Geodesic
Method of the optimal control in the solution of a variational problem
Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 6 (2016), pp. 28-39.

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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. 
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     author = {A. S. Ignat{\cyre}nko and B. E. Levitsii},
     title = {Method of the optimal control in the solution of a variational problem},
     journal = {Matemati\v{c}eska\^a fizika i kompʹ\^uternoe modelirovanie},
     pages = {28--39},
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     year = {2016},
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A. S. Ignatеnko; B. E. Levitsii. Method of the optimal control in the solution of a variational problem. Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 6 (2016), pp. 28-39. http://geodesic.mathdoc.fr/item/VVGUM_2016_6_a3/

[1] A.\;S. Ignatenko, B.\;E. Levitskiy, “Method of Optimal Control in the Solution of the Variational Problem for the Modules of Families of the Surfaces That Bend Around Obstacles in a Spherical Ring”, Tr. mat. tsentra im. N.\;I. Lobachevskogo, 13, 2002, 64–70

[2] L.\;S. Pontryagin, V.\;G. Boltyanskiy, The Mathematical Theory of Optimal Processes, Nauka Publ., M., 1983, 393 pp.