# How to find minima of such functional?

How can I find minima of the following functional?

$$J(u)=\int_0^1 (u^2(t)-x^2(t)) \,\mathrm dt \to min$$

$$\dot x(t)= u(t),\ t\in\ [t_0,T],\;x(0)=0$$

-
do you mean $\dot x$: \dot x? –  Git Gud Jan 14 at 15:28
yeah, thanks about it –  Javidan Jan 14 at 15:30
Do you want to find mínima in what space? –  Tomás Jan 14 at 16:00
in Real vector space –  Javidan Jan 14 at 16:01
@Javidan: I think people are asking you what space does the function $u$ live in. For example, it could be the space $C^1([0,1])$ of differentiable functions on the unit interval. Also, your $t \in [t_0, T]$ is confusing. Why is the interval different than the one implied in the integration bounds? I suggest you take some care to make the question more precise. Until then, I can only tell you that for the choice of space I suggested, the functional is always non-negative and attains minimum $0$ for certain trigonometric functions (this can be determined from Euler-Lagrange equations). –  Marek Jan 15 at 15:18