ON THE STRUCTURE OF THE SINGULAR SET OF A PIECEWISE SMOOTH MINIMAX SOLUTION OF THE HAMILTON–JACOBI–BELLMAN EQUATION

Aleksei S. Rodin     (Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences and Ural Federal University, Russian Federation)

Abstract


The properties of a minimax piecewise smooth solution of the Hamilton–Jacobi–Bellman equation are studied. It is known the Rankine–Hugoniot conditions are necessary and sufficient conditions for the points of nondifferentiability (singularity) of the minimax solution. We generalize this condition and describe the dimension of smooth manifolds contained in the singular set of the piecewise smooth solution in terms of state characteristics that come to this set. New structural properties of the singular set are obtained in the case where the Hamiltonian depends only on the impulse variable.


Keywords


Hamilton–Jacobi–Bellman equation, Minimax solution, Singular set, Piecewise smooth solution, Tangent space, Rankine–Hugoniot conditions

Full Text:

PDF

References


Dynamic programming. Princeton: Princeton Univ. Press, 1957. 392 p.

Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., and Mishchenko E.F. The mathematical theory of optimal processes. New York: Wiley, 1962.

Krasovskii N.N. Theory of motion control. Moscow: Nauka, 1968. 476 p. [in Russian].

Subbotina N.N., Kolpakova E.A., Tokmantsev T.B. and Shagalova L.G. The method of characteristics for Hamilton–Jacobi–Bellman equations. Ekaterinburg: RIO UrO RAN, 2013. 244 p. [in Russian].

Kolpakova E.A. The generalized method of characteristics in the theory of Hamilton–Jacobi equations and conservation laws // Trudy Inst. Mat. Mekh. UrO RAN 2010. Vol. 16, no. 5, P. 95-102. [in Russian]

Melikyan A.A. Generalized characteristics of first order PDEs: applications in optimal control and differential games. Boston: Birkhäuser, 1998.

Cannarsa P., Sinestrari C. Semiconcave functions, Hamilton–Jacobi equations and optimal control, 2004. 189 p.

Petrovskii I.G. Lectures on the theory of ordinary differential equations. Moscow: Mosk. Gos. Univ., 1984. 296 p. [in Russian].

Rockafellar R. Convex analysis. Princeton: Princeton Univ. Press, 1970. 470 p.

Subbotin A.I. Generalized solutions of first-order PDEs. The dynamical optimization perspective. New York: Birkhäuser, 1995. 336 p. DOI: 10.1007/978-1-4612-0847-1.

Crandall M.G. and Lions P.L. Viscosity solutions of Hamilton–Jacobi equations // Trans. Amer. Math. Soc. 1983. Vol. 277, no. 1. P. 1–42.

Subbotina N.N. and Kolpakova E.A. On the structure of locally Lipschitz minimax solutions of the Hamilton–Jacobi–Bellman equation in terms of classical characteristics // Proc. Steklov Inst. Math. 2010. Vol. 268, Suppl. 1, P. S222–S239.

Rodin A.S. On the Structure of the Singular Set of a Piecewise Smooth Minimax Solution of the Hamilton–Jacobi–Bellman Equation // Trudy Inst. Mat. Mekh. UrO RAN 2015. Vol. 21, no. 2, P. 198–205. [in Russian]




DOI: http://dx.doi.org/10.15826/umj.2016.1.006

Article Metrics

Metrics Loading ...

Refbacks

  • There are currently no refbacks.