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3.4.2 Numerical methods
In this section we propose a simple algorithm for the numerical solution of the optimal
control problem studied in the previous sections.
For this we notice that the optimality condition given in section 3.84 formally correspond to the necessary conditions for the point (U,(P1,P2),(Pl,P2)) to be a saddle point
of the Lagrangian functional [. to be given below, i.e. a solution for the problem
where
[.(m, (PhP2), (Pl,P2), (I-'hl-'2)) = J(m) -loT 10 ~lP1dxdt - 10 P10(X)P1(X,O)dx
- loT 10 a~2P2dxdt- 1oP20(X)P2(X,O)dx- loT 1o(U,VP1P1+{J1VP1,VP1
+k1P1Pl)dxdt -loT 1o(U,VP2P2 + {J2VP2,VP2 + k1P1P2 + ~k,p2P2)tkdt
+ IT(f I pl. I I. -hI P2dx)m;(t)dt + IT I -hI k,d.P2dxdt + (I-'l,P1) + (I-'2,P2)
Jo 1=1 1 Jp,
Jo Jo
This fact leads us to consider the Uzawa-like algorithm:
• At the beginning (I-'~, I-'g) is arbitrarily given .
• At iteration T we know (1-'1,1-'2) . Then we compute a saddle point for
(3.67)
(3.68)
This is equivalent to solve an unconstrained optimal control problem corresponding
to the same state equation as before, but replacing the cost function (3.39) by the
following
(3.70)
This problem can be solved by using gradient methods. According to optimality
conditions we have obtained, functional .r is differentiable and its gradient is given
by
V Y(m) = Uj(mi(t)) + I ~; I i; ~p1dx, j = 1, ... ,M) (3.71)
Then measures I-'i, i = 1,2 are updated by a gradient with projection method.
3.4.3 Remarks
From the mathematical point of view it is very interesting the case where each subset Pi
reduces to a point bi' In such a situation the right hand side of the equation for BOD has
to be replaced by
1 M
h Em;(t)5(x - b;)
i=l
(3.72)