Proving continuous dependence on initial conditions for flows

dynamical systems

I'm looking for assistance in solving a dynamical systems problem.

Considering the flow $\phi (t;x): \mathbb{R}\times \mathbb{R}^n \rightarrow \mathbb{R}^n$, that isn't necessarily associated to an ODE, for which I need to prove continuous dependence on initial conditions. I am given the following: let the initial condition $x_0 \in \mathbb{R}^n$ be given and show that for all $T > 0$ and $\delta > 0$ there is an $\epsilon > 0$ such that $||\phi(T;x_0)-\phi(T;\tilde{x_0})|| < \delta$ for all $\tilde{x_0}$ with $||x_0-\tilde{x_0}|| < \epsilon$.

I'm not really sure where to start with this problem. I believe it might need to be solved in three steps, proving first Lipschitz Dependence on Initial Conditions, then Smooth Dependence on Initial Conditions and finally Continuous Dependence on Parameters. However, I can't seem to make heads nor tails of it.

Any and all assistance in solving this problem would be greatly appreciated. Thanks in advance!

Best Answer

Firstofall, for us to guarantee properties of the flow you must have some extra information, I'll address here the case where $\phi(t,x)$ is the only solution of the system

$\begin{cases} x'(t)=f(t,x(t))\\ x(t_0)=x \end{cases}$$\star$

where $f:I\times U{\Bbb R^{n+1}}\rightarrow{\Bbb R^n}$ is a lipschitz function, that is $L=\mbox{sup}_{(t,x)\neq(t,y)\in I\times U}\frac{|| f(t,x)-f(t,y) ||} {||x-y||}<\infty$ is it's Lipschitz constant uniform of time.

We will use the following version of Gronwall's inequality

if $\psi\leq a +\int_0^t(b\psi(s)+c)ds$, then $\psi\leq a e^{bt}+\frac{c}{b}(e^{bt}-1)$

Then, $\star$ is equivalent to the following system $x(t)=x_0+\int^{t}_{0}f(s,x(s))ds$,

or $\phi(t,x)=x+\int^{t}_{0} \frac{\partial\phi(s,x(s))}{\partial t}ds$=$x(t)=x_0+\int^{t}_{0}f(s,\phi(s,x))ds$, $\star\star$

then $||\phi(t,x)-\phi(t,y)||\leq ||x-\tilde{x}||+\int^{t}_{0}||f(s,\phi(s,x))-f(s,\phi(s,y))||ds\leq ||x-\tilde{x}||+\int^{t}_{0}L||\phi(s,x)-\phi(s,y)||ds$

By the afore stated Gronwall's lemma $||\phi(t,x)-\phi(t,y)||\leq||x-\tilde{x}||e^{L|t-t_0|}$

This proofs your question for this case, since given $T\in I$ and any $\epsilon>0$(in your question it says to show there is some $\epsilon>0$ but any choice does there trick and I'm afraid it might be a mistake of the question), then choose $\delta = \epsilon/e^{L|T-t_0|}>0$

Then $||\phi(t,x)-\phi(t,y)||\leq||x-\tilde{x}||e^{L|t-t_0|}<\delta$ for $||x-\tilde{x}||<\epsilon$

For the dynamical systems where the flow is at least differentiable see https://math.stackexchange.com/a/528109/532993

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