Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-19T13:25:41.175Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear observers in reflexive Banach spaces

Published online by Cambridge University Press:  15 September 2003

Jean-François Couchouron  and
P. Ligarius
Jean-François Couchouron
Affiliation:
Université de Metz, INRIA Lorraine, Departement de Mathématiques, bâtiment A, Ile du Saulcy, 57045 Metz Cedex 01, France; [email protected].
P. Ligarius
Affiliation:
INSA of Rouen, Departement of Mathématiques, BP. 8, 76131 Mont-Saint-Aignan Cedex, France; [email protected].
Get access

Abstract

On an arbitrary reflexive Banach space, we build asymptotic observers for an abstract class of nonlinear control systems with possible compact outputs. An important part of this paper is devoted to various examples, where we discuss the existence of persistent inputs which make the system observable. These results make a wide generalization to a nonlinear framework of previous works on the observation problem in infinite dimension (see [11,18,22,26,27,38,40] and other references therein).

Keywords

Infinite dimensional systemsnonlinear systemsobserversregularly persistent inputscauchy problemmild solution.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 9 , January 2003 , pp. 67 - 103
Copyright
© EDP Sciences, SMAI, 2003

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

V. Barbu, Analysis and control of nonlinear infinite dimensional systems. Academic Press, Math. Sci. Engrg. 190 (1993).
Ball, J.M., Marsden, J.E. and Slemrod, M., Controllability of distributed bilinear systems. SIAM J. Control Optim. 20 (1982) 575-597. CrossRef
V. Barbu, Nonlinear semigroups and differential equations in Banach spaces. Noordhoff (1976).
A.S. Besicovitch, Almost periodic functions. Dover, New-York (1954).
P. Bénilan, Equations d'évolution dans un espace de Banach quelconque et applications, Thèse. Univ. Paris-XI, Orsay, France (1972).
Bénilan, P., Crandall, M.G. and Pazy, A., Bonnes solutions d'un problème d'évolution semi-linéaire. C. R. Acad. Sci. Paris 306 (1988) 527-530.
P. Bénilan, M.G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces. Preprint book (to appear).
Bounit, H. and Hammouri, H., Observer design for distributed parameter dissipative bilinear systems. Appl. Math. Comput. Sci. 8 (1998) 381-402.
H. Bounit, Contribution à la stabilisation et à la construction d'observateurs pour une classe de systèmes à paramètres distribués, Thesis. Univ. Claude Bernard, Lyon-I, France (1996).
H. Brezis,Analyse fonctionnelle. Masson, Paris, New-York, Barcelone, Milan, Mexico, Sao Paulo (1987).
Carmichael, N., Pritchard, A.J. and Quinn, M.D., State and parameter estimations for nonlinear systems. Appl. Math. Optim. 9 (1982) 133-161. CrossRef
Celle, F., Gauthier, J.P., Kazakos, D. and Sallet, G., Synthesis of nonlinear observers: A harmonic analysis approach. Math. System Theory 22 (1989) 291-322. CrossRef
Couchouron, J.-F. and Kamenski, M., An abstract topological point of view and a general averaging principle in the theory of differential inclusions. Nonlinear Anal. 42 (2000) 1101-1129. CrossRef
J.-F. Couchouron, Compactness theorems for abstract evolution problems (submitted).
Crandall, M.G., Nonlinear semigroups and evolution governed by accretive operators. Proc. Symp. Pure Math. 45 (1986) 305-337. CrossRef
R.F. Curtain and A.J. Pritchard, Infinite dimensional linear systems theory. Springer-Verlag, New York (1978).
D. Dochain, Contribution to the analysis and control of distributed parameter systems with application to (bio)chemical processes and robotics, Thesis. Univ. Cath. Louvain, Belgium (1994).
Dolecki, S. and Russel, L., A general theory of observation and control. SIAM J. Control Optim. 15 (1977) 185-220. CrossRef
N. El Alami, Analyse et commande optimale des systèmes bilineaires à paramètres distribués - Application aux procédés énergétiques, Thèse. Univ. de Perpignan, France (1986).
Gauthier, J.P. and Kupka, I., Observability and observers for nonlinear systems. SIAM J. Control Optim. 32 (1994) 975-994. CrossRef
Gauthier, J.P. and Kupka, I., Observability for systems with more outputs than inputs and asymptotic observers. Math. Z. 223 (1996) 47-78. CrossRef
Gauthier, J.P., Xu, C.Z. and Bounabat, A., An observer for infinite dimensional skew-adjoint bilinear systems. J. Math. Syst. Estim. Control 5 (1995) 1-20.
E. Hille and R.S. Philipps, Functional analysis and semi-groups. AMS colloquium publications, Vol. XXXI (1965).
T. Kato, Perturbation theory of linear operators. Springer-Verlag, New-York (1966).
Kato, T., Nonlinear evolution equations in Banach spaces. Proc. of Symp. Appl. Math. 17 (1965) 50-67. CrossRef
P. Ligarius, J.P. Gauthier and C.Z. Xu, A simple observer for distributed systems: Application on a heat exchanger. J. Math. Systems Estim. Control 8 (1998) 1-23 (retrieval code: 73494).
P. Ligarius, Observateurs de systèmes bilinéaires à paramètres répartis - Applications à un échangeur thermique, Thesis. Univ. of Rouen, France (1995).
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New-York (1983).
A. El Jai and A.J. Pritchard, Capteurs et actionneurs dans l'analyse des systèmes distribués. Masson (1986).
A.J. Pritchard, Introduction to semigroup theory. Springer-Verlag, Lecture Notes in Control Inform. Sci. 185 (1993) 1-22.
J. Prüss, On semilinear evolution equations in Banach spaces. J. Reine Angew. Math. 303/304 (1978) 144-158.
Slemrod, M., Feedback stabilization of a linear control system in Hilbert space with a priori bounded control. Math. Control Signal Syst. 2 (1989) 265-285. CrossRef
Sussman, H.J., Single input observability of continuous time systems. Math. Systems Theory 12 (1979) 371-393.
Sontag, E., On the observability of polynomial systems. SIAM J. Control Optim. 17 (1979) 139-151. CrossRef
R. Temam, Infinite dimensional dynamical systems in mechanics and physics. Springer-Verlag, New York, Appl. Math. Sci. (1988).
I.I. Vrabie, Compactness methods for nonlinear evolutions. John Wiley & Son, Pitman Monogr. Surveys Pures Appl. Math. 32 (1987).
W.M. Wonham, .Linear multivariable control, a geometric approach, 3rd Edn. Springer-Verlag, New York (1985).
Xu, C.Z., Ligarius, P. and Gauthier, J.P., An observer for infinite dimensional dissipative bilinear systems. Comput. Math. Appl. 29 (1995) 13-21. CrossRef
Xu, C.Z. and Gauthier, J.P., Analyse et commande d'un échangeur thermique à contre-courant. RAIRO APII 25 (1991) 377-396.
Exact, C.Z. Xu observability and exponential stability of infinite dimensional bilinear systems. Math. Control Signals Syst. 9 (1996) 73-93.