The optimal control problem $$ J(u) = \min \int_0^T \sqrt{u_1^2+u_2^2} dt $$ $$ \dot q = f(q,u),\quad q(0)=a\quad q(T)=b $$
($q\in\mathbb{R}^n$, $u\in\mathbb{R}^2$, $T$ is fixed and $f$ is smooth)
is equivalent to the optimal control problem $$ \tilde J(u) =\min \int_0^T (u_1^2+u_2^2) dt $$ $$ \dot q = f(q,u),\quad q(0)=a\quad q(T)=b $$ .
In which sense these problems are equivalent ? they have the same cost function : $J(u) = \tilde J(u)$ ? How to prove this equivalence (if you have a reference)?
If a constraint is added on the control like $|u|\le 1$ then the equivalence doesn't hold ?