Skip to main content

The riccati equation: When nonlinearity reduces to linearity

Part of the Lecture Notes in Mathematics book series (LNM,volume 1248)

Abstract

Riccati equations in R2 of the form dx/dt = a(x)(x) are thus of the following types: uncoupled scalar equations (relative to a suitable basis), equations based on a linear functional times the identity transformation (see generic example 3), two types of degeneracies represented by Eqns. (4.7) and (4.9), and the real and imaginary parts of a single complex Riccati equation.

Keywords

  • Banach Space
  • Riccati Equation
  • Banach Algebra
  • Fundamental Theorem
  • Quadratic System

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   29.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   39.95
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. W. A. Coppel, A survey of quadratic systems, J. Diff. Eqns. 2 (1966), 293–304.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. E. Hille, Lectures on Ordinary Differential Equations, Addison-Wesley, Reading, Mass., 1969.

    MATH  Google Scholar 

  3. E. Hille and R. S. Phillips, Functional Analysis and Semigroups, Amer. Math. Soc., Providence, R. I., 1957.

    MATH  Google Scholar 

  4. J. L. Lions, Contrôle optimale de systèmes gouvernés par des équations aux dérivées partielles, Études Mathématiques, Dunod and Gauthier-Villars, Paris, 1968.

    MATH  Google Scholar 

  5. W. T. Reid, Riccati Differential Equations, Academic Press, New York, 1972.

    MATH  Google Scholar 

  6. W. T. Reid, Sturmian Theory for Ordinary Differential Equations, Springer, New York, 1980 (prepared for publication by J. Burns, T. Herdman, and C. Ahlbrandt).

    CrossRef  MATH  Google Scholar 

  7. A. Vogt, A generalization of the Riccati equation (available from the author).

    Google Scholar 

  8. Yan Qian Ye, Some problems in the qualitative theory of ordinary differential equations, J. Diff. Eqns. 46 (1982), 153–164.

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1987 Springer-Verlag

About this paper

Cite this paper

Vogt, A. (1987). The riccati equation: When nonlinearity reduces to linearity. In: Gill, T.L., Zachary, W.W. (eds) Nonlinear Semigroups, Partial Differential Equations and Attractors. Lecture Notes in Mathematics, vol 1248. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0077425

Download citation

  • DOI: https://doi.org/10.1007/BFb0077425

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-17741-8

  • Online ISBN: 978-3-540-47791-4

  • eBook Packages: Springer Book Archive