Geodesic
Explicit solutions for a series of optimization problems with 2-dimensional control via convex trigonometry
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 494 (2020), pp. 86-92 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider a number of optimal control problems with 2-dimensional control lying in an arbitrary convex compact set $\Omega$. Solutions to these problems are obtained using methods of convex trigonometry. The paper includes (1) geodesics in the Finsler problem on the Lobachevsky hyperbolic plane; (2) left-invariant sub-Finsler geodesics on all unimodular 3D Lie groups (SU(2), SL(2), SE(2), SH(2)); (3) the problem of a ball rolling on a plane with a distance function given by $\Omega$; and (4) a series of “yacht problems” generalizing Euler’s elastic problem, the Markov–Dubins problem, the Reeds–Shepp problem, and a new sub-Riemannian problem on SE(2).
Keywords: sub-Finsler geometry, convex trigonometry, optimal control problem, Lobachevsky hyperbolic plane, unimodular 3D Lie groups, rolling ball, Euler’s elastica, yacht problems. 
@article{DANMA_2020_494_a19,
     author = {A. A. Ardentov and L. V. Lokutsievskiy and Yu. L. Sachkov},
     title = {Explicit solutions for a series of optimization problems with 2-dimensional control via convex trigonometry},
     journal = {Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleni\^a},
     pages = {86--92},
     year = {2020},
     volume = {494},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DANMA_2020_494_a19/}
}
TY  - JOUR
AU  - A. A. Ardentov
AU  - L. V. Lokutsievskiy
AU  - Yu. L. Sachkov
TI  - Explicit solutions for a series of optimization problems with 2-dimensional control via convex trigonometry
JO  - Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ
PY  - 2020
SP  - 86
EP  - 92
VL  - 494
UR  - http://geodesic.mathdoc.fr/item/DANMA_2020_494_a19/
LA  - ru
ID  - DANMA_2020_494_a19
ER  - 
%0 Journal Article
%A A. A. Ardentov
%A L. V. Lokutsievskiy
%A Yu. L. Sachkov
%T Explicit solutions for a series of optimization problems with 2-dimensional control via convex trigonometry
%J Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ
%D 2020
%P 86-92
%V 494
%U http://geodesic.mathdoc.fr/item/DANMA_2020_494_a19/
%G ru
%F DANMA_2020_494_a19
A. A. Ardentov; L. V. Lokutsievskiy; Yu. L. Sachkov. Explicit solutions for a series of optimization problems with 2-dimensional control via convex trigonometry. Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 494 (2020), pp. 86-92. http://geodesic.mathdoc.fr/item/DANMA_2020_494_a19/

[1] Lokutsievskii L.V., “Vypuklaya trigonometriya s prilozheniyami k subfinslerovoi geometrii”, Matem. sb., 210:8 (2019), 120–148 | DOI | MR | Zbl

[2] Gribanova I. A., “Kvazigiperbolicheskaya ploskost”, Sib. matem. zhurn., 40:2 (1999), 288–301 | MR | Zbl

[3] Jacobson N., Lie Algebras, Interscience, 1962 | MR | Zbl

[4] Agrachev A., Barilari D., “Sub-Riemanian structures on 3D Lie groups”, J. Dynamical and Control Systems, 18 (2012), 21–44 | DOI | MR | Zbl

[5] Busemann H., “The Isoperimetric Problem in the Minkowski Plane”, American J. of Mathematics, 69:4, Oct. (1947), 863–871 | DOI | MR | Zbl

[6] Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mischenko E.F., Matematicheskaya teoriya optimalnykh protsessov, Fizmatlit, M., 1961 | MR

[7] Agrachev A.A., Sachkov Yu.L., Geometricheskaya teoriya upravleniya, Fizmatlit, M., 2005

[8] Berestovskii V. N., “Geodezicheskie negolonomnykh levoinvariantnykh vnutrennikh metrik na gruppe Geizenberga i izoperimetriksy ploskosti Minkovskogo”, Sib. matem. zhurn., 35:1 (1994), 3–11 | Zbl

[9] Euler L., Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimitrici latissimo sensu accepti, Bousquet, Lausanne, 1744 | MR

[10] Markov A.A., “Neskolko primerov resheniya osobogo roda zadach o naibolshikh i naimenshikh velichinakh”, Soobsch. Kharkov. matem. obsch. Vtoraya ser., 1, 1889, 250–276

[11] Dubins L.E., “On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents”, American J. of Mathematics, 79:3 (1957), 497–516 | DOI | MR | Zbl

[12] Reeds J.A., Shepp L.A., “Optimal Paths for a Car That Goes Both Forwards and Backwards”, Pacific J. Math., 145:2 (1990), 367–393 | DOI | MR | Zbl

[13] Sachkov Yu.L., “Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane”, ESAIM: Control, Optimisation and Calculus of Variations, 17:2 (2011), 293–321 | DOI | MR | Zbl