An approach to constructing a minimax solution of the planar Dirichlet problem for the Bellman equation for the case of a closed non-convex boundary set with a non-empty interior has been developed. The approach is based on the constructions of the theory of alpha-sets, which allow revealing the features of non-convex sets and forming non-smooth solutions of first-order partial differential equations. The concepts characterizing the properties of weakly convex sets, metric projections, and Efimov–Stechkin support balls. are also involved. The paper describes a procedure for constructing a minimax solution in a situation where the edge set has a single pseudo-vertex, a special point on the edge set that causes the minimax solution to become singular. In some cases, the presence of a single pseudo-vertex on the edge set allows for the reparameterization of the entire edge set, which in turn provides a basis for constructing a minimax solution in a parametric form. In the paper, a procedure for the separation of the domain of consideration is presented and substantiated, which allows for the construction of a non-smooth solution of the boundary value problem on subdomains in a situation where the pseudo-vertex is a point of inflection on the boundary of the boundary set. The proposed construction is implemented explicitly using the example of a boundary set with a piecewise smooth boundary.
Keywords: Bellman equation, minimax solution, alpha set, metric projection, pseudo-vertex, set bisector, support ball, control, velocity
Received April 20, 2024
Revised June 08, 2026
Accepted June 15, 2026
Pavel Dmitrievich Lebedev, Cand. Sci (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620077 Russia, e-mail: pleb@yandex.ru
Aleksandr Aleksandrovich Uspenskii, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620077 Russia, e-mail: uspen@imm.uran.ru
REFERENCES
1. Oleinik O.A. Discontinuous solutions of non-linear differential equations. Uspekhi Mat. Nauk, 1957, vol. 12, no. 3 (75), pp. 3–73 (in Russian).
2. Kruzhkov S.N. Generalized solutions of the Hamilton–Jacobi equations of eikonal type. I. Formulation of the problems; existence, uniqueness and stability theorems; some properties of the solutions. Math. USSR-Sb., 1975, vol. 27, no. 3, pp. 406–446. https://doi.org/10.1070/SM1975v027n03ABEH002522
3. Crandall M.G., Lions P.L. Viscosity solutions of Hamilton–Jacobi equations. Trans. Amer. Math. Soc., 1983, vol. 277, no. 1, pp. 1–42. https://doi.org/10.2307/1999343
4. Lyapidevskiy V.YU., Teshukov V.M. Matematicheskiye modeli rasprostraneniya dlinnykh voln v neodnorodnoy zhidkosti [Mathematical models of long wave propagation in an inhomogeneous fluid]. Novosibirsk, Iz-vo SO RAN, 2000, 419 с. ISBN: 5-7692-0340-4 .
5. Kalise D., Kunisch K., Rao Z. (eds.) Hamilton–Jacobi–Bellman equations. Numerical methods and applications in optimal control. Berlin, Boston, De Gruyter Publ., 2018, 209 p. ISBN: 9783110543599 .
6. Covei D.P. Radial and non-radial solution structures for quasilinear Hamilton–Jacobi–Bellman equations in bounded settings. ArXiv:2511.06277, 2025, 19 p. https://doi.org/10.48550/arXiv.2511.06277
7. Krasovskii N.N., Subbotin A.I. Game-theoretical control problems. NY, Springer, 1988, 517 p. ISBN: 978-1-4612-8318-8 . Original Russian text was published in Krasovskii N. N., Subbotin A. I. Pozitsionnye differentsial’nye igry, Moscow, Nauka Publ., 1974, 456 p.
8. Subbotin A.I. Generalized solutions of first-order PDEs. The dynamical optimization perspective. Boston, Birkhäuser, 1995, 314 p. https://doi.org/10.1007/978-1-4612-0847-1 . Translated to Russian under the title Obobshchennyye resheniya uravnenii v chastnykh proizvodnykh pervogo poryadka. Perspektivy dinamicheskoy optimizatsii, Moscow, Izhevsk, In-t Komput. Issled. Publ., 2003, 336 p. ISBN: 5-93972-206-7 .
9. Dem’yanov V.F., Vasil’ev L.V. Nondifferentiable optimization. New York, Springer, 1985, 452 p. Original Russian text published in Dem’yanov V. F., Vasil’ev L. V. Nedifferentsiruyemaya optimizatsiya, Moscow, Nauka Publ., 1981, 384 p. ISBN: 978-0-387-90951-6 .
10. Ushakov V.N., Uspenskii A.A. Theorems on the separability of α-sets in Euclidean space. Proc. Steklov Inst. Math., 2017, vol. 299, iss. 1, pp. S231–S245. https://doi.org/10.1134/S0081543817090255
11. Efimov N.V., Stechkin S.B. Support properties of sets in Banach spaces and Chebyshev sets. Dokl. Akad. Nauk SSSR, 1959, vol. 127, no. 2, pp. 254–257 (in Russian).
12. Dubrovin B.A., Novikov S.P., Fomenko A.T. Sovremennaya geometriya. Metody i prilozheniya (2-ye izd., pererab.) [Modern geometry. Methods and applications] (2nd ed., revised). Moscow, Nauka Publ., 1986, 760 p.
13. Subbotin A.I. Minimaksmye neravenstva i uravneniya Gamil’tona—Yakobi [Minimax inequalities and Hamilton — Jacobi equations], Moscow, Nauka Publ., 1991, 216 p. ISBN: 5-02-000139-2 .
14. Lebedev P.D., Uspenskii A.A. Analytical and numerical construction of the optimal outcome function for a class of time-optimal problems. Comput. Math. Model., 2008, vol. 19, no. 4, pp. 375–386. https://doi.org/10.1007/s10598-008-9007-9
15. Lebedev P.D., Uspenskii P.D. Numerical-analytic construction of a generalized solution to the eikonal equation in the plane case. Sb. Math., 2024, vol. 215, iss. 9, pp. 1224–1248. https://doi.org/10.4213/sm9500e
16. Subbotina N.N., Shemyakin D.V. Method of characteristics in solving the Dirichlet problem for the eikonal equation in an unbounded domain. Trudy Inst. Mat. Mekh. UrO RAN, 2026, vol. 32, no. 2, pp. 231–241. https://doi.org/10.21538/0134-4889-2026-32-2-231-241
17. Borovskikh A.V. Eikonal equation for anisotropic media. J. Math. Sci., 2014, vol. 197, iss. 2, pp. 248–289. https://doi.org/10.1007/s10958-014-1714-5
18. Alimov A.R. Singularities of solutions of the eikonal equation. Diff. Equat., 2019, vol. 55, no. 10, pp. 1311–1316. https://doi.org/10.1134/S0012266119100069
19. Lebedev P.D., Uspenskii A.A. Construction of a nonsmooth solution in a time-optimal problem with a low order of smoothness of the boundary of the target set. Trudy Inst. Mat. Mekh. UrO RAN, 2019, vol. 25, no. 1, pp. 108–119 (in Russian). https://doi.org/10.21538/0134-4889-2019-25-1-108-119
20. Lebedev P.D., Uspenskii A.A. Analitycal methods for constructing the domain of differentiability of the minimax solution in a class of boundary value problems for a Hamilton type equation. Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2026, vol. 36, no. 1, pp. 23–41. https://doi.org/10.35634/vm260102
21. Bröcker Th., Lander L. Differentiable germs and catastrophes. Cambridge, New York, Cambridge Univ. Press, 1975, 179 p. https://doi.org/10.1017/cbo9781107325418 . Translated to Russian under the title Differentsiruyemyye rostki i katastrofy, Мoscow, Mir Publ., 1977, 208 p.
22. Uspenskii A.A. Calculation formulas for nonsmooth singularities of the optimal result function in a time-optimal problem. Proc. Steklov Inst. Math., 2015, vol. 291, iss. 1, pp. 239–254. https://doi.org/10.1134/S0081543815090163
23. Lebedev P.D., Uspenskii A.A. Conditions of transversality of branches of solution to nonlinear equation in speed problem with a circular indicatrix. Trudy Inst. Mat. Mekh. UrO RAN, 2008, vol. 14, no. 4, pp. 82–99 (in Russian).
24. Bruce J.W., Giblin P.J. Curves and singularities. A geometrical introduction to singularity theory. Cambridge, New York, Cambridge Univ. Press, 1984, 222 p. ISBN-10: 0521249457 . Translated to Russian under the title Krivyye i osobennosti. Geometricheskoye vvedeniye v teoriyu osobennostey, Мoscow, Mir Publ., 1988, 262 p. ISBN: 5-03-001194-3 .
25. Mestetskii L.M. Nepreryvnaya morfologiya binarnykh izobrazheniy: figury, skelety, tsirkulyary [Continuous morphology of binary images: figures, skeletons, circulars]. Moscow, Fizmatlit Publ., 2009, 288 p. ISBN: 978-5-9221-1050-1 .
26. Nackman L.R., Srinivasan V. Bisectors of linearly separable sets. Discrete Comput. Geom., 1991, vol. 6, pp. 263–275. https://doi.org/10.1007/BF02574688
27. Imai K., Kawamura A., Matoušek J., Reem D., Tokuyama T. Distance k-sectors exist. Comput. Geom., 2010, vol. 43, iss. 9, pp. 713–720. https://doi.org/10.1016/j.comgeo.2010.05.001
28. Efimov N.V., Stechkin S.B. Some properties of Chebyshev sets. Dokl. Akad. Nauk SSSR, 1958, vol. 118, no. 1, pp. 17–19 (in Russian).
29. Vlasov L.P. Approximative properties of sets in normed linear spaces. Russian Math. Surveys, 1973, vol. 28, iss. 6, pp. 1–66. https://doi.org/10.1070/RM1973v028n06ABEH001624
30. Attali D., Boissonnat J.D., Edelsbrunner H. Stability and computation of medial axes: a state-of-the-art report. In: Möller T., Hamann B., Russell R.D. (eds.) Mathematical foundations of scientific visualization, computer graphics, and massive data exploration. Ser. Mathematics and visualization. Berlin, Heidelberg, Springer, 2009, pp. 109–125. https://doi.org/10.1007/b106657_6
31. Białożyt A. The tangent cone, the dimension and the frontier of the medial axis. Nonlinear Differ. Equ. Appl., 2023, vol. 30, art. no. 27. https://doi.org/10.1007/s00030-022-00833-9
32. Uspenskii A.A., Lebedev P.D. Finding the value of the Chebyshev layer of a flat set using constructions of the theory of alpha sets and Efimov–Stechkin support balls. Proc. Steklov Inst. Math., 2025, vol. 331, iss. 1, pp. S71–S85. https://doi.org/10.1134/S0081543826600134
33. Goresky M., MacPherson R. Stratified Morse theory. Berlin, Heidelberg, Springer-Verlag, 1988, 272 p. https://doi.org/10.1007/978-3-642-71714-7 . Translated to Russian under the title Stratifitsirovannaya teoriya Morsa, Moscow, Mir Publ., 1991, 351 p. ISBN: 5-03-001850-6 .
Cite this article as: P.D. Lebedev, A.A. Uspenski. The procedure for separating and parameterizing the minimax solution in one class of boundary value problems for the Bellman equation. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 3, pp. 139–158.
