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- Yuri L. Sachkov
- 2006

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 nonlinear systems are all deeply rooted on a geometric view of control theory. It is perhaps surprising, and possibly… (More)

- Yuri L. Sachkov
- 2011

The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is considered. In the previous works [Moiseev and Sachkov, ESAIM: COCV, DOI: 10.1051/cocv/2009004; Sachkov, ESAIM: COCV, DOI: 10.1051/cocv/2009031], extremal trajectories were defined, their local and global optimality were studied. In this paper the global… (More)

The left-invariant sub-Riemannian problem on the group of motions of a plane is considered. Sub-Riemannian geodesics are parameterized 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.… (More)

- Yuri L. Sachkov
- 2010

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)

- Yuri L. Sachkov
- 2003

Flat sub-Riemannian structures are local approximations — nilpotentizations — 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. 1. Sub-Riemannian structures A sub-Riemannian geometry is a triple (M,∆, 〈·, ·〉), where… (More)

- Yuri L. Sachkov
- Discrete Mathematics & Theoretical Computer…
- 1997

First we recall definitions and state our problem. Let G be a real connected Lie group, L be its Lie algebra (i.e. the set of all right-invariant vector fields on G). For any A;B1; : : : ; Bm 2 L we consider the corresponding affine right-invariant system = fA+ m Xi=1 uiBi j 8i ui 2 Rg The attainable set A for the system is a subsemigroup of G generated by… (More)

- Yuri L. Sachkov
- 2013

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.

- Yuri L. Sachkov
- 2014

We consider the free nilpotent Lie algebra L with 2 generators, of step 4, and the corresponding connected simply connected Lie group G. We study the left-invariant sub-Riemannian structure on G defined by the generators of L as an orthonormal frame. We compute two vector field models of L by polynomial vector fields in R, and find an infinitesimal symmetry… (More)

We consider the sub-Riemannian length optimization problem on the group of motions of hyperbolic plane i.e. the special hyperbolic group SH(2). The system comprises of left invariant vector fields with 2 dimensional linear control input and energy cost functional. We prove the global controllability of control distribution and use Pontryagin Maximum… (More)

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