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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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    Added tag (calculus-of-variations), which is the branch of mathematics that deals with this sort of problem in general. – 2017-02-24
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    Yes, it is. Thanks! – 2017-02-24
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    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. – 2017-02-24
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    @Paul: this doesn't fulfill the last constraint. – 2017-02-24
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    @Paul It would be great to see how you arrived to this conclusion - teach a man how to fish... – 2017-02-24
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    @YvesDaoust, yes, will edit. – 2017-02-24
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    The basic idea is to (1) come up with the simplest function possible to meet the constraints, and (2) see what happens when you vary it. If you vary this function a little, you will introduce more "wiggle" into the function, increasing the value of what you are trying to minimize. – 2017-02-24
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    There are of course more rigorous ways of attacking a problem like this, but a little intuition can go a long way here. – 2017-02-24
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    @Paul In fact I hope to learn how to solve such problems by calculus of variations. But I don't know how to deal with the inequality constraint $f(x)\ge 0$. Could you please give me a brief explanation? Thanks! – 2017-02-24

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