If Zero solution is asymptotically stable, How to prove $\lim_{t \to +\infty} \|\Phi(t)\|=0$?

Asked Jan 03 '23 at 11:23

Active Jan 03 '23 at 14:01

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The system is $$\frac{d x}{d t} =A(t) x,\quad A(t)=(a_{ij}(t))_{n\times n}$$

$ \Phi(t)$ is the fundamental solution of the system. If zero solution is asymptotically stable, How to prove $$ \lim_{t \to +\infty} \|\Phi(t)\|=0 $$

I know the conclusion can prove the assumption. But I can't prove the above proposition by myself.

Asymptotic Stability of zero solution:

Suppose $x(t,t_0, x_0)=\Phi(t)\Phi^{-1}(t_0)x_0$ is the solution of the system, $$\forall \varepsilon >0, \exists \delta = \delta(\varepsilon), s.t. \|x_0\|< \delta \ \&\ t>t_0, \|x(t,t_0, x_0)\| \leq \varepsilon . $$

edited Jan 03 '23 at 14:01

asked Jan 03 '23 at 11:23

Nichts__97

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Don´t You mean $\lim_{\color{red}t\rightarrow \infty}...$? – Peter Melech Jan 03 '23 at 11:37
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You can get proper norm bars $|$ by using \| rather than ||. – joriki Jan 03 '23 at 11:45
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Depends entirely on your definition of asymptotically stable... – Hyperbolic PDE friend Jan 03 '23 at 11:51
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Can you write $x(t)$ using $\Phi(t)$ and $x_0$? It should hold for all possible $x_0$ – Arastas Jan 03 '23 at 13:57
I revised my question. – Nichts__97 Jan 04 '23 at 12:09

If Zero solution is asymptotically stable, How to prove $\lim_{t \to +\infty} \|\Phi(t)\|=0$?

0 Answers0