Search

A three-dimensional reachable set for a nonlinear controlled object "Dubins car" is investigated. The control is the angular velocity of rotation of the linear velocity vector. An integral quadratic constraint is imposed on the control. Based on the Pontryagin maximum principle, a description of the motions generating the boundary of the reachable set is given. The motions leading to the boundary are optimal Euler elasticae. Simulation results are presented. [ABSTRACT FROM AUTHOR]

The structure of the intersection of the sub-Riemannian sphere on the Cartan group with a 3-dimensional invariant manifold of main symmetries is described. [ABSTRACT FROM AUTHOR]

The structure of the intersection of the sub-Riemannian sphere on the Engel group with a two-dimensional invariant set of discrete symmetries is described: regularity, analytic properties, exp-log category, Whitney stratification, multiplicity of points, characterization in terms of abnormal trajectories, conjugate points and Maxwell points, and explicit expressions for the sub-Riemannian distance to singular points. [ABSTRACT FROM AUTHOR]

We describe all Carnot algebras with growth vector (2, 3, 5, 6), their normal forms, an invariant that separates them, and a change of basis that transforms such an algebra into a normal form. For each normal form, Casimir functions and symplectic foliations on the Lie coalgebra are computed. An invariant and normal forms of left-invariant (2, 3, 5, 6)-distributions are described. A classification, up to isometries, of all left-invariant sub-Riemannian structures on (2, 3, 5, 6)-Carnot groups is obtained. [ABSTRACT FROM AUTHOR]

For two-step free nilpotent Lie algebras, we describe symplectic foliations and Casimir functions. A left-invariant time-optimal problem is considered in which the set of admissible controls is given by a strictly convex compact set in the first layer of the Lie algebra that contains the origin in its interior. We describe integrals for the vertical subsystem of the Hamiltonian system of the Pontryagin maximum principle. The properties of solutions to this system for low ranks of the Poisson bivector are described. [ABSTRACT FROM AUTHOR]