Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-01T10:23:39.462Z Has data issue: false hasContentIssue false
  • English
  • Français

A special property of the matrix Riccati equation

Published online by Cambridge University Press:  17 April 2009

A.N. Stokes
A.N. Stokes
Affiliation:
Division of Environmental Mechanics, CSIRO, Canberra, ACT.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the domain of real symmetric matrices ordered by the positive definiteness criterion, the symmetric matrix Riccati differential equation has the unusual property of preserving the ordering of its solutions as the independent variable changes, Here is is shown that, subject to a continuity restriction, the Riccati equation is unique among comparable equations in possessing this property.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 10 , Issue 2 , April 1974 , pp. 245 - 253
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Athans, Michael, Falb, Peter L., Optimal control. An introduction to the theory and its applications (McGraw Hill, New York; Toronto, Ontario; London; 1966).Google Scholar
[2]Coppel, W.A., Disconjugacy (Lecture Notes in Mathematics, 220. Springer-Verlag, Berlin, Heidelberg, New York, 1971).CrossRefGoogle Scholar
[3]Fair, Wyman, “Continued fraction solution to the Riccati equation in a Banach algebra”, J. Math. Anal. Appl. 39 (1972), 318–323.CrossRefGoogle Scholar
[4]Kalman, R.E., “Contributions to the theory of optimal control”, Bol. Soc. Mat. Mexicana (2) 5 (1960), 102–119.Google Scholar
[5]Reid, William T., Riccati differential equations (Mathematics in Science and Engineering, 86. Academic Press, New York, London, 1972).Google Scholar
[6]Schumitzky, Alan. “On the equivalence between matrix Riccati equations and Fredholm resolvents”, J. Comput. System Sci. 2 (1968), 76–87.CrossRefGoogle Scholar
[7]Stokes, A.N., “Differential inequalities and the matrix Riccati equation”, (PhD Thesis, Australian National University, Canberra, 1972. See also, the abstract, Bull. Austral. Math. Soc. 9 (1973), 315–317).CrossRefGoogle Scholar
[8]Szarski, Jacek, Differential inequalities (Monografie Matematyczne, 43. PWN – Polish Scientific Publishers, Warszawa, 1965).Google Scholar
[9]Walter, Wolfgang, Differential- und Integral-Ungleichungen und ihre Anwendung bei Absahätzungs- und Eindeutigkeitsproblemen (Springer Tracts in Natural Philosophy, Ergebnisse der angewandten Mathematik, 2. Springer-Verlag, Berlin, Göttingen, Heidelberg, New York, 1964).CrossRefGoogle Scholar