On the circle criterion for boundary control systems in factor form

• Autorzy: Grabowski P.
• Języki publikacji: EN

Abstrakty

EN

In this paper we return to the origins of the circle criterion initiated by Irwin Sandberg nearly forthy years ago. A version of the Leray-Schauder alternative is applied to get an existence of an abstract Hammerstein output equation for the closed-loop system. This existence result completes Sandberg's method based on using the Banach fixed-point theorem. It is shown that the assertion of the circle criterion can be strengthened by adding a characterization of an asymptotic behaviour of the state trajectories. Results are being compared with a recent version of the circle criterion for boundary control systems in factor form. Some prospects for further studies are also suggested.

Słowa kluczowe

EN

infinite-dimensional control systems; semigroups; Lyapunov functionals; circle criterion

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2003
• Tom: Vol. 23
• Strony: 25--47
• Opis fizyczny: Bibliogr. 33 poz., rys.

Twórcy

autor

Grabowski P.

• Institute of Automatics, AGH University of Science and Technology, Cracow, Poland,

Bibliografia

• [1] Bartle R. G.: The Elements of Integration. New York, J. Wiley. 1966; Bartle R. G.: The Elements of Integration and Lebesgue Measure. New York, J. Wiley 1995 contains a corrected reprint as Chapters 1-10
• [2] Butzer P. L., Nessel R. J.: Fourier Analysis and Approximation. Vol. I: One -Dimensional Theory. Basel, Birkhauser 1971
• [3] Curtain R. F.: Regular linear systems and their reciprocals: applications to Riccati equations. Systems and Control Letters, 4 9 (2003), pp. 81-89
• [4] Curtain R. F.: Riccati equations for stable well-posed linear systems: the generic case, (to appear in SIAM Journal Control and Optimization)
• [5] Curtain R. F., Logemann H. and Staffans O.: Stability results of Popov-type for infinite - dimensional systems with applications to integral control. Journal of the London Mathematical Society, 86 (2003), pp. 779-816
• [6] Curtain R. F., Zwart H.: An Introduction to Infinite-Dimensional Linear Systems Theory. New York, Springer 1995
• [7] Hoffman K.: Banach Spaces of Analytic Functions. Englewood Cliffs, Prentice -Hall 1962
• [8] Grabowski P.: On the spectral - Lyapunov approach to parametric optimization of DPS. IMA Journal of Mathematical Control and Information, 7 (1990), pp. 317-338
• [9] Grabowski P.: Admissibility of observation functionals. International Journal of Control, 62, 1995, 1161-1173
• [10] Grabowski P., Callier F. M.: Admissibility of observation operators. Semigroup criteria of admissibility. Journal of Integral Equations and Operator Theory, 25 (1996), pp. 182-198
• [11] Grabowski P., Callier F.M.: Admissible observation operators. Duality of observation and control using factorizations. Dynamics of Continuous, Discrete and Impulsive Systems, 6 (1999), pp. 87-119
• [12] Grabowski P., Callier F.M.: On the circle criterion for boundary control systems in factor form: Lyapunov approach. Facultes Universitaires Notre - Dame de la Paix a Namur, Publications du Departement de Mathematique, Research Report 07 (2000), FUNDP, Namur, Belgium
• [13] Grabowski P., Callier F.M.: On the circle criterion for boundary control systems in factor form: Lyapunov stability and Lur'e equations. Facultes Universitaires Notre - Dame de la Paix a Namur, Publications du Departement de Mathematique, Research Report 05 (2002), FUNDP, Namur, Belgium
• [14] Grabowski P., Callier F. M.: Boundary control systems in factor form: Transfer functions and input-output maps. Integral Equations and Operator Theory, 41 (2001), pp. 1-37
• [15] Grabowski P., Callier F. M.: Circle criterion and boundary control systems in factor form: Input-output approach. International Journal of Applied Mathematics and Computer Science, 11 (2001), pp. 1387-1403
• [16] Halmos P. R., Sunder V. S.: Bounded Integral Operators on L2 Spaces. Berlin, Springer 1978
• [17] Hansen S., Weiss G.: The operator Carleson measure criterion for admissibility of control operators for diagonal semigroups on l2. Systems and Control Letters, 16 (1991), pp. 219-227
• [18] Jacob B., Partington J.R.: The Weiss conjecture on admissibility of observation operators for contraction semigroups. Journal of Integral Equations and Operator Theory, 40 (2001), pp. 231-243
• [19] Krasnosel'skii M. A.: Topological Methods in the Theory of Nonlinear Integral Equations, Moscow, GITTLI, 1956 (in Russian). English translation by A.H. Armstrong, New York, A Pergamon Press Book The Macmillan Co. 1964
• [20] Krasnosel'skii M. A., Lifshits E. A. and Sobolev A.V.: Positive Linear Systems The Method of Positive Operators. Moscow, Nauka, 1985 (in Russian). English translation by Jiirgen Appell, Berlin, Heldermann Verlag 1989
• [21] Kudrewicz J.: Stability of nonlinear systems with feedback. Automatics and Re mote Control, 25 (1964), pp. 1027-1037, translated from Avtomatika i Telemekhanika, 25 (1964). pp. 1145-1155
• [22] Kudrewicz J.: Frequency-domain methods in the theory of nonlinear dynamical systems. Warszawa, WNT 1970 (in Polish)
• [23] Logemann H., Curtain R. F.: Absolute stability results for well-posed infinite-dimensional systems with low-gain integral control. ESAIM: Control, Optimisation and Calculus of Variations, 5 (2000), pp. 395-424
• [24] Łojasiewicz S.: An Introduction to the Theory of Real Functions. Warsaw, PWN (in Polish) 1973 (First edition), English translation of the second Polish edition (1976), by G. H. Lawden, Chichester, Wiley-Interscience Publication 1988
• [25] Meehan M., O'Regan D.: Existence theory for nonlinear Fredholm and Volterra integral equations on half-open intervals. Nonlinear Analysis Ser. A: Theory, Methods and Applications, 35 (1999), pp. 355-387
• [26] Natanson I. P.: Theory of Functions of Real Variable. Second edition, Moscow, Gostekhizdat 1957. German translation edited by Karl Boegel, Berlin, Akademie-Verlag 1981
• [27] Pazy A.: Semigroups of Linear Operators and Applications to PDEs. Berlin, Springer 1983
• [28] Pruss J.: On the spectrum of Co-semigroup. Transactions of the AMS, 284 (1984), pp. 847-857
• [29] Sandberg I. W.: On the L2-boundedness of solutions of nonlinear functional equations. Bell System Technical Journal, 43 (1964), pp. 1581-1599
• [30]Sandberg I. W.: A frequency-domain condition for stability of feedback systems containing a single time-varying nonlinear element. Bell System Technical Journal, 43 (1964), pp. 1601-1608
• [31] Sandberg I. W.: A note on the application of contraction-mapping fixed point theorem to a class of nonlinear functional equations. SIAM Review, 7 (1965), pp. 199-204 or I. W. Sandberg, An observation concerning the application of the contraction-mapping fixed-point theorem, and a result concerning the norm-boundedness of solutions of nonlinear functional equations. Bell System Technical Journal, 44 (1965), pp. 1809-1812
• [32] Vainberg M. M.: Variational Methods for the Study of Nonlinear Operators. Mo scow: Gostekhizdat, 1956 (in Russian). Translated and supplemented by Amiel Feinstein, San Francisco-London-Amsterdam, Holden-Day Inc. 1964
• [33] Zemanian A. H.: Distribution Theory and Transform Analysis. New York, McGraw-Hill 1965