Motion Planning for Kinematic systems

In this paper, we present a general theory of motion planning for kinematic systems. This theory has been developed for long by one of the authors in a previous series of papers. It is mostly based upon concepts from subriemannian geometry. Here, we summarize the results of the theory, and we improve on, by developping in details an intricated case: the ball with a trailer, which corresponds to a distribution with flag of type 2,3,5,6. This paper is dedicated to Bernard Bonnard for his 60th birthday

Geometry and analysis of control-affine systems: motion planning, heat and Schr\uf6dinger evolution

This thesis is dedicated to two problems arising from geometric control theory, regarding control-affine systems $\dot q= f_0(q)+\sum_{j=1}^m u_j f_j(q)$, where $f_0$ is called the drift. In the first part we extend the concept of complexity of non-admissible trajectories, well understood for sub-Riemannian systems, to this more general case, and find asymptotic estimates. In order to do this, we also prove a result in the same spirit as the Ball-Box theorem for sub-Riemannian systems, in the context of control-affine systems equipped with the L1 cost. Then, in the second part of the thesis, we consider a family of 2-dimensional driftless control systems. For these, we study how the set where the control vector fields become collinear affects the diffusion dynamics. More precisely, we study whether solutions to the heat and Schr\uf6dinger equations associated with this Laplace-Beltrami operator are able to cross this singularity, and how its the presence affects the spectral properties of the operator, in particular under a magnetic Aharonov\u2013Bohm-type perturbation

Motion Planning for Kinematic Systems

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