Asymptotic behaviour of self-contracted planar curves and gradient orbits of convex functions

Aris Daniilidis (1, 2) , Olivier Ley (1, 3) , Stéphane Sabourau (1)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 UAB - Departament de Matemàtiques [Barcelona]
3 IRMAR - Institut de Recherche Mathématique de Rennes

We hereby introduce and study the notion of self-contracted curves, which encompasses orbits of gradient systems of convex and quasiconvex functions. Our main result shows that bounded self-contracted planar curves have a finite length. We also give an example of a convex function defined in the plane whose gradient orbits spiral infinitely many times around the unique minimum of the function.

Mots clés: Planar dynamical system; gradient trajectory; convex function; convex foliation; Lojasiewicz inequality

Domaines: Systèmes dynamiques [math.DS]; Optimisation et contrôle [math.OC]

Aris Daniilidis, Olivier Ley, Stéphane Sabourau. Asymptotic behaviour of self-contracted planar curves and gradient orbits of convex functions. Journal de Mathématiques Pures et Appliquées, 2010, 94 (2), pp.183-199. ⟨10.1016/j.matpur.2010.03.007⟩. ⟨hal-00315675⟩