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## cs代写|复杂网络代写complex network代考|SWITCHED SYSTEM THEORY

This section introduces the solutions of differential systems, MLFs, and stability theory under slow switching. For more detailed discussions, we refer the reader to Chapter 3 in [82].

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.

## 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., $\sigma(t)=\lim {t \searrow} t \sigma(t)$, 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|SWITCHED SYSTEM THEORY

$$\dot{x}(t)=f(t, x(t)), x(t) \in \mathbb{R}^n, t \in\left[t_0,+\infty\right),$$

$$|f(t, x(t))-f(t, y(t))| \leq L_0|x(t)-y(t)|,$$

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

$$\dot{x}(t)=f_p(t, x(t)), x(t) \in \mathbb{R}^n, p \in 1, \ldots, \kappa,$$

$$\dot{x}(t)=f_{\sigma(t)}(t, x(t)) .$$

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assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师