I have a math problem coming from physics.
How do you push an object $1$ m in $1$ sec under the force $F=-v^2$, so the energy used is minimal?
We have to minimize
$$\int_0^1\left(\frac{1}{2}a^2+v'(t)*v(t)+v(t)^3 \right) dt$$
under the constraints
\begin{align} v(0) &= a \\ \int_{0}^{1} v(t) dt &= 1 \\ v'(t) + v(t)^{2} &\ge 0\qquad for\, all \,\,t:\quad 0<t<1 \end{align}
My problem is that I can't get the inequality constraint working with the Euler-Lagrange equation in a meaningful way.
I have tried to guess some functions for $v(t)$, and if the mass of the object is $1$ kg, my record for the energy used is $1.31$ Joule.
Is it possible to break this record using calculus of variations? - or using other methods?