Advertisement

Dynamic Routing Problem in Distributed Parameter Setting Using Semigroup Theory

  • Pushkin Kachroo
  • Kaan M. A. Özbay
Chapter
Part of the Advances in Industrial Control book series (AIC)

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.

References

  1. 1.
    Kreyszig E (1989) Introductory functional analysis with applications, vol 81. Wiley, New YorkGoogle Scholar
  2. 2.
    Vidyasagar M (1992) Nonlinear systems analysis, 2nd edn. Prentice-Hall IncGoogle Scholar
  3. 3.
    Belleni-Morante A (1994) A concise guide to semigroups and evolution equations, vol 19. World ScientificGoogle Scholar
  4. 4.
    Adams RA, Fournier JJ (2003) Sobolev spaces, vol 140. Academic PressGoogle Scholar
  5. 5.
    Burns JA, Kang S (1991) A stabilization problem for Burgers’ Equation with unbounded control and observation. In: Estimation and control of distributed parameter systems. Springer, pp 51–72Google Scholar
  6. 6.
    Cole JD (1951) On a quasi-linear parabolic equation occurring in aerodynamics. Quart Appl Math 9(3):225–236MathSciNetCrossRefGoogle Scholar
  7. 7.
    Glimm J, Lax PD (1970) Decay of solutions of systems of nonlinear hyperbolic conservation laws. American Mathematical SocietyCrossRefGoogle Scholar
  8. 8.
    Hopf E (1950) The partial differential equation ut+ uux= \(\upmu \)xx. Communications on Pure and Applied mathematics 3(3):201–230MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kielhöfer H (1974) Stability and semilinear evolution equations in hilbert space. Arch Ration Mech Anal 57(2):150–165MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lax PD (1973) Hyperbolic systems of conservation laws and the mathematical theory of shock waves. Society for Industrial and Applied Mathematics, vol 11–16CrossRefGoogle Scholar
  11. 11.
    Maslov VP (1987) On a new principle of superposition for optimization problems. Russ Math Surv 42(3):43–54MathSciNetCrossRefGoogle Scholar
  12. 12.
    Burns JA, Kang S (1991) A control problem for burgers’ equation with bounded input/output. Nonlinear Dyn 2(4):235–262CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Electrical and Computer EngineeringUniversity of NevadaLas VegasUSA
  2. 2.Department of Civil and Urban EngineeringNew York UniversityBrooklynUSA

Personalised recommendations