How can I find minima of the following functional?
$$J(u)=\int_0^1 (u^2(t)-x^2(t)) \,\mathrm dt \to min$$
$$\dot x(t)= u(t),\ t\in\ [t_0,T],\;x(0)=0 $$
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do you mean $\dot x$: \dot x? – Git Gud Jan 14 '13 at 15:28
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yeah, thanks about it – Javidan Jan 14 '13 at 15:30
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Do you want to find mínima in what space? – Tomás Jan 14 '13 at 16:00
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in Real vector space – Javidan Jan 14 '13 at 16:01
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What is the domain of your functional $J$? – Tomás Jan 14 '13 at 16:03
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no domain is given. It's say: Find the minima $J(u)$.. – Javidan Jan 14 '13 at 16:08
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1@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 '13 at 15:18
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i am also confused. It's given by high school teacher at exam. Anyway, $J(u)$ can be not functional ? i have no ability to check its correctity for now – Javidan Jan 15 '13 at 17:28
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Are you sure the integral isn't $\int_0^\pi$ instead of $\int_0^1$, or something like that? – Antonio Vargas Jan 15 '13 at 18:38
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@AntonioVargas, sure. I copied all content of problem here – Javidan Jan 15 '13 at 20:32