Third order open mapping theorems and applications to the end-point map

Abstract

This paper is devoted to a third order study of the end-point map in sub-Riemannian geometry. We first prove third order open mapping results for maps from a Banach space into a finite dimensional manifold. In a second step, we compute the third order term in the Taylor expansion of the end-point map and we specialize the abstract theory to the study of length-minimality of sub-Riemannian strictly singular curves. We conclude with the third order analysis of a specific strictly singular extremal that is not length-minimizing.

New second-order optimality conditions in sub-Riemannian Geometry

“…Below we will discuss some conditions that can guarantee the existence of the infimum. They are related with the assumptions used in [BMP20] to derive third-order analog of Goh conditions. Later we will show that the existence of solutions of the characteristic OCP imposes strong conditions on underlying the sub-Riemannian trajectory.…”

“…We propose the name Monti's condition as this is one of the assumptions used in [BMP20] to derive the third-order optimality conditions. 3 This assumption is, however, a strong one.…”

“…It seems that for such curves, one needs to understand the geometry of third-order (and higher) order expansion of the extended end-point map. Some research in this direction was initiated in [BMP20], yet a fully satisfactory theory is yet to be developed.…”

“…Among the latter we must mention the derivation of the second-order necessary conditions for optimality [AS96] (known as Goh conditions) and a technical masterpiece of Hakavouri and Le Donne [HLD16] (se also [HLD18]) which ruled out corner-like singularities (some previous results in this direction are [LM08,Mon14b]). We are also aware of a very recent attempt to derive third-order necessary conditions of optimality [BMP20]. We refer to [Mon14a] for a discussion of the problem and its history.…”

Higher derivatives of the end-point map of a linear control system via adapted coordinates

“…As ). Finally, note that the concept of a jet allows to explain the understanding of second (and higher -see [BMP20]) derivatives in the spirit of Argachew. Namely, take a curve u s = u + s • u 1 + s 2 • u 2 + o(s 2 ) in Ω, and let us calculate the second Taylor expansion of End(u s ) in some local coordinate system:…”