Let's say I have an objective function $F$ with state variables $A,B,C$ and relative equations of motion, I can create the current time Hamiltonian with $H_C = F\{A,B,C\}+\alpha * \dot A+\beta * \dot B + \gamma \dot C$, where $\alpha, \beta, \gamma$ are the 3 co-state variables (I'm omitting here the control variables).

But what if I also have a constraint that in each moment in time I must have something like $aA + bB + cC = k$ (where $a,b,c,k$ are fixed parameters) ? How do I set this constraint in the hamiltonian ?

Of course I could simply use only the $A,B$ variables, but then I still have an equation of motion for $C$ (in my application $A$, $B$ and $C$ are different land use areas).

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    $\begingroup$ I think Dorfman might do a similar thing in this paper but I'm not absolutely certain because it's been a long time since I read it. Check it out and see if it helps. andrew.cmu.edu/course/88-737/optimal_control/papers/dorfman.pdf $\endgroup$
    – mark leeds
    Nov 26, 2023 at 2:07
  • $\begingroup$ In the meantime II have found this where the constraint is simply added to the Hamilton to maximise with an ancillary variable, a bit like with the Lagrangian. $\endgroup$
    – Antonello
    Nov 28, 2023 at 21:23
  • $\begingroup$ thanks. I'll check it out. $\endgroup$
    – mark leeds
    Nov 29, 2023 at 17:25


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