In this paper, a Mayer minimization problem with a free right endpoint for a nonlinear three-dimensional differential system with affine control is studied on a fixed time interval. For this problem, Lie brackets are defined that characterize the derivatives of the corresponding switching function up to and including the fourth order. For these Lie brackets, expansions over a set of linearly independent vector functions are considered. A type of expansion that can lead to the existence of a singular local second-order regimen for the optimal control in a minimization problem is studied in detail. Two specific examples of this problem are given in which this type of Lie bracket expansion holds.
Keywords: control affine differential system, Mayer problem, switching function, Lie bracket, singular regimen of local second order
Received November 18, 2025
Revised December 4, 2025
Accepted December 8, 2025
Funding Agency: The paper was published with the support of the Moscow Center for Fundamental and Applied Mathematics under the agreement no. 075-15-2025-345.
Evgenii Nikolaevich Khailov, Cand. Sci. (Phys.-Math.), Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow, 119992 Russia, e-mail: khailov@cs.msu.su
REFERENCES
1. Vasil’ev F.P. Metody optimizatsii [Optimization methods], Moscow, Faktorial Press, 2002, 824 p. ISBN: 5-88688-056-9 .
2. Zelikin M.I., Borisov V.F. Theory of chattering control with applications to astronautics, robotics, economics, and engineering. Boston, Birkhäuser, 1994, 244 p. https://doi.org/10.1007/978-1-4612-2702-1
3. Schättler H., Ledzewicz U. Geometric optimal control: theory, methods and examples. NН, Heidelberg, Dordrecht, London, Springer, 2012, 640 p. https://doi.org/10.1007/978-1-4614-3834-2
4. Gabasov R., Kirillova F.M. Osobyye optimal’nyye upravleniya [Special optimal controls]. Moscow, Librocom Publ., 2013, 256 p. ISBN: 978-5-397-03699-3 .
5. Kim D.P. Teoriya avtomaticheskogo upravleniya. T. 2. Mnogomernyye, nelineynyye, optimal’nyye i adaptivnyye sistemy: uchebnoye posobiye [Theory of automatic control. Vol. 2. Multidimensional, nonlinear, optimal and adaptive systems: a tutorial]. Moscow, Yurait Publ., 2016, 442 p. ISBN: 9785991686020 .
6. Zelikin M.I., Borisov V.F. Regimes with increasingly more frequent switchings in optimal control problems. Proc. Steklov Inst. Math., 1993, vol. 197, pp. 95–186.
7. Zhu J., Trélat E., Cerf M. Planar titling maneuver of a spacecraft: singular arcs in the minimum time problem and chattering. Discrete Contin. Dyn. Syst. Ser. B., 2016, vol. 21, no. 4, pp. 1347–1388. https://doi.org/10.3934/dcdsb.2016.21.1347
8. Melnikov N.B., Ronzhina M.I. Chattering extremals in control-affine stabilization problems. Russ. Math. Surv., 2024, vol. 79, iss. 5, pp. 931–933. https://doi.org/10.4213/rm10198e
9. Melnikov N.B., Ronzhina M.I. Chattering trajectories in stabilization problems for nonlinear control-affine systems. Tr. Inst. Mat. Mekh. UrO RAN, 2025, vol. 31, no. 1, pp. 138–153. https://doi.org/10.21538/0134-4889-2025-31-1-138-153
10. Lee E.B., Markus L. Foundations of optimal control theory. NY, London, Sydney, John Wiley & Sons, 1967, 576 p. ISBN: 0471522635 . Translated to Russian under the title Osnovy teorii optimal’nogo upravleniya, Moscow, Nauka Publ., 1972, 576 p.
11. Khailov E.N., Grigorenko N.L., Grigorieva E.V., Klimenkova A.D. Upravlyayemyye sistemy Lotki-Vol’terry v modelirovanii mediko-biologicheskikh protsessov [Controlled Lotka — Volterra systems in the modeling of biomedical processes], Moscow, MAKS PRESS Publ., 2021, 204 p. https://doi.org/10.29003/m2448.978-5-317-06681-9
12. Grigorieva E., Khailov E. Chattering and its approximation in control of psoriasis treatment. Discrete Contin. Dyn. Syst. Ser. B., 2019, vol. 24, iss. 5, pp. 2251–2280. https://doi.org/10.3934/dcdsb.2019094
Cite this article as: E.N. Khailov. On a singular regimen of local second order in a three-dimensional differential system affine in control. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 2, pp. 259–270.
