01220nas a2200121 4500008004100000245009100041210006900132260001000201520080700211653002801018100001601046856003601062 2014 en d00aThe curvature of optimal control problems with applications to sub-Riemannian geometry0 acurvature of optimal control problems with applications to subRi bSISSA3 aOptimal 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.10aSub-Riemannian geometry1 aRizzi, Luca uhttp://hdl.handle.net/1963/732101397nas a2200121 4500008004100000245006800041210006600109260001000175520100500185653002801190100002101218856003601239 2011 en d00aInvariants, volumes and heat kernels in sub-Riemannian geometry0 aInvariants volumes and heat kernels in subRiemannian geometry bSISSA3 aSub-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]).10aSub-Riemannian geometry1 aBarilari, Davide uhttp://hdl.handle.net/1963/6124