Continuity of a solution of a differential equation with respect to a time-dependent input?

Question

I am interested in a question of the following form, and I'm asking because the question seems so simple to pose (if not to answer) that I figure there must be some discipline interested in answering it. I am interested for some particular cases but I would like to know where it would fit into the study of differential equations more generally.

The question can be written like this: given $\epsilon > 0$, we want to find a $\delta > 0$...

$$ \lvert\lvert \phi(t)-\psi(t) \rvert\rvert < \delta $$ Such that we can say for the initial value problem $x'(t) = f(x(t),g(t)),\ x(t_0)=x_0$: $$ \lvert\lvert x(t)_{g(t) = \phi(t)}-x(t)_{g(t)=\psi(t)}\rvert \rvert<\epsilon $$ My case has all sorts of nice properties one might expect ($f \in C^\infty$, non-negativity of $x$, smoothness and boundedness of $g$, etc), but I figured that someone might recognize it more immediately if I posed it in greater generality.

It seems to have superficial similarities with structural stability (my problem came out of similar motivations to what structural stability intends to address), but in that case the input is not time-based, and with control theory, but in that case they're interested in (as far as I can tell) different kinds of questions.

Answer 1

If one assumes that $f$ is nice enough to be jointly Lipschitz continuous in both arguments, $$ \|f(x_1,g_1)-f(x_2,g_2)\|\le L\|x_1-x_2\|+M\|g_1-g_2\|, $$ then that directly results in a differential inequality that can be handled by the Grönwall lemma $$ \|\dot x_1(t)-\dot x_2(t)\|\le L\|x_1(t)-x_2(t)\|+M\|ϕ(t)-ψ(t)\| $$ which gives a bound $$ \|x_1(t)-x_2(t)\|\le \|x_1(0)-x_2(0)\|e^{Lt}+M\int_0^t e^{L(t-s)}\|ϕ(s)-ψ(s)\|\,ds $$ so that under the further assumptions $$ \|x_1(t)-x_2(t)\|\le Mδ\frac{e^{Lt}-1}{L}. $$ For a finite time interval $t\in [0,T]$ this bound can be made smaller than $ϵ$ by selecting $δ$ accordingly.