3
$\begingroup$

I need to minimize the following integral by varying parameter $\lambda$:

$$\int_0^\infty(f(x)-g(x,\lambda))^2dx$$

The functions $f(x)$ and $g(x,\lambda)$ are known and they satisfy $f(0)=g(0,\lambda)$ and $f(\infty)=g(\infty,\lambda)=0$. And the integral is convergent.

What is the corresponding Eular-Lagrange equation?

Thanks- mike

$\endgroup$
  • $\begingroup$ From what I understand, you use the EL equations to find an unknown function. If you know f and g and want to extremise the integral for $\lambda$ then this isn't a functional problem. $\endgroup$ – ClassicStyle May 20 '14 at 6:20
3
$\begingroup$

The "Euler-Lagrange" equation reads

$$ \int\limits_0^{\infty}(f-g)\dfrac{dg}{d\lambda}dx = 0. $$

Knowing $f$ and $g$ you can (hopefully) compute this integral and then solve for $\lambda$.

Notice that if $\exists \lambda_0$ such that $f(x)=g(x,\lambda_0)$ then $\lambda_0$ leads necessarily to a true minimum of your integral, whereas the equation above may be a local one.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.