Abstract
In this paper, the inputtostate 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 inputtostate 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 inputtostate stability of this coupled Lurie system is established. Two examples are provided to demonstrate the effectiveness of the theoretical results.
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, inputtostate 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 wellknown 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 onedimensional 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 wellknown 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 globalstability problem of equilibrium and developed a systematic approach that allows one to construct global Lyapunov functions for largescale coupled systems from building blocks of individual vertex systems. Later, this technique was appropriately developed and extended to some other coupled systems. In [6–8] several delayed coupled systems were discussed, and some sufficient conditions were obtained. Li et al. in [9–12] 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 discretetime coupled systems [13, 14], and Zhang et al. extended this technique to multidispersal coupled systems [15]. Besides, this technique is also applied to many practical applications, such as biological systems [16–18], neural networks [19, 20], and mechanical systems [20–23]. 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.
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 ndimensional 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 semidefinite (negative semidefinite). 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 mdimensional 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:
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
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).
Then coupling the vertex systems together, we obtain the following coupled control system:
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
In the proof of our main results, we need to find a global ISSLyapunov function for CCS (1). For the convenience of the proof, we now define vertex ISSLyapunov functions for CCS (1).
Definition 3
Set \(\{V_{i}(x_{i}), i\in\mathbb{L}\}\) is called a vertex ISSLyapunov function set for CCS (1) if every \(V_{i}(x_{i})\) is smooth and satisfies the following conditions:

Q1.
There exist positive constants \(\alpha_{i}\), \(\delta_{i}\), \(p\geq2\), such that
$$ \alpha_{i}x_{i}^{p}\leq V_{i}(x_{i}) \leq\delta_{i}x_{i}^{p},\quad x_{i} \in\mathbb{R}^{n_{i}}. $$ 
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
$$\dot{V}_{i}\bigl(x_{i}(t)\bigr)=\bigtriangledown V_{i}\bigl(x_{i}(t)\bigr) \Biggl[f_{i} \bigl(x_{i}(t),u_{i}\bigr)+\sum ^{l}_{j=1} a_{ij}P_{ij} \bigl(x_{i}(t),x_{j}(t),u_{j}\bigr) \Biggr]. $$ 
Q3.
Along each directed cycle \(C_{\mathcal{Q}}\) of weighted digraph \((\mathcal{G},D)\), there is
$$ \sum_{(s,r)\in E(C_{\mathcal{Q}})} F_{rs}(x_{r},x_{s}) \leq0. $$
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 ISSLyapunov 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
for \(x_{0}\in\mathbb{R}^{n}\) and \(u\in\mathbb{R}^{m}\).
Let
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
Let \(\tau=\inf\{t\geq t_{0}:V(x(t))\geq b+\varepsilon\}\). From conditions Q2 and Q3, we can obtain
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
This implies that
Denote
in which \(\delta>0\) is a certain constant. For simplicity, we write
then it is easy to see from condition Q1 that for \(t\geq t_{1}\),
and
Hence \(\alphax(t)^{p}\leq V(x(t))\leq\delta\chi(\u\)^{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
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
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
Therefore
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)
CCS (1) is robustly stable.

(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
$$ \bigl\vert x(t,x_{0},u) \bigr\vert \leq\beta_{1}\bigl( \vert x_{0} \vert ,t\bigr)+\beta_{2}\bigl( \Vert u_{T} \Vert ,tT\bigr)+\gamma \bigl( \bigl\Vert u^{\mathrm{T}} \bigr\Vert \bigr) $$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 \(u^{\mathrm{T}}=uu_{T}\).

(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)
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)
CCS (1) is weakly robustly stable.
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 wellstudied 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
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:
Before the main theorem, let us present some assumptions and two lemmas. The following fundamental assumptions for CLS (7) are given:

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) 
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) 
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
when \(y_{i}\geq\mu_{i}\),
when \(y_{i}\leq\mu_{i}\).
The proof of Lemma 2 can be seen in [4].
Since it is complex to construct a vertex ISSLyapunov function for CLS (7), we firstly construct a section of the vertex ISSLyapunov function in the following lemma. And then we give the entire vertex ISSLyapunov function for CLS (7) in the main theorem.
Lemma 3
Suppose that (7) satisfies assumptions A1A3. Definite a function
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
then there exists a \(\mathcal{K}\)function \(\gamma_{i}(\cdot)\) satisfying
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
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
for all \(1\leq i\leq l\), and note from (14) that
Because \(\sigma_{i}'(z)\leq\varepsilon_{i}/\sqrt{z}\), we can find a constant \(c_{i}>0\), independent of \(\varepsilon_{i}\), so that \(\sigma_{i}'(x_{i}^{\mathrm{T}}Q_{i}x_{i}) kx_{i}\leq c_{i}\varepsilon_{i}\) for all \(x_{i}\in\mathbb{R}^{n_{i}}\). And then we can obtain
Because of
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

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 \(\alpha _{i}(y_{i})\leqx_{i}/4k\) 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 \(y_{i}\leqx_{i}/4k\) and \(x_{i}\leq 4ky_{i}\), we can denote
$$ \sigma_{i}'\bigl(x_{i}^{\mathrm{T}}Q_{i}x_{i} \bigr)kx_{i}y_{i}\leq\frac {1}{4} \sigma_{i}'\bigl(x_{i}^{\mathrm{T}}Q_{i}x_{i} \bigr)x_{i}^{2} +4k^{2}y_{i}^{2} \pi_{i}\bigl(16\lambda_{i}k^{2}y_{i}^{2} \bigr). $$(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
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
for all \(x_{i}\in\mathbb{R}^{n_{i}}\). Thus it follows from (17) and (21) that, for all values of \(x_{i}\) and \(u_{i}\),
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
in which
and
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
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
which results in
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
with \(\beta_{i}(u_{i})=\delta_{i}u_{i}+2\phi_{i}^{1}(2u_{i})u_{i}\). Then we can find \(\xi_{i}>0\) satisfying
So we conclude that \(L_{i}(x_{i})\) is a vertex ISSLyapunov 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.

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

(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 LyapunovKrasovskii functionals which contain an exponential multiplier to solve the stabilization of an indirect control system.

(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,
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
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.
Example 2
We consider the ISS of a system described as follows:
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
To apply Theorem 3, we rewrite (23) as in (7), with
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
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.
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Qiao, Y., Huang, Y. & Chen, M. A graphtheoretic approach to global inputtostate stability for coupled control systems. Adv Differ Equ 2017, 129 (2017). https://doi.org/10.1186/s136620171129y
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Keywords
 inputtostate stability
 coupled control system
 Lyapunov function