Open Access

A graph-theoretic approach to global input-to-state stability for coupled control systems

Advances in Difference Equations20172017:129

DOI: 10.1186/s13662-017-1129-y

Received: 23 October 2016

Accepted: 6 March 2017

Published: 5 May 2017

Abstract

In this paper, the input-to-state stability for coupled control systems is investigated. A systematic method of constructing a global Lyapunov function for the coupled control systems is provided by combining graph theory and the Lyapunov method. Consequently, some novel global input-to-state stability principles are given. As an application to this result, a coupled Lurie system is also discussed. By constructing an appropriate Lyapunov function, a sufficient condition ensuring input-to-state stability of this coupled Lurie system is established. Two examples are provided to demonstrate the effectiveness of the theoretical results.

Keywords

input-to-state stability coupled control system Lyapunov function

1 Introduction

In recent years, coupled control systems (CCSs) have received considerable attention for their interesting characteristics from the mathematical point of view. The main interest has been focused on the investigation of the global dynamics of the systems, with a special emphasis on the study of stability. Meanwhile, input-to-state stability (ISS) for control systems has been extensively studied due to a wide range of applications in physics, biology, social science, neural networks, engineering fields, and artificial complex dynamical systems. For example, Sontag and Wang [1] showed the importance of the well-known Lyapunov sufficient condition for ISS and provided additional characterizations of the ISS property, including one in terms of nonlinear stability margins. Grüne [2] presented a new variant of the ISS property which is based on a one-dimensional dynamical system, showed the relation to the original ISS formulation, and described the characterizations by means of suitable Lyapunov functions. In [3], Angeli presented a framework for understanding such questions fully compatible with the well-known ISS approach and discussed applications of the newly introduced stability notions. In [4], Arcak and Teel analyzed ISS for the feedback interconnection of a linear block and a nonlinear element.

As far as we know, there are a lot of papers dealing with the ISS of individual control systems but few papers dealing with the ISS of CCSs. In general, the study of ISS for CCSs is complex, because it is very difficult to straightly construct an appropriate Lyapunov function for CCSs. However, in [5], Li and Shuai studied the global-stability problem of equilibrium and developed a systematic approach that allows one to construct global Lyapunov functions for large-scale coupled systems from building blocks of individual vertex systems. Later, this technique was appropriately developed and extended to some other coupled systems. In [68] several delayed coupled systems were discussed, and some sufficient conditions were obtained. Li et al. in [912] investigated the stochastic stability of coupled systems with both white noise and color noise. Moreover, by using this technique, Su et al. derived sufficient conditions ensuring global stability of discrete-time coupled systems [13, 14], and Zhang et al. extended this technique to multi-dispersal coupled systems [15]. Besides, this technique is also applied to many practical applications, such as biological systems [1618], neural networks [19, 20], and mechanical systems [2023]. Hence, the graph theory is a great method in the study of coupled systems.

Motivated by the above discussions, in this paper, we investigate the ISS of CCSs. A systematic method of constructing a global Lyapunov function for the CCSs is provided by combining graph theory and the Lyapunov method. Consequently, some novel global stability principles are given. As an application to this result, a coupled Lurie system is also discussed. By constructing an appropriate Lyapunov function, a sufficient condition ensuring the ISS of this coupled Lurie system is established. Finally, two examples and their numerical simulations are provided to demonstrate the effectiveness and correctness of the theoretical results.

The rest of the paper is organized as follows. In Section 2, some preliminaries and the problem description are presented. In Section 3, the main theorems and their rigorous proofs are described. Finally, in Section 4, an application to a coupled Lurie system is given, and the respective simulations are also given to demonstrate the effectiveness of our results.

2 Preliminaries and model formulation

Throughout the paper, unless otherwise specified, the following notations will be used. As we usually use, \(\mathbb{R}^{n}\) denotes the n-dimensional Euclidean space. Notations \(\mathbb{R}^{1}_{+}=[0,+\infty)\), \(\mathbb {Z}^{+}=\{1,2,\ldots\}\), \(\mathbb{L}=\{1,2,\ldots, l\}\), \(n=\sum_{i=1}^{l} n_{i}\), and \(m=\sum_{i=1}^{l} m_{i}\) for \(n_{i}, m_{i}\in \mathbb{Z}^{+}\) are used. For any \(x\in\mathbb{R}^{n}\), \(x^{\mathrm {T}}\) is its transpose and \(|x|\) is its Euclidean norm. Let \(\mathbb {R}^{n\times n}\) denote the set of \(n\times n\) real matrix space. For a matrix P, \(P\geq0\) (≤0) means that P is positive semi-definite (negative semi-definite). The symbol \(\psi_{1}\circ\psi_{2}\) stands for the composition of two functions \(\psi_{1}\) and \(\psi_{2}\). The gradient function of a function f is indicated by f. In an m-dimensional space, the symbol \(L_{\infty}^{m}\) indicates the set of all the functions which are endowed with essential supremum norm \(\|u\| =\sup\{|u(t)| \mid t\geq0\}\leq\infty\).

We recall some knowledge of graph theory that will be used in the rest of the paper. Define a weighted digraph \(\mathcal{G}=\{V,E,A\}\), in which set \(V=\{v_{1},v_{2},\ldots,v_{l}\}\) denotes l vertices of the graph, element \(e_{ij}\) of E denotes the arc leading from initial vertex j to terminal vertex i, and the element \(a_{ij}\) of a weighted adjacency matrix A denotes the weight of arc \(e_{ij}\). We denote \(a_{ij}>0\) if and only if there exists an arc from vertex i to vertex j in \(\mathcal{G}\), otherwise \(a_{ij}=0\), and we denote \(a_{ii}=0\) for all \(i\in\mathbb{L}\). Denote the digraph with weight matrix A as \((\mathcal{G},A)\). If a graph \(\mathcal{S}\) has the same vertex as \(\mathcal{G}\), we call it a subgraph of \(\mathcal{G}\). The weight \(W(\mathcal{S})\) of a subgraph \(\mathcal{S}\) is the product of the weights on all its arcs. If a connected subgraph has no cycle, it is a tree. We call \(v_{i}\) the root of the tree if vertex i of the tree is not a terminal vertex of any arcs and each of the remaining vertices is a terminal vertex of one arc. A subgraph Q is unicyclic when it is a disjoint union of rooted trees whose roots form a directed cycle. The Laplacian matrix of \(\mathcal{G}\) is defined as \(L=(b_{ij})_{l\times l}\), where \(b_{ij}=-a_{ij}\) for \(i\neq j\) and \(b_{ij}= \sum_{k\neq i}a_{ik}\) for \(i=j\).

The following lemma will be used in the proof of our main results.

Lemma 1

[5]

Assume \(l\geq2\). Then the following identity holds:
$$ \sum_{i,j=1}^{l}c_{i} a_{ij} F_{ij}(x_{i},x_{j})=\sum _{\mathcal{Q}\in \mathbb{Q}}W(\mathcal{Q})\sum_{(s,r)\in E(C_{\mathcal{Q}})}F_{rs}(x_{r}, x_{s}). $$
Here \(F_{ij} (x_{i},x_{j})\) are arbitrary functions for any \(1\leq i, j\leq l\), \(a_{ij}\) are elements of matrix A, \(\mathbb{Q}\) is the set of all spanning unicyclic graphs of \((\mathcal{G},A)\), \(W(\mathcal{Q})\) is the weight of \(\mathcal {Q}\), and \(C_{\mathcal{Q}}\) denotes the directed cycle of \(\mathcal {Q}\). And \(c_{i}\) denotes the cofactor of the ith diagonal element of L, in particular, if \((\mathcal{G},A)\) is strongly connected, then \(c_{i}>0\) for \(1\leq i \leq l\).

In the remainder of this section, we shall give the model formulation and state some definitions that will be used in the main results.

Given a digraph \((\mathcal{G},A)\) with l vertices (\(l\geq2\)) and \(A=(a_{ij})_{l\times l}\). A coupled control system can be constructed on \((\mathcal{G},A)\) by assigning each vertex its own dynamics and then coupling these vertex dynamics based on directed arcs in \((\mathcal{G},A)\). The details are as follows. Assume that the ith vertex dynamic is described by the control system
$$ \dot{x_{i}}(t)=f_{i}\bigl(x_{i}(t),u_{i} \bigr),\quad t\geq0, $$
where \(x_{i}\in\mathbb{R}^{n_{i}}\) denotes the value of vertex i, \(f_{i}:\mathbb{R}^{n_{i}}\times\mathbb{R}^{m_{i}}\rightarrow\mathbb {R}^{n_{i}}\) is continuously differentiable and satisfies \(f_{i}(0,0)=0\), function \(u_{i}:\mathbb{R}^{1}_{+}\rightarrow\mathbb {R}^{m_{i}}\) denotes the input of vertex i and it is measurable and locally essentially bounded. Assume that \(a_{ij}\geq0\) represents the effect factor from vertex j to vertex i and \(a_{ij} =0\) iff there exists no arc from j to i. Then we use function \(P_{ij}\) to describe the effect that subsystem j has on i and \(P_{ij}:\mathbb {R}^{n_{i}}\times\mathbb{R}^{n_{j}}\times\mathbb {R}^{m_{j}}\rightarrow\mathbb{R}^{n_{i}}\) is continuously differentiable and satisfies \(P_{ij}(0,0,0)=0\). For example, in a digraph with six vertices, we show the interaction in vertex j and vertex i (see Figure 1).
Figure 1

A coupled control system on a digraph with six vertices.

Then coupling the vertex systems together, we obtain the following coupled control system:
$$ \dot{x_{i}}(t)=f_{i}\bigl(x_{i}(t),u_{i} \bigr)+\sum^{l}_{j=1} a_{ij}P_{ij} \bigl(x_{i}(t),x_{j}(t),u_{j}\bigr), \quad i\in \mathbb{L},t\geq0. $$
(1)
Here we use \(x=(x_{1}^{\mathrm{T}},x_{2}^{\mathrm{T}},\ldots ,x_{l}^{\mathrm{T}})^{\mathrm{T}}\in\mathbb{R}^{n}\) to stand for the vector of state variables of (1), and denote by \(x(t)=x(t,x_{0},u)\) the solution of CCS (1) with initial state \(x_{0}=x(0)\) and input \(u=(u_{1}^{\mathrm {T}},u_{2}^{\mathrm{T}},\ldots,u_{l}^{\mathrm{T}})^{\mathrm{T}}\in L_{\infty}^{m}\).

To be more precise, we recall some definitions on the ISS of CCS (1). We refer to [1, 2] for definitions as follows.

Definition 1

A function \(\gamma:\mathbb{R}^{1}_{+}\rightarrow\mathbb{R}^{1}_{+}\) is a \(\mathcal{K}\)-function if it is continuous, strictly increasing, and \(\gamma(0)=0\). If a \(\mathcal{K}\)-function satisfies \(\gamma (s)\rightarrow\infty\) as \(s\rightarrow\infty\), we call it \(\mathcal{K}_{\infty}\)-function. A function \(\beta:\mathbb {R}^{1}_{+}\times\mathbb{R}^{1}_{+}\rightarrow\mathbb{R}^{1}_{+}\) is a \(\mathcal{K}\mathcal{\phi}\)-function if the function \(\beta(\cdot,t)\) is a \(\mathcal{K}\)-function for each fixed \(t\geq0\), and for each fixed \(s\geq0\), \(\beta(s,t)\) is decreasing to zero as \(t\rightarrow \infty\).

Definition 2

CCS (1) is called ISS if there exist a \(\mathcal{K\phi}\)-function \(\beta:\mathbb{R}^{1}_{+}\times \mathbb{R}^{1}_{+}\rightarrow\mathbb{R}^{1}_{+}\) and a \(\mathcal {K}\)-function γ such that for each input \(u\in L_{\infty}^{m}\) and \(x_{0}\in\mathbb{R}^{n}\), it holds that
$$ \bigl\vert x(t,x_{0},u) \bigr\vert \leq\beta\bigl( \vert x_{0} \vert ,t\bigr)+\gamma\bigl( \Vert u \Vert \bigr). $$
(2)

In the proof of our main results, we need to find a global ISS-Lyapunov function for CCS (1). For the convenience of the proof, we now define vertex ISS-Lyapunov functions for CCS (1).

Definition 3

Set \(\{V_{i}(x_{i}), i\in\mathbb{L}\}\) is called a vertex ISS-Lyapunov function set for CCS (1) if every \(V_{i}(x_{i})\) is smooth and satisfies the following conditions:
  1. Q1.
    There exist positive constants \(\alpha_{i}\), \(\delta_{i}\), \(p\geq2\), such that
    $$ \alpha_{i}|x_{i}|^{p}\leq V_{i}(x_{i}) \leq\delta_{i}|x_{i}|^{p},\quad x_{i} \in\mathbb{R}^{n_{i}}. $$
     
  2. Q2.
    There exist constants \(\xi_{i},d_{ij}\geq0\), functions \(F_{ij} (x_{i},x_{j})\), and \(\mathcal{K}\)-function \(\chi_{i}\) such that for any \(x_{i}\in\mathbb{R}^{n_{i}}\) and \(\mu_{i}\in\mathbb {R}^{m_{i}}\) satisfying \(\sum^{l}_{i=1}c_{i}\delta_{i}|x_{i}|^{p}\geq \sum^{l}_{i=1}c_{i}\delta_{i}|\chi_{i}(|\mu_{i}|)|^{p}\), where \(D=(d_{ij})_{l\times l}\) and \(c_{i}\) is the cofactor of the ith diagonal element of Laplacian matrix of \((\mathcal{G},D)\). Then we have
    $$ \dot{V}_{i}\bigl(x_{i}(t)\bigr)\leq-\xi_{i} \bigl\vert x_{i}(t) \bigr\vert ^{p}+\sum ^{l}_{j=1}d_{ij}F_{ij} \bigl(x_{i}(t),x_{j}(t)\bigr), $$
    in which
    $$\dot{V}_{i}\bigl(x_{i}(t)\bigr)=\bigtriangledown V_{i}\bigl(x_{i}(t)\bigr) \Biggl[f_{i} \bigl(x_{i}(t),u_{i}\bigr)+\sum ^{l}_{j=1} a_{ij}P_{ij} \bigl(x_{i}(t),x_{j}(t),u_{j}\bigr) \Biggr]. $$
     
  3. Q3.
    Along each directed cycle \(C_{\mathcal{Q}}\) of weighted digraph \((\mathcal{G},D)\), there is
    $$ \sum_{(s,r)\in E(C_{\mathcal{Q}})} F_{rs}(x_{r},x_{s}) \leq0. $$
     

3 Main results

In this section, the ISS of CCS (1) will be investigated. The approaches used in the proof of the main results are motivated by [1, 5].

Theorem 1

If CCS (1) admits a vertex ISS-Lyapunov function set \(\{ V_{i}(x_{i}), i\in\mathbb{L}\}\), and digraph \((\mathcal{G},D)\) is strongly connected, then the solution of CCS (1) is ISS.

Proof

In order to prove the conclusion, we need to find a \(\mathcal {K}\)-function \(\gamma(\cdot)\) and a \(\mathcal{K}\mathcal{\phi}\) function \(\beta(\cdot,\cdot)\) satisfying
$$ \bigl\vert x(t,x_{0},u) \bigr\vert \leq\beta\bigl( \vert x_{0} \vert ,t\bigr)+\gamma\bigl( \Vert u \Vert \bigr) $$
(3)
for \(x_{0}\in\mathbb{R}^{n}\) and \(u\in\mathbb{R}^{m}\).
Let
$$ V(x)=\sum^{l}_{i=1} c_{i}V_{i}(x_{i}), $$
in which \(c_{i}\) is the cofactor of the ith diagonal element of Laplacian matrix of \((\mathcal{G},D)\).
Consider a set: \(S=\{\eta: V(\eta)\leq b\}\), where \(b=\sum^{l}_{i=1} c_{i}\delta_{i}|\chi_{i}(|u_{i}|)|^{p}\). We can assert that if there exists \(t_{0}\geq0\) making \(x(t_{0})\in S\), then \(x(t)\in S\) for all \(t\geq t_{0}\). Suppose that this is not true, then there exist \(t>t_{0}\) and \(\varepsilon>0\) such that \(V(x(t))>b+\varepsilon\). We observe from condition Q1 that
$$ \sum^{l}_{i=1}c_{i} \delta_{i} \bigl\vert \chi_{i}\bigl( \vert u_{i} \vert \bigr) \bigr\vert ^{p}\leq V\bigl(x(t) \bigr)\leq\sum^{l}_{i=1}c_{i} \delta_{i} \bigl\vert x_{i}(t) \bigr\vert ^{p}. $$
(4)
Let \(\tau=\inf\{t\geq t_{0}:V(x(t))\geq b+\varepsilon\}\). From conditions Q2 and Q3, we can obtain
$$\begin{aligned} \dot{V}\bigl(x(t)\bigr) =&\sum^{l}_{i=1}c_{i} \dot{V_{i}}\bigl(x_{i}(t)\bigr) \\ \leq& -\sum^{l}_{i=1}c_{i} \xi_{i} \bigl\vert x_{i}(t) \bigr\vert ^{p}+ \sum^{l}_{i,j=1}c_{i}d_{ij}F_{ij} \bigl(x_{i}(t),x_{j}(t)\bigr) \\ =&-\sum^{l}_{i=1}c_{i} \xi_{i} \bigl\vert x_{i}(t) \bigr\vert ^{p}+ \sum_{Q\in\textit {Q}}W(Q)\sum_{(s,r)\in E(C_{Q})} F_{rs}\bigl(x_{r}(t),x_{s}(t)\bigr) \\ \leq&-\sum^{l}_{i=1}c_{i} \xi_{i} \bigl\vert x_{i}(t) \bigr\vert ^{p} \\ \leq&0. \end{aligned}$$
Therefore, \(V(x(t))\geq V(x(\tau))\) for some t in \((t_{0},\tau)\). This contradicts the minimality of τ, and hence \(x(t)\in S\) for all \(t\geq t_{0}\).
Now let \(t_{1}=\inf\{t\geq0;x(t)\in S\}\leq\infty\), then it follows from the above argument that
$$ V\bigl(x(t)\bigr)\leq\sum^{l}_{i=1} c_{i}\delta_{i} \bigl\vert \chi _{i}\bigl( \vert u_{i} \vert \bigr) \bigr\vert ^{p},\quad t\geq t_{1}. $$
This implies that
$$ \sum^{l}_{i=1} c_{i} \alpha_{i} \bigl\vert x_{i}(t) \bigr\vert ^{p}\leq V\bigl(x(t)\bigr)\leq\sum^{l}_{i=1} c_{i}\delta_{i} \bigl\vert \chi_{i}\bigl( \vert u_{i} \vert \bigr) \bigr\vert ^{p}. $$
Denote
$$ \chi\bigl( \Vert u \Vert \bigr)= \Biggl(\frac{1}{\delta}\sum ^{l}_{i=1} c_{i}\delta_{i} \bigl\vert \chi_{i}\bigl( \Vert u \Vert \bigr) \bigr\vert ^{p} \Biggr)^{\frac{1}{p}}, $$
in which \(\delta>0\) is a certain constant. For simplicity, we write
$$ \alpha= \Biggl\{ \sum^{l}_{i=1} \alpha_{i} \Biggr\} ^{1-\frac{p}{2}} \Bigl(\min_{i\in\mathbb{L}} \{c_{i}\alpha_{i}\} \Bigr)^{\frac{p}{2}}, $$
then it is easy to see from condition Q1 that for \(t\geq t_{1}\),
$$\begin{aligned} V\bigl(x(t)\bigr) =&\sum^{l}_{i=1} c_{i}V_{i}\bigl(x_{i}(t)\bigr) \\ \leq&\sum^{l}_{i=1} c_{i} \delta_{i} \bigl\vert \chi_{i}\bigl( \Vert u_{i} \Vert \bigr) \bigr\vert ^{p} \\ \leq&\sum^{l}_{i=1} c_{i} \delta_{i} \bigl\vert \chi_{i}\bigl( \Vert u \Vert \bigr) \bigr\vert ^{p} \\ =&\delta \bigl\vert \chi\bigl( \Vert u \Vert \bigr) \bigr\vert ^{p} \end{aligned}$$
and
$$\begin{aligned} V\bigl(x(t)\bigr) =&\sum^{l}_{i=1} c_{i}V_{i}\bigl(x_{i}(t)\bigr) \\ \geq& \sum^{l}_{i=1} c_{i} \alpha_{i} \bigl\vert x_{i}(t) \bigr\vert ^{p} \\ =&\sum^{l}_{j=1} c_{j} \alpha_{j}\sum^{l}_{i=1} \biggl[ \frac{c_{i}\alpha _{i}}{\sum^{l}_{k=1} c_{k}\alpha_{k}} \bigl\vert x_{i}(t) \bigr\vert ^{\frac{p}{2}} \biggr] \\ \geq&\sum^{l}_{j=1} c_{j} \alpha_{j} \Biggl[\sum^{l}_{i=1} \frac {c_{i}\alpha_{i}}{\sum^{l}_{k=1} c_{k}\alpha_{k}} \bigl\vert x_{i}(t) \bigr\vert ^{2} \Biggr]^{\frac{p}{2}} \\ \geq& \Biggl\{ \sum^{l}_{i=1} \alpha_{i} \Biggr\} ^{1-\frac{p}{2}} \Bigl(\min_{i\in\mathbb{L}} \{c_{i}\alpha_{i}\} \Bigr)^{\frac{p}{2}} \bigl\vert x(t) \bigr\vert ^{p}. \end{aligned}$$
Hence \(\alpha|x(t)|^{p}\leq V(x(t))\leq\delta|\chi(\|u\|)|^{p}\).
Since the digraph is strongly connected, it implies that \(c_{i}>0\), and then \(\alpha>0\), \(\delta>0\). Thus if we let \(\gamma(\|u\|)=(\delta/\alpha)^{\frac{1}{p}}\chi(\|u\| )\), we can obtain
$$ \bigl\vert x(t) \bigr\vert \leq\gamma\bigl( \Vert u \Vert \bigr), \quad t\geq t_{1}. $$
(5)
For \(t< t_{1}\), we have \(x(t)\mathbin{\bar{\in}}S\), which implies that \(\sum^{l}_{i=1}c_{i}\delta_{i}|x_{i}(t)|^{p}\geq\sum^{l}_{i=1}c_{i}\delta _{i}|\chi_{i}(|\mu_{i}|)|^{p}\). Consequently, from condition Q2, we can derive
$$\begin{aligned} \dot{V}\bigl(x(t)\bigr) \leq&-\sum^{l}_{i=1}c_{i} \xi_{i} \bigl\vert x_{i}(t) \bigr\vert ^{p}+ \sum^{l}_{i,j=1}c_{i}d_{ij}F_{ij} \bigl(x_{i}(t),x_{j}(t)\bigr) \\ \leq&-\sum^{l}_{i=1}c_{i} \xi_{i} \biggl(\frac{V_{i}(x_{i}(t))}{\alpha _{i}} \biggr) \\ \leq&-\min_{i\in\mathbb{L}} \biggl\{ \frac{c_{i}\xi_{i}}{\alpha _{i}} \biggr\} \sum ^{l}_{i=1}V_{i} \bigl(x_{i}(t)\bigr) \\ =&-\xi V\bigl(x(t)\bigr), \end{aligned}$$
where \(\xi=\min_{i\in\mathbb{L}} \{c_{i}\xi_{i}/\alpha_{i} \}\). The proof of Theorem 1 in [24] implies that there exists some \(\mathcal{K}\mathcal{\phi}\)-function \(\beta_{0}\) such that \(V(x(t))\leq\beta_{0}(V(x_{0}),t)\) for all \(t\leq t_{1}\). And letting \(\delta_{0}=\sum^{l}_{i=1}c_{i}\delta_{i}\), we can obtain from condition Q1 that
$$ V(x_{0})=\sum^{l}_{i=1}c_{i}V_{i} \bigl(x_{i}(0)\bigr)\leq\sum^{l}_{i=1}c_{i} \delta_{i} \bigl\vert x_{i}(0) \bigr\vert ^{p} \leq\sum^{l}_{i=1}c_{i}\delta _{i}|x_{0}|^{p}=\delta_{0}|x_{0}|^{p}. $$
Therefore
$$ \bigl\vert x(t) \bigr\vert \leq\beta\bigl( \vert x_{0} \vert ,t\bigr), $$
(6)
where \(\beta(r,t)=(\beta_{0}(\delta_{0}|r|^{p},t)/\alpha)^{\frac{1}{p}}\).

From (5) and (6), we can obtain \(|x(t)|\leq\beta (|x_{0}|,t)+\gamma(\|u\|)\) for all \(t\geq0\), that is, CCS (1) is ISS. □

In [1], the ISS for individual nonlinear control system was investigated by Sontag and Wang. Some classes of stability, like robust stability and weak robust stability for control systems, were investigated and some sufficient conditions were established to guarantee these stabilities. Motivated by [1], we have the following results.

Theorem 2

Let the conditions in Theorem  1 hold. Then:
  1. (1)

    CCS (1) is robustly stable.

     
  2. (2)
    There exist \(\mathcal{K}\mathcal{\phi}\)-functions \(\beta _{1}\), \(\beta_{2}\) and a \(\mathcal{K}\)-function γ such that, for any \(x_{0}\in\mathbb{R}^{n}\) and any input \(u\in L_{\infty}^{m}\), it holds that
    $$ \bigl\vert x(t,x_{0},u) \bigr\vert \leq\beta_{1}\bigl( \vert x_{0} \vert ,t\bigr)+\beta_{2}\bigl( \Vert u_{T} \Vert ,t-T\bigr)+\gamma \bigl( \bigl\Vert u^{\mathrm{T}} \bigr\Vert \bigr) $$
    for any \(0\leq T\leq t\), where \(u_{T}\) denotes the input for CCSs (1) when \(t=T\) and \(u^{\mathrm{T}}\) is defined by \(u^{\mathrm{T}}=u-u_{T}\).
     
  3. (3)

    For each \(\varepsilon>0\), there exists \(\delta>0\) such that \(|x(t,x_{0},u)|\leq\varepsilon\) for all inputs \(u\in L_{\infty}^{m}\) and initial states \(x_{0}\) with \(|x_{0}|\leq\delta\) and \(\|u\|\leq\delta\).

     
  4. (4)

    There exists a \(\mathcal{K}\)-function γ such that, for any \(r,\varepsilon>0\), there is \(T>0\) so that for every input \(u\in L_{\infty}^{m}\), it holds that \(|x(t,x_{0},u)|\leq\varepsilon+\gamma(\| u\|)\), whenever \(|x_{0}|\leq r\) and \(t\geq T\).

     
  5. (5)

    CCS (1) is weakly robustly stable.

     

4 An application to a coupled Lurie system

Now in order to illustrate the result of Theorem 1, let us apply this result to a coupled Lurie system (CLS). The absolute stability problem, formulated by Lurie and coworkers in the 1940s, has been a well-studied and fruitful area of research.

Assume that each vertex dynamic is described by a feedback interconnection of a linear block and a nonlinear element. To be simplified, \(x_{i}(t)\) and \(y_{i}(t)\) are denoted by \(x_{i}\) and \(y_{i}\), \(i=1,2,\ldots,l\). When a bounded input is set to every vertex system, it can be described as
$$\begin{aligned}& \dot{x_{i}} =A_{i}x_{i}+B_{i}\bigl(- \alpha_{i}(y_{i})+u_{i}\bigr), \\& y_{i} =K_{i}x_{i}, \end{aligned}$$
where \(x_{i}\in\mathbb{R}^{n_{i}}\), \(y_{i}\in\mathbb{R}^{m_{i}}\), \(K_{i}\in\mathbb{R}^{m_{i}\times n_{i}}\), \(A_{i}\in\mathbb {R}^{n_{i}\times n_{i}}\) is the personal state alteration matrix for the ith vertex system, \(B_{i}\in\mathbb{R}^{n_{i}\times m_{i}}\) is the feedback and input effect matrix, \(u_{i}\) denotes the input of vertex i, and \(\alpha_{i}(\cdot): \mathbb{R}^{m_{i}}\rightarrow\mathbb {R}^{m_{i}}\) is a feedback function. Let \(D_{j}\in\mathbb {R}^{n_{i}\times n_{j}}\) describe the effect that vertex system j has on i. Thus a CLS is obtained as follows:
$$ \begin{aligned} &\dot{x_{i}}=A_{i}x_{i}+B_{i} \bigl(-\alpha_{i}(y_{i})+u_{i}\bigr)+\sum ^{l}_{j=1}D_{j}x_{j}, \\ &y_{i}=K_{i}x_{i}. \end{aligned} $$
(7)
Before the main theorem, let us present some assumptions and two lemmas. The following fundamental assumptions for CLS (7) are given:
  1. A1:
    If \((A_{i},K_{i})\) is detectable and there exists matrix \(P_{i}=P_{i}^{\mathrm{T}}\geq0\) satisfying
    $$ A_{i}^{\mathrm{T}}P_{i}+P_{i}A_{i}+l \bigl(P_{i}^{\mathrm {T}}P_{i}+D_{i}^{\mathrm{T}}D_{i} \bigr)\leq0,\qquad K_{i}^{\mathrm{T}}=P_{i}B_{i}. $$
    (8)
     
  2. A2:
    If \(\varphi_{i}\) is a \(\mathcal{K}_{\infty}\)-function, and for all \(y_{i}\in\mathbb{R}^{m_{i}}\),
    $$ |y_{i}|\varphi_{i}\bigl( \vert y_{i} \vert \bigr)\leq y_{i}^{\mathrm{T}} \alpha_{i}(y_{i}). $$
    (9)
     
  3. A3:
    When \(|y_{i}|\geq\mu_{i}\), where \(\mu_{i}>0\)
    $$ \bigl\vert \alpha_{i}(y_{i}) \bigr\vert \leq y_{i}^{\mathrm{T}}\alpha_{i}(y_{i}). $$
    (10)
     
The results in this section and their proofs are motivated by [4].

Lemma 2

For CLS (7), there exist a constant \(\theta_{i}>0\) and a \(\mathcal{K}_{\infty}\)-function \(\eta_{i}(\cdot)\) satisfying
$$ \theta_{i}\bigl( \bigl\vert \alpha_{i}(y_{i}) \bigr\vert +|y_{i}|\bigr)\leq y_{i}^{\mathrm{T}} \alpha_{i}(y_{i}) $$
(11)
when \(|y_{i}|\geq\mu_{i}\),
$$ \eta_{i}\bigl( \vert y_{i} \vert \bigr) \vert y_{i} \vert ^{2}+\eta_{i}\bigl( \bigl\vert \alpha_{i}(y_{i}) \bigr\vert \bigr) \bigl\vert \alpha _{i}(y_{i}) \bigr\vert ^{2}\leq y_{i}^{\mathrm{T}}\alpha_{i}(y_{i}) $$
(12)
when \(|y_{i}|\leq\mu_{i}\).

The proof of Lemma 2 can be seen in [4].

Since it is complex to construct a vertex ISS-Lyapunov function for CLS (7), we firstly construct a section of the vertex ISS-Lyapunov function in the following lemma. And then we give the entire vertex ISS-Lyapunov function for CLS (7) in the main theorem.

Lemma 3

Suppose that (7) satisfies assumptions A1-A3. Definite a function
$$ \sigma_{i}(r)=\varepsilon_{i} \int_{0}^{r}\min \biggl\{ 1,\frac{1}{\sqrt {z}}, \pi_{i}(z) \biggr\} \, \mathrm{d}z, $$
where the constant \(\varepsilon_{i}>0\) and the \(\mathcal{K}\)-function \(\pi_{i}(\cdot)\) is to be specified. Let \(S_{i}(x_{i})=\sigma _{i}(x_{i}^{\mathrm{T}}Q_{i}x_{i})\), in which matrix \(Q_{i}^{\mathrm {T}}=Q_{i}>0\) satisfying
$$ (A_{i}-J_{i}K_{i})^{\mathrm {T}}Q_{i}+Q_{i}(A_{i}-J_{i}K_{i})+l \bigl(Q_{i}^{\mathrm {T}}Q_{i}+D_{i}^{\mathrm{T}}D_{i} \bigr)\leq-I, $$
(13)
then there exists a \(\mathcal{K}\)-function \(\gamma_{i}(\cdot)\) satisfying
$$ \dot{S_{i}}(x_{i})\leq-\gamma_{i}\bigl( \vert x_{i} \vert \bigr)+y_{i}^{\mathrm{T}}\alpha _{i}(y_{i})+\theta_{i}|u_{i}|+\sum _{j=1}^{l}F_{ij}(x_{i},x_{j}), $$
where \(F_{ij}(x_{i},x_{j})=x_{j}^{\mathrm{T}}D_{j}^{\mathrm {T}}D_{j}x_{j}-x_{i}^{\mathrm{T}}D_{i}^{\mathrm{T}}D_{i}x_{i}\).

Proof

Rewrite CLS (7) as
$$ \dot{x_{i}}=(A_{i}-J_{i}K_{i})x_{i}+J_{i}y_{i}+B_{i} \bigl(-\alpha _{i}(y_{i})+u_{i} \bigr)+\sum _{j=1}^{l}D_{j}x_{j} , $$
(14)
where \(J_{i}\) is chosen so that \(A_{i}-J_{i}K_{i}\) is a Hurwitz matrix. By the construction of \(S_{i}(\cdot)\), it is easy to see that \(S_{i}(\cdot)\) is positive definite and radially unbounded. Then we let \(k>0\) satisfy
$$2\max\bigl\{ \bigl\vert B_{i}^{\mathrm{T}}Q_{i}x_{i} \bigr\vert , \bigl\vert J_{i}^{\mathrm{T}}Q_{i}x_{i} \bigr\vert \bigr\} \leq k \vert x_{i} \vert $$
for all \(1\leq i\leq l\), and note from (14) that
$$ \dot{S_{i}}(x_{i})\leq\sigma_{i}' \bigl(x_{i}^{\mathrm{T}}Q_{i}x_{i}\bigr) \Biggl[2x_{i}^{\mathrm{T}}Q_{i} \Biggl((A_{i}-J_{i}K_{i})x_{i}+ \sum_{j=1}^{l}D_{j}x_{j} \Biggr) +k \vert x_{i} \vert \bigl( \bigl\vert \alpha_{i}(y_{i}) \bigr\vert + \vert y_{i} \vert + \vert u_{i} \vert \bigr) \Biggr]. $$
(15)
Because \(\sigma_{i}'(z)\leq\varepsilon_{i}/\sqrt{z}\), we can find a constant \(c_{i}>0\), independent of \(\varepsilon_{i}\), so that \(\sigma_{i}'(x_{i}^{\mathrm{T}}Q_{i}x_{i}) k|x_{i}|\leq c_{i}\varepsilon_{i}\) for all \(x_{i}\in\mathbb{R}^{n_{i}}\). And then we can obtain
$$ \dot{S_{i}}(x_{i})\leq\sigma_{i}' \bigl(x_{i}^{\mathrm {T}}Q_{i}x_{i} \bigr)2x_{i}^{\mathrm{T}}Q_{i} \Biggl((A_{i}-J_{i}K_{i})x_{i}+ \sum_{j=1}^{l}D_{j}x_{j} \Biggr) +c_{i}\varepsilon_{i}\bigl( \bigl\vert \alpha_{i}(y_{i}) \bigr\vert + \vert y_{i} \vert + \vert u_{i} \vert \bigr). $$
Because of
$$\begin{aligned}& 2\sum_{j=1}^{l}x_{i}^{\mathrm{T}}Q_{i}D_{j}x_{j} \\& \quad \leq \sum_{j=1}^{l} \bigl(Q_{i}^{\mathrm{T}}Q_{i}|x_{i}|^{2}+D_{j}^{\mathrm {T}}D_{j}|x_{j}|^{2} \bigr) \\& \quad = lQ_{i}^{\mathrm{T}}Q_{i}|x_{i}|^{2}+lD_{i}^{\mathrm {T}}D_{i}|x_{i}|^{2}+ \sum_{j=1}^{l}\bigl(D_{j}^{\mathrm {T}}D_{j}|x_{j}|^{2}-D_{i}^{\mathrm{T}}D_{i}|x_{i}|^{2} \bigr) \\& \quad = l\bigl(Q_{i}^{\mathrm{T}}Q_{i}+D_{j}^{\mathrm{T}}D_{j} \bigr)|x_{i}|^{2}+\sum_{j=1}^{l} \bigl(x_{j}^{\mathrm{T}}D_{j}^{T}D_{j}x_{j}-x_{i}^{\mathrm {T}}D_{i}^{\mathrm{T}}D_{i}x_{i} \bigr) \\& \quad = l\bigl(x_{i}^{\mathrm{T}}Q_{i}^{\mathrm{T}}Q_{i}x_{i}+x_{i}^{\mathrm {T}}D_{i}^{\mathrm{T}}D_{i}x_{i} \bigr)+\sum_{j=1}^{l}F_{ij}(x_{i},x_{j}), \end{aligned}$$
where \(F_{ij}(x_{i},x_{j})=x_{j}^{\mathrm{T}}D_{j}^{\mathrm {T}}D_{j}x_{j}-x_{i}^{\mathrm{T}}D_{i}^{\mathrm{T}}D_{i}x_{i}\), we can get according to (13) that
$$\begin{aligned} \dot{S_{i}}(x_{i}) \leq& \sigma_{i}'\bigl(x_{i}^{\mathrm {T}}Q_{i}x_{i} \bigr)x_{i}^{\mathrm{T}} \bigl((A_{i}-J_{i}K_{i})^{\mathrm {T}}Q_{i}+Q_{i}(A_{i}-J_{i}K_{i})+l \bigl(Q_{i}^{\mathrm {T}}Q_{i}+D_{i}^{\mathrm{T}}D_{i} \bigr) \bigr)x_{i} \\ &{} +c_{i}\varepsilon_{i}\bigl( \bigl\vert \alpha_{i}(y_{i}) \bigr\vert + \vert y_{i} \vert + \vert u_{i} \vert \bigr)+\sum _{j=1}^{l}F_{ij}(x_{i},x_{j}) \\ \leq& -\sigma_{i}'\bigl(x_{i}^{\mathrm{T}}Q_{i}x_{i} \bigr) \vert x_{i} \vert ^{2} +c_{i} \varepsilon_{i}\bigl( \bigl\vert \alpha_{i}(y_{i}) \bigr\vert + \vert y_{i} \vert + \vert u_{i} \vert \bigr)+\sum_{j=1}^{l}F_{ij}(x_{i},x_{j}). \end{aligned}$$
(16)
  • When \(|y_{i}|\geq\mu_{i}\), choosing \(\varepsilon_{i}=\theta_{i}/c_{i}\) and using (11), we have
    $$ \dot{S_{i}}(x_{i})\leq-\sigma_{i}' \bigl(x_{i}^{\mathrm {T}}Q_{i}x_{i} \bigr)|x_{i}|^{2}+y_{i}^{\mathrm{T}} \alpha_{i}(y_{i})+\theta _{i}|u_{i}|+ \sum_{j=1}^{l}F_{ij}(x_{i},x_{j}). $$
    (17)
  • When \(|y_{i}|\leq\mu_{i}\), we denote by \(\lambda_{i}\) the maximum eigenvalue of \(Q_{i}\). Considering the two cases \(|\alpha _{i}(y_{i})|\leq|x_{i}|/4k\) and \(|x_{i}|\leq4k|\alpha_{i}(y_{i})|\), and using \(\sigma_{i}'(z)\leq\pi_{i}(z)\), we can obtain
    $$ \sigma_{i}'\bigl(x_{i}^{\mathrm{T}}Q_{i}x_{i} \bigr)k \vert x_{i} \vert \bigl\vert \alpha_{i}(y_{i}) \bigr\vert \leq \frac{1}{4}\sigma_{i}' \bigl(x_{i}^{\mathrm{T}}Q_{i}x_{i}\bigr) \vert x_{i} \vert ^{2} +4k^{2} \bigl\vert \alpha_{i}(y_{i}) \bigr\vert ^{2} \pi_{i}\bigl(16\lambda_{i}k^{2} \bigl\vert \alpha _{i}(y_{i}) \bigr\vert ^{2}\bigr). $$
    (18)
    Considering the two cases \(|y_{i}|\leq|x_{i}|/4k\) and \(|x_{i}|\leq 4k|y_{i}|\), we can denote
    $$ \sigma_{i}'\bigl(x_{i}^{\mathrm{T}}Q_{i}x_{i} \bigr)k|x_{i}||y_{i}|\leq\frac {1}{4} \sigma_{i}'\bigl(x_{i}^{\mathrm{T}}Q_{i}x_{i} \bigr)|x_{i}|^{2} +4k^{2}|y_{i}|^{2} \pi_{i}\bigl(16\lambda_{i}k^{2}|y_{i}|^{2} \bigr). $$
    (19)
    We choose
    $$\pi_{i}(z)=\frac{1}{4k^{2}}\eta_{i} \biggl(\sqrt{ \frac{z}{16\lambda _{i}k^{2}}} \biggr), $$
    then substituting (18) and (19) into (16) yields
    $$\begin{aligned} \begin{aligned}[b] \dot{S_{i}}(x_{i})\leq{}&-\frac{1}{2} \sigma_{i}'\bigl(x_{i}^{\mathrm {T}}Q_{i}x_{i} \bigr) \vert x_{i} \vert ^{2}+\eta_{i}\bigl( \vert y_{i} \vert \bigr) \vert y_{i} \vert ^{2} +\eta_{i}\bigl( \vert \alpha_{i}y_{i} \vert \bigr) \vert \alpha_{i}y_{i} \vert ^{2} \\ &{}+\theta_{i}u_{i}+\sum _{j=1}^{l}F_{ij}(x_{i},x_{j}). \end{aligned} \end{aligned}$$
    (20)
From (12), (20), it implies that
$$ \dot{S_{i}}(x_{i})\leq-\frac{1}{2} \sigma_{i}'\bigl(x_{i}^{\mathrm {T}}Q_{i}x_{i} \bigr)|x_{i}|^{2}+y_{i}^{\mathrm{T}} \alpha_{i}(y_{i})+\theta _{i}|u_{i}|+ \sum_{j=1}^{l}F_{ij}(x_{i},x_{j}). $$
(21)
So, noting from (10) that \(\sigma_{i}'(z)=\varepsilon/z\) for sufficiently large z, we can find a \(\mathcal{K}_{\infty}\)-function \(\gamma_{i}(\cdot)\) such that
$$ \frac{1}{2}\sigma_{i}'\bigl(x_{i}^{\mathrm{T}}Q_{i}x_{i} \bigr)|x_{i}|^{2}\geq \gamma_{i}\bigl( \vert x_{i} \vert \bigr) $$
for all \(x_{i}\in\mathbb{R}^{n_{i}}\). Thus it follows from (17) and (21) that, for all values of \(x_{i}\) and \(u_{i}\),
$$ \dot{S_{i}}(x_{i})\leq-\gamma_{i}\bigl( \vert x_{i} \vert \bigr)+y_{i}^{\mathrm{T}}\alpha _{i}(y_{i})+\theta_{i}|u_{i}|+\sum _{j=1}^{l}F_{ij}(x_{i},x_{j}). $$
(22)
The proof is complete. □

Theorem 3

If CLS (7) satisfies assumptions A1, A2 and A3, then it is ISS.

Proof

Let \(L_{i}(x_{i})=x_{i}^{\mathrm{T}}P_{i}x_{i}+\sigma_{i}(x_{i}^{\mathrm {T}}Q_{i}x_{i})\), and let \(R_{i}(x_{i})=x_{i}^{\mathrm{T}}P_{i}x_{i}\), \(K_{i}=B_{i}^{\mathrm{T}}P_{i}\), then \(\nabla R_{i}=2x_{i}^{\mathrm{T}}P_{i}\). It follows that
$$\begin{aligned} \dot{R_{i}}(x_{i}) =&2x_{i}^{\mathrm{T}}P_{i} \Biggl(A_{i}x_{i}+B_{i}\bigl(- \alpha_{i}(y_{i})+u_{i}\bigr)+\sum _{j=1}^{l}D_{j}x_{j} \Biggr) \\ =&2x_{i}^{\mathrm{T}}P_{i}A_{i}x_{i}+2 \sum_{j=1}^{l}x_{i}^{\mathrm {T}}P_{i}D_{j}x_{j}+2x_{i}^{\mathrm{T}}P_{i}B_{i} \bigl(-\alpha_{i}(y_{i})+u_{i}\bigr), \end{aligned}$$
in which
$$ 2x_{i}^{\mathrm{T}}P_{i}B_{i}\bigl(- \alpha_{i}(y_{i})+u_{i}\bigr)\leq -2y_{i}^{\mathrm{T}}\alpha_{i}(y_{i})+2y_{i}^{\mathrm{T}}u_{i} $$
and
$$\begin{aligned}& 2\sum_{j=1}^{l}x_{i}^{\mathrm{T}}P_{i}D_{j}x_{j} \\& \quad = lP_{i}^{\mathrm{T}}P_{i}|x_{i}|^{2}+ \sum_{j=1}^{l}D_{j}^{\mathrm {T}}D_{j}|x_{j}|^{2} \\& \quad = l\bigl(P_{i}^{\mathrm{T}}P_{i}+D_{j}^{\mathrm{T}}D_{j} \bigr)|x_{i}|^{2}+\sum_{j=1}^{l} \bigl(D_{j}^{\mathrm{T}}D_{j}|x_{j}|^{2}-D_{i}^{\mathrm {T}}D_{i}|x_{i}|^{2} \bigr) \\& \quad = l\bigl(x_{i}^{\mathrm{T}}P_{i}^{\mathrm{T}}P_{i}x_{i}+x_{i}^{\mathrm {T}}D_{i}^{\mathrm{T}}D_{i}x_{i} \bigr)+\sum_{j=1}^{l}F_{ij}(x_{i},x_{j}). \end{aligned}$$
Noting that \(A_{i}^{\mathrm{T}}P_{i}+P_{i}A_{i}+l(P_{i}^{\mathrm {T}}P_{i}+D_{i}^{\mathrm{T}}D_{i})\leq0\), we get
$$ \dot{R_{i}}(x_{i})\leq-2y_{i}^{\mathrm{T}} \alpha _{i}\bigl(y_{i}+2y_{i}^{\mathrm{T}}u_{i} \bigr)+\sum_{j=1}^{l}F_{ij}(x_{i},x_{j}). $$
Considering the two cases \(|u_{i}|\leq\varphi_{i}(|y_{i}|)/2\) and \(|u_{i}|\geq\varphi_{i}(|y_{i}|)/2\) and using (9), we obtain the inequality
$$ 2y_{i}^{\mathrm{T}}u_{i}\leq2|y_{i}||u_{i}| \leq|y_{i}| \bigl\vert \varphi _{i}\bigl( \vert y_{i} \vert \bigr) \bigr\vert +2(\varphi_{i})^{-1} \bigl(2 \vert u_{i} \vert \bigr)|u_{i}|\leq y_{i}^{\mathrm{T}} \alpha_{i}(y_{i})+2( \varphi_{i})^{-1}\bigl(2 \vert u_{i} \vert \bigr)|u_{i}|, $$
which results in
$$ \dot{R_{i}}(x_{i})\leq-y_{i}^{\mathrm{T}} \alpha_{i}(y_{i})+2\varphi _{i}^{-1} \bigl(2 \vert u_{i} \vert \bigr)|u_{i}|+\sum _{j=1}^{l}F_{ij}(x_{i},x_{j}). $$
From Lemma 2, we know \(\dot{S_{i}}(x_{i})\leq-\gamma _{i}(|x_{i}|)+y_{i}^{\mathrm{T}}\alpha_{i}(y_{i})+\theta_{i}|u_{i}|+\sum_{j=1}^{l}F_{ij}(x_{i},x_{j})\). Then we have
$$ L_{i}(x_{i}) \leq-\gamma_{i}\bigl( \vert x_{i} \vert \bigr)+\beta_{i}\bigl( \vert u_{i} \vert \bigr) + \sum_{j=1}^{l}F_{ij}(x_{i},x_{j}) $$
with \(\beta_{i}(|u_{i}|)=\delta_{i}|u_{i}|+2\phi_{i}^{-1}(2|u_{i}|)|u_{i}|\). Then we can find \(\xi_{i}>0\) satisfying
$$ \dot{ L_{i}}(x_{i}) \leq-\xi_{i}|x_{i}|^{p} + \sum_{j=1}^{l}F_{ij}(x_{i},x_{j}). $$

So we conclude that \(L_{i}(x_{i})\) is a vertex ISS-Lyapunov function, then from Theorem 1, CLS (7) is ISS. □

Remark 1

In recent years, Lurie systems have been studied by many researchers [4, 25]. Particularly, compared with [25], the main differences are as follows.
  1. (i)

    This paper considers a coupled Lurie system, which is more complicated.

     
  2. (ii)

    This paper uses graph theory combining with the Lyapunov method to derive the ISS of the considered system. This technique does not need us solving any linear matrix inequality. Literature [25] proposed Lyapunov-Krasovskii functionals which contain an exponential multiplier to solve the stabilization of an indirect control system.

     
  3. (iii)

    In [25], time delay was considered, which is our further work.

     

Finally, two examples with their numerical simulations are provided to illustrate our results.

Example 1

Assume that there are three vertices and \(x_{i}=(x_{i1},x_{i2})^{\mathrm {T}}\in\mathbb{R}^{2}\). We now take the following coefficients for (7), and then take some numerical simulation. Here,
$$\begin{aligned}& A_{i}=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} -10 & 1 \\ 1 & -10 \end{array}\displaystyle \right ),\qquad B_{i}=\left ( \textstyle\begin{array}{c} 2 \\ 1 \end{array}\displaystyle \right ), \\& D_{1}=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} 2 & 0 \\ 1 & 1 \end{array}\displaystyle \right ),\qquad D_{2}=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} 1 & 1 \\ 1 & 1 \end{array}\displaystyle \right ),\qquad D_{3}=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} 2 & 1 \\ 0 & 1 \end{array}\displaystyle \right ). \end{aligned}$$
And let the input function \(u_{i}=0\), \(\alpha_{i}(y_{i})=iy_{i}\), \(i=1,2,3\). Moreover, we take \(\varphi_{i}(y_{i})=\frac{i}{2}y_{i}\) and \(\mu_{i}=1\) for \(i=1,2,3\). It is clear that conditions A2 and A3 are satisfied. If we let
$$ P_{1}=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} 3 & 2 \\ 2 & 3 \end{array}\displaystyle \right ),\qquad P_{2}=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} 1 & 1 \\ 1 & 4 \end{array}\displaystyle \right ),\qquad P_{3}=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} 2 & 1 \\ 1 & 3 \end{array}\displaystyle \right ), $$
by calculation, we get that condition A1 holds. Therefore, by Theorem 3, we derive that (7) is ISS. The respective simulation results are shown in Figure 2, which conforms the effectiveness of the developed results.
Figure 2

Solution to coupled Lurie system ( 7 ) with initial values \(\pmb{(x_{11},x_{12},x_{21},x_{22},x_{31}, x_{32})^{\mathrm {T}}=(0.5,-1,-2,2.5, -1.5,2)^{\mathrm{T}}}\) .

Example 2

We consider the ISS of a system described as follows:
$$ \begin{aligned} &\dot{x_{i}}=A_{i}x_{i}+B_{i}y_{i}+ \sum_{j=1}^{3}D_{j}x_{j}, \\ &\dot{y_{i}}=-K_{i}x_{i}-6y_{i}- \phi_{i}(y_{i})-u_{i}+\sum _{j=1}^{3}y_{j}, \end{aligned} \quad i=1,2,3, $$
(23)
with the input function \(u_{i}=C_{i}(x^{\mathrm{T}}_{i},y_{i})^{\mathrm {T}}\) in which \(C_{i}\) is a matrix and \(x_{i}=(x_{i1}, x_{i2})^{\mathrm {T}}\), function \(\phi_{i}(y_{i})=y_{i}^{3}\), and there exist matrices \(P_{i}^{\mathrm{T}}=P_{i}\geq0\) such that
$$ \begin{aligned} &A_{i}^{T}P_{i}+P_{i}A_{i}+3 \bigl(P_{i}^{T}P_{i}+D_{i}^{T}D_{i} \bigr)\leq0, \\ &K_{i}^{T}=P_{i}B_{i}, \end{aligned} \quad i=1,2,3. $$
(24)
To apply Theorem 3, we rewrite (23) as in (7), with
$$ A_{0i}=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} A_{i} & B_{i} \\ -K_{i} & -6 \end{array}\displaystyle \right ),\qquad B_{0i}=\left ( \textstyle\begin{array}{@{}c@{}} 0 \\ 1 \end{array}\displaystyle \right ),\qquad P_{0i}=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} P_{i} & 0 \\ 0 & 1 \end{array}\displaystyle \right ),\qquad D_{0i}=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} D_{i} & 0 \\ 0 & 1 \end{array}\displaystyle \right ) $$
and \(\alpha_{i}(y_{i})=3y_{i}+\phi_{i}(y_{i})\). Then \(\alpha _{i}(y_{i})\) satisfies conditions A2 and A3. And then, we let the values of \(A_{i}\), \(B_{i}\), \(D_{i}\), \(P_{i}\), \(i=1,2,3\), be the same as those in Example 1 and
$$ C_{1}=C_{2}=C_{3}=( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} 1 & 1 & 1 \end{array}\displaystyle ). $$
We can see that \(A_{0i}\), \(B_{0i}\), \(P_{0i}\), \(D_{0i}\) satisfy condition A1 because of (24). So, we conclude that system (23) is ISS. Figure 3 shows that the solution of system (23) is ISS.
Figure 3

Solution to coupled Lurie system ( 23 ) with initial values \(\pmb{(x_{11},x_{12},y_{1},x_{21},x_{22},y_{2},x_{31},x_{32},y_{3})^{\mathrm {T}}=(0.5,-1,1.75,2.5,-2,-0.78,2,-1.5,1.25)^{\mathrm{T}}}\) .

Declarations

Acknowledgements

We would like to thank the editors and the anonymous reviewers for carefully reading the original manuscript and for the constructive comments and suggestions to improve the presentation of this paper.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Shandong University, Weihai
(2)
Department of Mathematics, Harbin Institute of Technology (Weihai)
(3)
Department of Mathematics, Harbin Institute of Technology

References

  1. Sontag, E, Wang, Y: On characterizations of the input-to-state stability property. Syst. Control Lett. 24, 351-359 (1995) MathSciNetView ArticleMATHGoogle Scholar
  2. Grüne, L: Input-to-state dynamical stability and its Lyapunov function characterization. IEEE Trans. Autom. Control 47, 1499-1504 (2002) MathSciNetView ArticleGoogle Scholar
  3. Angeli, D: A Lyapunov approach to incremental stability properties. IEEE Trans. Autom. Control 47, 410-421 (2002) MathSciNetView ArticleGoogle Scholar
  4. Arcak, M, Teel, A: Input-to-state stability for a class of Lurie systems. Automatica 38, 1945-1949 (2002) MathSciNetView ArticleMATHGoogle Scholar
  5. Li, MY, Shuai, Z: Global-stability problem for coupled systems of differential equations on networks. J. Differ. Equ. 248, 1-20 (2010) MathSciNetView ArticleMATHGoogle Scholar
  6. Jin, T, Li, W, Feng, J: Outer synchronization of stochastic complex networks with time-varying delay. Adv. Differ. Equ. 2015, 359 (2015) MathSciNetView ArticleGoogle Scholar
  7. Wang, R, Li, W, Li, X: The almost sure stability of coupled system of stochastic delay differential equations on networks. Adv. Differ. Equ. 2015, 133 (2015) MathSciNetView ArticleGoogle Scholar
  8. Li, W, Zhang, X, Zhang, C: A new method for exponential stability of coupled reaction-diffusion systems with mixed delays: combining Razumikhin method with graph theory. J. Franklin Inst. Eng. Appl. Math. 352, 1169-1191 (2015) MathSciNetView ArticleMATHGoogle Scholar
  9. Wang, G, Li, W, Feng, J: Stability analysis of stochastic coupled systems on networks without strong connectedness via hierarchical approach. J. Franklin Inst. Eng. Appl. Math. 354, 1138-1159 (2017) MathSciNetView ArticleMATHGoogle Scholar
  10. Liu, Y, Li, W, Feng, J: Graph-theoretical method to the existence of stationary distribution of stochastic coupled systems. J. Dyn. Differ. Equ. (2017). doi:10.1007/s10884-016-9566-y Google Scholar
  11. Zhang, C, Li, W, Wang, K: Graph theory-based approach for stability analysis of stochastic coupled systems with Lévy noise on networks. IEEE Trans. Neural Netw. Learn. Syst. 26, 1689-1709 (2015) Google Scholar
  12. Li, W, Su, H, Wang, K: Global stability analysis for stochastic coupled systems on networks. Automatica 47, 215-220 (2011) MathSciNetView ArticleMATHGoogle Scholar
  13. Su, H, Li, W, Wang, K: Global stability analysis of discrete-time coupled systems on networks and its applications. Chaos 22, 033135 (2012) MathSciNetView ArticleMATHGoogle Scholar
  14. Su, H, Wang, P, Ding, X: Stability analysis for discrete-time coupled systems with multi-diffusion by graph-theoretic approach and its application. Discrete Contin. Dyn. Syst., Ser. B 21, 253-269 (2016) MathSciNetView ArticleMATHGoogle Scholar
  15. Zhang, C, Li, W, Wang, K: Graph-theoretic approach to stability of multi-group models with dispersal. Discrete Contin. Dyn. Syst., Ser. B 20, 259-280 (2015) MathSciNetView ArticleMATHGoogle Scholar
  16. Liu, M, Bai, C: Analysis of a stochastic tri-trophic food-chain model with harvesting. J. Math. Biol. 73, 597-625 (2016) MathSciNetView ArticleMATHGoogle Scholar
  17. Liu, M, Fan, M: Permanence of stochastic Lotka-Volterra systems. J. Nonlinear Sci. (2016). doi:10.1007/s00332-016-9337-2 Google Scholar
  18. Liu, M, Fan, M: Stability in distribution of a three-species stochastic cascade predator-prey system with time delays. IMA J. Appl. Math. (2016). doi:10.1093/imamat/hxw057 Google Scholar
  19. Li, W, Pang, L, Su, H, Wang, K: Global stability for discrete Cohen-Grossberg neural networks with finite and infinite delays. Appl. Math. Lett. 25, 2246-2251 (2012) MathSciNetView ArticleMATHGoogle Scholar
  20. Zhang, C, Li, W, Su, H, Wang, K: A graph-theoretic approach to boundedness of stochastic Cohen-Grossberg neural networks with Markovian switching. Appl. Math. Comput. 219, 9165-9173 (2013) MathSciNetMATHGoogle Scholar
  21. Li, W, Su, H, Wei, D, Wang, K: Global stability of coupled nonlinear systems with Markovian switching. Commun. Nonlinear Sci. Numer. Simul. 17, 2609-2616 (2012) MathSciNetView ArticleMATHGoogle Scholar
  22. Zhang, C, Li, W, Wang, K: A graph-theoretic approach to stability of neutral stochastic coupled oscillators network with time-varying delayed coupling. Appl. Math. Comput. 37, 1179-1190 (2014) MathSciNetMATHGoogle Scholar
  23. Zhang, X, Li, W, Wang, K: The existence and global exponential stability of periodic solution for a neutral coupled system on networks with delays. Appl. Math. Comput. 264, 208-217 (2015) MathSciNetGoogle Scholar
  24. Su, H, Qu, Y, Gao, S, Song, H, Wang, K: A model of feedback control system on network and its stability analysis. Commun. Nonlinear Sci. Numer. Simul. 18, 1822-1831 (2013) MathSciNetView ArticleMATHGoogle Scholar
  25. Shatyrko, A, Diblik, J, Khusainov, D, Ruzickova, M: Stabilization of Lur’e-type nonlinear control systems by Lyapunov-Krasovskii functionals. Adv. Differ. Equ. 2012, 229 (2012) MathSciNetView ArticleGoogle Scholar

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