Nonlinear observers in reflexive Banach spaces

Jean-François Couchouron; P. Ligarius

ESAIM: Control, Optimisation and Calculus of Variations (2003)

  • Volume: 9, page 67-103
  • ISSN: 1292-8119

Abstract

top
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).

How to cite

top

Couchouron, Jean-François, and Ligarius, P.. "Nonlinear observers in reflexive Banach spaces." ESAIM: Control, Optimisation and Calculus of Variations 9 (2003): 67-103. <http://eudml.org/doc/245793>.

@article{Couchouron2003,
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).},
author = {Couchouron, Jean-François, Ligarius, P.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {infinite dimensional systems; nonlinear systems; observers; regularly persistent inputs; cauchy problem; mild solution; infinite-dimensional systems; Cauchy problem},
language = {eng},
pages = {67-103},
publisher = {EDP-Sciences},
title = {Nonlinear observers in reflexive Banach spaces},
url = {http://eudml.org/doc/245793},
volume = {9},
year = {2003},
}

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
PB - EDP-Sciences
VL - 9
SP - 67
EP - 103
AB - 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).
LA - eng
KW - infinite dimensional systems; nonlinear systems; observers; regularly persistent inputs; cauchy problem; mild solution; infinite-dimensional systems; Cauchy problem
UR - http://eudml.org/doc/245793
ER -

References

top
  1. [1] V. Barbu, Analysis and control of nonlinear infinite dimensional systems. Academic Press, Math. Sci. Engrg. 190 (1993). Zbl0776.49005MR1195128
  2. [2] J.M. Ball, J.E. Marsden and M. Slemrod, Controllability of distributed bilinear systems. SIAM J. Control Optim. 20 (1982) 575-597. Zbl0485.93015MR661034
  3. [3] V. Barbu, Nonlinear semigroups and differential equations in Banach spaces. Noordhoff (1976). Zbl0328.47035MR390843
  4. [4] A.S. Besicovitch, Almost periodic functions. Dover, New-York (1954). Zbl0065.07102MR68029
  5. [5] P. Bénilan, Equations d’évolution dans un espace de Banach quelconque et applications, Thèse. Univ. Paris-XI, Orsay, France (1972). Zbl0246.47068
  6. [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. Zbl0635.34013
  7. [7] P. Bénilan, M.G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces. Preprint book (to appear). Zbl0635.34013
  8. [8] H. Bounit and H. Hammouri, Observer design for distributed parameter dissipative bilinear systems. Appl. Math. Comput. Sci. 8 (1998) 381-402. Zbl0910.93021MR1635810
  9. [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. [10] H. Brezis,Analyse fonctionnelle. Masson, Paris, New-York, Barcelone, Milan, Mexico, Sao Paulo (1987). Zbl0511.46001MR697382
  11. [11] N. Carmichael, A.J. Pritchard and M.D. Quinn, State and parameter estimations for nonlinear systems. Appl. Math. Optim. 9 (1982) 133-161. Zbl0502.93035MR680449
  12. [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. Zbl0691.93005MR1027553
  13. [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. Zbl0972.34049MR1780458
  14. [14] J.-F. Couchouron, Compactness theorems for abstract evolution problems (submitted). Zbl1008.47057
  15. [15] M.G. Crandall, Nonlinear semigroups and evolution governed by accretive operators. Proc. Symp. Pure Math. 45 (1986) 305-337. Zbl0637.47039MR843569
  16. [16] R.F. Curtain and A.J. Pritchard, Infinite dimensional linear systems theory. Springer-Verlag, New York (1978). Zbl0389.93001MR516812
  17. [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. [18] S. Dolecki and L. Russel, A general theory of observation and control. SIAM J. Control Optim. 15 (1977) 185-220. Zbl0353.93012MR451141
  19. [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. [20] J.P. Gauthier and I. Kupka, Observability and observers for nonlinear systems. SIAM J. Control Optim. 32 (1994) 975-994. Zbl0802.93008MR1280224
  21. [21] J.P. Gauthier and I. Kupka, Observability for systems with more outputs than inputs and asymptotic observers. Math. Z. 223 (1996) 47-78. Zbl0863.93008MR1408862
  22. [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. Zbl0832.93008MR1646283
  23. [23] E. Hille and R.S. Philipps, Functional analysis and semi-groups. AMS colloquium publications, Vol. XXXI (1965). Zbl0078.10004
  24. [24] T. Kato, Perturbation theory of linear operators. Springer-Verlag, New-York (1966). Zbl0148.12601MR203473
  25. [25] T. Kato, Nonlinear evolution equations in Banach spaces. Proc. of Symp. Appl. Math. 17 (1965) 50-67. Zbl0173.17104MR184099
  26. [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). Zbl0924.93006MR1651284
  27. [27] P. Ligarius, Observateurs de systèmes bilinéaires à paramètres répartis – Applications à un échangeur thermique, Thesis. Univ. of Rouen, France (1995). 
  28. [28] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New-York (1983). Zbl0516.47023MR710486
  29. [29] A. El Jai and A.J. Pritchard, Capteurs et actionneurs dans l’analyse des systèmes distribués. Masson (1986). Zbl0627.93001
  30. [30] A.J. Pritchard, Introduction to semigroup theory. Springer-Verlag, Lecture Notes in Control Inform. Sci. 185 (1993) 1-22. Zbl0829.47033MR1208263
  31. [31] J. Prüss, On semilinear evolution equations in Banach spaces. J. Reine Angew. Math. 303/304 (1978) 144-158. Zbl0398.34057MR514677
  32. [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. Zbl0676.93057MR997216
  33. [33] H.J. Sussman, Single input observability of continuous time systems. Math. Systems Theory 12 (1979) 371-393. Zbl0422.93019MR541865
  34. [34] E. Sontag, On the observability of polynomial systems. SIAM J. Control Optim. 17 (1979) 139-151. Zbl0409.93013MR516861
  35. [35] R. Temam, Infinite dimensional dynamical systems in mechanics and physics. Springer-Verlag, New York, Appl. Math. Sci. (1988). Zbl0662.35001MR953967
  36. [36] I.I. Vrabie, Compactness methods for nonlinear evolutions. John Wiley & Son, Pitman Monogr. Surveys Pures Appl. Math. 32 (1987). Zbl0721.47050MR932730
  37. [37] W.M. Wonham, .Linear multivariable control, a geometric approach, 3rd Edn. Springer-Verlag, New York (1985). Zbl0609.93001MR770574
  38. [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. Zbl0829.93006MR1321254
  39. [39] C.Z. Xu and J.P. Gauthier, Analyse et commande d’un échangeur thermique à contre-courant. RAIRO APII 25 (1991) 377-396. Zbl0741.93064
  40. [40] C.Z. Xu, Exact observability and exponential stability of infinite dimensional bilinear systems. Math. Control Signals Syst. 9 (1996) 73-93. Zbl0862.93007MR1410049

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.