Optimal shape design for 2D heat equations in large time

Emmanuel Trélat (1, 2) , Can Zhang (3, 1) , Enrique Zuazua (4, 1)

1 SU - Sorbonne Université
2 CaGE - Control And GEometry
3 School of Mathematics and Statistics [Wuhan]
4 Departamento de Matemáticas [Madrid]

In this paper, we investigate the asymptotic behavior of optimal designs for the shape optimization of 2D heat equations in long time horizons. The control is the shape of the domain on which heat diffuses. The class of 2D admissible shapes is the one introduced by Sverák in [29], of all open subsets of a given bounded open set, whose complementary sets have a uniformly bounded number of connected components. Using a Γ-convergence approach, we establish that the parabolic optimal designs converge as the length of the time horizon tends to infinity, in the complementary Hausdorff topology, to an optimal design for the corresponding stationary elliptic equation.

Mots clés: Shape optimization; Long-time behavior; Γ-convergence; Heat equation; Elliptic equation

Domaines: Optimisation et contrôle [math.OC]

Emmanuel Trélat, Can Zhang, Enrique Zuazua. Optimal shape design for 2D heat equations in large time. Pure and Applied Functional Analysis, 2018, 3 (1), pp.255--269. ⟨hal-01442997⟩