Mathematics
Method of the optimal control in the solution of a variational problem
A. S. Ignatånko,
B. E. Levitsii Kuban State University, Krasnodar
Abstract:
The paper provides a complete solution for the variational problem of finding a revolution surface of minimum area in the metric
$|x|^{-n+1}$, corresponding extreme metric for
$p$-module of family of surfaces that separate
boundary components of a spherical ring.
The surface area in the
$n$-dimensional Euclidean space
$R^n$, defined by the rotation of the curve
$\gamma$ around the polar axis, calculated in the metric
$\frac {1}{|x|^{n-1}}$,
$x \in R^n$,
$n \geq 3$, expressed by the formula
\begin{equation*}
S(\gamma) = (n-1)\omega_{n-1} \int_{t_0}^{t_1} \sin^{n-2} \varphi (t) \sqrt{(\varphi^{'}(t))^2+(\rho^{'} (t))^2} dt,
\end{equation*}
where
$\omega_n$ is a volume of
$n$-dimensional sphere of radius 1,
$\gamma$ is the curve of the family of planar piecewise-smooth curves, given by the parametric equation
$z(t)=e^{\rho (t) + i \varphi (t)}$,
$t \in [t_0,t_1]$,
is lying in the closed set $\overline{B_r} = \{ z: r \leq |z| \leq r(1+\delta), \varphi \in [ \varphi_0, \varphi_1 ] \}$,
$( 0< \varphi_0 < \varphi_1 \leq \pi)$ and is connecting the point
$z(t_0)=r(1+\delta)e^{i\varphi_0}$
and the point
$z(t_1)=r(1+\delta_1)e^{i\varphi_1}$,
$0 \leq \delta_1 \leq \delta$.
The problem is to find the infimum of the functional
$S(\gamma)$ in the described class of curves with natural condition that we consider only curves for which in the points of differentiability
$\varphi^{'}(t) \geq 0$ and
$\rho^{'}(t) \leq 0$. The method of optimal controls by L. Pontryagin [2] is applied for search for optimal trajectories. The properties of the hyperelliptic integral of a special type,
arising in the solution of the variational problem, were investigated.
Keywords:
minimal surface, surface of revolution, method of the optimal control, optimal trajectory, hyperelliptic integral.
UDC:
517.53:517.977
BBK:
22.161.5
DOI:
10.15688/jvolsu1.2016.6.3