Minimize the sum of a function with respect to that function

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Hello everyone,
my problem is related to look for the function that minimizes the sum, that is the following: Given and J(x) satifying the generalized Bellmann equation, solve
subject to certain constrains.
Now, if the problem would be the minimization with respect to x, then I would use fmincon, however in this case I don't know what to do. Can anybody help me, please?

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