For the following optimization problem: find the extremal values of $$ I(x) = \int_a^b F(t,u,x) dt$$ where $x:[a,b]\rightarrow\mathbb{R}$ is a continuous function and $u$ is the primitive of $x$, one can find the solution with Euler Lagrange equations.
Can similar equations be given for the situation where the function $x$ is only Riemann integrable?
A specific example would be like this: let $X$ be the set of integrable functions $x:[0,1]\rightarrow[0,\infty)$ such that $\int_0^1 x(t) dt=1$. Find $$ \inf_{x\in X} \int_0^1 x^2(t) dt $$ Of course this particular problem can be easily solved by Cauchy inequality: $$ \int_0^1 x^2(t) dt \geq \left(\int_0^1 x(t)dt\right)^2 = 1$$ but such a solution isn't always available. However, if it can be assumed that $x$ is continuous, then if $u(w)=\int_0^w x(t)dt$, it holds that $u(0)=0$, $u(1)=1$, and the problem becomes $$ \inf \int_0^1 u'^2 dt $$ for which one can write the Euler Lagrange equation $2u''=0$, which gives $u'=c$, and from the boundary conditions it follows that $u(t)=t$, and $x(t)=u'(t)=1$, which is the same result given by Cauchy. So in the end, the optimal value was obtained for a continuous function $x$. Is this always like this? Meaning that the Euler Lagrange equations can always be used in similar problems? If so, why?