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Geometrical methods have had a profound impact in the development of modern nonlinear control theory. Fundamental results such as the orbit theorem, feedback linearization, disturbance decoupling or the various controllability tests for non-linear systems are all deeply rooted on a geometric view of control theory. It is perhaps surprising, and possibly… (More)
The left-invariant sub-Riemannian problem on the group of motions of a plane is considered. Sub-Riemannian geodesics are parametrized by Jacobi's functions. Discrete symmetries of the problem generated by reflections of pendulum are described. The corresponding Maxwell points are characterized, on this basis an upper bound on the cut time is obtained .
Flat sub-Riemannian structures are local approximations — nilpo-tentizations — of sub-Riemannian structures at regular points. Lie algebras of symmetries of flat maximal growth distributions and sub-Riemannian structures of rank two are computed in dimensions 3, 4, and 5. A sub-Riemannian geometry is a triple (M, ∆, ·, ··), where M is a smooth man-ifold, ∆… (More)
The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is studied. Local and global optimality of extremal trajectories is characterized. Lower and upper bounds on the first conjugate time are proved. The cut time is shown to be equal to the first Maxwell time corresponding to the group of discrete symmetries… (More)
The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is considered. were defined, their local and global optimality were studied. In this paper the global structure of the exponential mapping is described. On this basis an explicit characterization of the cut locus and Maxwell set is obtained. The optimal… (More)
Known and new results on controllability of right-invariant systems on solvable Lie groups are presented and discussed. The main ideas and technique used are outlined, illustrating examples are given. Some open questions are suggested.
The aim of this paper is to present some recent results on controllability of right-invariant systems on Lie groups. From the Lie-theoretical point of view, we study conditions under which subsemigroups generated by half-planes in the Lie algebra of a Lie group coincide with the whole Lie group.
We compute two vector field models of the Carnot algebra with the growth vector (2, 3, 5, 8), and an infinitesimal symmetry of the corresponding sub-Riemannian structure.
, title " Controllability and optimal control for invariant control systems on Lie groups and homogeneous spaces " , " Generalized solutions to Laplace equation " .