Conjugate and cut time in the sub-riemannian problem on the group of motions of a plane

Sachkov, Yuri L.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 4, pp. 1018-1039.

The left-invariant sub-riemannian problem on the group of motions (rototranslations) of a plane SE(2) is studied. Local and global optimality of extremal trajectories is characterized. Lower and upper bounds on the first conjugate time are proved. The cut time is shown to be equal to the first Maxwell time corresponding to the group of discrete symmetries of the exponential mapping. Optimal synthesis on an open dense subset of the state space is described.

Classification : 49J15, 93B29, 93C10, 53C17, 22E30
Mots clés : optimal control, sub-riemannian geometry, differential-geometric methods, left-invariant problem, group of motions of a plane, rototranslations, conjugate time, cut time

[1] A.A. Agrachev, Exponential mappings for contact sub-Riemannian structures. J. Dyn. Control Syst. 2 (1996) 321-358. | Zbl

[2] A.A. Agrachev, Geometry of optimal control problems and Hamiltonian systems, in Nonlinear and Optimal Control Theory, Lect. Notes Math. CIME 1932, Springer Verlag (2008) 1-59. | Zbl

[3] A.A. Agrachev and Y.L. Sachkov, Control Theory from the Geometric Viewpoint. Springer-Verlag, Berlin (2004). | Zbl

[4] A.A. Agrachev, U. Boscain, J.P. Gauthier and F. Rossi, The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups. J. Funct. Anal. 256 (2009) 2621-2655. | Zbl

[5] G. Citti and A. Sarti, A cortical based model of perceptual completion in the roto-translation space. J. Math. Imaging Vis. 24 (2006) 307-326. | Zbl

[6] C. El-Alaoui, J.P. Gauthier and I. Kupka, Small sub-Riemannian balls on ${ℝ}^3$. J. Dyn. Control Syst. 2 (1996) 359-421. | Zbl

[7] V. Jurdjevic, Geometric Control Theory. Cambridge University Press (1997). | Zbl

[8] J.P. Laumond, Nonholonomic motion planning for mobile robots, Lecture Notes in Control and Information Sciences 229. Springer (1998).

[9] I. Moiseev and Y.L. Sachkov, Maxwell strata in sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV (2009), doi:10.1051/cocv/2009004. | Numdam

[10] J. Petitot, The neurogeometry of pinwheels as a sub-Riemannian contact structure. J. Physiology - Paris 97 (2003) 265-309.

[11] J. Petitot, Neurogéometrie de la vision - Modèles mathématiques et physiques des architectures fonctionnelles. Éditions de l'École polytechnique, France (2008).

[12] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko, The mathematical theory of optimal processes. Wiley Interscience (1962). | Zbl

[13] Y.L. Sachkov, Exponential mapping in generalized Dido's problem. Mat. Sbornik 194 (2003) 63-90 (in Russian). English translation in Sbornik: Mathematics 194 (2003). | Zbl

[14] Y.L. Sachkov, Discrete symmetries in the generalized Dido problem. Matem. Sbornik 197 (2006) 95-116 (in Russian). English translation in Sbornik: Mathematics, 197 (2006) 235-257. | Zbl

[15] Y.L. Sachkov, The Maxwell set in the generalized Dido problem. Matem. Sbornik 197 (2006) 123-150 (in Russian). English translation in Sbornik: Mathematics 197 (2006) 595-621. | Zbl

[16] Y.L. Sachkov, Complete description of the Maxwell strata in the generalized Dido problem. Matem. Sbornik 197 (2006) 111-160 (in Russian). English translation in: Sbornik: Mathematics 197 (2006) 901-950. | Zbl

[17] Y.L. Sachkov, Maxwell strata in Euler's elastic problem. J. Dyn. Control Syst. 14 (2008) 169-234. | Zbl

[18] Y.L. Sachkov, Conjugate points in Euler's elastic problem. J. Dyn. Control Syst. 14 (2008) 409-439. | Zbl

[19] Y.L. Sachkov, Cut locus and optimal synthesis in sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV (submitted).

[20] A.V. Sarychev, The index of second variation of a control system. Matem. Sbornik 113 (1980) 464-486 (in Russian). English translation in Math. USSR Sbornik 41 (1982) 383-401. | Zbl

[21] A.M. Vershik and V.Y. Gershkovich, Nonholonomic Dynamical Systems - Geometry of distributions and variational problems (in Russian), in Itogi Nauki i Tekhniki: Sovremennye Problemy Matematiki, Fundamental'nyje Napravleniya 16, VINITI, Moscow (1987) 5-85. English translation in Encyclopedia of Math. Sci. 16, Dynamical Systems 7, Springer Verlag. | Zbl

[22] E.T. Whittaker and G.N. Watson, A Course of Modern Analysis. An introduction to the general theory of infinite processes and of analytic functions; with an account of principal transcendental functions. Cambridge University Press, Cambridge (1996). | Zbl

[23] S. Wolfram, Mathematica: a system for doing mathematics by computer. Addison-Wesley, Reading, USA (1991). | Zbl