Dynamical systems on weighted lattices: general theory

Abstract

In this work a theory is developed for unifying large classes of nonlinear discrete-time dynamical systems obeying a superposition of a weighted maximum or minimum type. The state vectors and input-output signals evolve on nonlinear spaces which we call complete weighted lattices and include as special cases the nonlinear vector spaces of minimax algebra. Their algebraic structure has a polygonal geometry. Some of the special cases unified include max-plus, max-product, and probabilistic dynamical systems. We study problems of representation in state and input-output spaces using lattice monotone operators, state and output responses using nonlinear convolutions, solving nonlinear matrix equations using lattice adjunctions, stability and controllability. We outline applications in state-space modeling of nonlinear filtering; dynamic programming (Viterbi algorithm) and shortest paths (distance maps); fuzzy Markov chains; and tracking audio-visual salient events in multimodal information streams using generalized hidden Markov models with control inputs.