The error bounds for an ordinary differential equation: \begin{equation} \dot{x}(t) = f(x(t)) \end{equation} with respect to initial conditions $x(t_0) = x_0$, $\hat{x}(t_0)=\hat{x}_0$ \begin{equation} \left|\left|\hat{x}(t)-x(t)\right|\right| \leqslant e^{L\left|t-t_0\right|}\left|\left|\hat{x}(0)-x(0)\right|\right| \end{equation} where L is a Lipschitz constant of $f$ with respect to $x$. For a proof, see e.g. Stoer, Bulirsch.
I have read in an article [Arnold] that for a system: \begin{equation} \dot{x}(t) = f(x(t), u(t)) \end{equation} hold a similar bounds with respect to different input signals $u(t)$ and $\tilde{u}(t)$: \begin{equation} \left|\left|\tilde{x}(t)-x(t)\right|\right| \leqslant \left(Ce^{L\left|t-t_0\right|}-1\right)\max_{s\in[t_0,t]}\left|\left|\tilde{u}(s)-u(s)\right|\right| \end{equation} where L is a Lipschitz constant of $f$ with respect to $x$. Initial conditions are assumed to be equal $\tilde{x}(0) = x(0) = x_0$. An input signal $\tilde{u}(t)$ is assumed to be a polynomial approximation of $u(t)$.
I would like to ask for a help with proving this statement. I went through the following steps: \begin{align} &x(t) = x_0 + \int^t_{t_0}f(x(s), u(s)) \mathrm{ds}\\ &\tilde{x}(t) = x_0 + \int^t_{t_0}f(\tilde{x}(s), \tilde{u}(s)) \mathrm{ds}\\ &\tilde{x}(t) - x(t) = \int^t_{t_0} \left[f(\tilde{x}(s), \tilde{u}(s)) - f(x(s), u(s)) \right] \mathrm{ds}\\ &\left|\left|\tilde{x}(t) - x(t)\right|\right| \leqslant \int^t_{t_0} \left|\left|f(\tilde{x}(s), \tilde{u}(s)) - f(x(s), u(s)) \right|\right| \mathrm{ds} \end{align} I get stuck at this step. I am not sure whether I am supposed to use Lipschitz condition: \begin{equation} \left|\left|f(\tilde{x}(t), U(t)) - f(x(t), U(t)) \right|\right| \leqslant L\left|\left|\tilde{x}(t) - x(t)\right|\right| \end{equation} and whether this is a correct formulation of Lipschitz condition with respect to x or whether I need to somehow use the fact about polynomial approximation. Any assistance is appreciated.
