Construction of a solution to a velocity problem in the case of violation of the smoothness of the curvature of the target set boundary

P. D. Lebedev ; A. A. Uspenskii

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 53 (2019), pp. 98-114.

Résumé

For the development of analytical and numerical algorithms for constructing nonsmooth solutions of optimal control problems, procedures are proposed for constructing scattering curves for a single class of control velocity problems. We consider the reduction problems for a minimal time of solutions of a dynamical system with a circular velocity vectogram for the case where the target set is generally nonconvex, and its boundary has points at which the curvature smoothness is violated. These points are referred to as pseudovertices, the characteristic points of the target set, which are responsible for the occurrence of the singularity of the optimal result function. When forming a proper reparameterization (in this case, taking into account the geometry of the velocity vector diagram) of the arc of the boundary of the target set containing a pseudovertex, the scattering curve is constructed as an integral curve. Moreover, the initial conditions of the corresponding Cauchy problem are determined by the properties of the pseudovertex. One of the numerical characteristics of the pseudovertex, the pseudovertex marker, determines the initial velocity of the material point describing a smooth portion of the scattering curve. This approach to the identification and construction (in analytical or numerical form) of singular curves was previously substantiated for a number of cases of a target boundary that are different in the order of smoothness. It should be emphasized that the case considered in this paper is the most specific, in particular, because of the revealed connection between the dynamic problem and the problem of polynomial algebra. It is proved that the pseudovertex marker is the nonpositive root of some third-order polynomial whose coefficients are determined by the one-sided derivatives of curvatures of the pseudovertex of the target set. The effectiveness of the developed theoretical methods and numerical procedures is illustrated by specific examples.

Détail

Keywords: velocity problem, dispersing curve, bisector of a set, pseudovertex, optimal result function, curvature.

Bibliographie

[1] Lebedev P. D., Uspenskii A. A., “Analytical and numerical construction of the optimal outcome function for a class of time-optimal problems”, Computational Mathematics and Modeling, 19:4 (2008), 375–386 | DOI | MR | Zbl

[2] Lebedev P. D., Uspenskii A. A., “Geometry and asymptotics of wavefronts”, Russian Mathematics, 52:3 (2008), 24–33 | DOI | MR | MR | Zbl

[3] Subbotin A. I., Generalized solutions of first order PDEs: the dynamical optimization perspective, Birkhäuser, Boston, 1995, xii+314 pp. | DOI | MR | MR

[4] Kruz̆kov 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, 27:3 (1975), 406–446 | DOI | MR | Zbl

[5] Dem'yanov V. F., Vasil'ev L. V., Non-differentiable optimization, Nauka, M., 1981, 384 pp. | MR

[6] Isaacs R., Differential games, John Wiley and Sons, N.Y., 1965 | MR | Zbl

[7] Arnold V. I., Singularities of caustics and wave fronts, Springer, Dordrecht, 1990 | DOI | MR | MR

[8] Sedykh V. D., “On the topology of wave fronts in spaces of low dimension”, Izvestiya: Mathematics, 76:2 (2012), 375–418 | DOI | DOI | MR | Zbl

[9] Lebedev P. D., Uspenskii A. A., “Construction of the optimal result function and dispersing lines in time-optimal problems with a nonconvex target set”, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 22, no. 2, 2016, 188–198 (in Russian) | DOI

[10] 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”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 28:1 (2018), 59–73 (in Russian) | DOI | MR | Zbl

[11] Rashevskii P. K., A course in differential geometry, URSS, M., 2003 | MR

[12] Uspenskii A. A., Lebedev P. D., “On the set of limit values of local diffeomorphisms in wavefront evolution”, Proceedings of the Steklov Institute of Mathematics, 272, suppl. 1 (2011), 255–270 | DOI | MR | Zbl

[13] Ushakov V. N., Uspenskii A. A., Lebedev P. D., “Geometry of singular curves for one class of velocity”, Vestnik Sankt-Peterburgskogo Universiteta. Seriya 10. Prikladnaya Matematika. Informatika. Protsessy Upravleniya, 2013, no. 3, 157–167 (in Russian)

[14] Uspenskii A. A., Lebedev P. D., “Construction of the optimal outcome function for a time-optimal problem on the basis of a symmetry set”, Automation and Remote Control, 70:7 (2009), 1132–1139 | DOI | MR | Zbl

[15] Vinberg E. B., Algebra of polynomials, Prosveshchenie, M., 1980

[16] 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

[17] 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”, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 25, no. 1, 2019, 108–119 (in Russian) | DOI