Optimal control for the forward-backward heat coupled systems

(z1, z2) ∈ (L2(Q))2, (v1, v2) ∈ (Uad), whereUad is a nonempty closed convex of the Hilbert space of controlsL2(Q). This coupled problem represents the distribution of the heat by conduction in the Ω material during time T , knowing that the heat on the edge of the solid is null, as the one of instant 0 is. Let us recall that the conduction is the action of progressively transmitting the heat. The source (v1, v2) of the coupled heat system applied to Ω then induces a heat pair solution (z1, z2) inside it. Nevertheless, such a forward–backward problem is ill-posed, since it does not exist a solution belonging to the imposed space for any source, in general. That is why engineers use approximate methods in order to determine the adequate source. We therefore propose a theoretical method which permits to find the ‘best’ source-heat pair, solution of a corresponding optimal control problem.