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ВИДЕОТЕКА |
Международная молодежная конференция «Геометрия и управление»
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Almost-Riemannian Geometry of the Two-Sphere Ivan Beschastnyi PhD student, Pereslavl-Zalessky, Russia |
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Аннотация: Consider the two following vector fields on \begin{equation*} f_1(x)= x\times e_2, \qquad f_2(x) = x\times\sqrt{1-a^2}e_1, \qquad x\in\mathbb{R}^3, \qquad |x| = 1, \end{equation*} where Vector fields \begin{equation*} \Delta_x = \operatorname{span}\{f_1(x),f_2(x)\}. \end{equation*} It's easy to see that \begin{equation*} \Delta_x + [\Delta,\Delta]_x = T_xS^2. \end{equation*} Assume that there is a scalar product \begin{equation*} g(f_i,f_j)=\delta_{ij}, \qquad i,j=1,2. \end{equation*} A triple In the talk the problem of finding minimal curves of this structure will be discussed. This problem can be formulated as an optimal control problem: \begin{gather*} \dot{x} = u_1f_1(x) + u_2f_2(x), \\[5pt] x,\omega\in \mathbb{R}^3, \qquad |x|=1, \\[5pt] (u_1,u_2)\in\mathbb{R}^2, \qquad a\in(0,1), \\[5pt] x(0) = \gamma_0, \qquad x(T)=x_T, \\[5pt] \int_0^T \sqrt{u_1^2+u_2^2}\,dt\rightarrow\min. \end{gather*} We'll give a full parameterization of the geodesics and show how this problem is connected with the sub-Riemannian problems on SO(3). We'll also give description of Maxwell sets and bounds on the cut time. Язык доклада: английский |