On the circle criterion for boundary control systems in factor form: Lyapunov stability and Lur'e equations

ESAIM - Control, Optimisation and Calculus of Variations

Volumn 12, Issue 1, 2006, Pages 169-197

  Grabowski, Piotr ^a   Callier, Frank M ^b  

^a AGH UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Poland)

^b UNIVERSITY OF NAMUR   (Belgium)

Author keywords

Circle criterion; Infinite dimensional control systems; Lyapunov functional; Semigroups

Indexed keywords

BOUNDARY CONDITIONS; FEEDBACK CONTROL; FUNCTION EVALUATION; MATHEMATICAL OPERATORS; PROBLEM SOLVING;

CIRCLE CRITERION; INFINITE-DIMENSIONAL CONTROL SYSTEMS; LYAPUNOV FUNCTIONAL; SEMIGROUPS;

CONTROL SYSTEM ANALYSIS;

EID: 33645784354     PISSN: 12928119     EISSN: 12623377     Source Type: Journal    
DOI: 10.1051/cocv:2005027     Document Type: Article

References (45)

1. W. Arendt and C.J.K. Batty, Tauberian theorems and stability of one-parameter semigroups. Trans. Amer. Math. Soc. 306 (1988) 837-862.
2. A.V. Balakrishnan, On a generalization of the Kalman-Yacubovič lemma. Appl. Math. Optim. 31 (1995) 177-187.
3. F. Bucci, Frequency domain stability of nonlinear feedback systems with unbounded input operator. Dynam. Contin. Discrete Impuls. Syst. 7 (2000) 351-368.
4. F.M. Callier and J. Winkin, LQ-optimal control of infinite-dimensional systems by spectral factorization. Automatica 28 (1992) 757-770.
5. R.F. Curtain, Linear operator inequalities for strongly stable weakly regular linear systems. Math. Control Signals Syst. 14 (2001) 299-337.
6. R.F. Curtain, Regular linear systems and their reciprocals: application to Riccati equations. Syst. Control Lett. 49 (2003) 81-89.
7. R.F. Curtain, Riccati equations for stable well-posed linear systems: The generic case. SIAM J. Control Optim. 42 (2003) 1671-1702.
8. R.F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory. Heidelberg, Springer (1995).
9. R.F. Curtain, H. Logemann and O. Staffans, Stability results of Popov-type for infinite - dimensional systems with applications to integral control. Proc. London Math. Soc. 86 (2003) 779-816.
10. H. Górecki, S. Fuksa, P. Grabowski and A.Korytowski, Analysis and Synthesis of Time-Delay Systems. Warsaw & Chichester: PWN and J. Wiley (1989).
11. P. Grabowski, On the spectral - Lyapunov approach to parametric optimization of DPS. IMA J. Math. Control Inform. 7 (1990) 317-338.
12. P. Grabowski, The LQ controller problem: an example. IMA J. Math. Control Inform. 11 (1994) 355-368.
13. P. Grabowski, On the circle criterion for boundary control systems in factor form. Opuscula Math. 23 (2003) 1-25.
14. P. Grabowski and F.M. Callier, Admissible observation operators. Duality of observation and control using factorizations. Dynamics Continuous, Discrete Impulsive Systems 6 (1999) 87-119.
15. P. Grabowski and F.M. Callier, On the circle criterion for boundary control systems in factor form: Lyapunov approach. Facultés Universitaires Notre-Dame de la Paix à Namur, Publications du Département de Mathématique, Research Report 07 (2000), FUNDP, Namur, Belgium.
16. P. Grabowski and F.M. Callier, Boundary control systems in factor form: Transfer functions and input-output maps. Integral Equations Operator Theory 41 (2001) 1-37.
17. P. Grabowski and F.M. Callier, Circle criterion and boundary control systems in factor form: Input-output approach. Internat. J. Appl. Math. Comput. Sci. 11 (2001) 1387-1403.
18. P. Grabowski and F.M. Callier, On the circle criterion for boundary control systems in factor form; Lyapunov stability and Lur'e equations. Facultés Universitaires Notre-Dame de la Paix à Namur, Publications du Département de Mathématique, Research Report 05 (2002), FUNDP, Namur, Belgium.
19. U. Grenander and G. Szegö, Toeplitz Forms and Their Application, Berkeley: University of California Press (1958).
20. K. Hoffman, Banach Spaces of Analytic Functions. Englewood Cliffs: Prentice-Hall (1962).
21. I. Lasiecka and R. Triggiani, Differential and Algebraic Riccati Equations with Application to Boundary/Point Control Problems: Continuous Theory and Approximation Theory. Lect. Notes Control Inform. Sci. 164 (1991) 1-160.
22. I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Part I: Abstract Parabolic Systems, Cambridge: Cambridge University Press, Encyclopedia Math. Appl. 74 (2000).
23. I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Part II: Abstract Hyperbolic-Like Systems over a Finite Time Horizon, Cambridge: Cambridge University Press, Encyclopedia Math. Appl. 75 (2000).
24. A.L. Likhtarnikov and V.A. Yacubovich, The frequency domain theorem for continuous one-parameter semigroups, IZVESTIJA ANSSSR. Seria matematicheskaya. 41 (1977) 895-911 (in Russian).
25. H. Logemann and R.F. Curtain, Absolute stability results for well-posed infinite-dimensional systems with low-gain integral control. ESAIM: COCV5 (2000) 395-424.
26. J.-Cl. Louis and D.Wexler, The Hilbert space regulator problem and operator Riccati equation under stabilizability. Annales de la Société Scientifique de Bruxelles 105 (1991) 137-165.
27. Yu. Lyubich and Vû Quôc Phong, Asymptotic stability of linear differential equations in Banach spaces. Studia Math. 88 (1988) 37-41.
28. E. Noldus, On the stability of systems having several equilibrium points. Appl. Sci. Res. 21 (1969) 218-233.
29. E. Noldus, A. Galle and L. Jasson, The computation of stability regions for systems with many singular points. Intern. J. Control 17 (1973) 641-652.
30. E. Noldus, New direct Lyapunov-type method for studying synchronization problems. Automatica 13 (1977) 139-151.
31. A.A. Nudel'man and P.A. Schwartzman, On the existence of solution of some operator inequalities. Sibirsk. Mat. Zh. 16 (1975) 563-571 (in Russian).
32. J.C. Oostveen and R.F. Curtain, Riccati equations for strongly stabilizable bounded linear systems. Automatica 34 (1998) 953-967.
33. L. Pandolfi, Kalman-Popov-Yacubovich theorem: an overview and new results for hyperbolic control systems. Nonlinear Anal. Theor. Methods Appl. 30 (1997) 735-745.
34. L. Pandolfi, Dissipativity and Lur'e problem for parabolic boundary control system, Research Report, Dipartamento di Matematica, Politecnico di Torino 1 (1997) 1-27;
35. SIAM J. Control Optim. 36 (1998) 2061-2081.
36. L. Pandolfi, The Kalman-Yacubovich-Popov theorem for stabilizable hyperbolic boundary control systems. Integral Equations Operator Theory 34 (1999) 478-493.
37. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. New York, Springer-Verlag (1983).
38. D. Salamon, Realization theory in Hilbert space. Math. Systems Theory 21 (1989) 147-164.
39. O.J. Staffans, Quadratic optimal control of stable well-posed linear systems through spectral factorization. Math. Control Signals Systems 8 (1995) 167-197.
40. O.J. Staffans and G. Weiss, Transfer functions of regular linear systems, Part II: The system operator and the Lax-Phillips semigroup. Trans. Amer. Math. Soc. 354 (2002) 3229-3262.
41. M. Vidyasagar, Nonlinear Systems Analysis. 2nd Edition, Englewood Cliffs NJ, Prentice-Hall (1993).
42. G. Weiss, Transfer functions of regular linear systems. Part I: Characterization of regularity. Trans. AMS 342 (1994) 827-854.
43. M. Weiss, Riccati Equations in Hilbert Spaces: A Popov function approach, Ph.D. Thesis, Rijksuniversiteit Groningen, The Netherlands (1994).
44. M. Weiss and G. Weiss, Optimal control of stable weakly regular linear systems. Math. Control Signals Syst. 10 (1997) 287-330.
45. R.M. Young, An Introduction to Nonharmonic Fourier Series. New York, Academic Press (1980).