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Let $f(x)$ be a function on $[0,+\infty)$. I want to solve the following functional problem:

$\min L(f)=\int_0^{\infty} (f'(x))^2\mathrm dx$

subject to

$f(0)=a$

$f(x)\ge 0,\forall x$

$\int_0^\infty f(x)\mathrm dx=1$

where $a>0$. Is there any analytic solution to this problem? If not, an efficient numerical approximation will also be helpful.

Thanks for help!

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  • $\begingroup$ Added tag (calculus-of-variations), which is the branch of mathematics that deals with this sort of problem in general. $\endgroup$ – GEdgar Feb 24 '17 at 13:09
  • $\begingroup$ Yes, it is. Thanks! $\endgroup$ – wwtian Feb 24 '17 at 13:12
  • $\begingroup$ The minimum will be given by $f(x)=-(a^2/2) x+a$ for $0 \leq x \leq 1/a$, $=0$ otherwise (assuming you don't need $C^2$). It is straightforward to show that this is in fact the minimum. If you do need $C^2$, then there is no minimum. $\endgroup$ – Paul Feb 24 '17 at 13:18
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    $\begingroup$ @Paul: this doesn't fulfill the last constraint. $\endgroup$ – Yves Daoust Feb 24 '17 at 13:20
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    $\begingroup$ @Paul It would be great to see how you arrived to this conclusion - teach a man how to fish... $\endgroup$ – TZakrevskiy Feb 24 '17 at 13:21

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