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Abstrakty
The modern method of analysis of the distributed parameter systems relies on the transformation of the dynamical model to an abstract differential equation on an appropriately chosen Banach or, if possible, Hilbert space. A linear dynamical model in the form of a first order abstract differential equation is considered to be well-posed if its right-hand side generates a strongly continuous semigroup. Similarly, a dynamical model in the form of a second order abstract differential equation is well-posed if its right-hand side generates a strongly continuous cosine family of operators. Unfortunately, the presence of a feedback leads to serious complications or even excludes a direct verification of assumptions of the Hille-Phillips-Yosida and/or the Sova-Fattorini Theorems. The class of operators which are similar to a normal discrete operator on a Hilbert space describes a wide variety of linear operators. In the papers [12, 13] two groups of similarity criteria for a given hybrid closed-loop system operator are given. The criteria of the first group are based on some perturbation results, and of the second, on the application of Shkalikov's theory of the Sturm-Liouville eigenproblems with a spectral parameter in the boundary conditions. In the present paper we continue those investigations showing certain advanced applications of the Shkalikov's theory. The results are illustrated by feedback control systems examples governed by wave and beam equations with increasing degree of complexity of the boundary conditions.
Czasopismo
Rocznik
Tom
Strony
45--97
Opis fizyczny
Bibliogr. 38 poz., rys., tab.
Twórcy
autor
- AGH University of Science and Technology, Institute of Automatics, al. Mickiewicza 30, B1, rm. 314, 30-059 Cracow, Poland, pgrab@ia.agh.edu.pl
Bibliografia
- [1] Bailey T., Hubbard J.E., Distributed piezoelectric polymer active vibration control of a cantilever beam. AIAA JOURNAL on GUIDANCE CONTROL and DYNAMICS. 1985. 8. 605-611.
- [2] Birkhoff G. D., On the asymptotic character of the solution of certain linear differential equation containing a parameter. TRANSACTIONS of the AMERICAN MATHEMATICAL SOCIETY. 1908. 9. 219-231.
- [3] Birkhoff G. D., Boundary value and expansion problems of ordinary linear differential equations. TRANSACTIONS of the AMERICAN MATHEMATICAL SOCIETY. 1908. 9. 373-395.
- [4] Chen G., Krantz S. G., Ma D. W., Wayne C. E., West H. H., The Euler-Bernoulli beam equation with boundary energy dissipation. In: Sung J. Lee (Ed.) Operator Methods for Optimal Control Problems. LECTURE NOTES in PURE and APPLIED MATHEMATICS. 1987. 108. 67-96. New York Marcel Dekker Inc.
- [5] Conrad F., Stabilization of beams by positive feedback control. SIAM JOURNAL of CONTROL and OPTIMIZATION. 1990. 28. 2. 423-437.
- [6] Conrad F., Morgul O., On the stabilization of a flexible beam with a tip mass. SIAM JOURNAL of CONTROL and OPTIMIZATION. 1997. To appear.
- [7] Curtain R. F., Oostveen J. C, Absolute stability of collocated systems. Submitted to ICOTA'98, Perth: Australia. 1998.
- [8] Fattorini H. O., Second Order Linear Differential Equations in Banach Spaces. Amsterdam, North Holland. 1985.
- [9] Gokhberg I. C, Krein M. G., Introduction to the Theory of Non-Self-Adjoint Operators. Providence, AMS. 1969.
- [10] Górecki H., Fuksa S., Grabowski P., Korytowski A., Analysis and Synthesis of Time Delay Systems. Warsaw and Chichester, PWN and J.Wiley & Sons. 1989.
- [11] Grabowski P., Spectral and Lyapunov Methods in the Analysis of Infinite -Dimensional Feedback Systems. ZESZYTY NAUKOWE AGH, s. Automatyka. 1991. 58. 1-189 (in Polish).
- |12] Grabowski P., Well-posedness and stability analysis of hybrid feedback systems. JOURNAL of MATHEMATICAL SYSTEMS, ESTIMATION and CONTROL, 1996. 6. 121-124 (summary). Full electronic manuscript - retrieval code 15844.
- [13] Grabowski P., Spectral approach to well-posedness and stability analysis of hybrid feedback systems. In: Wajs W., Grabowski P. (Eds.), Studies in Automatics, 1996. Kraków: Wydawnictwa AGH. 104-139.
- [14] Halmos P., A Hilbert Space Problem Book. Princeton, Van Nostrand. 1967.
- [15] Huang F., Characteristic condition of exponential stability of linear dynamical systems in H-spaces. ANNALS of DIFFERENTIAL EQUATIONS. 1985. 1. 43-56.
- [16] Huang F.; Strong asymptotic stability of linear dynamical systems in Banach spaces. JOURNAL of DIFFERENTIAL EQUATIONS. 1993. 104. 307-324.
- [17] Janas J., On unbounded hyponormal operators. Pt. I. ARKIV fur MATHEMATIK. 1989. 27. 273-281; Pt. II. INTEGRAL EQUATIONS and OPERATOR THEORY. 1992. 15. 470-478.
- [18] Kato T., Perturbation Theory for Linear Operators. New York, Springer. 1966.
- [19] Kim J.U., Renardy Y., Boundary control of the Timoshenko beam. SIAM JOURNAL of CONTROL and OPTIMIZATION. 1987. 25. 417-429.
- [20] Krall A.M., Asymptotic stability of the Euler-Bernoulli beam with boundary control. JOURNAL of MATHEMATICAL ANALYSIS and APPLICATIONS. 1989. 137. 1. 288-295.
- [21] Levan N., Stabilizability of two classes of contraction semigroups. JOURNAL of OPTIMIZATION THEORY and APPLICATIONS. 1993. 76. 111-130.
- [22] Littman W., Markus L., Stabilization of a hybrid system of elasticity by feedback boundary damping. ANNALI di MATEMATICA PURA ed APPLICATA. 1988. 152. 281-330.
- [23] Mifdal A., Etude de la stabilisation forte et uniforme de systeme hybride. These, Universite de Henri Poincare de Nancy. 1997.
- [24] Morgul O., Dynamic boundary control of the Timoshenko beam. AUTOMATICA. 1992. 28. 6. 1255-1260.
- [25] Morgul O., A dynamic control law for the wave equation. AUTOMATICA. 1994. 30. 11. 1785-1792.
- [26] Morgul O., Rao B.P., Conrad F., On the stabilization of a cable with a trip mas. IEEE TRANSACTIONS on AUTOMATIC CONTROL. 1994. 39. 10. 2140-2145.
- [27] Oostveen J. C, Curtain R. F., Riccati Equations for Strongly Stabilizable Bounded Linear Systems. Preprint 1998.
- [28] Pazy A., Semigroups of Linear Operators and Applications to PDEs. Berlin. Springer. 1983.
- [29] Pruss J., On the spectrum of C0-semigroup. TRANSACTIONS of the AMS. 1984. 284. 847-857.
- [30] Rebarber R. L., Spectral determination for a cantilever beam. IEEE TRANSACTIONS on AUTOMATIC CONTROL. 1989. 34. 5. 502-510.
- [31] Rideau P., Controle d'un assemblage de pontres flexibles par des capteurs actionneurs ponctuels: etude du spectre du systeme. These, Ecole Nationale Superieure des Mines de Paris, Sophia-Antipolis. 1985.
- [32] Schechter M., Principles of Functional Analysis. New York, Academic Press. 1971.
- [33] Shkalikov A., Boundary problem for ordinary differential operators with parameter in the boundary conditions. JOURNAL of SOVIET MATHEMATICS. 1986. 33. 1311-1342.
- [34] Slemrod M., Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control. MATHEMATICS of CONTROL SIGNALS and SYSTEMS. 1989. Vol. 2. 265-285.
- [35] Tamarkin J., On some general problems for linear ordinary differential equations. St. Peterburg. M.N. Frolova Press. 1917.
- [36] Tamarkin J., On some general problems for linear ordinary differential equations and expansion of an arbitrary function in series of fundamental functions. MATHEMATISCHE ZEITSCHRIFT. 1927. 27. 1. 1-54.
- [37] Weidmann J., Linear Operators in Hilbert Spaces. New York, Springer. 1980.
- [38] Weiss G., Weak Lp-stability of linear semigroups on Hilbert space implies EXS. JOURNAL of DIFFERENTIAL EQUATIONS. 1988. 76. 513-523.
Typ dokumentu
Bibliografia
Identyfikator YADDA
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