Abstract
We consider left-invariant optimal control problems on connected Lie groups. The Pontryagin maximum principle gives necessary optimality conditions. Namely, the extremal trajectories are the projections of trajectories of the corresponding Hamiltonian system on the cotangent bundle of the Lie group. The Maxwell points (i.e., the points where two different extremal trajectories meet each other) play a key role in the study of optimality of extremal trajectories. The reason is that an extremal trajectory cannot be optimal after a Maxwell point. We introduce a general construction for Maxwell points depending on the algebraic structure of the Lie group.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Agrachev, D. Barilari, and U. Boscain, A Comprehensive Introduction to Sub-Riemannian Geometry: From the Hamiltonian Viewpoint (Cambridge Univ. Press, Cambridge, 2020), Cambridge Stud. Adv. Math. 181.
A. A. Agrachev and Yu. L. Sachkov, Geometric Control Theory (Fizmatlit, Moscow, 2005). Engl. transl.: Control Theory from the Geometric Viewpoint (Springer, Berlin, 2004), Encycl. Math. Sci. 87.
A. A. Ardentov, “Multiple solutions in Euler’s elastic problem,” Autom. Remote Control 79 (7), 1191–1206 (2018) [transl. from Avtom. Telemekh., No. 7, 22–40 (2018)].
V. I. Arnol’d and A. B. Givental’, “Symplectic geometry,” in Dynamical Systems–4 (VINITI, Moscow, 1985), Itogi Nauki Tekh., Ser.: Sovrem. Probl. Mat., Fundam. Napravl. 4, pp. 5–135. Engl. transl. in Dynamical Systems IV: Symplectic Geometry and Its Applications (Springer, Berlin, 1990), Encycl. Math. Sci. 4, pp. 1–138.
J. E. Marsden, R. Montgomery, and T. Ratiu, Reduction, Symmetry, and Phases in Mechanics (Am. Math. Soc., Providence, RI, 1990), Mem. AMS 88 (436).
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems (Springer, New York, 1999).
A. V. Podobryaev, “Symmetric extremal trajectories in left-invariant optimal control problems,” Russ. J. Nonlinear Dyn. 15 (4), 569–575 (2019).
A. V. Podobryaev, “Symmetries in left-invariant optimal control problems,” Sb. Math. 211 (2), 275–290 (2020) [transl. from Mat. Sb. 211 (2), 125–140 (2020)].
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes (Fizmatgiz, Moscow, 1961; Pergamon, Oxford, 1964).
L. Rizzi and U. Serres, “On the cut locus of free, step two Carnot groups,” Proc. Am. Math. Soc. 145 (12), 5341–5357 (2017).
Yu. L. Sachkov, “Discrete symmetries in the generalized Dido problem,” Sb. Math. 197 (2), 235–257 (2006) [transl. from Mat. Sb. 197 (2), 95–116 (2006)].
Yu. L. Sachkov, “The Maxwell set in the generalized Dido problem,” Sb. Math. 197 (4), 595–621 (2006) [transl. from Mat. Sb. 197 (4), 123–150 (2006)].
Yu. L. Sachkov, “Complete description of the Maxwell strata in the generalized Dido problem,” Sb. Math. 197 (6), 901–950 (2006) [transl. from Mat. Sb. 197 (6), 111–160 (2006)].
Funding
This work is supported by the Russian Science Foundation under grant 17-11-01387-P.
Author information
Authors and Affiliations
Corresponding author
Additional information
Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2021, Vol. 315, pp. 202–210 https://doi.org/10.4213/tm4223.
Rights and permissions
About this article
Cite this article
Podobryaev, A.V. Construction of Maxwell Points in Left-Invariant Optimal Control Problems. Proc. Steklov Inst. Math. 315, 190–197 (2021). https://doi.org/10.1134/S008154382105014X
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S008154382105014X