A Paradox in the Approximation of Dirichlet Control Problems in Curved Domains

In this talk we consider the following optimal control problem (P)  minJ(u) = ∫ Ω L(x, yu(x)) dx+ N 2 ∫ Γ u(x) dσ(x) subject to (yu, u) ∈ (L∞(Ω) ∩H(Ω))× L(Γ), α ≤ u(x) ≤ β for a.e. x ∈ Γ, where Γ is a smooth manifold, yu is the state associated to the control u, given by a solution of the Dirichlet problem { −∆y + a(x, y) = 0 in Ω, y = u on Γ. (1) To solve the problem (P) numerically, it is usually necessary to approximate Ω by a (typically polygonal) new domain Ωh. The difference between the solutions of both infinity dimensional control problems, one formulated in Ω and the second in Ωh, was studied in [1], where an error of order O(h) was proved. In [2], the numerical approximation of the problem defined in Ω was considered. The authors used a finite element method such that Ωh was the polygon formed by the union of all triangles of the mesh of parameter h. They proved an error of order O(h) for the difference between continuous and discrete optimal controls. Here we show that the estimate obtained in [1] cannot be improved, which leads to the paradox that the numerical solution is a better approximation of the optimal control than the exact one obtained just by changing the domain Ω to Ωh.