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

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.