# Changing the quality of the transient process in a nonlinear system (in Mathematica)

I urgently need advice and help.

I have a system of differential equations like this:

$$\begin{cases} \frac{dx}{dt} = y[t] \cdot \alpha \cdot sin(\omega t) + \frac{d}{dt}(\alpha \cdot sin(\omega t)) \\ \frac{dy}{dt} + h \cdot y(t) = \frac{d}{dt}(e^{-(x[t] - 2)^2}) \end{cases}$$

Parameters: $$\alpha = 0.3, h = 1, \omega = 2 \pi 0.5, x(0)=1/4, y(0)=0$$

It corresponds to the following structural scheme:

The code that simulates such a system is shown below:

 ClearAll["Global*"]

pars = {\[Alpha]1 = 0.3, h1 = 1, \[Omega]1 = 2 Pi 0.5}

extr = Exp[-(x[t] - 2)^2]

sys =
NDSolve[{x'[t] ==
y[t] \[Alpha]1 Sin[\[Omega]1 t] +
D[\[Alpha]1 Sin[\[Omega]1 t], t],
y'[t] + h1 y[t] == D[extr, t], x[0] == 1/4, y[0] == 0},
x, {t, 0, 500}]


The numerical solution is presented below:

Plot[{Evaluate[x[t] /. sys]}, {t, 0, 150}, PlotRange -> Full,
PlotPoints -> 50]
`

It can be seen that the transition process is a transition from the initial point to the final one with a certain character.

I need to change this character i.e. make the transition from one point to another exponentially. Like this:

What are the ways to solve this problem? What to do, add a regulator or manipulate the system of differential equations?