Vol. 25, no. 1, 2019
Procedures for the construction of an optimal result function have been developed for a planar time-optimal control problem with a circular velocity vectorgram and nonconvex compact target set whose boundary has smoothness 1 or 2. Pseudovertices, which are characteristic points of the boundary of the target set defining the character of the singularity of this function, are studied. Differential dependences for smooth segments of the singular set are revealed, which allows to consider and construct them as arcs of integral curves. The necessary conditions for the existence of pseudovertices are found and formulas for the projections of points of the singular set in neighborhoods of pseudovertices are obtained. The proposed procedures are implemented in the form of computational algorithms. Their efficiency is illustrated by examples of the numerical solution of optimal-time control problems with different orders of smoothness of the boundaries of the target sets. Visualization of the results is performed.
Keywords: time-optimal problem, singular set, dispersing curve, optimal result function, pseudo-vertex, symmetry set
Received December 5, 2018
Revised February 7, 2019
Accepted February 11, 2019
Funding Agency: This work was supported by Resolution No. 211 of March 16, 2013, of the Government of the Russian Federation (agreement no. 02.A03.21.0006).
Pavel Dmitrievich Lebedev, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620002 Russia, e-mail: pleb@yandex.ru
Aleksandr Aleksandrovich Uspenskii, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620002 Russia, e-mail: uspen@imm.uran.ru
Cite this article as: P.D. Lebedev, A.A. Uspenskii. Construction of a nonsmooth solution in a time-optimal problem with a low order of smoothness of the boundary of the target set, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 1, pp. 108-119.
REFERENCES
1. Krasovskii N.N., Subbotin A.I. Game-theoretical control problems. N Y: Springer, 1988, 517 p. ISBN: 978-1-4612-8318-8 . Original Russian text published in Krasovskii N.N., Subbotin A.I. Pozitsionnye differentsial’nye igry. Moscow, Nauka Publ., 1974, 456 p.
2. 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. doi: 10.1007/s10598-008-9007-9
3. Lebedev P.D., Uspenskii A.A. Geometry and asymptotics of wavefronts. Russian Mathematics (Izvestiya VUZ. Matematika), 2008, vol. 52, no. 3, pp. 24–33. doi: 10.3103/S1066369X08030031
4. Ushakov V.N., Uspenskii A.A., Lebedev P.D. Geometry of singular curves for one class of velocity. Vestn. S.-Petersburg Univer. Ser. 10. Prikl. Mat. Inform. Prots. Upr., 2013, no. 3, pp. 157–167 (in Russian).
5. Uspenskii A.A. , Lebedev P.D. Construction of the optimal outcome function for a time-optimal problem on the basis of a symmetry set. Autom. Remote Control, 2009, vol. 70, no. 7, pp. 1132–1139. doi: 10.1134/S0005117909070054
6. Lebebev P.D., Uspenskii A.A., Ushakov V.N. Construction of nonsmooth solutions in one class of velocity problems. Constructive Nonsmooth Analysis and Related Topics (dedicated to the memory of V.F.Demyanov), CNSA 2017 — Proc. 10 July 2017, article no. 7973981, pp. 185–188. doi: 10.1109/CNSA.2017.7973981
7. Isaacs R. Differential games. N Y: John Wiley and Sons, 1965, 384 p. ISBN: 0471428604 . Translated to Russian under the title Differentsial’nye igry. Moscow, Mir Publ., 1967, 479 p.
8. Subbotin A.I. Generalized solutions of first-order PDEs. The dynamical optimization perspective. Basel, BirkhЈauser, 1995. 314 p. doi: 10.1007/978-1-4612-0847-1 . Translated to Russian under the title Obobshchennye resheniya uravnenii v chastnykh proizvodnykh pervogo poryadka: Perspektivy dinamicheskoi optimizatsii. Moscow, Izhevsk: Inst. Komp’yuter. Issled. Publ., 2003. 336 p.
9. 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, Mathematics of the USSR-Sbornik, 1975, vol. 27, no. 3, pp. 406–446. doi: 10.1070/SM1975v027n03ABEH002522
10. Dem’yanov V.F., Vasil’ev L.V. Nondifferentiable optimization. N Y: Springer-Verlag, 1985, 452 p. ISBN: 978-0-387-90951-6 . Original Russian text published in Dem’yanov V. F. , Vasil’ev L.V. Nedifferentsiruemaya optimizatsiya. Moscow: Nauka Publ., 1981, 384 p.
11. Sedykh V.D. On the topology of cooriented wave fronts in spaces of small dimensions. Mosc. Math. J., 2011, vol. 11, no. 3, pp. 583–598.
12. Sedykh,Vyacheslav D. On the topology of wave fronts in spaces of low dimension. Izvestiya: Mathematics, 2012, vol. 76, no. 2, pp. 375–418. doi: 10.1070/IM2012v076n02ABEH002588
13. Arnold V.I. Singularities of caustics and wave fronts. Springer Netherlands, 1990, Math. Appl., vol. 62, 259 p. doi: 10.1007/978-94-011-3330-2 . Translated to Russian under the title Osobennosti kaustik i volnovyh frontov. Moscow: FAZIS Publ., 1996, 334 p.
14. Giblin P.J., Warder J.P. Evolving evolutoids. American Mathematical Monthly, 2014, vol. 121, no. 10, pp. 871–889. doi: 10.4169/amer.math.monthly.121.10.871
15. Giblin P.J., Reeve G. Centre symmetry sets of families of plane curves. Demonstratio Mathematica, 2015, vol. 48, no. 2, pp. 167–192. doi: 10.1515/dema-201-0016
16. Mestetskiy L.M. Nepreryvnaja morfologija binarnyh izobrazhenij: figury, skelety, cirkuljary (Continuous morphology of binary images. Figures, skeletons, circular). Moscow: Fizmatlit Publ., 2009, 288 p. ISBN: 978-5-9221-1050-1 .
17. Rashevskii P.K. Kurs differencial’noj geometrii [A Course in Differential Geometry]. Moscow: URSS Publ., 2003, 432 p. ISBN: 5-354-00294-X .
18. Leichtweiss K. Konvexe Mengen. Berlin: Springer, 1980, 330 p. ISBN: 978-3-540-09071-7 . Translated to Russian under the title Vypuklye mnozhestva. Moscow: Nauka Publ., 1985, 335 p.
19. Lebedev P.D., Uspenskii A.A. Program for constructing wave fronts and functions of the Euclidean distance to a compact nonconvex set. Certificate of state registration of the computer program no. 2017662074, October 27, 2017.
20. Preparata F.P., Shamos M.I. Computational geometry: An introduction. N Y: Springer-Verlag, 1988, 400 p. doi: 10.1007/978-1-4612-1098-6
