Euclidean distance to a closed set as a minimax solution of the Dirichlet problem for the Hamilton-Jacobi equation

A. A. Uspenskii ; P. D. Lebedev

Vestnik rossijskih universitetov. Matematika, Tome 23 (2018) no. 124, pp. 797-804

Résumé

A combined (jointing analytical methods and computational procedures) approach to the construction of solutions in a class of boundary-value problems for a Hamiltonian-type equation is proposed. In the class of problems under consideration, the minimax (generalized) solution coincides with the Euclidean distance to the boundary set. The properties of this function are studied depending on the geometry of the boundary set and the differential properties of its boundary. Methods are developed for detecting pseudo-vertices of a boundary set and for constructing singular solution sets with their help. The methods are based on the properties of local diffeomorphisms and use partial one-sided limits. The effectiveness of the research approaches developed is illustrated by the example of solving a planar timecontrol problem for the case of a nonconvex target set with boundary of variable smoothness.

Détail

Mots-clés : Euclidean distance
Keywords: Hamilton-Jacobi equation, Dirichlet problem, minimax solution, optimal result function, velocity, singular set, local diffeomorphism.

Bibliographie

[1] A. I. Subbotin, Generalized Solutions of First-Order PDEs. The Dynamical Optimization Perspective, Birkhauser, Boston, 1995

[2] N. N. Krasovskii, A. I. Subbotin, Positional Differential Games, Nauka Publ., Moscow, 1974, 456 pp. (In Russian)

[3] P. D. Lebedev, A. A. Uspenskii, V. N. Ushakov, “Construction of a minimax solution for an eikonal-type equation”, Proceedings of the Steklov Institute of Mathematics, 14:2 (2008), 182–191 (In Russian)

[4] V. F. Dem'yanov, L. V. Vasil'yev, Nondifferentiable Optimization, Springer-Verlag, New York, 1985

[5] J. W. Bruce ,P. J. Giblin, Curves and Singularities, Cambridge University Press, Cambridge, 1984

[6] A. A. Uspenskii, P. D. Lebedev, “Transversality conditions for solution branches of a nonlinear equation in a time-optimal problem with circular indicatrix”, Proceedings of the Steklov Institute of Mathematics, 14:4 (2008), 82–100 (In Russian)

[7] A. A. Uspenskii, P. D. Lebedev, “On the set of limit values of local diffeomorphisms in wavefront evolution”, Proceedings of the Steklov Institute of Mathematics, 16:1 (2010), 175–185 (In Russian)

[8] A. A. Uspenskii, P. D. Lebedev, “The construction of singular curves for generalized solutions of eikonal-type equations with a curvature break in the boundary of the boundary set”, Proceedings of the Steklov Institute of Mathematics, 22:1 (2016), 282–293 (In Russian)

[9] A. A. Uspenskii, “Necessary conditions for the existence of pseudovertices of the boundary set in the Dirichlet problem for the eikonal equation”, Proceedings of the Steklov Institute of Mathematics, 21:1 (2015), 250–263 (In Russian)

[10] A. A. Uspenskii, P. D. Lebedev, “Identification of the singularity of the generalized solution of the Dirichlet problem for an eikona type equation under the conditions of minimal smoothness of a boundary set”, The Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 28:1 (2018), 59–73 (In Russian)

[11] A. A. Uspenskii, “Calculation formulas for nonsmooth singularities of the optimal result function in a time-optimal problem”, Proceedings of the Steklov Institute of Mathematics, 20:3 (2014), 276–290 (In Russian)

[12] V. N. Ushakov, A. A. Uspenskii, A. G. Malev, “An estimate of the stability defect for a positional absorption set subjected to discriminant transformations”, Proceedings of the Steklov Institute of Mathematics, 17:2 (2011), 209–224 (In Russian)

[13] A. A. Uspenskii, P. D. Lebedev, “Procedures for Calculating the Nonconvexity Measures of a Plane Set”, Computational Mathematics and Mathematical Physics, 49:3 (2009), 431–440 (In Russian)