Is there analytic solution to this functional problem?

Question

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!