### cs代写|复杂网络代写complex network代考|Solutions of differential systems

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## cs代写|复杂网络代写complex network代考|Solutions of differential systems

Consider the system
$$\dot{x}(t)=f(t, x(t)), x(t) \in \mathbb{R}^{n}, t \in\left[t_{0},+\infty\right),$$
where $f(t, x(t)):\left[t_{0},+\infty\right) \times \mathbb{R}^{n} \mapsto \mathbb{R}^{n}$. Denote by $x_{0}$ the initial value $x\left(t_{0}\right)$. A classical solution for the Cauchy problem of $(2.5)$ with $x\left(t_{0}\right)=x_{0}$ on $\left[t_{0}, T\right]$ is a continuously differentiable map $x(t):\left[t_{0}, T\right] \mapsto \mathbb{R}^{n}$ that satisfies (2.5). According to the well-known Peano’s theorem, one knows that if the function $f$ is continuous in a neighborhood of $t_{0}, x_{0}$, system (2.5) has at least one classical solution defined in a neighborhood of $t_{0}, x_{0}$. To proceed, the concept of Lipschitz condition is introduced.

Definition $2.2$ [27] A function $f(t, x(t)):\left[t_{0},+\infty\right) \times \mathbb{R}^{n} \mapsto \mathbb{R}^{m}$ is said to be globally Lipschitz in $x(t)$ uniformly over $t$ if there exists a positive scalar $L_{0}$ such that
$$|f(t, x(t))-f(t, y(t))| \leq L_{0}|x(t)-y(t)|$$
for all $(t, x(t))$ and $(t, y(t))$.
Theorem 2.1 [27] If $f(t, x(t)):\left[t_{0},+\infty\right) \times \mathbb{R}^{n} \mapsto \mathbb{R}^{n}$ is continuous in $t$ and globally Lipschitz in $x(t)$ uniformly over $t$, then, for all $x_{0} \in \mathbb{R}^{n}$, there exists a unique classical solution of (2.5) over the time interval $\left[t_{0},+\infty\right)$ with initial condition $x_{0}$.

However, since our view is toward systems with switching, the assumption that the function $f$ is continuous in both $t$ and $x(t)$ is too restrictive. The following example shows that, if the function is discontinuous, then classical solution of (2.5) might not exist.

Example 2.1 [27] Discontinuous Vector Field with Nonexistence of Classical Solutions: Consider the function $f(t, x(t)):[0,+\infty) \times \mathbb{R} \mapsto \mathbb{R}$ defined by
$$f(t, x(t))= \begin{cases}-1, & x(t)>0 \ 1, & x(t) \leq 0\end{cases}$$
with initial value $x(0)=0$. It is obviously that the function $f$ is discontinuous at $x(t)=0$. Suppose there exists a classical solution $x(t):[0, T) \mapsto \mathbb{R}$ that satisfies (2.7). Then $\dot{x}(0)=f(0, x(0))=f(0,0)=1$ which implies that, for sufficiently small $t>0, x(t)>0$, and hence $\dot{x}(t)=f(t, x(t))=-1$. But this contradicts the fact that $t \mapsto \dot{x}(t)$ is continuous. Hence, there is no classical solution starting from zero.

It turns out that for the existence and uniqueness result to hold, it is sufficient to demand that $f$ is piecewise continuous in $t[82]$. So we consider the Carathéodory’s solution $x(\cdot)$ that is given by
$$x(t)=x_{0}+\int_{t_{0}}^{t} f(s, x(s)) d s .$$
Note that (2.8) satisfies the differential equation (2.5) almost everywhere.

## cs代写|复杂网络代写complex network代考|Multiple Lyapunov functions

To proceed, the notion of time dependent switching is introduced.
As a special kind of hybrid dynamic system, switched system has been studied for quite some time by researchers from applied mathematics, systems and control fields. Roughly speaking, a switched system is a dynamic system that consists of a number of subsystems and a switching rule that determines switches among these subsystems. Suppose the switched system is generated by the following family of subsystems
$$\dot{x}(t)=f_{p}(t, x(t)), x(t) \in \mathbb{R}^{n}, p \in{1, \ldots, \kappa},$$
together with a switching signal $\sigma(t):\left[t_{0},+\infty\right) \mapsto{1, \ldots, \kappa}$. Note that $\sigma(t)$ is a piecewise constant function that switches at the switching time instants $t_{1}, t_{2}, \ldots$, and is constant on the time interval $\left[t_{k}, t_{k+1}\right), k=0,1, \ldots .$ In this book, we assume $\sigma(t)$ is right continuous, i.e., $\sigma(t)=\lim {t} t \sigma(t)$, and $\inf {k \in \mathbb{N}}\left(t_{k+1}-t_{k}\right) \geq \tau_{m}$ for some given positive scalar $\tau_{m}$ where inf represents the infimum. Please see Figure $2.2$ for an example. Thus the switched systems with time-dependent switching signal $\sigma(t)$ can be described by the equation
$$\dot{x}(t)=f_{\sigma(t)}(t, x(t)) .$$
According to Theorem 2.1, each subsystem has a unique solution over arbitrary interval $\left[t_{k}, t_{k+1}\right), k=0,1, \ldots$, with arbitrary initial value $x\left(t_{k}\right) \in \mathbb{R}^{n}$ if the function $f_{p}$, for each $p=1, \ldots, \kappa$, is globally Lipschitz in $x(t)$ uniformly over $t$. Thus the switched system (2.10) is well defined for arbitrary switching signal $\sigma(t)$ defined above and any given initial value $x\left(t_{0}\right) \in \mathbb{R}^{n}$. Throughout this chapter, we assume that such a globally Lipschitz condition holds for the subsystems, and thus the well-definedness of the switched system is guaranteed. We further assume that $f_{p}\left(t, \mathbf{0}{n}\right)=\mathbf{0}{n}$ for each $p=1, \ldots, \kappa$. Thus, the zero vector is an equilibrium point of the switched system (2.10). Next, some stability notions for the zero equilibrium point of switched systems are introduced.

## cs代写|复杂网络代写complex network代考|Stability under slow switching

We firstly introduce the notion of dwell time. If there exist $\tau_{M} \geq \tau_{m}>0$ such that $\tau_{m} \leq t_{i+1}-t_{i} \leq \tau_{M}<+\infty$ for $i=0,1, \ldots$, then $\tau_{m}$ is called the dwell time of the switching signal $\sigma(t)$ (see Figure $2.4$ for a simple illustration). In the sequel, we assume that the switching signal $\sigma(t)$ always satisfies the condition that $\tau_{m} \leq t_{i+1}-t_{i} \leq \tau_{M}<+\infty$ for $i=0,1, \ldots .$

Theorem 2.2 [82] Suppose all subsystems in the family $(2.10)$ with $p \in{1, \ldots, \kappa}$ are globally exponentially stable, and there exists a Lyapunov function $V_{p}(t, x(t))$ : $\left[t_{0},+\infty\right) \times \mathbb{R}^{n} \mapsto[0,+\infty)$ for each $p \in{1, \ldots, \kappa}$ such that

(1) $a_{p}|x(t)|^{2} \leq V_{p}(t, x(t)) \leq b_{p}|x(t)|^{2}$;
(2) $\frac{\partial V_{p}(t, x(t))}{\partial t}+\frac{\partial V_{p}(t, x(t))}{x(t)} f_{p}(t, x(t)) \leq-c_{p}|x(t)|^{2}$,
with $a_{p}, b_{p}$, and $c_{p}$ being positive scalars. Then the switched system (2.10) is globally exponentially stable if the dwell time
$$\tau_{m}>\frac{\ln \gamma}{\rho}, \gamma=\frac{\max {p=1, \ldots, \kappa} b{p}}{\min {p=1, \ldots, \kappa} a{p}}, \rho=\min \left{\frac{c_{1}}{b_{1}}, \ldots, \frac{c_{\kappa}}{b_{\kappa}}\right} .$$

## cs代写|复杂网络代写complex network代考|Solutions of differential systems

X˙(吨)=F(吨,X(吨)),X(吨)∈Rn,吨∈[吨0,+∞),

|F(吨,X(吨))−F(吨,是(吨))|≤大号0|X(吨)−是(吨)|

F(吨,X(吨))={−1,X(吨)>0 1,X(吨)≤0

X(吨)=X0+∫吨0吨F(s,X(s))ds.

## cs代写|复杂网络代写complex network代考|Multiple Lyapunov functions

X˙(吨)=Fp(吨,X(吨)),X(吨)∈Rn,p∈1,…,ķ,

X˙(吨)=Fσ(吨)(吨,X(吨)).

## cs代写|复杂网络代写complex network代考|Stability under slow switching

(1) 一个p|X(吨)|2≤在p(吨,X(吨))≤bp|X(吨)|2;
(2) ∂在p(吨,X(吨))∂吨+∂在p(吨,X(吨))X(吨)Fp(吨,X(吨))≤−Cp|X(吨)|2,

\tau_{m}>\frac{\ln \gamma}{\rho}, \gamma=\frac{\max {p=1, \ldots, \kappa} b{p}}{\min {p=1 , \ldots, \kappa} a{p}}, \rho=\min \left{\frac{c_{1}}{b_{1}}, \ldots, \frac{c_{\kappa}}{b_ {\kappa}}\right} 。\tau_{m}>\frac{\ln \gamma}{\rho}, \gamma=\frac{\max {p=1, \ldots, \kappa} b{p}}{\min {p=1 , \ldots, \kappa} a{p}}, \rho=\min \left{\frac{c_{1}}{b_{1}}, \ldots, \frac{c_{\kappa}}{b_ {\kappa}}\right} 。

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