Piecewise function with form determined by current value

Question

I would like to create a piecewise function that determines the dynamics of a variable $x_t$ as follows:

Given a starting value $x_0$ and a parameter $\hat{x}$:

xHat = 0.10;
x0 = 0.15;

Let $x_t$ evolve according to $x_t=e^{-1.5 t} x_0$ if $x_t$ is above $\hat{x}$, according to $x_t=e^{-1.2 t} x_0 + 0.0027(1 - e^{-1.2 t})$ once $x_t$ goes below $\hat{x}$ and in general switch between the two evolutions based on the current value of $x_t$. I have set it up as follows:

xP[t_] := E^(-1.5*t)*x0
xN[t_] := E^(-1.2*t)*x0 + 0.0027*(1 - E^(-1.2*t))
myfun = Piecewise[{{xP[t], xP[t] > xHat}, {xN[t], xN[t] < xHat}}]
Plot[{xP[t], xN[t], myfun}, {t, 0, 1}, PlotRange -> All]

But this does not work. Here is the plot:

Instead I would expect the line to start on the blue trajectory, and then jump to the yellow one when the function goes below 0.1.

How should I correctly define the piecewise function?

Answer 1

It is not clear what you want. Your piecewise function does not define a function value for x approx. between 0.27 and 0.34. Therefore myfun uses the default value of zero.

If you want that myfun switches from xP[t] to xN[t] if xP[t]<xHat, you must write:

myfun[t_] = Piecewise[{{xP[t], xP[t] > xHat}, {xN[t], xP[t] < xHat}}];
Plot[{xP[t], xN[t], myfun[t]}, {t, 0, 1}, PlotRange -> All]