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Solve an optimal control problem with the Wolfram Language?

Iuval Clejan
Posted 8 years ago

I have Mathematica version 8, but will gladly get the latest version if it is necessary to accomplish the following goal: find two control functions that maximize a certain functional in the control functions and 2 other functions subject to inequality constraints on the control functions and also non-linear differential equation constraints involving all 4 functions. If this is not possible, I will settle for solving for only one control function. Can someone please give me an idea of what must be done?
POSTED BY: Iuval Clejan
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There are some demonstration on this topic on the Wolfram Demonstrations Project, e.g. Moon Landing Simulation. I would think that they may give you an insight on your problem.
POSTED BY: Valeriu Ungureanu
Iuval Clejan
Iuval Clejan
Posted 8 years ago

This doesn't work if I have another inequality to satisfy, such as y>0. In that case we can change the variable to y=(1+Tanh[s])/2 or s=InvTanh[2y-1)
POSTED BY: Iuval Clejan
Updating Name
Updating Name
Posted 8 years ago

I think I see: s is just a change of variable from y, not a dynamic variable with a separate Hamilton Jacobi equation.
POSTED BY: Updating Name
Iuval Clejan
Iuval Clejan
Posted 8 years ago

And s is a dynamic variable? So a new term in the Lagrangian would be lambda (y-1+s^2) for satisfying y<1?
POSTED BY: Iuval Clejan
Iuval Clejan
Iuval Clejan
Posted 8 years ago

I don't understand how a slack term leads to an inequality. For example if I want variable y<1 then what term would you add to Lagrangian?
POSTED BY: Iuval Clejan

expr >= 0 should be changed to expr - s^2 == 0. expr <= 0 should be changed to expr + s^2 == 0
POSTED BY: Frank Kampas

Inequality constraints can be converted to equality constraints with the addition of slack or surplus terms, which are generally the square of a new variable, to insure they stay positive.
POSTED BY: Frank Kampas
Iuval Clejan
Iuval Clejan
Posted 8 years ago

I don't currently have Courant and Hilbert but I'm guessing that they are talking of a way to ensure the differential equation constraints. Of course the Lagrange multipliers are time-dependent. They are variables which have their own Hamilton Jacobi equation, which is just the dif eq constraints, and then they appear as dependent variables in the other H-J equations. The thing that concerns me is not the differential equation constraints (which can be handled with Lagrange multipliers), but the inequality constraints. I suppose I can introduce some auxiliary function into the Lagrangian which gives a high penalty for going outside the domain allowed by the inequality?

POSTED BY: Iuval Clejan

I have been able to do some constrained functional optimizations in Mathematica by using time-dependent Lagrange multipliers, as described in Courant & Hilbert, Methods of Mathematical Physics, in Chapter, Section 7, Variational Problems With Subsidiary Conditions. Unfortunately, I cannot at this time locate one of those calculations to post here.
POSTED BY: Frank Kampas
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