}{*{20}{c}} {{\omega _0}} \\ {{\omega _1}} \\ \vdots \\ {{\omega _m}} \\ \end{array}} \right] = \widehat W_m\left[ {\begin{array}{*{20}{c}} {\int_{t - h}^t {\dot x(\tau ){\mathop{\rm d}\nolimits} \tau } } \\ {\frac{1}{h}\int_{t - h}^t {\dot x(\tau )(\tau - t + h){\mathop{\rm d}\nolimits} \tau } } \\ \vdots \\ {\frac{1}{{{h^m}}}\int_{t - h}^t {\dot x(\tau ){{(\tau - t + h)}^m}{\mathop{\rm d}\nolimits} \tau } } \\ \end{array}} \right] = \widehat W_m\left[ {\begin{array}{*{20}{c}} {{y_0}(t)} \\ {{y_1}(t)} \\ \vdots \\ {{y_m}(t)} \\ \end{array}} \right] \end{equation}$ $

(3.6)

$\begin{equation} \left[ {\begin{array}{*{20}{c}} {{\nu _0}} \\ {{\nu _1}} \\ \vdots \\ {{\nu _{m - 1}}} \\ \end{array}} \right] = \widehat{\bar W}_m\left[ {\begin{array}{*{20}{c}} {\frac{1}{h}\int_{t - h}^t {\dot x(\tau )(\tau - t + h){\mathop{\rm d}\nolimits} \tau } } \\ {\frac{1}{{{h^2}}}\int_{t - h}^t {\dot x(\tau ){{(\tau - t + h)}^2}{\mathop{\rm d}\nolimits} \tau } } \\ \vdots \\ {\frac{1}{{{h^m}}}\int_{t - h}^t {\dot x(\tau ){{(\tau - t + h)}^m}{\mathop{\rm d}\nolimits} \tau } } \\ \end{array}} \right] = \widehat{\bar W}_m\left[ {\begin{array}{*{20}{c}} {{y_1}(t)} \\ {{y_2}(t)} \\ \vdots \\ {{y_m}(t)} \\ \end{array}} \right] \end{equation}$ $

(3.7)

We define extra-states $ \chi (t) $ and $ \xi (t) $ as follows

$\begin{equation} \chi (t) = \left[ {\begin{array}{*{20}{c}} {x(t)} \\ {\left[ {\begin{array}{*{20}{c}} {{y_1}(t)} \\ \vdots \\ {{y_m}(t)} \\ \end{array}} \right]} \\ \end{array}} \right]{, _{}} \quad \xi (t) = \left[ {\begin{array}{*{20}{c}} {\left[ {\begin{array}{*{20}{c}} {x(t)} \\ {x(t - h)} \\ \end{array}} \right]} \\ {\left[ {\begin{array}{*{20}{c}} {{y_1}(t)} \\ \vdots \\ {{y_m}(t)} \\ \end{array}} \right]} \\ \end{array}} \right] \end{equation}$ $

(3.8)

The extra-states $ \chi (t) $ can be expressed by $ \xi (t) $

$\begin{equation} \chi (t) = B\xi (t) \end{equation}$ $

(3.9)

where

$\begin{equation} B = \left[ {\begin{array}{*{20}{c}} {{e_1}} \\ {{e_3}} \\ {{e_4}} \\ \vdots \\ {{e_m}} \\ \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\left[ {\begin{array}{*{20}{c}} {{I_n}} & {{0_n}} \\ \end{array}} \right]} & {} \\ {} & {{I_{nm}}} \\ \end{array}} \right] \end{equation}$ $

(3.10)

where $ {e_k} = [{\underbrace { $\begin{array}{*{20}{c}} 0 & 0 & 0 \\ \end{array}$ }_{k - 1}}{I}\underbrace { $\begin{array}{*{20}{c}} 0 & 0 & 0 \\ \end{array}$ }_{m + 2 -k}] $ denotes the k-th row coefficient of $ \xi (t) $, $ I_n $ and $ 0_n $ denote the identity and zeros matrix with dimensions $ n \times n $, respectively.

And the system (3.1) can be rewritten as

$\begin{equation} \dot x(t) = {A_e}\xi (t) \end{equation}$ $

(3.11)

where $ {A_e} = A{e_1} + {A_h}{e_2} $.

The time derivative of $ \chi (t) $ can be obtained as follows

$\begin{equation} \dot \chi (t) = C\xi (t) \end{equation}$ $

(3.12)

where

$ C = \left[ {

$\begin{array}{*{20}{c}} {\left[ {\begin{array}{*{20}{c}} A & {{A_h}} \\ \end{array}} \right]} & {{0_{n \times nm}}} \\ M & { - \frac{1}{h}\Lambda } \\ \end{array}$ } \right] $

$ M = \left[ {

$\begin{array}{*{20}{c}} {A - \frac{1}{h}I} & {{A_h} + \frac{1}{h}I} \\ A & {{A_h}} \\ A & {{A_h}} \\ \vdots & \vdots \\ A & {{A_h}} \\ \end{array}$ } \right]{, _{}} \quad \Lambda = \left[ {

$\begin{array}{*{20}{c}} 0 & {} & {} & {} & {} \\ {2I} & 0 & {} & {} & {} \\ {} & {3I} & 0 & {} & {} \\ {} & {} & \ddots & \ddots & {} \\ {} & {} & {} & {mI} & 0 \\ \end{array}$ } \right] $

According to (3.6) and (3.8), we have

$\begin{equation} \left[ {\begin{array}{*{20}{c}} {{\omega _0}} \\ {{\omega _1}} \\ \vdots \\ {{\omega _m}} \\ \end{array}} \right] = \widehat W_m\left[ {\begin{array}{*{20}{c}} {{y_0}(t)} \\ {{y_1}(t)} \\ \vdots \\ {{y_m}(t)} \\ \end{array}} \right] = \widehat W_m\left[ {\begin{array}{*{20}{c}} {x(t) - x(t - h)} \\ {{y_1}(t)} \\ \vdots \\ {{y_m}(t)} \\ \end{array}} \right] = \widehat W_m\Psi \xi (t) \end{equation}$ $

(3.13)

where

$ \Psi = \left[ {

$\begin{array}{*{20}{c}} {{I_n}} & { - {I_n}} & 0 \\ 0 & 0 & {{I_{nm}}} \\ \end{array}$ } \right] $

With similar method, we have following according to (3.7) and (3.8)

(3.14)

where $ \bar \Psi = \left[{ $\begin{array}{*{20}{c}} {{0_{nm \times n}}} & {{0_{nm \times n}}} & {{I_{nm}}} \\ \end{array}$ } \right] $

In order to analysis the stability of the system (3.1), we consider the following Lyapunov-Krasovskii functional (LKF) candidates

$\begin{equation} V = \left[ \begin{array}{l} \chi {(t)^{\rm T}}P\chi (t) + \int_{t - h}^t {{x^{\rm T}}(\tau )Qx(\tau ){\mathop{\rm d}\nolimits} \tau } \\ + h\int_{t - h}^t {\int_\theta ^t {{{\dot x}^{\rm T}}(\tau )R\dot x(\tau )} {\mathop{\rm d}\nolimits} \tau {\mathop{\rm d}\nolimits} \theta } + \int_{t - h}^t {\int_\gamma ^t {\int_\theta ^t {{{\dot x}^{\rm T}}(\tau )S\dot x(\tau )} {\mathop{\rm d}\nolimits} \tau {\mathop{\rm d}\nolimits} \theta {\mathop{\rm d}\nolimits} } } \gamma \\ \end{array} \right] \end{equation}$ $

(3.15)

Taking the time derivative of $ V(t) $ yields

$\begin{equation} \begin{aligned} \dot V(t) & = \left[ \begin{array}{l} {\chi ^{\rm T}}(t)P\dot \chi (t) + {{\dot \chi }^{\rm T}}(t)P\chi (t) \\ + {x^{\rm T}}(t)Qx(t) - {x^{\rm T}}(t - h)Qx(t - h) \\ + {h^2}{{\dot x}^{\rm T}}(t)R\dot x(t) - h\int_{t - h}^t {{{\dot x}^{\rm T}}(\tau )R\dot x(\tau ){\mathop{\rm d}\nolimits} \tau } \\ + \frac{{{h^2}}}{2}{{\dot x}^{\rm T}}(t)S\dot x(t) - \int_{t - h}^t {\int_\theta ^t {{{\dot x}^{\rm T}}(\tau )S\dot x(\tau )} {\mathop{\rm d}\nolimits} \tau {\mathop{\rm d}\nolimits} \theta } \\ \end{array} \right] \\ &\le \xi _{}^{\mathop{\rm T}\nolimits} (t)\left[ \begin{array}{l} B_{}^{\mathop{\rm T}\nolimits} PC + C_{}^{\mathop{\rm T}\nolimits} PB + e_1^{\mathop{\rm T}\nolimits} Q{e_1} - e_2^{\mathop{\rm T}\nolimits} Q{e_2} + {h^2}A_e^{\mathop{\rm T}\nolimits} R{A_e} + \frac{1}{2}{h^2}A_e^{\mathop{\rm T}\nolimits} S{A_e}\\ - \Psi _{}^{\rm T}\widehat W_{m}^{\rm T}\Omega_m(R) \widehat W_m\Psi - \bar \Psi _{}^{\rm T}\widehat{\bar W}_{m}^{\rm T}\bar \Omega_m(S) \widehat{\bar W}_m\bar \Psi \\ \end{array} \right]\xi (t) \\ & \lt 0 \\ \end{aligned} \end{equation}$ $

(3.16)

Recalling that (2.34) and (2.55), following inequalities are employed to yield the upper bound of $ \dot V(t) $

$\begin{equation} h\int_{t - h}^t {{{\dot x}^{\rm T}}(\tau )R\dot x(\tau ){\mathop{\rm d}\nolimits} \tau } \ge \varpi _{}^{\rm T}\Omega_m(R) \varpi = \xi _{}^{\rm T}(t)\left( {\Psi _{}^{\rm T}\widehat W_{m}^{\rm T}\Omega_m(R) \widehat W_m\Psi } \right)\xi (t) \end{equation}$ $

(3.17)

$\begin{equation} \int_{t - h}^t {\int_\theta ^t {{{\dot x}^{\mathop{\rm T}\nolimits} }(\tau )S\dot x(\tau ){\mathop{\rm d}\nolimits} \tau {\mathop{\rm d}\nolimits} \theta } } \ge \bar \nu _{}^{\rm T}\bar \Omega_m(S) \bar \nu = \xi _{}^{\rm T}(t)\left( {\bar \Psi _{}^{\rm T}\widehat{\bar W}_{m}^{\rm T}\bar \Omega_m(S) \widehat{\bar W}_m\bar \Psi } \right)\xi (t) \end{equation}$ $

(3.18)

This complete the proof.

3.2. Examples

Example 1: We consider the well-known delay dependent stable system (3.1) with following coefficient matrices as given in ^[29]:

$ A = \left[ {

$\begin{array}{*{20}{c}} { - 2} & 0 \\ 0 & { - 0.9} \\ \end{array}$ } \right]{, _{}} \quad {A_h} = \left[ {

$\begin{array}{*{20}{c}} { - 1} & 0 \\ { - 1} & { - 1} \\ \end{array}$ } \right] $

Using delay sweeping techniques the maximum allowable delay $ {h_{\max }} = 6.1725 $ can be obtained. Also many recent papers provide different results using Jensen inequality, Wirtinger-based inequality, and so on. The allowable maximum delays are shown in Table 1. We observe that the upper bounds obtained by our proposed inequalities are significantly better than those in other literatures.

Table 1. The maximum allowable delay.

Theorems	$ h_{\text{max}} $	Number of variables
Sun et al. (2010)^[24]	4.47	$ 1.5{n^2} + 1.5n $
Park, Ko, and Jeong (2011)^[28]	5.02	$ 18{n^2} + 18n $
Ariba, Gouaisbaut, and Johansson (2010)^[42]	5.12	$ 7{n^2} + 4n $
Seuret and Gouaisbaut (2013)^[18]	6.059	$ 3{n^2} + 2n $
Hien and Trinh (2015)^[43]	6.16	$ 19.5n^2+4.5n $
Liu and Seuret (2017) Theorem 1^[38]	6.1664	$ 79.5{n^2} + 4.5n $
Theorem 1 (m=0)	4.472	$ 1.5{n^2} + 1.5n $
Theorem 1 (m=1)	6.059	$ 3.5{n^2} + 2.5n $
Theorem 1 (m=2)	6.167	$ 6{n^2} + 3n $
Theorem 1 (m=3)	6.1719	$ 9.5{n^2} + 3.5n $
Theorem 1 (m=4)	6.1725	$ 14{n^2} + 4n $

| Show Table

DownLoad: CSV

Example 2: We consider the dynamics of machining chatter with following coefficient matrices as firstly studied in ^[36]:

$ A = \left[ {

$\begin{array}{*{20}{c}} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ { - 10 - K} & {10} & 0 & 0 \\ 5 & { - 15} & 0 & { - 0.25} \\ \end{array}$ } \right]{, _{}} \quad {A_h} = \left[ {

$\begin{array}{*{20}{c}} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ { - K} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array}$ } \right] $

where $ K $ denotes a parameter.

It's obviously that the system is stable with $ K $ less than some upper bound. Here we try to the upper bound in various delays. It's shown that Lemma 1 and Lamme 2 yield more stability region than those derived from Jensen and Wirtinger-based Lemma, as illustrated in Figure 1. When the parameter $ K \le 0.295 $, the system is still stable even the delay is very large, such as $ h = 500 $.

Figure 1. Allowable upper $ K $ with variable delay $ h $.

DownLoad: Full-Size Img PowerPoint

4. Applications to systems with time-varying delays

4.1. Systems with time-varying delays

Let us consider the following system with interval time-varying delay:

$\begin{equation} \begin{array}{l} \dot x(t) = Ax(t) + {A_h}x(t - h(t)) \\ x(t) = \varphi (t){, _{}} \quad t \in [ - {h_2}{, _{}}0] \\ \end{array} \end{equation}$ $

(4.1)

where $ x(t) \in {\textbf{R}^n} $ denotes the state vector of the system with $ n $ dimensions, $ A $ and $ A_h $ are real known constant matrices with appropriate dimensions, the continuously differentiable functions $ h(t) $ and $ \varphi (t) $ denote the system's time-varying delay and the initial condition, respectively.

Assumption 1: The delay function $ h(t) $ and its differential $ \dot h(t) $ both have finite bounds, i.e., there exist scales $ {h_2} \ge {h_1} > 0 $ and $ {\mu _1} \le {\mu _2} \le 1 $ such that

$\begin{equation} \left\{ \begin{array}{l} 0 \lt {h_1} \le h(t) \le {h_2} \\ {\mu _1} \le \dot h(t) \le {\mu _2} \le 1 \\ \end{array} \right. \end{equation}$ $

(4.2)

Theorem 2: The system (4.1) is asymptotically stable if there exist matrices $ P > 0 $, $ Q_1 > 0 $, $ Q_2 > 0 $, $ Q_3 > 0 $, $ R_1 > 0 $, $ R_2 > 0 $, $ R_3 > 0 $, and $ S_1 > 0 $, $ S_2 > 0 $, $ S_3 > 0 $ such that the following conditions hold^[41]:

$\begin{equation} \Phi = \left[ \begin{array}{l} {B_2^{\rm T}}PC_2 + {C_2^{\rm T}}PB_2 + e_1^{\mathop{\rm T}\nolimits} ({Q_1} + {Q_2} + {Q_3}){e_1} - e_3^{\mathop{\rm T}\nolimits} {Q_1}{e_3} - e_4^{\mathop{\rm T}\nolimits} {Q_2}{e_4} - (1 - {\mu _2})e_2^{\mathop{\rm T}\nolimits} {Q_3}{e_2} \\ + {h_1}A_e^{\mathop{\rm T}\nolimits} {R_1}{A_e} - \frac{1}{{{h_1}}}\Psi _1^{\rm T}{\widehat W_m^{\mathop{\rm T}\nolimits} }{\Omega _1(R_1)}\widehat W_m{\Psi _1} + {h_2}A_e^{\mathop{\rm T}\nolimits} {R_2}{A_e} - \frac{1}{{{h_2}}}\Psi _2^{\rm T}{\widehat W_m^{\mathop{\rm T}\nolimits} }{\Omega _2(R_2)}\widehat W_m{\Psi _2} \\ + {h_2}A_e^{\mathop{\rm T}\nolimits} {R_3}{A_e} - \frac{{(1 - {\mu _2})}}{{{h_2}}}\Psi _3^{\rm T}{\widehat W_m^{\mathop{\rm T}\nolimits} }{\Omega _3(R_3)}\widehat W_m{\Psi _3} + \frac{{h_1^2}}{2}A_e^{\mathop{\rm T}\nolimits} {S_1}{A_e} - \bar \Psi _1^{\rm T}{\widehat{\bar W}_m^{\mathop{\rm T}\nolimits} }{{\bar \Omega }_1(S_1)}\widehat{\bar W}_m{{\bar \Psi }_1} \\ + \frac{{h_2^2}}{2}A_e^{\mathop{\rm T}\nolimits} {S_2}{A_e} - \bar \Psi _2^{\rm T}{\widehat{\bar W}_m^{\mathop{\rm T}\nolimits} }{{\bar \Omega }_2(S_2)}\widehat{\bar W}_m{{\bar \Psi }_2} + \frac{{h_2^2}}{2}A_e^{\mathop{\rm T}\nolimits} {S_3}{A_e} - (1 - {\mu _2})\bar \Psi _3^{\rm T}{\widehat{\bar W}_m^{\mathop{\rm T}\nolimits} }{{\bar \Omega }_3(S_3)}\widehat{\bar W}_m{{\bar \Psi }_3} \\ \end{array} \right] \lt 0 \end{equation}$ $

(4.3)

where the notations in (4.2) are intermediate variables that defined properly in previous and in the process of proof, which can be found as $ B_2 $ in (4.8), $ C_2 $ in (4.11), $ e_1, e_2, e_3, e_4 $ in (3.10), $ h_1, h_2 $ in (4.6), $ \mu_2 $ in (4.16), $ A_e $ in (3.11), $ \Psi_1, \Psi_2, \Psi_3 $ in (4.14), $ \bar \Psi_1, \bar \Psi_2, \bar \Psi_3 $ in (4.14), $ W_m $ in (2.7), $ \widehat W_m $ in (2.29), $ \widehat {\bar{W}}_m $ in (3.7), $ \Omega_1, \Omega_2, \Omega_3 $ in (4.13), $ \bar \Omega_m $ in (3.18), $ \bar \Omega_1, \bar \Omega_2, \bar \Omega_3 $ in (4.13).

Proof: If the delay $ h $ is varying with time $ t $, then we can develop from (3.3)

$\begin{equation} \begin{aligned} \frac{{\mathop{\rm d}\nolimits} }{{{\mathop{\rm d}\nolimits} t}}{y_k}(t) & = \frac{{\partial {y_k}(t)}}{{\partial t}} + \frac{{\partial {y_k}(t)}}{{\partial h}}\frac{{\partial h}}{{\partial t}} \\ & = \dot x(t) - \frac{k}{h}{y_{k - 1}}(t) - \frac{{k\dot h}}{h}\left( {{y_k}(t) - {y_{k - 1}}(t)} \right) \\ & = \dot x(t) - \frac{{(1 - \dot h)k}}{h}{y_{k - 1}}(t) - \frac{{\dot hk}}{h}{y_k}(t) \\ \end{aligned} \end{equation}$ $

(4.4)

$\begin{equation} \begin{aligned} \frac{{\mathop{\rm d}\nolimits} }{{{\mathop{\rm d}\nolimits} t}}{y_1}(t) & = \dot x(t) - \frac{{(1 - \dot h)}}{h}{y_0}(t) - \frac{{\dot hk}}{h}{y_1}(t) \\ & = \left[ {A - \frac{{(1 - \dot h)}}{h}I} \right]x(t) + \left[ {{A_h} + \frac{{(1 - \dot h)}}{h}I} \right]x(t - h) - \frac{{\dot hk}}{h}{y_1}(t) \\ \end{aligned} \end{equation}$ $

(4.5)

If $ h = {h_1} $ or $ h = {h_2} $ is a constant variable, (3.3) yields

$\begin{equation} \begin{aligned} \frac{{\mathop{\rm d}\nolimits} }{{{\mathop{\rm d}\nolimits} t}}{{\hat y}_k}({h_i}, t) & = \dot x(t) - \frac{k}{{{h_i}}}{{\hat y}_{k - 1}}({h_i}, t) \\ & = Ax(t) + {A_h}(t - h) - \frac{k}{{{h_i}}}{{\hat y}_{k - 1}}({h_i}, t) \\ \frac{{\mathop{\rm d}\nolimits} }{{{\mathop{\rm d}\nolimits} t}}{{\hat y}_1}({h_i}, t) & = \dot x(t) - \frac{1}{{{h_i}}}{{\hat y}_0}({h_i}, t) \\ & = \left( {A - \frac{1}{{{h_i}}}I} \right)x(t) + {A_h}x(t - h) + \frac{1}{{{h_i}}}x(t - {h_i}) \\ (i = 1{, _{}}2) \\ \end{aligned} \end{equation}$ $

(4.6)

We introduce the following extra-states $ {\hat \chi _m}(t) $ and $ {\hat \xi _m}(t) $ as follows

$\begin{equation} {\hat \chi _m}(t) = \left[ {\begin{array}{*{20}{c}} {x(t)} \\ {\begin{array}{*{20}{c}} {\left[ {\begin{array}{*{20}{c}} {{y_1}(t)} \\ \vdots \\ {{y_m}(t)} \\ \end{array}} \right]} \\ {\left[ {\begin{array}{*{20}{c}} {{{\hat y}_1}({h_1}, t)} \\ \vdots \\ {{{\hat y}_m}({h_1}, t)} \\ \end{array}} \right]} \\ {\left[ {\begin{array}{*{20}{c}} {{{\hat y}_1}({h_2}, t)} \\ \vdots \\ {{{\hat y}_m}({h_2}, t)} \\ \end{array}} \right]} \\ \end{array}} \\ \end{array}} \right]{, _{}}{ \quad }{\hat \xi _m}(t) = \left[ {\begin{array}{*{20}{c}} {\left[ {\begin{array}{*{20}{c}} {x(t)} \\ {x(t - h)} \\ {x(t - {h_1})} \\ {x(t - {h_2})} \\ \end{array}} \right]} \\ {\begin{array}{*{20}{c}} {\left[ {\begin{array}{*{20}{c}} {{y_1}(t)} \\ \vdots \\ {{y_m}(t)} \\ \end{array}} \right]} \\ {\left[ {\begin{array}{*{20}{c}} {{{\hat y}_1}({h_1}, t)} \\ \vdots \\ {{{\hat y}_m}({h_1}, t)} \\ \end{array}} \right]} \\ {\left[ {\begin{array}{*{20}{c}} {{{\hat y}_1}({h_2}, t)} \\ \vdots \\ {{{\hat y}_m}({h_2}, t)} \\ \end{array}} \right]} \\ \end{array}} \\ \end{array}} \right] \end{equation}$ $

(4.7)

The extra-states can be expressed by $ {\hat \xi _m}(t) $

$\begin{equation} {\hat \chi _m}(t) = {B_2}{\hat \xi _m}(t) \end{equation}$ $

(4.8)

where

$\begin{equation} {B_2}(h) = \left[ {\begin{array}{*{20}{c}} {\left[ {\begin{array}{*{20}{c}} {{I_n}} & {{0_n}} & {{0_n}} & {{0_n}} \\ \end{array}} \right]} & {} & {} & {} \\ {} & {{I_{nm}}} & {} & {} \\ {} & {} & {{I_{nm}}} & {} \\ {} & {} & {} & {{I_{nm}}} \\ \end{array}} \right] \end{equation}$ $

(4.9)

And the system (3.1) can be rewritten as

$\begin{equation} \dot x = {A_e}{\hat \xi _m}(t) \end{equation}$ $

(4.10)

where $ {A_e} = \left[{ $\begin{array}{*{20}{c}} A & {{A_h}} & {{0_{n \times (nm + 2n)}}} \end{array}$ } \right] $

The time derivative of $ \hat \chi_m (t) $ can be obtained as follows

$\begin{equation} \dot {\hat \chi}_m (t) = C_2(h, \dot h){\hat \xi _m}(t) \end{equation}$ $

(4.11)

where

$ C_2(h, \dot h) = \left[ {

$\begin{array}{*{20}{c}} {\left[ {\begin{array}{*{20}{c}} A & {{A_d}} & {{0_n}} & {{0_n}} \\ \end{array}} \right]} & {} & {} & {} \\ {{M_0}} & { - \frac{{(1 - \dot h)}}{h}\Lambda - \frac{{\dot h}}{h}\Gamma } & {} & {} \\ {{M_1}} & {} & { - \frac{1}{{{h_1}}}\Lambda } & {} \\ {{M_2}} & {} & {} & { - \frac{1}{{{h_2}}}\Lambda } \\ \end{array}$ } \right] $

where

$\begin{array}{l} \Lambda = \left[ {\begin{array}{*{20}{c}} 0 & {} & {} & {} & {} \\ {2I} & 0 & {} & {} & {} \\ {} & {3I} & 0 & {} & {} \\ {} & {} & \ddots & \ddots & {} \\ {} & {} & {} & {mI} & 0 \\ \end{array}} \right]{, _{}}{ \quad }\Gamma = \left[ {\begin{array}{*{20}{c}} I & {} & {} & {} & {} \\ {} & {2I} & {} & {} & {} \\ {} & {} & {3I} & {} & {} \\ {} & {} & {} & \ddots & {} \\ {} & {} & {} & {} & {mI} \\ \end{array}} \right] \\\\ {M_0} = \left[ {\begin{array}{*{20}{c}} {A - \frac{{(1 - \dot h)}}{h}I} & {{A_h} + \frac{{(1 - \dot h)}}{h}I} & {{0_n}} & {{0_n}} \\ A & {{A_h}} & {{0_n}} & {{0_n}} \\ \vdots & \vdots & \vdots & \vdots \\ A & {{A_h}} & {{0_n}} & {{0_n}} \\ \end{array}} \right] \\ \end{array}$ $

$ {M_1} = \left[ {

$\begin{array}{*{20}{c}} {A - \frac{1}{{{h_1}}}I} & {{A_h}} & {\frac{1}{{{h_1}}}{I_n}} & {{0_n}} \\ A & {{A_h}} & {{0_n}} & {{0_n}} \\ \vdots & \vdots & \vdots & \vdots \\ A & {{A_h}} & {{0_n}} & {{0_n}} \\ \end{array}$ } \right]{, _{}}{ \quad }{M_2} = \left[ {

$\begin{array}{*{20}{c}} {A - \frac{1}{{{h_2}}}I} & {{A_h}} & {{0_n}} & {\frac{1}{{{h_2}}}{I_n}} \\ A & {{A_h}} & {{0_n}} & {{0_n}} \\ \vdots & \vdots & \vdots & \vdots \\ A & {{A_h}} & {{0_n}} & {{0_n}} \\ \end{array}$ } \right] \\ $

In order to analysis the stability of the system (4.1), we consider the following Lyapunov-Krasovskii functional (LKF) candidates

$\begin{equation} V(t) = \sum\limits_{k = 1}^{10} {{V_k}(t)} \end{equation}$ $

(4.12)

where

$\begin{aligned} {V_1}(t) & = \hat \chi {(t)^{\rm T}}P\hat \chi (t) \\ {V_2}(t) & = \int_{t - {h_1}}^t {{x^{\rm T}}(s){Q_1}x(s){\mathop{\rm d}\nolimits} s} \\ {V_3}(t) & = \int_{t - {h_2}}^t {{x^{\rm T}}(s){Q_2}x(s){\mathop{\rm d}\nolimits} s} \\ {V_4}(t) & = \int_{t - h(t)}^t {{x^{\rm T}}(s){Q_3}x(s){\mathop{\rm d}\nolimits} s} \\ {V_5}(t) & = \int_{t - {h_1}}^t {\int_s^t {{{\dot x}^{\rm T}}(u){R_1}\dot x(u)} {\mathop{\rm d}\nolimits} u{\mathop{\rm d}\nolimits} s} \\ {V_6}(t) & = \int_{t - {h_2}}^t {\int_s^t {{{\dot x}^{\rm T}}(u){R_2}\dot x(u)} {\mathop{\rm d}\nolimits} u{\mathop{\rm d}\nolimits} s} \\ {V_7}(t) & = \int_{t - h(t)}^t {\int_s^t {{{\dot x}^{\rm T}}(u){R_3}\dot x(u)} {\mathop{\rm d}\nolimits} u{\mathop{\rm d}\nolimits} s} \\ {V_8}(t) & = \int_{t - {h_1}}^t {\int_\theta ^t {\int_s^t {{{\dot x}^{\rm T}}(u){S_1}\dot x(u)} {\mathop{\rm d}\nolimits} u{\mathop{\rm d}\nolimits} s{\mathop{\rm d}\nolimits} \theta } } \\ {V_9}(t) & = \int_{t - {h_2}}^t {\int_\theta ^t {\int_s^t {{{\dot x}^{\rm T}}(u){S_2}\dot x(u)} {\mathop{\rm d}\nolimits} u{\mathop{\rm d}\nolimits} s{\mathop{\rm d}\nolimits} \theta } } \\ {V_{10}}(t) & = \int_{t - h(t)}^t {\int_\theta ^t {\int_s^t {{{\dot x}^{\rm T}}(u){S_3}\dot x(u)} {\mathop{\rm d}\nolimits} u{\mathop{\rm d}\nolimits} s{\mathop{\rm d}\nolimits} \theta } } \\ \end{aligned}$ $

Taking the time derivative of $ {V_k}(t) $ yields

$\begin{equation} \begin{aligned} {{\dot V}_1}(t) & = \hat \chi_m {{(t)}^{\rm T}}P\dot {\hat \chi}_m (t) + \dot {\hat \chi}_m {{(t)}^{\rm T}}P\hat \chi_m (t) \\ & = {{\hat \xi_m }^{\rm T}}(t)\left( {{B^{\rm T}}PC + {C^{\rm T}}PB} \right)\hat \xi_m (t) \\ {{\dot V}_2}(t) & = {x^T}(t){Q_1}x(t) - {x^{\rm T}}(t - {h_1}){Q_1}x(t - {h_1}) \\ & = {{\hat \xi_m }^{\rm T}}(t)\left( {e_1^{\mathop{\rm T}\nolimits} {Q_1}{e_1} - e_3^{\mathop{\rm T}\nolimits} {Q_1}{e_3}} \right)\hat \xi_m (t) \\ {{\dot V}_3}(t) & = {x^{\rm T}}(t){Q_2}x(t) - {x^{\rm T}}(t - {h_2}){Q_2}x(t - {h_2}) \\ & = {{\hat \xi_m }^{\rm T}}(t)\left( {e_1^{\mathop{\rm T}\nolimits} {Q_2}{e_1} - e_4^{\mathop{\rm T}\nolimits} {Q_2}{e_4}} \right)\hat \xi_m (t) \\ {{\dot V}_4}(t) & = {x^{\rm T}}(t){Q_3}x(t) - (1 - \dot h){x^{\rm T}}(t - h){Q_3}x(t - h) \\ & = {{\hat \xi_m }^{\rm T}}(t)\left[ {e_1^{\mathop{\rm T}\nolimits} {Q_3}{e_1} - (1 - \dot h)e_2^{\mathop{\rm T}\nolimits} {Q_3}{e_2}} \right]\hat \xi_m (t) \\ {{\dot V}_5}(t) & = {h_1}{{\dot x}^{\rm T}}(t){R_1}\dot x(t) - \int_{t - {h_1}}^t {{{\dot x}^{\rm T}}(s){R_1}\dot x(s){\mathop{\rm d}\nolimits} s} \\ &\le {{\hat \xi_m }^{\rm T}}(t)\left( {{h_1}A_e^{\mathop{\rm T}\nolimits} {R_1}{A_e} - \frac{1}{{{h_1}}}\Psi _1^{\rm T}{{\widehat W_m}^{\mathop{\rm T}\nolimits} }{\Omega _1(R_1)}\widehat W_m{\Psi _1}} \right)\hat \xi_m (t) \\ {{\dot V}_6}(t) & = {h_2}{{\dot x}^{\rm T}}(t){R_2}\dot x(t) - \int_{t - {h_2}}^t {{{\dot x}^{\rm T}}(s){R_2}\dot x(s){\mathop{\rm d}\nolimits} s} \\ &\le {{\hat \xi_m }^{\rm T}}(t)\left( {{h_2}A_e^{\mathop{\rm T}\nolimits} {R_2}{A_e} - \frac{1}{{{h_2}}}\Psi _2^{\rm T}{{\widehat W_m}^{\mathop{\rm T}\nolimits} }{\Omega _2(R_2)}\widehat W_m{\Psi _2}} \right)\hat \xi_m (t) \\ {{\dot V}_7}(t) & = h(t){{\dot x}^{\rm T}}(t){R_3}\dot x(t) - (1 - \dot h)\int_{t - h(t)}^t {{{\dot x}^{\rm T}}(s){R_3}\dot x(s){\mathop{\rm d}\nolimits} s} \\ &\le {{\hat \xi_m }^{\rm T}}(t)\left( {hA_e^{\mathop{\rm T}\nolimits} {R_3}{A_e} - \frac{{1 - \dot h}}{h}\Psi _3^{\rm T}{{\widehat W_m}^{\mathop{\rm T}\nolimits} }{\Omega _3(R_3)}\widehat W_m{\Psi _3}} \right)\hat \xi_m (t) \\ {{\dot V}_8}(t) & = \frac{{h_1^2}}{2}{{\dot x}^{\rm T}}(t){S_1}\dot x(t) - \int_{t - {h_1}}^t {\int_s^t {{{\dot x}^{\rm T}}(u){S_1}\dot x(u)} {\mathop{\rm d}\nolimits} u{\mathop{\rm d}\nolimits} s} \\ &\le {{\hat \xi_m }^{\rm T}}(t)\left( {\frac{{h_1^2}}{2}A_e^{\mathop{\rm T}\nolimits} {S_1}{A_e} - \bar \Psi _1^{\rm T}{{\widehat{\bar W}_m}^{\mathop{\rm T}\nolimits} }{{\bar \Omega }_1(S_1)}\widehat{\bar W}_m{{\bar \Psi }_1}} \right)\hat \xi_m (t) \\ {{\dot V}_9}(t) & = \frac{{h_2^2}}{2}{{\dot x}^{\rm T}}(t){S_2}\dot x(t) - \int_{t - {h_2}}^t {\int_s^t {{{\dot x}^{\rm T}}(u){S_2}\dot x(u)} {\mathop{\rm d}\nolimits} u{\mathop{\rm d}\nolimits} s} \\ &\le {{\hat \xi_m }^{\rm T}}(t)\left( {\frac{{h_2^2}}{2}A_e^{\mathop{\rm T}\nolimits} {S_2}{A_e} - \bar \Psi _2^{\rm T}{{\widehat{\bar W}_m}^{\mathop{\rm T}\nolimits} }{{\bar \Omega }_2(S_2)}\widehat{\bar W}_m{{\bar \Psi }_2}} \right)\hat \xi_m (t) \\ {{\dot V}_{10}}(t) & = \frac{{h_{}^2}}{2}{{\dot x}^{\rm T}}(t){S_3}\dot x(t) - (1 - \dot h)\int_{t - {h_1}}^t {\int_s^t {{{\dot x}^{\rm T}}(u){S_1}\dot x(u)} {\mathop{\rm d}\nolimits} u{\mathop{\rm d}\nolimits} s} \\ &\le {{\hat \xi_m }^{\rm T}}(t)\left( {\frac{{{h^2}}}{2}A_e^{\mathop{\rm T}\nolimits} {S_3}{A_e} - (1 - \dot h)\bar \Psi _3^{\rm T}{{\widehat{\bar W}_m}^{\mathop{\rm T}\nolimits} }{{\bar \Omega }_3(S_3)}\widehat{\bar W}_m{{\bar \Psi }_3}} \right)\hat \xi_m (t) \\ \end{aligned} \end{equation}$ $

(4.13)

where

$\begin{equation} \begin{array}{l} {\Psi _1} = \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{I_n}} & {{0_n}} & { - {I_n}} & {{0_n}} & {{0_{nm}}} \\ \end{array}} \\ {{{\bar \Psi }_1}} \\ \end{array}} \right]{, _{}} \quad {{\bar \Psi }_1} = \left[ {\begin{array}{*{20}{c}} {{0_{nm \times 4n}}} & {{0_{nm}}} & {{I_{nm}}} & {{0_{nm}}} \\ \end{array}} \right] \\ {\Psi _2} = \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{I_n}} & {{0_n}} & {{0_n}} & { - {I_n}} & {{0_{nm}}} \\ \end{array}} \\ {{{\bar \Psi }_2}} \\ \end{array}} \right]{, _{}} \quad {{\bar \Psi }_2} = \left[ {\begin{array}{*{20}{c}} {{0_{nm \times 4n}}} & {{0_{nm}}} & {{0_{nm}}} & {{I_{nm}}} \\ \end{array}} \right] \\ {\Psi _3} = \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{I_n}} & { - {I_n}} & {{0_n}} & {{0_n}} & {{0_{nm}}} \\ \end{array}} \\ {{{\bar \Psi }_3}} \\ \end{array}} \right]{, _{}} \quad {{\bar \Psi }_3} = \left[ {\begin{array}{*{20}{c}} {{0_{nm \times 4n}}} & {{I_{nm}}} & {{0_{nm}}} & {{0_{nm}}} \\ \end{array}} \right] \\ \end{array} \end{equation}$ $

(4.14)

Thus the sum of $ {\dot V_k}(t) $, $ k = 1{, _{}}2{, ^{}} \cdots {, _{}}10 $ yields

$\begin{equation} \begin{aligned} \dot V(t) & = {\xi ^T}(t)\underbrace {\left[ \begin{array}{l} {B_2^{\rm T}}PC_2 + {C_2^{\rm T}}PB_2 + e_1^{\mathop{\rm T}\nolimits} ({Q_1} + {Q_2} + {Q_3}){e_1} - e_3^{\mathop{\rm T}\nolimits} {Q_1}{e_3} - e_4^{\mathop{\rm T}\nolimits} {Q_2}{e_4} - (1 - \dot h)e_2^{\mathop{\rm T}\nolimits} {Q_3}{e_2} \\ + {h_1}A_e^{\mathop{\rm T}\nolimits} {R_1}{A_e} - \frac{1}{{{h_1}}}\Psi _1^{\rm T}{\widehat W_m^{\mathop{\rm T}\nolimits} }{\Omega _1(R_1)}\widehat W_m{\Psi _1} + {h_2}A_e^{\mathop{\rm T}\nolimits} {R_2}{A_e} - \frac{1}{{{h_2}}}\Psi _2^{\rm T}{\widehat W_m^{\mathop{\rm T}\nolimits} }{\Omega _2(R_2)}\widehat W_m{\Psi _2} \\ + hA_e^{\mathop{\rm T}\nolimits} {R_3}{A_e} - \frac{{(1 - \dot h)}}{h}\Psi _3^{\rm T}{\widehat W_m^{\mathop{\rm T}\nolimits} }{\Omega _3(R_3)}\widehat W_m{\Psi _3} + \frac{{h_1^2}}{2}A_e^{\mathop{\rm T}\nolimits} {S_1}{A_e} - \bar \Psi _1^{\rm T}{\widehat{\bar W}_m^{\mathop{\rm T}\nolimits} }{{\bar \Omega }_1(S_1)}\widehat{\bar W}_m{{\bar \Psi }_1} \\ + \frac{{h_2^2}}{2}A_e^{\mathop{\rm T}\nolimits} {S_2}{A_e} - \bar \Psi _2^{\rm T}{\widehat{\bar W}_m^{\mathop{\rm T}\nolimits} }{{\bar \Omega }_2(S_2)}\widehat{\bar W}_m{{\bar \Psi }_2} + \frac{{h_{}^2}}{2}A_e^{\mathop{\rm T}\nolimits} {S_3}{A_e} - (1 - \dot h)\bar \Psi _3^{\rm T}{\widehat{\bar W}_m^{\mathop{\rm T}\nolimits} }{{\bar \Omega }_3(S_3)}\widehat{\bar W}_m{{\bar \Psi }_3} \\ \end{array} \right]}_{\Xi (h{, _{}}\dot h)}\xi (t) \\ & \lt 0 \end{aligned} \end{equation}$ $

(4.15)

Notice that $ \Xi (h{, _{}}\dot h) \le \Xi ({h_2}{, _{}}{\mu _2}) $ for all $ h \in [{h_1}{, _{}}{h_2}] $ and $ \dot h \in [{\mu _1}{, _{}}{\mu _2}] $, we can develop that $ \dot V(t) \le {\xi ^T}(t)\Phi \xi (t) < 0 $, where

$\begin{equation} \Phi = \Xi ({h_2}, {\mu _2}) = \left[ \begin{array}{l} {B_2^{\rm T}}PC_2 + {C_2^{\rm T}}PB_2 + e_1^{\mathop{\rm T}\nolimits} ({Q_1} + {Q_2} + {Q_3}){e_1} - e_3^{\mathop{\rm T}\nolimits} {Q_1}{e_3} - e_4^{\mathop{\rm T}\nolimits} {Q_2}{e_4} - (1 - {\mu _2})e_2^{\mathop{\rm T}\nolimits} {Q_3}{e_2} \\ + {h_1}A_e^{\mathop{\rm T}\nolimits} {R_1}{A_e} - \frac{1}{{{h_1}}}\Psi _1^{\rm T}{\widehat W_m^{\mathop{\rm T}\nolimits} }{\Omega _1(R_1)}\widehat W_m{\Psi _1} + {h_2}A_e^{\mathop{\rm T}\nolimits} {R_2}{A_e} - \frac{1}{{{h_2}}}\Psi _2^{\rm T}{\widehat W_m^{\mathop{\rm T}\nolimits} }{\Omega _2(R_2)}\widehat W_m{\Psi _2} \\ + {h_2}A_e^{\mathop{\rm T}\nolimits} {R_3}{A_e} - \frac{{(1 - {\mu _2})}}{{{h_2}}}\Psi _3^{\rm T}{\widehat W_m^{\mathop{\rm T}\nolimits} }{\Omega _3(R_3)}\widehat W_m{\Psi _3} + \frac{{h_1^2}}{2}A_e^{\mathop{\rm T}\nolimits} {S_1}{A_e} - \bar \Psi _1^{\rm T}{\widehat{\bar W}_m^{\mathop{\rm T}\nolimits} }{{\bar \Omega }_1(S_1)}\widehat{\bar W}_m{{\bar \Psi }_1} \\ + \frac{{h_2^2}}{2}A_e^{\mathop{\rm T}\nolimits} {S_2}{A_e} - \bar \Psi _2^{\rm T}{\widehat{\bar W}_m^{\mathop{\rm T}\nolimits} }{{\bar \Omega }_2(S_2)}\widehat{\bar W}_m{{\bar \Psi }_2} + \frac{{h_2^2}}{2}A_e^{\mathop{\rm T}\nolimits} {S_3}{A_e} - (1 - {\mu _2})\bar \Psi _3^{\rm T}{\widehat{\bar W}_m^{\mathop{\rm T}\nolimits} }{{\bar \Omega }_3(S_3)}\widehat{\bar W}_m{{\bar \Psi }_3} \\ \end{array} \right] \end{equation}$ $

(4.16)

This complete the proof.

4.2. Examples

Example 1: We also consider the well-known delay dependent stable system (4.1) with following coefficient matrices as given in ^[29]:

$\begin{equation} A = \left[ {\begin{array}{*{20}{c}} { - 2} & 0 \\ 0 & { - 0.9} \\ \end{array}} \right]{, _{}} \quad {A_h} = \left[ {\begin{array}{*{20}{c}} { - 1} & 0 \\ { - 1} & { - 1} \\ \end{array}} \right] \end{equation}$ $

(4.17)

The delay rate bounds $ {\mu _1} = - \mu $, $ {\mu _2} = \mu $. We herein calculate the allowable upper bound $ {h_2} $ for various delay rate $ \mu $ via Theorem 2, as illustrate in Figure 2. It's shown that $ {h_2} $ deceases continuously with delay rate $ \mu $ growing.

Figure 2. Allowable upper $ {h_2} $ with variable delay $ \mu $.

DownLoad: Full-Size Img PowerPoint

The allowable upper bounds $ {h_2} $ varying with given $ \mu $ are shown in Table 2. We observe that the upper bounds obtained by Theorem 2 are significantly better than others. Theorem 1 provides the least conservative results.

Table 2. Allowable upper bound $ {h_2} $ for different $ \mu $ (example 1).

	$ \mu $
Methods	0.1	0.2	0.5	0.8	Number of variables
Fridman and Uri (2002)^[44]	3.604	3.033	2.008	1.364	$ 5.5{n^2} + 1.5n $
He et al. (2007)^[16]	3.605	3.039	2.043	1.492	$ 3{n^2} + 3n $
Park and Ko (2007)^[45]	3.658	3.163	2.337	1.934	$ 11.5{n^2} + 4.5n $
Ariba and Gouaisbaut (2009)^[13]	4.794	3.995	2.682	1.957	$ 22{n^2} + 8n $
Zeng et al. (2013) (N=2)^[17]	4.466	3.657	2.375	1.987	$ 11.5{n^2} + 3.5n $
Zeng et al. (2013) (N=3)^[17]	4.628	3.766	2.442	2.079	$ 17{n^2} + 5n $
Seuret and Gouaisbaut (2013)^[18]	4.703	3.834	2.420	2.137	$ 10{n^2} + 3n $
Zeng et al. (2015)^[33]	4.788	4.060	3.055	2.615	$ 65{n^2} + 11n $
Theorem 2 (m=2)	5.791	5.496	5.123	4.906	$ 14.5{n^2} + 4.5n $

| Show Table

DownLoad: CSV

For simulation, let the time-varying delay $ h(t) = 3 + 2\cos (0.25t) $, which means that $ {h_1} = 1 $, $ {h_2} = 5 $, $ {\mu _1} = - 0.5 $, and $ {\mu _2} = 0.5 $. The initial condition of the system is chosen as $ {\rm{x}}(0) = {[1{, _{}}-1]^{\mathop{\rm T}\nolimits} } $. The time history of system states is illustrated in Figure 3. As our expectation, both states asymptotically converge to zero despite the previous vibration.

Figure 3. Time history of system states.

DownLoad: Full-Size Img PowerPoint

Example 2: Consider the time-varying delay system (4.1) with the following parameters ^[33]:

$\begin{equation} A = \left[ {\begin{array}{*{20}{c}} 0 & 1 \\ { - 1} & { - 1} \\ \end{array}} \right]{, _{}} \quad {A_h} = \left[ {\begin{array}{*{20}{c}} 0 & 0 \\ 0 & { - 1} \\ \end{array}} \right] \end{equation}$ $

(4.18)

When the delay is constant ($ \mu = 0 $), the analytical upper bound can be obtain $ {h_{\max }} = \pi $. The improvement of our approach is shown in Table 3. It's verified that the advantage of Theorem 2 is over the results in other literatures.

Table 3. Allowable upper bound $ {h_2} $ for different $ \mu $ (example 2).

	$ \mu $
Methods	0.1	0.2	0.5	0.8	Number of variables
Park and Ko (2007)^[45]	1.99	1.81	1.75	1.61	$ 11.5{n^2} + 4.5n $
Kim (2011)^[46]	2.52	2.17	2.02	1.62	$ 49{n^2} + 3n $
Zeng et al. (2015)^[33]	3.03	2.57	2.41	1.93	$ 65{n^2} + 11n $
Theorem 2 (m=2)	3.136	3.04	2.95	2.90	$ 14.5{n^2} + 4.5n $

| Show Table

DownLoad: CSV

5. Conclusions

New single and double integral inequalities with arbitrary approximation order are developed through the use of shifted Legendre polynomials and Cholesky decomposition. These two inequalities encompass several former well-known integral inequities, such as Jensen inequality, Wirtinger-based inequality, auxiliary function-based integral inequalities, and bring new less-conservative stability criteria by employing proper Lyapunov-Krasovskii functionals. Several numerical examples have been provided which show large improvements compared to existing results in both constant and time-varying delay systems.

Acknowledgments

The authors would like to thank the anonymous reviewers for their constructive comments that have greatly improved the quality of this paper.

This work is supported by Shanghai Nature Science Fund under contract No. 19ZR1426800, Shanghai Jiao Tong University Global Strategic Partnership Fund (2019 SJTU-UoT), WF610561702, and Shanghai Jiao Tong University Young Teachers Initiation Programme, AF4130045.

Conflict of interest

All authors declare no conflicts of interest in this paper.

References

[1]	P. Antonelli and P. Marcati, A model of synchronization over quantum networks, J. Phys. A, 50 (2017), 315101, 19 pp.
[2]	R. Bellman, Stability Theory of Differential Equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953.
[3]	Magnus and Fer expansions for matrix differential equations: The convergence problem. J. Phys. A (1998) 31: 259-268.
[4]	The Magnus expansion and some of its applications. Phys. Rep. (2009) 470: 151-238.
[5]	S.-H. Choi, J. Cho and S.-Y. Ha, Practical quantum synchronization for the Schrödinger-Lohe system, J. Phys. A, 49 (2016), 205203, 17 pp.
[6]	S.-H. Choi and S.-Y. Ha, Quantum synchronization of the Schrödinger-Lohe model, J. Phys. A, 47 (2014), 355104, 16 pp.
[7]	Dynamical system approach to synchronization of the coupled Schrödinger-Lohe system. Quart. Appl. Math. (2017) 75: 555-579.
[8]	H. Huh, S.-Y. Ha and D. Kim, Emergent behaviors of the Schrödinger-Lohe model on cooperative-competitive networks, J. Differential Equations, 263 (2017), 8295–8321.
[9]	H. Huh, S.-Y. Ha and D. Kim, Asymptotic behavior and stability for the Schrödinger-Lohe model, J. Math. Phys., 59 (2018), 102701, 21 pp.
[10]	M. A. Lohe, Quantum synchronization over quantum networks, J. Phys. A, 43 (2010), 465301, 20 pp.
[11]	M. A. Lohe, Non-Abelian Kuramoto model and synchronization, J. Phys. A, 42 (2009), 395101, 25 pp.
[12]	On the exponential solution of differential equations for a linear operator. Comm. Pure Appl. Math. (1954) 7: 649-673.

This article has been cited by:

Julee Shahni, Randhir Singh, Numerical solution and error analysis of the Thomas–Fermi type equations with integral boundary conditions by the modified collocation techniques, 2024, 441, 03770427, 115701, 10.1016/j.cam.2023.115701

Reader Comments

Your name:*

Email:*
© 2019 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

Networks and Heterogeneous Media

1.2 1.8

Metrics

Article views(2217) PDF downloads(392) Cited by(0)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Networks and Heterogeneous Media

Remarks on the Schrödinger-Lohe model

Related Papers:

Abstract

1. Introduction

2. Shifted Legendre polynomials-based single and double integral inequalities

2.1. Shifted Legendre polynomials for single integral

2.2. Shifted Legendre polynomials for double integral

2.3. Shifted Legendre polynomials-based single integral inequality

2.4. Shifted Legendre polynomials-based double integral inequality

3. Applications to systems with constant delays

3.1. Systems with constant delays

3.2. Examples

4. Applications to systems with time-varying delays

4.1. Systems with time-varying delays

4.2. Examples

5. Conclusions

Acknowledgments

Conflict of interest

References

This article has been cited by:

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

Abstract

1. Introduction

2. Shifted Legendre polynomials-based single and double integral inequalities

2.1. Shifted Legendre polynomials for single integral

2.2. Shifted Legendre polynomials for double integral

2.3. Shifted Legendre polynomials-based single integral inequality

2.4. Shifted Legendre polynomials-based double integral inequality

3. Applications to systems with constant delays

3.1. Systems with constant delays

3.2. Examples

4. Applications to systems with time-varying delays

4.1. Systems with time-varying delays

4.2. Examples

5. Conclusions

Acknowledgments

Conflict of interest

References