A note on the equivalence of a strongly convex function and its induced contractive differential equation. (August 2022)

• Record Type: Journal Article
• Authors: Fitzsimmons, Maxwell; Liu, Jun
• Is Part Of: Automatica. Volume 142(2022)
• Publication Date: 2022-08
• Subjects: Contraction theory -- Lyapunov methods -- Convex optimization; Automatic control -- Periodicals; Automation -- Periodicals; 629.805
• DOI: 10.1016/j.automatica.2022.110349
• Languages: English
• ISSNs: 0005-1098

Abstract

Abstract: A strongly convex function naturally induces a gradient flow that is contractive. This paper is a short investigation on when the converse to the previous statement holds. That is, given a differential equation that is contractive, does there exist a strongly convex function that induces the differential equation? We show that, if sufficient smoothness of the vector field is assumed, then the contractivity of such a differential equation with a symmetric Jacobian is equivalent to the existence of a strongly convex function which induces the differential equation as its gradient flow.