Hölder equivalence of the value function for control-affine systems∗. (8th August 2014)

• Record Type: Journal Article
• Authors: Prandi, Dario
• Is Part Of: ESAIM. Volume 20:Number 4(2014:Oct.)
• Page Start: 1224
• Page End: 1248
• Publication Date: 2014-08-08
• Subjects: Control-affine systems, -- time-dependent systems, -- sub-Riemannian geometry, -- value function, -- Ball–Box theorem, -- nilpotent approximation; System analysis -- Periodicals; Calculus of variations -- Periodicals; Mathematical analysis -- Periodicals; Mathematical optimization -- Periodicals; Control theory -- Periodicals; 515.64
• DOI: 10.1051/cocv/2014014
• Languages: English
• ISSNs: 1292-8119

Abstract

Abstract : We prove the continuity and the Hölder equivalence w.r.t. an Euclidean distance of the value function associated with the L 1 cost of the control-affine system q̇ = f 0 ( q ) + ∑ j=1 m uj f j ( q ), satisfying the strong Hörmander condition. This is done by proving a result in the same spirit as the Ball–Box theorem for driftless (or sub-Riemannian) systems. The techniques used are based on a reduction of the control-affine system to a linear but time-dependent one, for which we are able to define a generalization of the nilpotent approximation and through which we derive estimates for the shape of the reachable sets. Finally, we also prove the continuity of the value function associated with the L 1 cost of time-dependent systems of the form q̇ = ∑j=1 m uj fj t (q) .