Nonlinear observers in reflexive Banach spaces
ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 67-103.

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).

DOI : 10.1051/cocv:2003001
Classification : 47H020, 47H06, 93B07, 93C20, 93C25
Mots-clés : infinite dimensional systems, nonlinear systems, observers, regularly persistent inputs, cauchy problem, mild solution 
@article{COCV_2003__9__67_0,
     author = {Couchouron, Jean-Fran\c{c}ois and Ligarius, P.},
     title = {Nonlinear observers in reflexive {Banach} spaces},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {67--103},
     publisher = {EDP-Sciences},
     volume = {9},
     year = {2003},
     doi = {10.1051/cocv:2003001},
     mrnumber = {1957091},
     zbl = {1069.47511},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2003001/}
}
TY  - JOUR
AU  - Couchouron, Jean-François
AU  - Ligarius, P.
TI  - Nonlinear observers in reflexive Banach spaces
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2003
SP  - 67
EP  - 103
VL  - 9
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2003001/
DO  - 10.1051/cocv:2003001
LA  - en
ID  - COCV_2003__9__67_0
ER  - 
%0 Journal Article
%A Couchouron, Jean-François
%A Ligarius, P.
%T Nonlinear observers in reflexive Banach spaces
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2003
%P 67-103
%V 9
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv:2003001/
%R 10.1051/cocv:2003001
%G en
%F COCV_2003__9__67_0
Couchouron, Jean-François; Ligarius, P. Nonlinear observers in reflexive Banach spaces. ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 67-103. doi : 10.1051/cocv:2003001. http://www.numdam.org/articles/10.1051/cocv:2003001/

[1] V. Barbu, Analysis and control of nonlinear infinite dimensional systems. Academic Press, Math. Sci. Engrg. 190 (1993). | MR | Zbl

[2] J.M. Ball, J.E. Marsden and M. Slemrod, Controllability of distributed bilinear systems. SIAM J. Control Optim. 20 (1982) 575-597. | MR | Zbl

[3] V. Barbu, Nonlinear semigroups and differential equations in Banach spaces. Noordhoff (1976). | MR | Zbl

[4] A.S. Besicovitch, Almost periodic functions. Dover, New-York (1954). | MR | Zbl

[5] P. Bénilan, Equations d'évolution dans un espace de Banach quelconque et applications, Thèse. Univ. Paris-XI, Orsay, France (1972).

[6] P. Bénilan, M.G. Crandall and A. Pazy, Bonnes solutions d'un problème d'évolution semi-linéaire. C. R. Acad. Sci. Paris 306 (1988) 527-530. | Zbl

[7] P. Bénilan, M.G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces. Preprint book (to appear).

[8] H. Bounit and H. Hammouri, Observer design for distributed parameter dissipative bilinear systems. Appl. Math. Comput. Sci. 8 (1998) 381-402. | MR | Zbl

[9] 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).

[10] H. Brezis,Analyse fonctionnelle. Masson, Paris, New-York, Barcelone, Milan, Mexico, Sao Paulo (1987). | MR | Zbl

[11] N. Carmichael, A.J. Pritchard and M.D. Quinn, State and parameter estimations for nonlinear systems. Appl. Math. Optim. 9 (1982) 133-161. | MR | Zbl

[12] F. Celle, J.P. Gauthier, D. Kazakos and G. Sallet, Synthesis of nonlinear observers: A harmonic analysis approach. Math. System Theory 22 (1989) 291-322. | MR | Zbl

[13] J.-F. Couchouron and M. Kamenski, An abstract topological point of view and a general averaging principle in the theory of differential inclusions. Nonlinear Anal. 42 (2000) 1101-1129. | MR | Zbl

[14] J.-F. Couchouron, Compactness theorems for abstract evolution problems (submitted). | Zbl

[15] M.G. Crandall, Nonlinear semigroups and evolution governed by accretive operators. Proc. Symp. Pure Math. 45 (1986) 305-337. | MR | Zbl

[16] R.F. Curtain and A.J. Pritchard, Infinite dimensional linear systems theory. Springer-Verlag, New York (1978). | MR | Zbl

[17] 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).

[18] S. Dolecki and L. Russel, A general theory of observation and control. SIAM J. Control Optim. 15 (1977) 185-220. | MR | Zbl

[19] N. El Alami, Analyse et commande optimale des systèmes bilineaires à paramètres distribués - Application aux procédés énergétiques, Thèse. Univ. de Perpignan, France (1986).

[20] J.P. Gauthier and I. Kupka, Observability and observers for nonlinear systems. SIAM J. Control Optim. 32 (1994) 975-994. | MR | Zbl

[21] J.P. Gauthier and I. Kupka, Observability for systems with more outputs than inputs and asymptotic observers. Math. Z. 223 (1996) 47-78. | MR | Zbl

[22] J.P. Gauthier, C.Z. Xu and A. Bounabat, An observer for infinite dimensional skew-adjoint bilinear systems. J. Math. Syst. Estim. Control 5 (1995) 1-20. | MR | Zbl

[23] E. Hille and R.S. Philipps, Functional analysis and semi-groups. AMS colloquium publications, Vol. XXXI (1965). | Zbl

[24] T. Kato, Perturbation theory of linear operators. Springer-Verlag, New-York (1966). | MR | Zbl

[25] T. Kato, Nonlinear evolution equations in Banach spaces. Proc. of Symp. Appl. Math. 17 (1965) 50-67. | MR | Zbl

[26] 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). | MR | Zbl

[27] P. Ligarius, Observateurs de systèmes bilinéaires à paramètres répartis - Applications à un échangeur thermique, Thesis. Univ. of Rouen, France (1995).

[28] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New-York (1983). | MR | Zbl

[29] A. El Jai and A.J. Pritchard, Capteurs et actionneurs dans l'analyse des systèmes distribués. Masson (1986). | Zbl

[30] A.J. Pritchard, Introduction to semigroup theory. Springer-Verlag, Lecture Notes in Control Inform. Sci. 185 (1993) 1-22. | MR | Zbl

[31] J. Prüss, On semilinear evolution equations in Banach spaces. J. Reine Angew. Math. 303/304 (1978) 144-158. | MR | Zbl

[32] M. Slemrod, Feedback stabilization of a linear control system in Hilbert space with a priori bounded control. Math. Control Signal Syst. 2 (1989) 265-285. | MR | Zbl

[33] H.J. Sussman, Single input observability of continuous time systems. Math. Systems Theory 12 (1979) 371-393. | MR | Zbl

[34] E. Sontag, On the observability of polynomial systems. SIAM J. Control Optim. 17 (1979) 139-151. | MR | Zbl

[35] R. Temam, Infinite dimensional dynamical systems in mechanics and physics. Springer-Verlag, New York, Appl. Math. Sci. (1988). | MR | Zbl

[36] I.I. Vrabie, Compactness methods for nonlinear evolutions. John Wiley & Son, Pitman Monogr. Surveys Pures Appl. Math. 32 (1987). | MR | Zbl

[37] W.M. Wonham, .Linear multivariable control, a geometric approach, 3rd Edn. Springer-Verlag, New York (1985). | MR | Zbl

[38] C.Z. Xu, P. Ligarius and J.P. Gauthier, An observer for infinite dimensional dissipative bilinear systems. Comput. Math. Appl. 29 (1995) 13-21. | MR | Zbl

[39] C.Z. Xu and J.P. Gauthier, Analyse et commande d'un échangeur thermique à contre-courant. RAIRO APII 25 (1991) 377-396. | Zbl

[40] C.Z. Xu, Exact observability and exponential stability of infinite dimensional bilinear systems. Math. Control Signals Syst. 9 (1996) 73-93. | MR | Zbl

Cité par Sources :