Minimizing the functional using Euler-Lagrange and Cauchy matrix

Question

I'm given this problem where I need to minimize $$I(U) = \int_0^1 u'(x)^2dx$$ with $u(0) =a_0, u(1) = a_1$ among all functions that satisfy $$0 = \int_0^1 u(x)\cos(b_i x)dx, \quad i = 1,2,\cdots,N.$$

And I have no idea how to set up my E-L equation, please help.

Answer 1

The functional $I$ is defined as

$$I[u] = \int_a^b F(x,u,u') \, dx=\int_a^b [u'(x)]^2 \, dx.$$

There are $N$ constraints, for $i = 1, \ldots, N$,

$$K_i[u] = \int_a^b G_i(x,u,u') \, dx=\int_a^b u(x)\cos(b_ix) \, dx=0.$$

Introduce the Lagrangian

$$L[u] = I[u] + \sum_{i=1}^N \lambda_iK_i[u].$$

A necessary condition for an extremal is the Euler-Lagrange equation

$$F_u - \frac{d}{dx}F_{u'} + \sum_{i=1}^N \lambda_i\left[(G_i)_u - \frac{d}{dx}(G_i)_{u'}\right]= 0.$$

This reduces to

$$2u''(x)+ \sum_{i=1}^N \lambda_i\cos(b_ix)= 0.$$