We consider a series of optimal control problems with 2-dimensional control lying in an arbitrary convex compact set Ω. The considered problems are well studied for the case when Ω is a unit disc, but barely studied for arbitrary Ω. We derive extremals to these problems in general case by using machinery of convex trigonometry, which allows us to do this identically and independently on the shape of Ω. The paper describes geodesics in (i) the Finsler problem on the Lobachevsky hyperbolic plane; (ii) left-invariant sub-Finsler problems on all unimodular 3D Lie groups (SU(2), SL(2), SE(2), SH(2)); (iii) the problem of rolling ball on a plane with distance function given by Ω; (iv) a series of “yacht problems” generalizing Euler’s elastic problem, Markov-Dubins problem, Reeds-Shepp problem and a new sub-Riemannian problem on SE(2); and (v) the plane dynamic motion problem.
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Keywords: Sub-Finsler geometry, convex trigonometry, optimal control problem, Lobachevsky hyperbolic plane, unimodular 3D Lie group, yacht problems, rolling ball
@article{COCV_2021__27_1_A34_0,
author = {Ardentov, A. A. and Lokutsievskiy, L. V. and Sachkov, Yu. L.},
title = {Extremals for a series of {sub-Finsler} problems with 2-dimensional control via convex trigonometry},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021024},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021024/}
}
TY - JOUR AU - Ardentov, A. A. AU - Lokutsievskiy, L. V. AU - Sachkov, Yu. L. TI - Extremals for a series of sub-Finsler problems with 2-dimensional control via convex trigonometry JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021024/ DO - 10.1051/cocv/2021024 LA - en ID - COCV_2021__27_1_A34_0 ER -
%0 Journal Article %A Ardentov, A. A. %A Lokutsievskiy, L. V. %A Sachkov, Yu. L. %T Extremals for a series of sub-Finsler problems with 2-dimensional control via convex trigonometry %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021024/ %R 10.1051/cocv/2021024 %G en %F COCV_2021__27_1_A34_0
Ardentov, A. A.; Lokutsievskiy, L. V.; Sachkov, Yu. L. Extremals for a series of sub-Finsler problems with 2-dimensional control via convex trigonometry. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 32. doi: 10.1051/cocv/2021024
[1] , Rolling balls and octonions. Proc. Steklov Inst. Math. 258 (2007) 13–22.
[2] and , Sub-Riemanian structures on 3D Lie groups. J. Dyn. Control Syst. 18 (2012) 21–44.
[3] , and , A Comprehensive Introduction to Sub-Riemannian Geometry. Cambridge University Press (2019).
[4] and , Control Theory from the Geometric Viewpoint. Springer-Verlag, Berlin (2004).
[5] , Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it, Leningrad State Univ. Annals [Uchenye Zapiski] Math. Ser. 6 (1939) 3–35.
[6] and , Extremal trajectories for bounded velocity mobile robots. Proceedings 2002 IEEE International Conference on Robotics and Automation (Cat. No.02CH37292) 2 (2002) 1747–1752.
[7] and , Time Optimal Trajectories for Bounded Velocity Differential Drive Vehicles. Int. J. Robotics Res. 21 (2002) 199–217.
[8] and , Sub-Riemannian geometry. Progr. Math. 144 (1996).
[9] , Homogeneous manifolds with an intrinsic metric. II. Sib. Math. J. 30 (1989) 180–191.
[10] , Geodesics of nonholonomic left-invariant intrinsic metrics on the Heisenberg group and isoperimetric curves on the Minkowski plane. Sib. Math. J. 35 (1994) 1–8.
[11] , , Shapes of spheres of special nonholonomic left-invariant intrinsic metrics on some Lie groups. Sib. Math. J. 42 (2001) 613–628.
[12] and , Geodesics in the sub-Riemannian problem on the group SO(3). Sbornik Math. 207 (2016) 29–56.
[13] and , Invariant Carnot-Caratheodory metrics on S3, SO(3), SL(2) and Lens Spaces. SIAM J. Control Optim. 47 (2008) 1851–1878.
[14] , , and , Curve cuspless reconstruction via sub-Riemannian geometry. ESAIM: COCV 20 (2014) 748–770.
[15] , and , Nonisotropic 3-level quantum systems: complete solutions for minimum time and minimumenergy. Discrete Contin. Dyn. Syst. Ser. B 5 (2005) 957–990.
[16] and , On the rate of convergence to the asymptotic cone for nilpotent groups and subFinsler geometry. Proc. Natl. Acad. Sci. USA 110 (2013) 19220–19226.
[17] , The Isoperimetric Problem in the Minkowski Plane. Am. J. Math. 69 (1947) 863–871.
[18] , and , Extremal trajectories and Maxwell strata in sub-Riemannian problem on group of motions of pseudo-Euclidean plane. JDCS 20 (2014) 341–364.
[19] , and , Maxwell strata and conjugate points in sub-Riemannian problem on the Lie group SH(2). JDCS 22 (2016) 747–770.
[20] , and , Cut locus and optimal synthesis in sub-Riemannian problem on the lie group SH(2). JDCS 23 (2017) 155–195.
[21] , , and , Minimum wheel-rotation paths for differential-drive mobile robots. Proceedings 2006 IEEE International Conference on Robotics and Automation (2006) 1616–1623.
[22] , , , and , Stochastic optimization of sailing trajectories in an upwind regatta. J. Oper. Res. Soc. 66 (2015) 807–821.
[23] , On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents. Am. J. Math. 79 (1957) 497–516.
[24] , Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimitrici latissimo sensu accepti. Lausanne: Bousquet (1744).
[25] and , A Markov Decision Process Model for Strategic Decision Making in Sailboat Racing. Advances in Artificial Intelligence. Canadian AI. In vol. 6657 of Lecture Notes in Computer Science. Springer, Berlin, Heidelberg (2011) 110–121.
[26] , Differential equations with discontunuous righthand sides. Kluwer (1988).
[27] , The quasihyperbolic plane. Sib. Math. J. 40 (1999) 245–257.
[28] , Lie Algebras. Interscience (1962).
[29] , The geometry of the plate-ball problem. Arch. Rat. Mech. Anal. 124 (1993) 305–328.
[30] and , Rolling Problems on Spaces of Constant Curvature. Lagrangian and Hamiltonian Methods for Nonlinear Control 2006, edited by et al. In Vol. 366 of Lecture Notes in Control and Information Sciences (2007).
[31] , Convex trigonometry with applications to sub-Finsler geometry. Sb. Math. 210 (2019) 1179–1205.
[32] , Explicit formulae for geodesics in left invariant sub-Finsler problems on Heisenberg groups via convex trigonometry. To appear J. Dyn. Control Syst..
[33] , Some examples of the solution of a special kind of problem on greatest and least quantities, Soobshch. Kharkovsk. Mat. Obshch. 1 (1887) 250–276 (in Russian).
[34] and , Maxwell strata in sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV 16 (2010) 380–399.
[35] , A tour of subriemannnian geometries, their geodesics and applications. Am. Math. Soc. (2002).
[36] and , Optimising Yacht Routes under Uncertainty. Proceedings of the 15th Chesapeake Sailing Yacht Symposium, Annapolis, Maryland, USA. P2001-5.
[37] , , and , The mathematical theory of optimal processes. Wiley Interscience (1962).
[38 , Optimal routing in sailing. Proceeding of the conference Sports Physics, June 8-10, Palaiseau, France (2016).
[39] and , Optimal paths for a car that goes both forwards and backwards. Pacific J. Math. 145 (1990) 367–393.
[40] , Convex Analysis. Princeton University Press (1997).
[41] , Conjugate and cut time in sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV 16 (2010) 1018–1039.
[42] , Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV 17 (2011) 293–321.
[43] , and , The symplectic structure of the primary visual cortex. Biol. Cybernet. 98 (2008) 33–48.
[44] , A generalization of the trigonometric functions. Am. Math. Monthly 66 (1959) 879–884.
[45] and , Nonholonomic dynamical systems. geometry of distributions and variational problems. Itogi Nauk. i Tekhn. Sovrem. Probl. Mat. Fund. Napravl. Dinam. Sistem. 7 (1986) 5–85 (In Russian).
[46] , and , Some generalized trigonometric sine functions and their applications. Appl. Math. Sci. 6 (2012) 6053–6068.
[47] , and , Continuous wind speed models based on stochastic differential equations. Appl. Energy 104 (2013) 42–49.
Cité par Sources :
Section 6 was written by A.A. Ardentov. Sections 1–3, 5, 7 were written by L.V. Lokutsievskiy. Section 4 was written by Yu.L. Sachkov. All results in this paper are products of authors collaborative work.
The work of A.A. Ardentov is supported by the Russian Science Foundation under grant 17-11-01387-p and performed in Ailamazyan Program Systems Institute of Russian Academy of Sciences.
The work of L.V. Lokutsievskiy is supported by the Russian Science Foundation under grant 20-11-20169 and performed in Steklov Mathematical Institute of Russian Academy of Sciences.
The work of Yu.L. Sachkov is supported by the Russian Foundation for Basic Research, project number 19-31-51023.
