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Calculus of variations with integral constraints

Usagi

Member
Jun 26, 2012
46


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

Indicium Physicus
Staff member
Jan 26, 2012
4,191
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

Member
Jun 26, 2012
46
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.