Im trying to find whether an analytical solution can be given for the following integral:
$$\int_0^\infty \dot{f}(t) \cdot u(\dot{f}(t)-a) \cdot e^{-s t} dt.$$
Expressing the solution of course in terms of:
$$ F(s) = \int_0^\infty f(t) e^{-s t} dt,$$
and of course $f$ fulfills whichever is needed of it and $u$ is the Heaviside step function.
This all comes from a two body system. This term normally would be of the form $C \cdot \dot{z}$ (a damper), but I'm trying to make a model of "knee speed", in which the constant $C$ takes a certain value for small speeds ($\dot{z}$), and another for large speeds (in our case $\dot{z} >a$). When you input this, instead of the usual transform of the speed $\dot{z}$ you get a product of the form above. A Heaviside function appears and it's not just a function of time as usual in this kinds of problems.
In conclusion, I'd like an analytical approach if possible (which I'm not even sure it is), and yes, I know I can just take the original system and numerically solve it without recurring to Laplace, and honestly at this point it'd be easier, but I was interested in the problem anyway. Thanks in advance for any comments.