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I came across the following "functional" at work:

$$ \Pi [b]=\int_0^\infty\int_0^{\lambda b(v,\lambda)} vf(v,\lambda) \; dv \; d\lambda $$

it's part of an optimization problem that tries to find $b$, subject to some constraints on $b$.

I'm not familiar with that type of integral, where the solution function is actually in one of the bounds of the integral. Is there a specific name for that type of integral? Would the calculus of variations address that type of optimization problem? Or is there a field of functional analysis (calculus?) that would address it?


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Are you integrating with respect to $v$ or $\lambda$? Surely it can't be with respect to both? – Bruno Joyal Dec 1 '11 at 3:51
Yeah, I'm afraid it's with respect to both... But you are right, I need to fix the formula :-) Thanks! - the second integral is from 0 to +infinity. – Frank Dec 1 '11 at 4:04

The functional

$$ \Pi [b]=\int_0^\infty\int_0^{\lambda b(v,\lambda)} vf(v,\lambda) \; dv \; d\lambda $$

is not well defined, due to the function $b(v,\lambda)$ appearing in the integration. For example, if we consider $b(v,\lambda)=v$ then

$$ \Pi [b]=\int_0^\infty\int_0^{\lambda v} v f(v,\lambda) \; dv \; d\lambda, $$

which is ill-defined, due to $\int_0^{\lambda v} v f(v,\lambda) \; dv$.

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