Minimising an integral with unspecified endpoints

Question

The problem is to find the equation that minimises the following functional: $$ J[y] = \int_0^1 \frac{1}{2}(y')^2 +yy'+y'+y \ dx. $$ The endpoints are not specified.

So far I have calculated the solution of the Euler Lagrange equation to be $$ y(x) = C_1x+C_2+\frac{x^2}{2}, $$ I am unsure of how to proceed with no other information.

Answer 1

There are two options, both of which lead to the conclusion that this functional has no stationary points.

One option is to use the natural boundary condition $(L_{y'})'(a)=0$ for each endpoint at which the function is unconstrained. In the present case we have $(L_{y'})'=y''+y'$, so this yields the contradictory conditions $1+C_1=0$ at $x=0$ and $2+C_1=0$ at $x=1$.

Alternatively, you can calculate the value of the functional as a function of the parameters $C_1$ and $C_2$ and minimize it using ordinary calculus. This would be a tedious process if you carried it out in detail, but you can avoid the effort by noting that the value of the functional is linear in $C_2$ with non-zero first-order coefficient, so it doesn't have a minimum with respect to $C_2$.

Answer 2

The natural boundary condition is $L_{y'}=0$ or $y'+y+1=0$ which must hold at both $x=0$ and $x=1$. With your solution this leads to the two equations $C_1+C_2+1=0$ and $2C_1+C_2=-5/2$ for $C_1$ and $C_2$.