A “bounded” constraint in a variational problem

Is there any standard approach to solve the following kind of variational problem?

Maximize $F=\int_0 ^1 L(x,y,y')dx$ subject to the constraint $|\int_0 ^1 M(x,y,y')dx| \lt k$ where $y$ is the function of $x$ to be solved for.

I can think immediately of maximizing $\int_0 ^1 Ldx + \lambda(\int_0^1 Mdx-k)^2$ but don't know whether that will yield the right solution. I can't seem to find this kind of problem solved in the usual references, e.g. Bruce Van Brunt's Calculus of Variations.

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How about first looking for an unconstrained maximum of $F$ and seeing if it satisfies the constraint? Then try the constrained problems $\max F \,| \int M = k$ and $\max F \,| \int M = -k$ separately. –  kiwi Apr 7 '12 at 18:47