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Conjugate and cut loci of a two-sphere of revolution with application to optimal control. (English) Zbl 1184.53036
A result of Sinclair and Tanaka states: Given a smooth metric \(dr^2+ m^2(r^2)d\theta^2\) on \(S^2\), where \(r\) is the angular variable along the meridian, \(\theta\) is the angle of revolution, the reflective symmetry condition \(m(2a- r)= m(r)\) holds, and the Gaussian curvature is monotone along ameridian, then (away from the pole) the cut locus of a point is a simple branch on the antipodal parallel. The authors of the present paper derive the same conclusion with sharper assumptions. These also give global optimal results is orbital transfer and for Lindblad equations in quantum control.

MSC:
53C20 Global Riemannian geometry, including pinching
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
49K15 Optimality conditions for problems involving ordinary differential equations
70Q05 Control of mechanical systems
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References:
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