Idempotent/tropical analysis, the Hamilton-Jacobi and Bellman equations

Details Idempotent/tropical analysis, the Hamilton-Jacobi and Bellman equations Idempotent/tropical analysis, the Hamilton-Jacobi and Bellman equations

2012-03-02

arxiv.org/abs/1203.0522v1

Mathematics

Rings and Algebras

70 pages, 5 figures, CIME lectures (2011), to be published in Lecture Notes in Mathematics (Springer)

Scientific paper

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 Schr\"{o}dinger 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.

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