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Smooth optimal synthesis for infinite horizon variational problems

Andrei A. Agrachev; Francesca C. Chittaro — 2009

ESAIM: Control, Optimisation and Calculus of Variations

We study Hamiltonian systems which generate extremal flows of regular variational problems on smooth manifolds and demonstrate that negativity of the generalized curvature of such a system implies the existence of a global smooth optimal synthesis for the infinite horizon problem. We also show that in the Euclidean case negativity of the generalized curvature is a consequence of the convexity of the Lagrangian with respect to the pair of arguments. Finally, we give a generic classification for...

Sub-Riemannian Metrics: Minimality of Abnormal Geodesics versus Subanalyticity

Andrei A. Agrachev; Andrei V. Sarychev — 2010

ESAIM: Control, Optimisation and Calculus of Variations

We study sub-Riemannian (Carnot-Caratheodory) metrics defined by noninvolutive distributions on real-analytic Riemannian manifolds. We establish a connection between regularity properties of these metrics and the lack of length minimizing abnormal geodesics. Utilizing the results of the previous study of abnormal length minimizers accomplished by the authors in [Annales IHP. , p. 635-690] we describe in this paper two classes of the germs of distributions (called 2-generating and medium fat) such...

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