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Optimal control of the sphere $$S^n$$ rolling on $$E^n$$. (English) Zbl 1064.49021
Summary: This paper investigates the optimal control problem concerning the sphere $$S^{n}$$ rolling without slipping on the $$n$$-dimensional Euclidean space $$E^n$$, $$n \geq 2$$. The differential equations governing the behaviour of the sphere constitute a sub-Riemannian distribution on the Lie group $$G = \mathbb R^n\times SO_{n+1}$$. Minimizing over the lengths of paths traced by the point of contact of the sphere yields an optimal control problem which is exploited using Noether’s theorem to derive a family of integrals of motion for the system. This family is then employed to prove that all optimizing trajectories are projections of normal extremals. The Lax form of the extremal equations gives rise to an additional family of integrals which is reminiscent of the Manakov integrals of motion for the free rigid body problem. Both families are required to show complete integrability if $$n =4$$.

##### MSC:
 49K15 Optimality conditions for problems involving ordinary differential equations 70F25 Nonholonomic systems related to the dynamics of a system of particles 70E18 Motion of a rigid body in contact with a solid surface 70Q05 Control of mechanical systems 37J60 Nonholonomic dynamical systems 93C85 Automated systems (robots, etc.) in control theory
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