Advanced Search

Fix two points $x,\overline{x}\in S^2$ and two directions (without orientation) $\eta ,\overline{\eta }$ of the velocities in these points. In this paper we are interested to the problem of minimizing the cost $J\left[\gamma \right]={\int }_0^T\left({}_{\gamma \left(t\right)}(\dot{\gamma }\left(t\right),\dot{\gamma }\left(t\right))+K_{\gamma \left(t\right)}^2{}_{\gamma \left(t\right)}(\dot{\gamma }\left(t\right),\dot{\gamma }\left(t\right))\right)\mathrm{d}t$ along all smooth curves starting from x with direction η and ending in $\overline{x}$ with direction $\overline{\eta }$. Here g is the standard Riemannian metric on S ² and...

We consider the problem of minimizing ${\int }_0^{\ell }\sqrt{{\xi }^2+K^2\left(s\right)}\mathrm{d}s$ ∫ 0 ℓ ξ 2 + K 2 ( s )   d s for a planar curve having fixed initial and final positions and directions. The total length is free. Here is the arclength parameter, () is the curvature of the curve and > 0 is a fixed constant. This problem comes from a model of geometry of vision due to Petitot, Citti and Sarti. We study existence of local and global minimizers for this problem. We prove that if for a certain choice of boundary conditions there is no...

In the framework of the Authors' research papers devoted to studies of mathematical systems with mixed structures, various extensions of nonlinear integral and integro-differential equations of Volterra and Picone's types play a very important role. The Authors, continuing their previous paper published in the same «Rendiconti Lincei» and concerned with the existence, the unicity and the stability of a new extension of Volterra nonlinear integro-differential equations, deal in what follows with...