Stability analysis, or, Can we prove this limit to be zero? Let's think about this ODE 
$$
\dot{y}(t) = \gamma \left(g(t) - y(t)\right),\quad \gamma > 0,
$$
where $g(t)$ is a Lipschitz continuous function. It can be seen that the value of $y(\cdot)$ goes with the value of $g(\cdot)$. And increasing the value of $\gamma$ make this more sensitive (or to say, make this reaction faster). We can see from intuition that when $\gamma$ is large enough and this system has run for a while, $y(t)$ should be very close to $g(t)$. And now, I have two request.
1. Please give a stability analysis of this system for when $\gamma$ goes to infinity.
2. Or furthermore, can you prove that when $\gamma$ goes to infinity, the error $E(t) = g(t) - y(t)$ goes to zero? (i.e., if $t_1 > 0$, can you prove $\lim_{\gamma \rightarrow \infty}E(t_1) = 0$?)
If we solve the ODE directly, we obtain
$$
y(t) = y(0)e^{-\gamma t} + \gamma e^{-\gamma t} \int_{0}^{t}g(t)e^{\gamma t'}\text{d}t'.
$$
Then, we have
$$
\lim_{\gamma \rightarrow \infty}E(t_1) = \lim_{\gamma \rightarrow \infty} \left(g(t_1) - y(t_1)\right) = \lim_{\gamma\rightarrow\infty}\left[g(t_{1})-\gamma e^{-\gamma t}\int_{0}^{t_{1}}g(t)e^{\gamma t'}\text{d}t'\right].
$$
I can't see how this limit can be proved zero. Please give some stability analysis of the ODE for when $\gamma$ goes to infinity. Or if you could and if the conjecture is right, please prove the limit to be zero.
Any help is appreciated.
 A: I think i have a proof, here $M$ is the Lipschitz constant for $g$. 
Write $E(t)=g(t)-\gamma e^{-\gamma t}\int_0^t e^{\gamma \tau} g(\tau) d\tau$
$lim_{\gamma->\infty} ||E(t)|| = lim_{\gamma->\infty} ||\gamma e^{-\gamma t}\int_0^t e^{\gamma \tau} g(t)d\tau-\gamma e^{-\gamma t}\int_0^t e^{\gamma \tau} g(\tau)d\tau$||
=>
R.H.S=$lim_{\gamma->\infty} ||\gamma e^{-\gamma t}\int_0^t e^{\gamma \tau} [g(t)-g(\tau)]d\tau ||$
Now by Lipschitz, $||g(t)-g(\tau)||<M|t-\tau|$
=>
R.H.S $\leq lim_{\gamma->\infty} ||\gamma e^{-\gamma t}\int_0^t e^{\gamma \tau}M |t-\tau|d\tau|| $
Remember $t>\tau$=>
R.H.S $\leq lim_{\gamma->\infty} ||M\gamma e^{-\gamma t}[\int_0^t te^{\gamma \tau}d\tau-\int_0^t\tau e^{\gamma \tau}d\tau]|| $
Integrate first integral and apply integration by parts on the second integral to obtain
$lim_{\gamma->\infty} ||M\gamma e^{-\gamma t}[(t/\gamma)e^{\gamma t}-(t/\gamma)e^{\gamma t}-\int_0^t (1/\gamma)e^{\gamma \tau}d\tau]|| $
=>
$lim_{\gamma->\infty} ||M\gamma e^{-\gamma t}[ (1/\gamma^2)e^{\gamma t}]|| $
R.H.S $= 0$
Note: In the last two steps, I have only kept the slowest decaying term (as $\gamma->\infty$). There are two other terms but they die fast exponentially with factor $\gamma t$.
