Recent content by petterson

Thanks very much for this idea, Stephen - I'll have to think about this for a little bit.

Yes I mean ##I(f) \le k##. In the original problem ## x## is between 0 and 1. The boundary ## f^{-1}(0) ## comes from a substitution. We also restrict to ##f(x) \ge 0##. I'm sorry for all the confusion (I thought there might be more of a standard technique that could help me) . Below is the...

Sorry I wasn't quite clear with ##f'(x)## and the upper bound - from the setup of the problem it is a given that ## k ## is an upper bound on the integral. Typically we find ## f'(x) < 0## (in related problems) - it would be nicer not to impose it as a constraints but we may easily do so if it...

We consider only the case ##f^{-1}(0) > 0 ##. Typically, the solution would be such that ##f'(x) < 0##. The problem is also such that ##k## is always an upper bound on the integral. If I choose ## f^{-1}(0)## equal to a given constant ##C## , I think I can find a stationary point with the...

Hi, I'm trying to solve the following problem ##\max_{f(x)} \int_{f^{-1}(0)}^0 (kx- \int_0^x f(u)du) f'(x) dx##. I have only little experience with calculus of variations - the problem resembles something like ## I(x) = \int_0^1 F(t, x(t), x'(t),x''(t))dt## but I don't know about the...

I was able to simplify my problem to the following equation: $$ F(g(b)) - F(b) + F'(b)h(b) + F''(b) s(b) - k = 0 $$ ##g(b)## is a decreasing function, k a constant. If necessary, one could also assume that ##g(b)## is an affine function. Does the delay differential equation framework apply...

Thanks andrewkirk. Doing what you proposed the equation looks nicer, I obtain the following: $$ F'(g(b)) g'(b) [h(b) - v(g(b))] + F(g(b)) h'(b) + F''(b)s(b) + F'(b) [s'(b) + v(b) - h(b)] - F(b) h'(b) + h'(b) = 0 $$ There is still the issue with those nested function ##F(g(b))## and...

Hi, I was wondering if anyone had some advice on how to solve the following equation for ## F(b)##: $$ F(g(b)) h(b) + F'(b) s(b) - F(b)h(b) + h(b) + \int_{g(b)}^{b} v(x) F'(x) dx = 0 $$ Any hints on how to tackle this would be highly appreciated. Thank you!