Abstract:
A Finsler analog of the Lobachevsky plane is the Lie group of proper affine transformations of the real line with a left-invariant Finsler structure generated by a convex compact set in the Lie algebra with the origin in its interior. We consider the isoperimetric problem on this Lie group, with the volume form also taken to be left-invariant. This problem is formulated as an optimal control problem. Applying the Pontryagin maximum principle, we find the optimal isoperimetric loops in an explicit form in terms of convex trigonometry functions. We also present a generalized isoperimetric inequality in a parametric form.
Citation:
V. A. Myrikova, “An Isoperimetric Problem on the Lobachevsky Plane with a Left-Invariant Finsler Structure”, Optimal Control and Dynamical Systems, Collected papers. On the occasion of the 95th birthday of Academician Revaz Valerianovich Gamkrelidze, Trudy Mat. Inst. Steklova, 321, Steklov Math. Inst., Moscow, 2023, 223–236; Proc. Steklov Inst. Math., 321 (2023), 208–221
\Bibitem{Myr23}
\by V.~A.~Myrikova
\paper An Isoperimetric Problem on the Lobachevsky Plane with a Left-Invariant Finsler Structure
\inbook Optimal Control and Dynamical Systems
\bookinfo Collected papers. On the occasion of the 95th birthday of Academician Revaz Valerianovich Gamkrelidze
\serial Trudy Mat. Inst. Steklova
\yr 2023
\vol 321
\pages 223--236
\publ Steklov Math. Inst.
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4325}
\crossref{https://doi.org/10.4213/tm4325}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4643643}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2023
\vol 321
\pages 208--221
\crossref{https://doi.org/10.1134/S0081543823020153}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85170851207}
Citation in format AMSBIB
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