Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



An $ L^2(\Omega )$-based algebraic approach to boundary stabilization for linear parabolic systems

Author: Takao Nambu
Journal: Quart. Appl. Math. 62 (2004), 711-748
MSC: Primary 93D15; Secondary 35K50, 47N70, 93C20
DOI: https://doi.org/10.1090/qam/2104271
MathSciNet review: MR2104271
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the stabilization problem of linear parabolic boundary control systems. While the control system is described by a pair of standard linear differential operators $ \left( {L, \tau } \right)$, the corresponding semigroup generator generally admits no Riesz basis of eigenvectors. Very little information on the fractional powers of this generator is needed. In this sense our approach has enough generality as a prototype to be used for other types of parabolic systems. We propose in this paper a unified algebraic approach to the stabilization of a variety of parabolic boundary control systems.

References [Enhancements On Off] (What's this?)

  • [1] H. Amann, Feedback stabilization of linear and semilinear parabolic systems, LNPAM vol. 116, Marcel Dekker, New York, 1989 MR 1009387
  • [2] C. I. Byrnes, D. S. Gilliam, V. I. Shubov, and G. Weiss, Regular linear systems governed by a boundary controlled heat equation, J. Dynamical and Control Systems 8, 341-370 (2002) MR 1914447
  • [3] R. F. Curtain, Finite dimensional compensators for parabolic distributed systems with unbounded control and observation, SIAM J. Control Optim. 22, 255-276 (1984) MR 732427
  • [4] D. Fujiwara, Concrete characterization of the domain of fractional powers of some elliptic differential operators of the second order, Proc. Japan Acad. Ser. A Math. Sci. 43, 82-86 (1967) MR 0216336
  • [5] J. S. Gibson and A. Adamian, Approximation theory for linear quadratic-Gaussian optimal control of flexible structures, SIAM J. Control. Optim. 29, 1-37 (1991) MR 1088217
  • [6] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, New York, 1983 MR 737190
  • [7] P. Grisvard, Caractérisation de quelques espaces d'interpolation, Arch. Rational Mech. Anal. 25, 40-63 (1967) MR 0213864
  • [8] S. Itô, Diffusion Equations, Amer. Math. Soc., Providence, 1992
  • [9] T. Kato, A generalization of the Heinz inequality, Proc. Japan Acad. Ser. A Math. Sci. 37, 305-308 (1961) MR 0145345
  • [10] I. Lasiecka and R. Triggiani, Feedback semigroups and cosine operators for boundary feedback parabolic and hyperbolic equations, J. Differential Equations 47, 246-272 (1983) MR 688105
  • [11] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, vol. I, Springer-Verlag, New York, 1972 MR 0350177
  • [12] D. G. Luenberger, Observers for multivariable systems, IEEE Trans. Automat. Control, AC-11, 190-197 (1966) MR 0441429
  • [13] T. Nambu, On stabilization of partial differential equations of parabolic type: Boundary observation and feedback, Funkcial. Ekvac. 28, 267 298 (1985) MR 852115
  • [14] -, An extension of stabilizing compensators for boundary control systems of parabolic type, J. Dynamics and Differential Equations 1, 327-346 (1989) MR 1020709
  • [15] -, Characterization of the domain of fractional powers of a class of elliptic differential operators with feedback boundary conditions, J. Differential Equations 136, 294-324 (1997) MR 1448827
  • [16] -, An algebraic method of stabilization for a class of boundary control systems of parabolic type, J. Dynamics and Differential Equations 13, 59-85 (2001) MR 1822212
  • [17] D. Salamon, Infinite dimensional linear systems with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc. 300, 383-431 (1987) MR 876460
  • [18] E. C. Titchmarsh, The Theory of Functions, The Clarendon Press, Oxford, 1939 MR 3155290
  • [19] G. Weiss and R. F. Curtain, Dynamic stabilization of regular linear systems, IEEE Trans. Automat. Control AC-42, 4-21 (1997) MR 1439361
  • [20] W. M. Wonham, On pole assignment in multi-input controllable linear systems, IEEE Trans. Automat. Control AC-12, 660-665 (1967)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 93D15, 35K50, 47N70, 93C20

Retrieve articles in all journals with MSC: 93D15, 35K50, 47N70, 93C20

Additional Information

DOI: https://doi.org/10.1090/qam/2104271
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society