On smoothness conditions and selection of the edge of a scattering surface in one class of 3D time-optimal problems

P. D. Lebedev ; A. A. Uspenskii

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 63 (2024), pp. 37-48.

Résumé

A class of time-optimal problems in three-dimensional space with a spherical velocity vectogram is considered. The target set is a parametrically defined smooth curve. Numerical and analytical approaches to constructing the bisector of the target set — the scattering surface — in the time-optimal problem are proposed. The algorithms are based on formulas for the edge points of the scattering surface, written in terms of curve invariants. It is shown that these points form the edge of the bisector and lie at the centers of the touching spheres to the curve. A theorem on sufficient conditions for the smoothness of a scattering surface is proven. The equations of the tangent plane to the bisector are found for those points from which exactly two optimal trajectories emerge. An example of solving a time-optimal problem in the form of a set of level surfaces of the optimal result function is given, highlighting the surface of their non-smoothness.

Détail

Keywords: time-optimal problem, scattering surface, bisector, pseudovertex, extreme point, optimal result function
Mots-clés : tangent plane

Bibliographie

[1] Isaacs R., Differential games, John Wiley and Sons, New York, 1965 | Zbl

[2] Lebedev P.D., Uspenskii A.A., “Combined algorithms for constructing a solution to the time-optimal problem in three-dimensional space based on the selection of extreme points of the scattering surface”, Ural Mathematical Journal, 8:2 (2022), 115–126 | DOI | MR | Zbl

[3] Lebedev P.D., Uspenskii A.A., “Analytic-numerical approach to construction of minimax solution to the Hamilton–Jacobi equation in three-dimensional space”, Journal of Mathematical Sciences, 262:3 (2022), 291–300 | DOI | MR | Zbl

[4] Giblin P., “Symmetry sets and medial axes in two and three dimensions”, The mathematics of surfaces IX, Proceedings of the Ninth IMA Conference on the Mathematics of Surfaces, Springer, London, 2000, 306–321 | DOI | MR | Zbl

[5] Lebedev P.D., Uspenskii A.A., Ushakov V.N., “Construction of a minimax solution for an eikonal-type equation”, Proceedings of the Steklov Institute of Mathematics, 263:suppl. 2 (2008), S191–S201 | DOI | MR | Zbl

[6] Shcherbakov R.N., Pichurin L.F., Differentials help geometry, Prosveshchenie, Moscow, 1982

[7] Subbotin A.I., Generalized solutions of first order PDEs. The dynamical optimization perspective, Boston, Birkhäuser, 1995 | DOI | MR

[8] Dem’yanov V.F., Vasil’ev L.V., Non-differentiable optimization, Nauka, Moscow, 1981

[9] Giblin P., Reeve G., “Centre symmetry sets of families of plane curves”, Demonstratio Mathematica, 48:2 (2015), 167–192 | DOI | MR | Zbl

[10] Sotomayor J., Siersma D., Garcia R., “Curvatures of conflict surfaces in Euclidean 3-space”, Banach Center Publications, 50 (1999), 277–285 | DOI | MR | Zbl

[11] Uspenskii A.A., “Calculation formulas for nonsmooth singularities of the optimal result function in a time-optimal problem”, Proceedings of the Steklov Institute of Mathematics, 291:suppl. 1 (2015), 239–254 | DOI | MR

[12] Nigmedzyanova A.M., Differential geometry. Part 1: Differential geometry of curves, Kazan University, Kazan, 2014

[13] 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 Instituta Matematiki i Mekhaniki UrO RAN, 25:1 (2019), 108–119 (in Russian) | DOI | MR

[14] Uspenskii A.A., Lebedev P.D., “On the structure of the singular set of solutions in one class of 3D time-optimal control problems”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki, 31:3 (2021), 471–486 (in Russian) | DOI | MR | Zbl

[15] Lebedev P.D., Uspensky A.A., A program for constructing a solution to a time-optimal problem in three-dimensional space with a spherical velocity vectogram and a non-convex target set, Certificate of state registration of the computer program No. 2022666123, September 07, 2022.

[16] Ushakov V.N., Ershov A.A., “Estimation of the growth of the degree of nonconvexity of reachable sets in terms of $\alpha$-sets”, Doklady Mathematics, 102:3 (2020), 532–537 | DOI | DOI | MR | Zbl

[17] Ushakov V. N., Ershov A.A., Matviychuk A.R., “On estimating the degree of nonconvexity of reachable sets of control systems”, Proceedings of the Steklov Institute of Mathematics, 315 (2021), 247–256 | DOI | DOI | MR | Zbl