%0 Thesis
%D 2014
%T The curvature of optimal control problems with applications to sub-Riemannian geometry
%A Luca Rizzi
%K Sub-Riemannian geometry
%X Optimal control theory is an extension of the calculus of variations, and deals with the optimal behaviour of a system under a very general class of constraints. This field has been pioneered by the group of mathematicians led by Lev Pontryagin in the second half of the 50s and nowadays has countless applications to the real worlds (robotics, trains, aerospace, models for human behaviour, human vision, image reconstruction, quantum control, motion of self-propulsed micro-organism). In this thesis we introduce a novel definition of curvature for an optimal control problem. In particular it works for any sub-Riemannian and sub-Finsler structure. Related problems, such as comparison theorems for sub-Riemannian manifolds, LQ optimal control problem and Popp's volume and are also investigated.
%I SISSA
%G en
%U http://hdl.handle.net/1963/7321
%1 7367
%2 Mathematics
%4 1
%# MAT/03 GEOMETRIA
%$ Submitted by Luca Rizzi (lrizzi@sissa.it) on 2014-05-23T12:10:38Z
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%] Introduction
1) The curvature of optimal control problems
2) Comparison theorems for conjugate points in sub-Riemannian geometry
3) On conjugate times of LQ optimal control problems
4) A formula for Popp's volume in sub-Riemannian geometry
%0 Thesis
%D 2011
%T Invariants, volumes and heat kernels in sub-Riemannian geometry
%A Davide Barilari
%K Sub-Riemannian geometry
%X Sub-Riemannian geometry can be seen as a generalization of Riemannian geometry under non-holonomic constraints. From the theoretical point of view, sub-Riemannian geometry is the geometry underlying the theory of hypoelliptic operators (see [32, 57, 70, 92] and references therein) and many problems of geometric measure theory (see for instance [18, 79]). In applications it appears in the study of many mechanical problems (robotics, cars with trailers, etc.) and recently in modern elds of research such as mathematical models of human behaviour, quantum control or motion of self-propulsed micro-organism (see for instance [15, 29, 34])\\r\\nVery recently, it appeared in the eld of cognitive neuroscience to model the\\r\\nfunctional architecture of the area V1 of the primary visual cortex, as proposed by Petitot in [87, 86], and then by Citti and Sarti in [51]. In this context, the sub-Riemannian heat equation has been used as basis to new applications in image reconstruction (see [35]).
%I SISSA
%G en
%U http://hdl.handle.net/1963/6124
%1 6005
%2 Mathematics
%3 Functional Analysis and Applications
%4 -1
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