Optimal Synthesis in a Radially Symmetric Grushin Space

Abstract

We study a class of radially symmetric Grushin-type structures on ${\mathbb{R}}^3$ generated by $X={\partial }_x,Y={\partial }_y,Z_f=f(r){\partial }_z$, with $r$ the cylindrical radius, and $f$ positive away from the vertical axis, increasing, and unbounded. Under a natural monotonicity assumption on the elasticity $\eta (r)=r{f}^{\prime }(r)/f(r)$, we obtain the optimal synthesis from singular points. For profiles satisfying an additional monotonicity constraint on the radial effective potential which ensures that geodesics are either radially periodic, circular or planar straight lines, we demonstrate a continuous, globally defined cut time candidate for Riemannian geodesics, which coincides with the radial period, evidently a Maxwell time, for all radially periodic geodesics. We then specialize to the homogeneous family $f(r)=r^{\alpha }$, $\alpha >1$, and determine the full optimal synthesis from Riemannian points. The cut time is shown to coincide with the complete radial period. Finally, we identify the resulting cut locus, which forms a pair of properly embedded two-dimensional regions on the cylinder through the initial point. Finally, we obtain a necessary condition for the constructed Maxwell time to be the true cut time, and rule it out as the global cut mechanism for the full class under consideration.

arXiv comment43 Pages, 5 Figures, 1 table. Major revision. The manuscript has been extensively rewritten, with substantial new results and proofs. Comments Welcome