# 数学代写|常微分方程代写ordinary differential equation代考|MATH53

## 数学代写|常微分方程代写ordinary differential equation代考|Stability of Linear Systems

A linear homogeneous differential equation has a rest point at the origin. We will use our results about the solutions of constant coefficient homogeneous linear differential equations to study the stability of this rest point. The next result is fundamental.

Theorem 2.34. Suppose that $A$ is an $n \times n$ (real) matrix. The following statements are equivalent:
(1) There is a norm ||$_a$ on $\mathbb{R}^n$ and a real number $\lambda>0$ such that for all $v \in \mathbb{R}^n$ and all $t \geq 0$,
$$\left|e^{t A} v\right|_a \leq e^{-\lambda t}|v|_a .$$
(2) If ||$_g$ denotes a norm on $\mathbb{R}^n$, there is a constant $C \geq 1$ and a real number $\lambda>0$ such that for all $v \in \mathbb{R}^n$ and all $t \geq 0$,
$$\left|e^{t A} v\right|_g \leq C e^{-\lambda t}|v|_g .$$
(3) Every eigenvalue of A has negative real part.
Moreover, if $-\lambda$ exceeds the largest of all the real parts of the eigenvalues of $A$, then $\lambda$ can be taken to be the decay constant in (1) or (2). Also, if every eigenvalue of $A$ has negative real part, then the zero solution of $\dot{x}=A x$ is asymptotically stable.
Proof. We will show that $(1) \Rightarrow(2) \Rightarrow(3) \Rightarrow(1)$.
To show (1) $\Rightarrow$ (2), let ||$_a$ be the norm in statement (1) and ||$_g$ the norm in statement (2). Because these norms are defined on the finite dimensional vector space $\mathbb{R}^n$, they are equivalent. In particular, there are constants $K_1>0$ and $K_2>0$ such that for all $x \in \mathbb{R}^n$ we have
$$K_1|x|_g \leq|x|_a \leq K_2|x|_g .$$

## 数学代写|常微分方程代写ordinary differential equation代考|Stability of Nonlinear Systems

Theorem $2.34$ states that the zero solution of a constant coefficient homogeneous linear system is asymptotically stable if the spectrum of the coefficient matrix lies in the left half of the complex plane. The principle of linearized stability states that the same result is true for steady state solutions of nonlinear equations provided that the system matrix of the linearized system along the steady state solution has its spectrum in the left half plane. As stated, this principle is not a theorem. However, in this section we will formulate and prove a theorem on linearized stability which is strong enough for most applications. In particular, we will prove that a rest point of an autonomous differential equation $\dot{x}=f(x)$ in $\mathbb{R}^n$ is asymptotically stable if all eigenvalues of the Jacobian matrix at the rest point have negative real parts. Our stability result is also valid for some nonhomogeneous nonautonomous differential equations of the form
$$\dot{x}=A(t) x+g(x, t), \quad x \in \mathbb{R}^n$$
where $g: \mathbb{R}^n \times \mathbb{R} \rightarrow \mathbb{R}^n$ is a smooth function.
A fundamental tool used in our stability analysis is the formula, called the variation of constants formula, given in the next proposition.

Proposition $2.37$ (Variation of Constants Formula). Consider the initial value problem
$$\dot{x}=A(t) x+g(x, t), \quad x\left(t_0\right)=x_0$$
and let $t \mapsto \Phi(t)$ be a fundamental matrix solution for the homogeneous system $\dot{x}=A(t) x$ that is defined on some interval $J_0$ containing $t_0$. If $t \mapsto$ $\phi(t)$ is the solution of the initial value problem defined on some subinterval of $J_0$, then
$$\phi(t)=\Phi(t) \Phi^{-1}\left(t_0\right) x_0+\Phi(t) \int_{t_0}^t \Phi^{-1}(s) g(\phi(s), s) d s$$

# 常微分方程代考

## 数学代写|常微分方程代写ordinary differential equation代考|Stability of Linear Systems

(1) 存在范数 $|_a$ 上 $\mathbb{R}^n$ 和一个实数 $\lambda>0$ 这样对于所有人 $v \in \mathbb{R}^n$ 和所有 $t \geq 0$ ，
$$\left|e^{t A} v\right|_a \leq e^{-\lambda t}|v|_a .$$
(2) 如果 $|_g$ 表示规范 $\mathbb{R}^n$ ，有一个常数 $C \geq 1$ 和一个实数 $\lambda>0$ 这样对于所有人 $v \in \mathbb{R}^n$ 和所有 $t \geq 0$ ，
$$\left|e^{t A} v\right|_g \leq C e^{-\lambda t}|v|_g .$$
(3) A的每个特征值都有负实部。

$$K_1|x|_g \leq|x|_a \leq K_2|x|_g .$$

## 数学代写|常微分方程代写ordinary differential equation代考|Stability of Nonlinear Systems

$$\dot{x}=A(t) x+g(x, t), \quad x \in \mathbb{R}^n$$

$$\dot{x}=A(t) x+g(x, t), \quad x\left(t_0\right)=x_0$$

$$\phi(t)=\Phi(t) \Phi^{-1}\left(t_0\right) x_0+\Phi(t) \int_{t_0}^t \Phi^{-1}(s) g(\phi(s), s) d s$$

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