Polygons as optimal shapes with convexity constraint

Jimmy Lamboley (1, 2) , Arian Novruzi (3)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 Department of Mathematics and Statistics [Ottawa]

In this paper, we focus on the following general shape optimization problem: $$ \min\{J(\Om),\ \Om\ convex,\ \Om\in\mathcal S_{ad}\}, $$ where $\mathcal S_{ad}$ is a set of 2-dimensional admissible shapes and $J:\mathcal{S}_{ad}\rightarrow\R$ is a shape functional. Using a specific parameterization of the set of convex domains, we derive some extremality conditions (first and second order) for this kind of problem. Moreover, we use these optimality conditions to prove that, for a large class of functionals (satisfying a concavity like property), any solution to this shape optimization problem is a polygon.

Mots clés: Shape optimization; convex constraint; optimality conditions

Domaines: Optimisation et contrôle [math.OC]

Jimmy Lamboley, Arian Novruzi. Polygons as optimal shapes with convexity constraint. SIAM Journal on Control and Optimization, 2009, 48 (5), pp.3003-3025. ⟨hal-00362329⟩