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Using optimal control theory, I am trying to find the optimal paths of d and k such that they will maximize profit as given by lc:

lc = 
  -35/2 d[t] (d[t] - 6/10) (d[t] - 9/10) - 20k[t] (k[t] - 5/10) (k[t] - 1) + 
    d[t]k[t] - u[t]^2 - w[t]^2 + md[t]u[t] +  mk[t]w[t]

with d'[t] = u[t] and k'[t] = w[t].

This gives me the following:

DSolve[{
   d[0] == a,
   k[0] == b,
   md[20] E^(-r 20) == 0,
   mk[20] E^(-r 20) == 0,
   d'[t] == D[lc, md[t]],
   k'[t] == D[lc, mk[t]],
   md'[t] == -D[lc, d[t]] + r md[t],
   mk'[t] == -D[lc, k[t]] + r mk[t],
   D[lc, u[t]] == 0,
   D[lc, w[t]] == 0} /. r -> 1/2, 
{d[t], k[t], u[t], w[t], md[t], mk[t]}, t]

But this just returns the input. What am I doing wrong?

My ultimate goal with this is to be able to obtain d'[t] (=u[t]) given d[0] and k[0]. A symbolic solution in terms of a and b would be best, but giving numerical values for a and b and just getting a numerical value for d'[0] would also suffice.

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I guess it's too nonlinear for DSolve. If you have clues about a and b, maybe try NDSolve instead. –  Silvia Dec 20 '13 at 3:38

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