# Dynamic Routing Problem in Distributed Parameter Setting Using Semigroup Theory

## Abstract

The aim of this chapter is to present the background information on semigroup theory and evolution equations and motivate their use in the design of controllers, especially feedback controllers, for the dynamic traffic routing and assignment problems. The chapter presents the highway model in the semigroup context and also provides a starting point for traffic routing control design using the semigroup theory. In order to present this framework, this chapter presents the fundamentals of the applicable functional analysis from the ground up. It reviews the topics of topological spaces, vector spaces, metric spaces, normed linear spaces, banach and Hilbert spaces, semigroups and finally how the routing problem can be viewed in this framework as an abstract differential controlled equation.

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