Pavel D. Lebedev     (Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russian Federation; Ural Federal University, 19 Mira str., Ekaterinburg, 620002, Russian Federation)
Alexander A. Uspenskii     (Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russian Federation)


A class of time-optimal control problems in three-dimensional space with a spherical velocity vector is considered. A smooth regular curve \(\Gamma\) is chosen as the target set. We distinguish pseudo-vertices that are characteristic points on \(\Gamma\) and responsible for the appearance of a singularity in the function of the optimal result. We reveal analytical relationships between pseudo-vertices and extreme points of a singular set belonging to the family of bisectors. The found analytical representation for the extreme points of the bisector is taken as the basis for numerical algorithms for constructing a singular set. The effectiveness of the developed approach for solving non-smooth dynamic problems is illustrated by an example of numerical-analytical construction of resolving structures for the time-optimal control problem.


Time-optimal problem, Dispersing surface, Bisector, Pseudo-vertex, Extreme point, Curvature, Singular set, Frenet-Serret frame (TNB frame)

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  1. Arnold V.I. Singularities of Caustics and Wave Fronts. Dordrecht: Springer, 1990. 259 p. DOI: 10.1007/978-94-011-3330-2
  2. Dem’yanov V.F., Vasil’ev L.V. Nedifferenciruemaya optimizaciya [Non-Differentiable Optimization], Moscow: Nauka, 1981. 384 p. (in Russian)
  3. Giblin P.J. Symmetry sets and medial axes in two and three dimensions. In: The Mathematics of Surfaces IX. Cipolla R., Martin R. (eds.). London: Springer, 2000. P. 306–321. DOI: 10.1007/978-1-4471-0495-7_18
  4. Isaacs R. Differential games. N.Y.: John Wiley and Sons, 1965. 384 p.
  5. Kružkov 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. P. 406–446. (in Russian) DOI: 10.1070/SM1975v027n03ABEH002522
  6. 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. P. 375–386. DOI: 10.1007/s10598-008-9007-9
  7. Lebedev P.D., Uspenskii A.A. Construction of scattering curves in one class of time-optimal control problems with leaps of a target set boundary curvature. Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., 2020. Vol. 55. P. 93–112. (in Russian) DOI: 10.35634/2226-3594-2020-55-07
  8. Lebedev P.D., Uspenskii A.A. Program for constructing a solution to the tome-optimal problem in three-dimensional space with a spherical velocity vectogram and a nonconvex target set. Certificate of state registration of the computer program, no. 2022666810, September 07, 2022.
  9. Poznyak E.G., Shikin E.V. Differencial’naya geometriya: pervoe znakomstvo. [Differential Geometry: the First Acquaintance]. Moscow: MSU, 1990. 384 p.
  10. Scherbakov R.N., Pichurin L.F. Differencialy pomogayut geometrii [Differentials Help Geometry]. Moscow: Prosveschenie, 1982. 192 p. (in Russian)
  11. Sedykh V.D. On Euler characteristics of manifolds of singularities of wave fronts. Funct. Anal. Appl., 2012. Vol. 46, No. 1. P. 77–80. DOI: 10.1007/s10688-012-0012-6
  12. Sedykh V.D. Topology of singularities of a stable real caustic germ of type \(E_6\). Izv. Math., 2018. Vol. 82, No. 3. P. 596–611. DOI: 10.1070/IM8643
  13. Siersma D. Properties of conflict sets in the plan. Banach Center Publ., 1999. Vol. 50. P. 267–276. DOI: 10.4064/-50-1-267-276
  14. Sotomayor J., Siersma D., Garcia R. Curvatures of conflict surfaces in Euclidean 3-space. Banach Center Publ., 1999. Vol. 50. P. 277–285. DOI: 10.4064/-50-1-277-285
  15. Subbotin A.I. Generalized Solutions of First Order PDEs: The Dynamical Optimization Perspective. Boston: Birkhäuser, 1995. XII+314 p. DOI: 10.1007/978-1-4612-0847-1
  16. Ushakov V.N., Ershov A.A., Matviychuk A.R. On Estimating the degree of nonconvexity of reachable sets of control systems. Proc. Steklov Inst. Math., 2021, Vol. 315. P. 247–256. DOI: 10.1134/S0081543821050199
  17. Ushakov V.N., Uspenskii A.A., Lebedev P.D. Construction of a minimax solution for an Eikonal-type equation. Proc. Steklov Inst. Math., 2008. Vol. 263, Suppl. 2. P. 191–201. DOI: 10.1134/S0081543808060175
  18. 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, Suppl. 1. P. 239–254. DOI: 10.1134/S0081543815090163
  19. Uspenskii A.A., Lebedev P.D. Identification of the singularity of the generalized solution of the Dirichlet problem for an Eikonal type equation under the conditions of minimal smoothness of a boundary set. Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2018. Vol. 28, No. 1. P. 59–73. (in Russian) DOI: 10.20537/vm180106


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