# Lyapunov theorem for LTV systems

Consider continuous-time LTV systems of the form
$$\dot{x}=A \mid(t) x(t)$$
Let $P(t)$ be a time-varying matrix that satisfies
$$\eta I \leq P(t) \leq \rho I \quad \eta, \rho>0$$
as well as
$$A(t)^{T} P(t)+P(t) A(t)+\dot{P}(t) \leq 0$$
Then the LTV system is uniformly stable. If, in addition,
$$A(t)^{T} P(t)+P(t) A(t)+\dot{P}(t) \leq-\nu I \quad \nu>0$$
then the LTV system is uniformly exponentially stable.

# UAS is equivalent to UES for LTV systems

Theorem 2: UAS is equivalent to UES
The following two statements are equivalent for LTV systems 1. $\dot{x}=A(t) x$ is uniformly asymptotically stable (UAS)
2. $\dot{x}=A(t) x$ is uniformly exponentially stable (UES)
It is obvious that UES $\Longrightarrow$ UAS. What about the converse?

# Proof of UAS $\Longrightarrow$ UES

Let’s assume system is UAS. By definition there exist $\gamma, T>0$ such that
$$\begin{gathered} \left\|\Phi\left(t, t_{0}\right)\right\| \leq \gamma \text { for all } t, t_{0} t \geq t_{0} \\ \left\|\Phi\left(t, t_{0}\right)\right\| \leq \frac{1}{2} \text { for all } t, t_{0}, t \geq t_{0}+T \end{gathered}$$
Now
\begin{aligned} \left\|\Phi\left(t_{0}+k T, t_{0}\right)\right\| &=\left\|\Phi\left(t_{0}+k T, t_{0}+(k-1) T\right) \cdots \Phi\left(t_{0}+T, t_{0}\right)\right\| \\ & \leq\left\|\Phi\left(t_{0}+k T, t_{0}+(k-1) T\right)\right\| \cdots\left\|\Phi\left(t_{0}+T, t_{0}\right)\right\| \\ & \leq \frac{1}{2} \cdots \frac{1}{2}=\frac{1}{2^{k}} \end{aligned}

Therefore
\begin{aligned} \left\|\Phi\left(t, t_{0}\right)\right\| &=\left\|\Phi\left(t, t_{0}+k T\right) \Phi\left(t_{0}+k T, t_{0}\right)\right\| \\ & \leq\left\|\Phi\left(t, t_{0}+k T\right)\right\|\left\|\Phi\left(t_{0}+k T, t_{0}\right)\right\| \\ & \leq \gamma \frac{1}{2^{k}}=2 \gamma\left(\frac{1}{2}\right)^{k+1} \\ & \leq 2 \gamma\left(\frac{1}{2}\right)^{\frac{t-t_{0}}{T}} \\ &=2 \gamma e^{-\ln (2)\left(\frac{t-t_{0}}{T}\right)} \end{aligned}
Let $M=2 \gamma$ and $\mu=\frac{\ln 2}{T}$ to obtain
$$\left\|\Phi\left(t, t_{0}\right)\right\| \leq M e^{-\mu\left(t-t_{0}\right)}$$

# Converse theorem of Lyapunov for LTV

Theorem 3: Converse Lyapunov theorem for UES
Assume that the LTV system
$$\dot{x}=A(t) x$$
is UES and assume $\|A(t)\| \leq \alpha .$ Then there exists Lyapunov function $V(t, x)=x^{T} P(t) x$ where
$$P(t)=\int_{t}^{+\infty} \Phi^{T}(\tau, t) \Phi(\tau, t) d \tau$$
satisfying all conditions of the UES Lyapunov theorem

# Stability implies Robustness

Consider the linear, time-invariant system
$$\dot{x}=A x$$
which is exponentially stable. Now consider the perturbed LTV system
$$\dot{x}=(A+F(t)) x \quad\|F(t)\|_{2}<\beta$$
Is the perturbed LTV system also uniform exponentially stable? For what values of $\beta$ ?

Since $\dot{x}=A x$ is exponentially stable, we can solve the Lyapunov equation
$$A^{T} P+P A=-I$$
Therefore $V(x)=x^{T} P x$ is a Lyapunov function for the LTI system. Can we use this as a Lyapunov function for the perturbed LTV system? To
show that the LTV systems is UES, $V(x)$ must satisfy for $\eta, \rho, \nu>0$
$$\begin{gathered} \eta\|x\|_{2}^{2} \leq V(x) \leq \rho\|x\|_{2}^{2} \quad \eta, \rho>0 \\ \dot{V} \leq-\nu\|x\|_{2}^{2} \quad \nu>0 \end{gathered}$$
From the Raleigh-Ritz inequality we already have
$$\lambda_{\min }(P) x^{T} x \leq x^{T} P x \leq \lambda_{\max }(P) x^{T} x$$

\begin{aligned} \dot{V} &=\dot{x}^{T} P x+x^{T} P \dot{x}=x^{T}(A+F(t))^{T} P x+x^{T} P(A+F(t)) x \\ &=x^{T}\left(A^{T} P+P A\right) x+x^{T}(F(t) P+P F(t)) x \\ &=-x^{T} x+x^{T}(F(t) P+P F(t)) x \\ &=-\nu x^{T} x ? \end{aligned}
Now the perturbation term can be bounded as follows
$$x^{T}(F(t) P+P F(t)) x \leq 2\|F(t)\|_{2}\|P\|_{2} x^{T} x \leq 2 \beta \lambda_{\max }(P) x^{T} x$$
Therefore $\dot{V} \leq-x^{T} x+2 \beta \lambda_{\max }(P) x^{T} x$. Thus if
$$-1+2 \beta \lambda_{\max }(P)=\nu<0 \Longrightarrow \dot{V} \leq-\nu x^{T} x$$
This proves the following result

##### Math 130A，Ordinary Differential Equations

Introduction to probability and stochastic porcesses

Theory 太多 …Practice题目有点hold 不住？