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 Rus. J. Nonlin. Dyn., 2020, Volume 16, Number 1, Pages 195–208 (Mi nd706)

Mathematical problems of nonlinearity

Asymptotics of Extremal Controls in the Sub-Riemannian Problem on the Group of Motions of Euclidean Space

A. P. Mashtakov, A. Yu. Popov

Ailamazyan Program Systems Institute of RAS, Pereslavl-Zalessky, Yaroslavl Region, 152020 Russia

Abstract: We consider a sub-Riemannian problem on the group of motions of three-dimensional space. Such a problem is encountered in the analysis of 3D images as well as in describing the motion of a solid body in a fluid. Mathematically, this problem reduces to solving a Hamiltonian system the vertical part of which is a system of six differential equations with unknown functions $u_1, \ldots, u_6$. The optimality consideration arising from the Pontryagin maximum principle implies that the last component of the vector control $\bar{u}$, denoted by $u_6$, must be constant. In the problem of the motion of a solid body in a fluid, this means that the fluid flow has a unique velocity potential, i.e., is vortex-free. The case ($u_6 = 0$), which is the most important for applications and at the same time the simplest, was rigorously studied by the authors in 2017. There, a solution to the system was found in explicit form. Namely, the extremal controls $u_1, \ldots, u_5$ were expressed in terms of elliptic functions. Now we consider the general case: $u_6$ is an arbitrary constant. In this case, we obtain a solution to the system in an operator form. Although the explicit form of the extremal controls does not follow directly from these formulas (their calculation requires the inversion of some nontrivial operator), it allows us to construct an approximate analytical solution for a small parameter $u_6$. Computer simulation shows a good agreement between the constructed analytical approximations and the solutions computed via numerical integration of the system.

Keywords: Hamiltonian system, Pontryagin maximum principle, sub-Riemannian, Lie group Received

 Funding Agency Grant Number Russian Science Foundation 17-11-01387 This work was supported by the Russian Science Foundation under grant 17-11-01387 and performed at the Ailamazyan Program Systems Institute of the Russian Academy of Sciences.

DOI: https://doi.org/10.20537/nd200115

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MSC: 93C10, 93C15
Accepted:12.12.2019
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Citation: A. P. Mashtakov, A. Yu. Popov, “Asymptotics of Extremal Controls in the Sub-Riemannian Problem on the Group of Motions of Euclidean Space”, Rus. J. Nonlin. Dyn., 16:1 (2020), 195–208

Citation in format AMSBIB
\Bibitem{MasPop20}
\by A. P. Mashtakov, A. Yu. Popov
\paper Asymptotics of Extremal Controls in the Sub-Riemannian Problem on the Group of Motions of Euclidean Space
\jour Rus. J. Nonlin. Dyn.
\yr 2020
\vol 16
\issue 1
\pages 195--208
\mathnet{http://mi.mathnet.ru/nd706}
\crossref{https://doi.org/10.20537/nd200115}
\elib{https://elibrary.ru/item.asp?id=43279267}
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