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
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
W. A. Coppel, A survey of quadratic systems, J. Diff. Eqns. 2 (1966), 293–304.
E. Hille, Lectures on Ordinary Differential Equations, Addison-Wesley, Reading, Mass., 1969.
E. Hille and R. S. Phillips, Functional Analysis and Semigroups, Amer. Math. Soc., Providence, R. I., 1957.
J. L. Lions, Contrôle optimale de systèmes gouvernés par des équations aux dérivées partielles, Études Mathématiques, Dunod and Gauthier-Villars, Paris, 1968.
W. T. Reid, Riccati Differential Equations, Academic Press, New York, 1972.
W. T. Reid, Sturmian Theory for Ordinary Differential Equations, Springer, New York, 1980 (prepared for publication by J. Burns, T. Herdman, and C. Ahlbrandt).
A. Vogt, A generalization of the Riccati equation (available from the author).
Yan Qian Ye, Some problems in the qualitative theory of ordinary differential equations, J. Diff. Eqns. 46 (1982), 153–164.
Editor information
Editors and Affiliations
Rights 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
