Abstract:
Necessary and sufficient conditions for the minimax solution to the Cauchy problem for the Hamilton–Jacobi–Bellman equation are obtained as viability conditions for classical characteristics inside the graph of the minimax solution. Using this property, a representative formula for a one-dimensional conservation law in terms of classical characteristics is derived. An estimate of the numerical integration of the characteristic system is presented and errors of numerical realizations of representative formulas are determined for the conservation law and its potential equal to the minimax solution of the Hamilton–Jacobi–Bellman equation.
Citation:
N. N. Subbotina, E. A. Kolpakova, “On the structure of locally Lipschitz minimax solutions of the Hamilton–Jacobi–Bellman equation in terms of classical characteristics”, Trudy Inst. Mat. i Mekh. UrO RAN, 15, no. 3, 2009, 202–218; Proc. Steklov Inst. Math. (Suppl.), 268, suppl. 1 (2010), S222–S239
\Bibitem{SubKol09}
\by N.~N.~Subbotina, E.~A.~Kolpakova
\paper On the structure of locally Lipschitz minimax solutions of the Hamilton--Jacobi--Bellman equation in terms of classical characteristics
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2009
\vol 15
\issue 3
\pages 202--218
\mathnet{http://mi.mathnet.ru/timm416}
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\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2010
\vol 268
\issue , suppl. 1
\pages S222--S239
\crossref{https://doi.org/10.1134/S0081543810050160}
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This publication is cited in the following 8 articles:
Ivan Yegorov, Peter M. Dower, “Perspectives on Characteristics Based Curse-of-Dimensionality-Free Numerical Approaches for Solving Hamilton–Jacobi Equations”, Appl Math Optim, 83:1 (2021), 1
Rodin A.S. Shagalova L.G., “Bifurcation Points of the Generalized Solution of the Hamilton-Jacobi-Bellman Equation”, IFAC PAPERSONLINE, 51:32 (2018), 866–870
A. A. Uspenskii, “Neobkhodimye usloviya suschestvovaniya psevdovershin kraevogo mnozhestva v zadache Dirikhle dlya uravneniya eikonala”, Tr. IMM UrO RAN, 21, no. 1, 2015, 250–263
A. S. Rodin, “O strukture singulyarnogo mnozhestva kusochno-gladkogo minimaksnogo resheniya uravneniya Gamiltona — Yakobi — Bellmana”, Tr. IMM UrO RAN, 21, no. 2, 2015, 198–205
A. A. Uspenskii, “Derivatives by virtue of diffeomorphisms and their applications in control theory and geometrical optics”, Proc. Steklov Inst. Math. (Suppl.), 293, suppl. 1 (2016), 238–253
I. Yegorov, Y. Todorov, “Synthesis of optimal control in a mathematical model of tumour–immune dynamics”, Optim Control Appl Methods, 36:1 (2015), 93
N. N. Subbotina, E. A. Kolpakova, “Method of characteristics for optimal control problems and conservation laws”, Journal of Mathematical Sciences, 199:5 (2014), 588–595
E. A. Kolpakova, “Obobschennyi metod kharakteristik v teorii uravnenii Gamiltona–Yakobi i zakonov sokhraneniya”, Tr. IMM UrO RAN, 16, no. 5, 2010, 95–102
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