1,143 research outputs found
Control of Nonholonomic Systems and Sub-Riemannian Geometry
Lectures given at the CIMPA School "Geometrie sous-riemannienne", Beirut,
Lebanon, 201
On Regularity of Abnormal Subriemannian Geodesics
We prove the smoothness of abnormal minimizers of subriemannian manifolds of
step 3 with a nilpotent basis. We prove that rank 2 Carnot groups of step 4
admit no strictly abnormal minimizers. For any subriemannian manifolds of step
less than 7, we show all abnormal minimizers have no corner type singularities,
which partly generalize the main result of Leonardi-Monti.Comment: This paper has been withdrawn by the author due to a crucial
computation error in (F_t^1)_sta
Computations involving differential operators and their actions on functions
The algorithms derived by Grossmann and Larson (1989) are further developed for rewriting expressions involving differential operators. The differential operators involved arise in the local analysis of nonlinear dynamical systems. These algorithms are extended in two different directions: the algorithms are generalized so that they apply to differential operators on groups and the data structures and algorithms are developed to compute symbolically the action of differential operators on functions. Both of these generalizations are needed for applications
Exponential stabilization of driftless nonlinear control systems using homogeneous feedback
This paper focuses on the problem of exponential stabilization of controllable, driftless systems using time-varying, homogeneous feedback. The analysis is performed with respect to a homogeneous norm in a nonstandard dilation that is compatible with the algebraic structure of the control Lie algebra. It can be shown that any continuous, time-varying controller that achieves exponential stability relative to the Euclidean norm is necessarily non-Lipschitz. Despite these restrictions, we provide a set of constructive, sufficient conditions for extending smooth, asymptotic stabilizers to homogeneous, exponential stabilizers. The modified feedbacks are everywhere continuous, smooth away from the origin, and can be extended to a large class of systems with torque inputs. The feedback laws are applied to an experimental mobile robot and show significant improvement in convergence rate over smooth stabilizers
Local properties of almost-Riemannian structures in dimension 3
A 3D almost-Riemannian manifold is a generalized Riemannian manifold defined
locally by 3 vector fields that play the role of an orthonormal frame, but
could become collinear on some set \Zz called the singular set. Under the
Hormander condition, a 3D almost-Riemannian structure still has a metric space
structure, whose topology is compatible with the original topology of the
manifold. Almost-Riemannian manifolds were deeply studied in dimension 2. In
this paper we start the study of the 3D case which appear to be reacher with
respect to the 2D case, due to the presence of abnormal extremals which define
a field of directions on the singular set. We study the type of singularities
of the metric that could appear generically, we construct local normal forms
and we study abnormal extremals. We then study the nilpotent approximation and
the structure of the corresponding small spheres. We finally give some
preliminary results about heat diffusion on such manifolds
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