My brother asked me a question which I didn't know the answer to.
Are there theorems about existence, uniqueness and stability of solutions of ODEs of the followin type
$$ \frac{d^2 y}{dt^2} = f(t,y,\frac{dy}{dt})H(g(y)), $$
where $f$ and $g$ are Lipschitz functions and $H$ is the Heaviside function?
If there are no general theorems that could apply, how does one analyze these problems for given $f,g$?
Here is one possible approach.
Choose smooth approximations of $H$. For example consider a smooth function
$$ \eta:\mathbb{R}\to [0,\infty)$$
such that
$$\eta (t)=0,\;\;\forall t\leq 0$$
$$\eta(t)=1,\;\;\forall t\geq 1.$$
Now define
$$H_\varepsilon(t)= \eta(t/\varepsilon)$$
Then
$$\lim_{\varepsilon\searrow 0} H_\varepsilon (t) =H(t),\;\;\forall t\neq 0$$.
Now investigate the above equation with $H$ replaced with $H_\varepsilon$ an hope that you understand what happens when $\varepsilon \searrow 0$.
On the other hand, there is a large class of evolution equations with discountinuous right-hand side described by the so called maximally monotone operators. For 2nd order equations such as yours, one of the most influential papers is
Barbu, Viorel Existence theorems for a class of two point boundary problems. J. Differential Equations 17 (1975), 236–257.
For the general theory of evolution equations involving maximal monotone operators the best sources are
Barbu, Viorel Nonlinear differential equations of monotone types in Banach spaces. Springer Monographs in Mathematics. Springer, New York, 2010. x+272 pp.
Brezis, Haim Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. (French) North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. vi+183 pp
Your brother needs to provide more specific questions that he wants answered. But here are a few thoughts.
Suppose you want to prove existence and uniqueness of the initial value problem. The initial data will imply a value for $H(g(y))$. You can use the standard theorem to solve the initial value problem and extend the solution until $g(y) = 0$. At that point, check the sign of $g'(y)y'$. If it is nonzero, then you can just use the limiting values of $y$ and $y'$ to solve the initial value problem starting from that point. The fact that $y$ and $y'$ are being extended continuously implies that the ODE is being solved at least in the weak sense, and that's usually all that is needed.
If $g(y)$ does not change sign when it vanishes, things get more complicated. But that's something your brother should worry about only if he really does encounter it.
