Hostname: page-component-84b7d79bbc-7nlkj Total loading time: 0 Render date: 2024-07-30T09:04:52.787Z Has data issue: false hasContentIssue false
  • English
  • Français

Stabilization of control processes in Hilbert spaces

Published online by Cambridge University Press:  14 November 2011

L. Pandolfi
L. Pandolfi
Affiliation:
University of Florence, Italy
Get access Rights & Permissions [Opens in a new window]

Synopsis

In this paper we study the stabilization problem for non autonomous control processes in Hilbert spaces. We prove that a stabilizing feedback exists if and only if an associated Riccati equation has a bounded solution which is symmetric and positive definite.

An application to control processes with delays in control is presented.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 81 , Issue 3-4 , 1978 , pp. 247 - 258
Copyright
Copyright © Royal Society of Edinburgh 1978

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Delfour, M. C., McCalla, C. and Mitter, S. K.Stability and the infinite-time quadratic cost problem for linear hereditary differential systems. SIAM J. Control 13 (1975), 48–88.CrossRefGoogle Scholar
2Bensoussan, A., Delfour, M. and Mitter, S. K.Optimal control of linear integral equations with a quadratic cost function: the infinite time problem. Université de Montreal, Centre de Recherches Mathématiques, Report 454 (1977).Google Scholar
3Conti, R. Linear Differential Equations and Control. Institutiones Matematicae, 1. (London: Academic Press, 1976).Google Scholar
4Curtain, R. F.The infinite dimensional Riccati equation with applications to affine hereditary differential systems. SIAM J. Control 13 (1975), 1130–1143.CrossRefGoogle Scholar
5Curtain, R. F. and Pritchard, A. J.The infinite dimensional Riccati Equation. J. Math. Anal. Appl. 47 (1974), 43–57.CrossRefGoogle Scholar
6Datko, R.Extending a Theorem of A. M. Liapunov to Hilbert spaces. J. Math. Anal. Appl. 32 (1970), 610–616.CrossRefGoogle Scholar
7Datko, R.A linear control problem in an abstract Hilbert space. J. Differential Equations 9 (1971), 346–359.CrossRefGoogle Scholar
8Kato, T.Perturbation theory of linear operators (Berlin: Springer, 1966).Google Scholar
9Riesz, F. and Nagy, B.Leçons d'Analyse Fonctionnelle (Paris: Gauthier, 1957).Google Scholar
10Treves, F.Topological Vector Spaces, Distributions and Kernels. Pure and Appl. Math. 25 (1967).Google Scholar