Optimality results in orbit transfer

Conjugate and cut loci of a two-sphere of revolution with application to optimal control

“…This provides a bridge between Riemannian geometry and optimal control, with promising extensions to the singular case and the Zermelo navigation problem on Riemannian manifolds [2]. Secondly, we give a neat proof of the structure of the conjugate and cut loci in coplanar orbital transfer, which were previously computed in [4] using the explicit parameterization of the extremal flow for each initial point where it is sufficient to consider the half period mapping. In the tangential case, they were obtained using a normal form and numerical simulations.…”

Convexity of injectivity domains on the ellipsoid of revolution: The oblate case

“…This allows to interpretate the parameter as the ratio μ between the minor and major axes of the conformal ellipsoid. 2 …”

“…Because of symmetries, convexity has only to be checked on a quarter of the curve, α ∈ [0, π/2]. The boundary of the injectivity domain on the cotangent space isα → T (p θ ) (p θ , p ϕ ),so the curvature condition is expressed as a sign condition on the quantity1 Also independently introduced in[5] 2. Surprisingly, the Gauss curvature of these metrics is nondecreasing from the North the pole to the equator (which ensures a simple structure of cut loci [10, Main Theorem]) if and only if μ is again greater or equal to…”

Bifurcations of the conjugate locus

“…Of particular relevance to this paper is the so-called "last geometric statement of Jacobi", which asserts (among other things) that the conjugate locus of a non-umbilic point on the triaxial ellipsoid has precisely 4 cusps (see [20] for a historical sketch and list of references). This conjecture was recently proved by Itoh and Kiyohara [13], and a renewed interest in the conjugate and cut locus can be seen in the recent papers providing formal studies ( [11], [24], [14], [15]), simulations ( [23], [8], [21], [22], [20]) and applications ( [6], [5], [3], [4], [7]). It is no surprise that the papers which focused on the triaxial ellipsoid and surfaces of revolution made heavy use of the fact that the geodesic flow on those surfaces is (Liouville) integrable.…”