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Differential equation system in Optimization Toolbox

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Artyom
Artyom 2012-4-30
Hi, everyone.
I'm trying to solve dynamic programming problem. I have differential equation system like that one:
dydt = zeros(2,1);
dydt(1) = z(1);
dydt(2) = z(2)-z(1);
With constraints:
z>=0;
z(1)+z(2)<=x(2);
z(2)<=x(1);
x(2)>=0;
I must maximize x(2) on period T=5.
How can I optimize DES and find z with Optimization Toolbox?
Thank you.
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Richard Brown
Richard Brown 2012-4-30
Are the z variables constrained to take on integer values?
Artyom
Artyom 2012-4-30
No, it can take any values. Except constraints of course.

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回答（1 个）

Richard Brown
Richard Brown 2012-5-1
Your problem is a linear program in the arrays x1, x2, z1, z2. Because the RHS of the ODEs is piecewise constant, the problem can be reformulated as a series of difference equations. You therefore have the following linear equality constraints:
k = 0, ..., 4:
x1[k+1] = x1[k] + z1[k]
x2[k+1] = x2[k] + z2[k] - z1[k]
where x1[0] and x2[0] are known initial conditions
And you have the following linear inequality constraints:
k = 0, ..., 4:
z1[k] >= 0
z2[k] >= 0
z1[k] + z2[k] <= x2[k]
z2[k] <= x1[k]
z2[k] - z1[k] >= -x2[k]
Your cost is also linear:
c(z1, z2, x1, x2) = x2[5]
All of this defines a linear program that you can solve with linprog. All you require is a little bookkeeping to formulate the constraint matrices (and conversion to ones-based indexing)

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