Chapter 35 - Pseudo-Additive Measures and Their Applications

Abstract

This chapter discusses the pseudoadditive measures and and the corresponding integrals, which give a base for pseudoanalysis. The pseudoadditive measures are applied in optimization problems, nonlinear partial differential equations, nonlinear difference equations, optimal control, and fuzzy systems.Pseudoanalysis uses many mathematical tools from different fields, such as functional equations, variational calculus, measure theory, functional analysis, optimization theory, and semiring theory. The advantage of the pseudoanalysis is that the problems from many different fields are covered with one theory and unified methods. This approach gives solutions in such a form that are not achieved by other theories. In some cases, it enables nonlinear equations to obtain exact solutions in the similar form as for linear equations. Some obtained principles such as the pseudolinear superposition principle allows transferring methods of linear equations to nonlinear equations. Pseudointegral that is defined as the limits of the corresponding idempotent Riemannian sums is also elaborated in the chapter.