Here is my own attempt. What do you think? :
$$\min J[y]=\int_0^3y'^3\;dx,\;y(0)=0,\;y(3)=1$$
$$F=y'^3,\;\;F_{y'}=3y'^2$$
Our task is to find a piecewise smooth extremal if such exists. If such an extremal exists, then at the point $x=c$, where the piecewise smooth extremal is not continuously differentiable, it must be that:
$$(\text{First Weierstrass-Erdmann condition})$$
$$F_{y_1'}|_{x=c-0}=F_{y_2'}|_{x=c+0}$$
that is:
$$3y_1'^2|_{x=c-0}=3y_2'^2|_{x=c+0}$$
or
$$y_1'^2|_{x=c-0}=y_2'^2|_{x=c+0},$$
where I have marked $y_1, y_2$ as the the two separate pieces of the function $y$. If $y_1=y_2$, then we have a only a smooth solution. Another possibility is that $y_1=-y_2$ or vice versa.
The other Weierstrass-Erdmann condition obviously hold for the case, when $y_1=y_2$, so lets check if they hold for $y_1=-y_2$ :
$$(\text{Second Weierstrass-Erdmann condition})$$
$$(F-y_1'F_{y_1'})|_{x=c-0}=(F-y_2'F_{y_2'})|_{x=c+0}$$
$$(y_1'^3-y_1'3y_1'^2)|_{x=c-0}=(y_2'^3-y_2'3y_2'^2)|_{x=c+0}$$
or
$$\left((-y_2)'^3-(-y_2)'3(-y_2)'^2\right)|_{x=c-0}=(y_2'^3-y_2'3y_2'^2)|_{x=c+0}$$
$$\left(-y_2'^3+3y_2'^3\right)|_{x=c-0}=\left(y_2'^3-3y_2'^3\right)|_{x=c+0}$$
$$2y_2'^3|_{x=c-0}=-2y_2'^3|_{x=c+0}$$
$$y_2'^3|_{x=c-0}=-y_2'^3|_{x=c+0}$$
The last equality holds only if $$y_2'=0,$$
which would mean that $y_1=C, y_2=-C$, which is not a piecewise smooth curve and the boundary conditions don't hold either for this possibility.
This means that there are only smooth solutions for this problem:
Euler's equation:
$$F_{y'}=3y'^2=C$$
$$y'=\pm C$$
$$y=Cx+D$$
from boundary conditions we get
$$y=\frac{1}{3}x$$
