# Liapunov Stability

For Cauchy Problem, $I$ is a bounded interval, and $y(t)\in C^n(I)$,

$y^{(n)}=f(t,y(t))$,

where $t\in I$, I wonder the stability for this problem. Thus I looked up the definition for the stability in the sense of Liapunov(zero stable).

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Note: I didn’t give the Cauchy data for this problem, we had assumed all the initial data were given.

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Definition: The stability means:

for any perturbation $(\delta_0,\delta(t))$ satisfying

$|\delta_0|<\epsilon, |\delta(t)|<\epsilon,\forall t\in I$

with $\epsilon>0$ sufficient small to ensure that the solution $z(t)$ to the perturbed problem does exist., then

$\exists C>0$ independent of $\epsilon$ such that $|y(t)-z(t)|$

Claim 1: If our force term $f(t,y(t))$ is uniformly continuous, then we can assure the stability(Liapunov).

Proof: Let $\omega(t)=z(t)-y(t)$, we have

$\omega^{(n)}(t)=f(t,z)-f(t,y)+\delta(t)$.

Hence,

$\displaystyle\omega(t)=\delta_0(t)+\int\int\int\dots\int_{\Omega} [f(s,z(s))-f(s,y(s))]\mathrm{d}s+\int\int\int\dots\int_{\Omega}\delta(s)\mathrm{d}s$

Thanks to the uniform-continuity.

$\displaystyle|\omega(t)|\le \delta_0+\int\int\int\dots\int_{\Omega}L|z(s)-y(s)|\mathrm{d}s+\int\int\int\dots\int_{\Omega}\delta(s)\mathrm{d}s$

Which is,

$\displaystyle|\omega(t)|\le\delta_0+\int\int\int\dots\int_{\Omega}L|\omega(s)|\mathrm{d}s+\int\int\int\dots\int_{\Omega}\delta(s)\mathrm{d}s$

equivalent to:

$\displaystyle|\omega(t)|\le L\int\int\int\dots\int_{\Omega} |\omega(s)|\mathrm{d}s+(1+meas(\Omega))\epsilon$

By Gronwall lemma,
$\displaystyle|\omega(t)|\le (1+meas(\Omega))\epsilon \exp{(L\cdot meas(\Omega))}$

Where $\Omega=[0,t]^n$.

Here $C=(1+K)\exp(LK)$, $K=\max_{t\in I} t^n$.

Here I have to make some comment: the initial data, we say with pertubation $f^{(k)}=\delta_k(t)$, thus the integral need another term $\displaystyle\frac{T^n-1}{T-1}\epsilon$, which doesn’t matter much on this problem.