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Idempotent/Tropical Analysis, the Hamilton–Jacobi and Bellman Equations

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Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2074))

Abstract

Tropical and idempotent analysis with their relations to the Hamilton–Jacobi and matrix Bellman equations are discussed. Some dequantization procedures are important in tropical and idempotent mathematics. In particular, the Hamilton–Jacobi–Bellman equation is treated as a result of the Maslov dequantization applied to the Schrödinger equation. This leads to a linearity of the Hamilton–Jacobi–Bellman equation over tropical algebras. The correspondence principle and the superposition principle of idempotent mathematics are formulated and examined. The matrix Bellman equation and its applications to optimization problems on graphs are discussed. Universal algorithms for numerical algorithms in idempotent mathematics are investigated. In particular, an idempotent version of interval analysis is briefly discussed.

In dear memory of my beloved wife Irina.

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Acknowledgements

The author is sincerely grateful to V.N. Kolokoltsov, V.P. Maslov, S.N. Sergeev, A.N. Sobolevski and A.V. Tchourkin for valuable suggestions, help and support. This work is supported by the RFBR grant 12-01-00886-a and RFBR/CNRS grant 11-01-93106.

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  1. Institute for Information Transmission Problems, B. Karetnyi per. 19/1, Moscow, 127994, Russia

    Grigory L. Litvinov

  2. National Research University Higher School of Economics, Moscow, Russia

    Grigory L. Litvinov

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Litvinov, G.L. (2013). Idempotent/Tropical Analysis, the Hamilton–Jacobi and Bellman Equations. In: Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications. Lecture Notes in Mathematics(), vol 2074. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36433-4_4

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