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The Kalman-Yakubovich-Popov theorem for stabilizable hyperbolic boundary control systems

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Abstract

In this paper we present a version of the Kalman-Yakubovich-Popov theorem for a class of boundary control systems of hyperbolic type. Unstable, controllable systems are considered and stabilizability withunbounded feedbacks is permitted.

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References

  1. Clements D.J., Anderson B.D.O.,Singular optimal control, the linear quadratic problem, Lecture Notes Control Inf. Sci. 5 Springer-Verlag, Berlin, 1978.

    Google Scholar 

  2. Arov D.Z., Nudel'man M.A., Passive linear stationary dynamical scattering systems with continuous time,Integ. Equat. Oper. Th.,24 1–45, 1996.

    Google Scholar 

  3. Balakrishnan A.V., On a generalization of the Kalman-Yakubovich Lemma,Applied Mathematics and Optimization,31, 177–187, 1995.

    Google Scholar 

  4. Boyd S., Balakrishnan V., Kabamba P., A bisection method for computing theH norm of a transfer matrix and related problems,Math. Control Signals Systems,2 207–219, 1989.

    Google Scholar 

  5. Churilov, A.N., On the solvability of matrix inequalities,Mat. Zamietki,36, 725–732, 1984.

    Google Scholar 

  6. Curtain, R.F., The Kalman-Yakubovich-Popov Lemma for Pritchard-Salamon systems,System & Control Lett.,27 67–72, 1996; Correction to the Kalman-Yakubovich-Popov Lemma for Pritchard-Salamon systems,Systems & Control Lett.,28 237–238, 1996.

    Google Scholar 

  7. Datko, R., Extending a theorem of A. M. Lyapunov to Hilbert spaces,J. Math. Anal. and Appl. 32 610–616, 1970.

    Google Scholar 

  8. Faibusovich, L.E., Matrix Riccati inequality: existence of solutions,Systems & Control Lett.,9 59–64, 1987.

    Google Scholar 

  9. Fattorini O. Boundary control systems,SIAM J. Control,6, 349–385, 1968.

    Google Scholar 

  10. Flandoli F., Lasiecka I., Triggiani R., Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler-Bernoulli equations,Annali Mat. Pura Appl.,153 307–382, 1988.

    Google Scholar 

  11. Gantmacher, F.R.Theory of matrices, vol. 1, Chelsea Pub. Co., New York, 1960.

    Google Scholar 

  12. Grabowski, P., On the spectral Lyapunov approach to parametric optimization of distributed parameter systems,IMA J. Cont. Optimization,7, 317–338, 1990.

    Google Scholar 

  13. Kalman R.E., Contribution to the theory of optimal control,Bol. Soc. Mat. Mexicana,6 102–119, 1960.

    Google Scholar 

  14. Kalman R.E., Lyapunov functions for the problem of Lur'e in automatic control,Proc. Nat. Acad. Sci. U.S.A.,49, 201–205, 1963.

    Google Scholar 

  15. Lasiecka I., Triggiani R., Algebraic Riccati equations arising in boundary/point control: a review of theoretical and numerical results, inPerspective in control theory, Jakubczyk B., Malanowski K., Respondek W. eds., Birkhauser, Boston, 1990: Part 1:Continuous case, pp. 175–210; Part 2:Approximation theory, 211–235.

    Google Scholar 

  16. Lasiecka I., Triggiani R.,Differential and algebraic Riccati equations with applications to boundary/point control problems: continuous theory and approximation theory, Lecture Notes Control Inf. Sci., 164 Springer-Verlag, Berlin, 1991.

    Google Scholar 

  17. Likhtarnikov A.L., Yakubovich V.A., The frequency theorem for equations of evolutionary type,Siberian J of Mathematics,17, 1069–1085, 1976.

    Google Scholar 

  18. Louis J-Cl., Wexler D., The Hilbert space regulator problem and operator Riccati equation under stabilizability,Annales de la Soc. Scient. de Bruxelles,105, 137–165, 1991.

    Google Scholar 

  19. Molinari B.P., Nonegativity of a quadratic functional,SIAM J. Control,13 792–806, 1975.

    Google Scholar 

  20. Nudel'man, A.A. Schwartzman, N.A., On the existence of the solutions to certain operatorial inequalities, (in russian)Sib. Math. Z.,16 562–571, 1975.

    Google Scholar 

  21. Pandolfi, L., The Kalman-Yakubovich-Popov Theorem: an overview and new results for hyperbolic control systems,Nonlinear Analysis TMA,30, 735–745, 1997.

    Google Scholar 

  22. Pandolfi, L., Dissipativity and Lur'e problem for parabolic boundary control systems, in print,SIAM J. Control Optim.

  23. Rantzer, A., On the Kalman-Yakubovich-Popov LemmaSystems & Control Lett.,28 7–10, 1996.

    Google Scholar 

  24. Russel D.L., Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions,SIAM Review,20, 639–739, 1978.

    Google Scholar 

  25. Staffans O.J., Quadratic optimal control of well-posed linear systems, in print,SIAM J. Control Optim.

  26. Weiss M., Weiss G., Optimal control of stable weakly regular systems,Math. Control Signals Systems,10 287–330, 1997.

    Google Scholar 

  27. Willems, J.C., On the existence of a nonpositive solution to the Riccati equation,IEEE Trans. Automat. Control,AC 19 592–593, 1974.

    Google Scholar 

  28. Yakubovich V.A., Solution of certain matrix inequalities occurring in control theory,Dok. Akad. Nauk. SSSR 143, 1304–1307, 1962.

    Google Scholar 

  29. Yakubovich V.A., The frequency theorem in control theory,Siberian J of Mathematics,14 384–419, 1973.

    Google Scholar 

  30. Yakubovich V.A., A frequency theorem for the case in which the state and control spaces are Hilbert spaces, with an applications to some problems in the synthesis of optimal controls. I,Siberian J of Mathematics,15, 457–476 1974; II,Siberian J of Mathematics,16 828–845, 1975.

    Google Scholar 

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Paper partially supported by the Italian MINISTERO DELLA RICERCA SCIENTIFICA E TECNOLOGICA within the program of GNAFA-CNR and by NATO CRG program SA.5-2-05 (CRG940161).

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Pandolfi, L. The Kalman-Yakubovich-Popov theorem for stabilizable hyperbolic boundary control systems. Integr equ oper theory 34, 478–493 (1999). https://doi.org/10.1007/BF01272886

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  • DOI: https://doi.org/10.1007/BF01272886

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