Calculus of variations with integral constraints

[Usagi]

Both p(x,y) and q(x,y) are probability density functions, q(x,y) is an already known density function, my job is to minimise C[p,q] with respect to 3 conditions, they are listed in the red numbers, 1, 2, 3. Setting up the lagrange function and simplifying it up to equation (21) is fine with me, however I am lost when they mention "calculus of variations" as I have not studied, I assume (22) follows on from the calculus of variation technique they used, I was wondering where I can read about calculus of variations to help me solve problems like this with integral constraints? Thanks!

 

[Ackbach]

My favorite CoV book is John Troutman's Variational Calculus and Optimal Control: Optimization with Elementary Convexity. There are loads of other good books out there, though. Some of them assume you have familiarity with functional analysis/Lebesgue integration, and some of them don't. Troutman's book is a good one because he assumes very little: basically multivariable calculus, and perhaps linear algebra.

 

[Usagi]

Thanks Ackbach, I've had a read of Troutman's book, it is indeed very insightful however there isn't much on integral constraints and optimisation of multi-integral functions, do you have any ideas how to solve the above optimization problem? Cheers.