### cs代写|复杂网络代写complex network代考|Consensus tracking of CNSs with first-order nonlinear

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## cs代写|复杂网络代写complex network代考|dynamics and directed switching topologies

This chapter studies the consensus tracking of CNSs with first-order nonlinear dynamics and directed switching topologies. This chapter begins by overviewing some previous works and indicating our motivations. Section $5.2$ studies the case with Lipschitz type nonlinear dynamics without assuming that each possible network topology has a directed spanning tree. Specifically, this section proposes an algorithm for selecting the pinned nodes such that the graph contains a directed spanning tree. Section $5.3$ studies the case with Lorenz type nonlinear dynamics under directed fixed topology as well as directed switching topologies, where the Lorenz systems include the Chen and Lü systems. Finally, some simulations are given to validate the obtained theoretical results.

According to whether the final synchronization states depend on the initial value or not, synchronization in CNSs can be generally categorized into local synchronization [98, 102] and global synchronization [96]. Compared with the local synchronization, the global synchronization means that the network synchronization can be achieved under any given initial conditions, thus is more favorable in practical applications. In [96], a distance between the nodes’ states and the synchronization manifold was introduced, based on which a new methodology was proposed to investigate the global synchronization of coupled systems. Later, general algebraic connectivity was proposed in [218] to study the global synchronization as well as local synchronization problems in strongly connected networks. Global synchronization for coupled linear systems via state or output feedback control was studied in [224]. In $[179,204]$, global synchronization for a class of CNSs with sampling-data coupling was

addressed. For the case that the considered networks are not strongly connected or even do not contain any directed spanning tree, the pinning synchronization problem arises $[74,176,178,216]$. Pinning synchronization in scale-free and small-world complex networks were addressed in [178] and [176], respectively. Later, local and global pinning synchronization in random and scale-free networks were studied in [74]. It is worth noting that global synchronization is actually consensus tracking by regarding the target system in the network as a leader. In [216], pinning synchronization of undirected CNSs was further addressed. Without assuming the network topology is undirected or strongly connected, it was proved in [20] that a single controller can pin a coupled CNS to its homogeneous trajectory under some suitable conditions. Global pinning synchronization for a class of CNSs has been investigated in [66] under a $V$-stability framework. However, it is previously assumed in the aforementioned literature that each possible network topology contains a directed spanning tree with the leader being the root node. This indicates that each agent in the considered network can be influenced by the leader directly or indirectly. In some real cases, the aforementioned condition is hard or even impossible to verify.

Motivated by the aforementioned works on consensus tracking (i.e., global pinning synchronization) of CNSs, this chapter aims to solve the consensus tracking problem for a class of switched CNSs where some possible network topologies may not contain any directed spanning tree. By using a combined tool from $M$-matrix theory and stability analysis of switched systems, a new kind of topology-dependent MLFs for the switched networks is constructed. Theoretical analysis indicates that the consensus tracking in such a CNS can be achieved if some carefully selected nodes are pinned such that the network topology contains a directed spanning tree rooted at the target node frequently enough as the network evolves with time. Without causing any confusion, global pinning synchronization is referred as consensus tracking in the subsequent analysis in this chapter.

## cs代写|复杂网络代写complex network代考|Model formulation

Suppose that the considered CNS consists of $N$ nodes, the dynamics of agent $i$ are given by
$$\dot{x}{i}(t)=f\left(x{i}(t), t\right)+\alpha \sum_{j=1}^{N} a_{i j}(t)\left(x_{j}(t)-x_{i}(t)\right)$$
where $x_{i}(t)=\left[x_{i 1}(t), \ldots, x_{i n}(t)\right]^{T} \in \mathbb{R}^{n}$ for $i=1, \ldots, N$ represent the states of agent $i, \alpha>0$ is the coupling strength, and $\mathcal{A}(t)=\left[a_{i j}(t)\right]{N \times N}$ is the adjacency matrix of graph $\mathcal{G}(\mathcal{A}(t))$ at time $t$. Throughout this chapter, the derivatives of all functions at switching time points should be considered as their right-hand derivatives. According to the definition of Laplacian matrix for a graph, it follows from (5.1) that $$\dot{x}{i}(t)=f\left(x_{i}(t), t\right)-\alpha \sum_{j=1}^{N} l_{i j}(t) x_{j}(t),$$
where $\mathcal{L}(t)=\left[l_{i j}(t)\right]_{N \times N}$ is the Laplacian matrix of graph $\mathcal{G}(\mathcal{A}(t))$.

The control goal in this section is to design pinning controllers for some appropriately selected agents in (5.2) such that the states of each agent in the considered network will approach $s(t)$ when $t$ approaches $+\infty$, i.e., $\lim {t \rightarrow \infty}\left|x{i}(t)-s(t)\right|=0$, for all $i=1, \ldots, N$ and arbitrarily given initial conditions, where
$$\dot{s}(t)=f(s(t), t) .$$
Here, $s(t)$ may be an equilibrium point, a periodic orbit, or even a chaotic orbit. Motivated by the works in [74], pinning network (5.2) by using linear controllers $-\alpha c_{i}(t)\left(x_{i}(t)-s(t)\right)$ to agent $i$ leads to
$$\dot{x}{i}(t)=f\left(x{i}(t), t\right)-\alpha \sum_{j=1}^{N} l_{i j}(t) x_{j}(t)-\alpha c_{i}(t)\left(x_{i}(t)-s(t)\right)$$
where $c_{i}(t) \in{0,1}$ and $c_{i}(t)=1$ if and only if agent $i$ of (5.2) is pinned at time $t$.
Let $e_{i}(t)=x_{i}(t)-s(t), i=1, \ldots, N$. It thus follows from (5.4) that
$$\dot{e}{i}(t)=f\left(x{i}(t), t\right)-f(s(t), t)-\alpha \sum_{j=1}^{N} l_{i j}(t) e_{j}(t)-\alpha c_{i}(t) e_{i}(t)$$
By taking the target system (5.3) as a virtual leader of the CNS under consideration, one may get the augmented network topology $\mathcal{G}(\widetilde{\mathcal{A}}(t))$ consisting of $N+1$ agents. Labeling the index of the virtual agent as 0 , the Laplacian matrix $\widetilde{L}(t)$ of the augmented network topology $\mathcal{G}(\widetilde{\mathcal{A}}(t))$ can be written as:
$$\tilde{\mathcal{L}}(t)=\left[\begin{array}{cc} 0 & \mathbf{0}{N}^{T} \ \mathbf{P}(t) & \overline{\mathcal{L}}(t) \end{array}\right] \in \mathbb{R}^{(N+1) \times(N+1)}$$ $$\overline{\mathcal{L}}(t)=\left[\begin{array}{cccc} \sum{j \in \mathcal{N}{1}} a{1 j}(t) & -a_{12}(t) & \ldots & -a_{1 N}(t) \ -a_{21}(t) & \sum_{j \in \mathcal{N}{2}} a{2 j}(t) & \ldots & -a_{2 N}(t) \ \vdots & \vdots & \ddots & \vdots \ -a_{N 1}(t) & -a_{N 2}(t) & \ldots & \sum_{j \in \mathcal{N}{N}} a{N j}(t) \end{array}\right]$$

## cs代写|复杂网络代写complex network代考|Main results

Based on the analysis in the last section, one has that, for each $s \in \mathcal{P}, \mathcal{G}\left(\widetilde{\mathcal{A}}^{s}\right)$ contains a directed spanning tree rooted at agent 0 . Denote the Laplacian matrix of $\mathcal{G}\left(\widetilde{\mathcal{A}}^{s}\right)$ by $\widetilde{\mathcal{L}}^{s}$. Without loss of generality, let
$$\begin{gathered} \tilde{\mathcal{L}}^{s}=\left[\begin{array}{cc} 0 & \mathbf{0}{N}^{T} \ \mathbf{P}^{s} & \overline{\mathcal{L}}^{s} \end{array}\right] \in \mathbb{R}^{(N+1) \times(N+1)}, \ \overline{\mathcal{L}}^{s}=\left[\begin{array}{cccc} \sum{j \in \mathcal{N}{1}} a{1 j}^{s} & -a_{12}^{s} & \cdots & -a_{1 N}^{s} \ -a_{21}^{s} & \sum_{j \in \mathcal{N}{2}} a{2 j}^{s} & \cdots & -a_{2 N}^{s} \ \vdots & \vdots & \ddots & \vdots \ -a_{N 1}^{s} & -a_{N 2}^{s} & \cdots & \sum_{j \in \mathcal{N}{N}} a{N j}^{s} \end{array}\right], \end{gathered}$$
where $\mathbf{P}^{s}=-\left[a_{10}^{s}, \ldots, a_{N 0}^{s}\right]^{T}, a_{i 0}^{s}=c_{i}^{s}$, and $a_{i 0}^{s}=1$ if agent $i$ in graph $\mathcal{G}\left(\mathcal{A}^{s}\right)$ is pinned, $i=1, \ldots, N$. According to the condition that, for each $s \in \mathcal{P}, \mathcal{G}\left(\widetilde{\mathcal{A}}^{s}\right)$ contains a directed spanning tree, then $\overline{\mathcal{L}}^{s}$ is a nonsingular $M$-matrix. Then, by using Lemma $2.15$, we can get some positive definite matrices $\left(\bar{\Phi}^{\sigma\left(\bar{t}{\rho}\right)} \overline{\mathcal{L}}^{\sigma\left(\bar{t}{\rho}\right)}+\left(\overline{\mathcal{L}}^{\sigma\left(\bar{t}{\rho}\right)}\right)^{T} \bar{\Phi}^{\sigma\left(\bar{t}{\rho}\right)}\right)$ by letting $\bar{\Phi}^{\sigma\left(\bar{t}{\rho}\right)}=\operatorname{diag}\left{\phi{1}^{\sigma\left(\bar{t}{\rho}\right)}, \ldots, \phi{N}^{\sigma\left(\bar{t}{\rho}\right)}\right}$ with $\phi^{\sigma\left(\bar{t}{\rho}\right)}=\left[\phi_{1}^{\sigma\left(\bar{t}{\rho}\right)}, \ldots, \phi{N}^{\sigma\left(\bar{t}{\rho}\right)}\right]^{T}$ satisfies $\left(\overline{\mathcal{L}}^{\sigma\left(\bar{t}{\rho}\right)}\right)^{T} \phi^{\sigma\left(\bar{t}{\mu}\right)}=\mathbf{1}{N} .$

For notational brevity, one may let
where $\mathcal{Q}{\text {sub }}^{t{\min }^{p}}=\left{\sigma(t): t \in\left[t_{\min }^{\rho}, \bar{t}{\rho+1}\right)\right}, \tilde{\lambda}{0}^{i}$ is the smallest eigenvalue of $\overline{\mathcal{L}}^{i}+$ $\left(\bar{\Phi}^{\sigma\left(\bar{t}{\rho}\right)}\right)^{-1}\left(\overline{\mathcal{L}}^{i}\right)^{T} \bar{\Phi}^{\sigma\left(\bar{t}{\rho}\right)}$. Furthermore, let
$$\mu=\max {i \neq j, i, j \in \mathcal{P}} \frac{\phi{\max }^{i}}{\phi_{\min }^{j}},$$
where $\phi_{\min }^{s}=\min {r=1, \ldots, N} \phi{r}^{s}, \phi_{\max }^{s}=\max {r=1, \ldots, N} \phi{r}^{s}$, for each $s \in \mathcal{P}$.
Based on the above analysis, one may get the following theorem which summarizes the main results of this section.

## cs代写|复杂网络代写complex network代考|Model formulation

X˙一世(吨)=F(X一世(吨),吨)+一个∑j=1ñ一个一世j(吨)(Xj(吨)−X一世(吨))

X˙一世(吨)=F(X一世(吨),吨)−一个∑j=1ñl一世j(吨)Xj(吨),

The control goal in this section is to design pinning controllers for some appropriately selected agents in (5.2) such that the states of each agent in the considered network will approachs(吨)什么时候吨方法+∞， IE，林吨→∞|X一世(吨)−s(吨)|=0， 对全部一世=1,…,ñ并且任意给定初始条件，其中

s˙(吨)=F(s(吨),吨).

X˙一世(吨)=F(X一世(吨),吨)−一个∑j=1ñl一世j(吨)Xj(吨)−一个C一世(吨)(X一世(吨)−s(吨))

## cs代写|复杂网络代写complex network代考|Main results

where\mathcal{Q}{\text {sub }}^{t{\min }^{p}}=\left{\sigma(t): t \in\left[t_{\min }^{\rho} , \bar{t}{\rho+1}\right)\right}, \波浪号{\lambda}{0}^{i}\mathcal{Q}{\text {sub }}^{t{\min }^{p}}=\left{\sigma(t): t \in\left[t_{\min }^{\rho} , \bar{t}{\rho+1}\right)\right}, \波浪号{\lambda}{0}^{i}是的最小特征值大号¯一世+ (披¯σ(吨¯ρ))−1(大号¯一世)吨披¯σ(吨¯ρ). 此外，让

μ=最大限度一世≠j,一世,j∈磷φ最大限度一世φ分钟j,

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