Advertisement

Annali di Matematica Pura ed Applicata

, Volume 143, Issue 1, pp 47–100 | Cite as

Finite rank, relatively bounded perturbations of semigroups generators

Part II: Spectrum and riesz basis assignment with applications to feedback systems
  • I. Lasiecka
  • R. Triggiani
Article

Summary

This paper is motivated by, and ultimately directed to, boundary feedback partial differential equations of both parabolic and hyperbolic type, defined on a bounded domain. It is written, however, in abstract form. It centers on the (feedback) operator AF=A+P; A the infinitesimal generator of a s.c. semigroup on H; P an Abounded, one dimensional range operator (typically nondissipative), so that P=(A·, a)b, for a, b ∈ H. While Part I studied the question of generation of a s.c. semigroup on H by AF and lack thereof, the present Part II focuses on the following topics: (i) spectrum assignment of AF, given A and a ∈ H, via a suitable vector b ∈ H; alternatively, given A, via a suitable pair of vectors a, b ∈ H; (ii) spectrality of AF—and lack thereof—when A is assumed spectral (constructive counterexamples include the case where P is bounded but the eigenvalues of A have zero gap, as well as the case where P is genuinely Abounded). The main result gives a set of sufficient conditions on the eigenvalues {λn} of A, the given vector a ∈ H and a given suitable sequence {εn} of nonzero complex numbers, which guarantee the existence of a suitable vector b ∈ H such that AF possesses the following two desirable properties: (i) the eigenvalues of AF are precisely equal to λnn; (ii) the corresponding eigenvectors of AF form a Riesz basis (a fortiori, AF is spectral). While finitely many εn′s can be preassigned arbitrarily, it must be however that εn → 0 « sufficiently fast ». Applications include various types of boundary feedback stabilization problems for both parabolic and hyperbolic partial differential equations. An illustration to the damped wave equation is also included.

Keywords

Infinitesimal Generator Finite Rank Hyperbolic Type Feedback Stabilization Present Part 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [B-B.1]
    P. Butzer - H.Berens,Semigroups of Operators and Approximations, Springer-Verlag, 1967.Google Scholar
  2. [C.1]
    G. Chen,Control and Stabilization for the wave equation, SIAM J. Control,17 (1979), pp. 66–81.Google Scholar
  3. [C-H.1]
    R. Courant -D. Hilbert,Methods of Mathematical Physics, I, Interscience Pubs., John Wiley, New York, 1953.Google Scholar
  4. [C-L.1
    E. A. Coddington -N. Levison,Theory of Ordinary Differential Equations, McGraw Hill, New York, 1955.Google Scholar
  5. [D.1]
    N. Dunford,A survey of the theory of spectral operators, Bulletin Amer. Math. Soc.,64, 5 (Sept. 1958), pp. 217–274.Google Scholar
  6. [D-S.1]
    N. Dunford -J. Schwartz,Linear operators, I (1958), II (1963) and III (1971), Interscience Pubs., John Wiley, New York.Google Scholar
  7. [K.1]
    T.Kato,Perturbation theory of linear operators, Springer-Verlag, 1966.Google Scholar
  8. [L-T.1]
    I. Lasiecka -R. Triggiani,Hyperbolic equations with Dirichlet boundary feedback via position vector: Regularity and almost stabilization, Parts I, II, III, Applied Mathematics and Optimization I:8 (1981), pp. 1–37; II:8 (1982), pp. 103–130; III:8 (1982), pp. 199–221.Google Scholar
  9. [L-T.2]
    I. Lasiecka -R. Teiggiani,Stabilization and structural assignment of Dirichlet boundary feedback parabolic equations, SIAM Control & Optimization,21 (1983), pp. 766–803.Google Scholar
  10. [L-T.3]
    I. Lasiecka -R. Triggiani,Structural assignment of Neumann boundary feedback parabolic equations: the unbounded case in the feedback loop, Annali Matematica Pura e Applicata (IV), Vol. XXXII (1982), pp. 131–175.Google Scholar
  11. [L-T.4]
    I. Lasiecka -R. Triggiani,Stabilization of Neumann boundary feedback of parabolic equations: the case of trace in the feedback loop, Applied Mathematics and Optimization,10 (1983), pp. 307–350.Google Scholar
  12. [L-T.5]
    I. Lasiecka -R. Triggiani,Dirichlet boundary stabilization of the wave equation with damping feedback, J. Mathematical Analysis and Applic, Vol.97 (1983), pp. 112–130.Google Scholar
  13. [L-T.6]
    I. Lasiecka -R. Triggiaki,Nondissipative boundary stabilization of hyperbolic equations with boundary observation, J. de Mathematiques Pures et Appliquees,63 (1984), pp. 59–80.Google Scholar
  14. [L-T.7]
    I. Lasiecka -R. Triggiani,Feedback semigroups and cosine operators for boundary feedback parabolic and hyperbolic equations, J. Diff. Eqts.,47, 2 (1983), pp. 246–272.Google Scholar
  15. [R.1]
    D. L. Russell,Canonical forms and spectral determination for a class of hyperbolic distributed parameter control systems, J. Mathematical Analysis & Applic.,62, 1 (1978), pp. 186–225.Google Scholar
  16. [R.2]
    D. L.Russell,Differential delay equations as canonical forms for controlled hyperbolic systems with applications to spectral assignment, Control theory of systems governed by partial differential Equations, Academic Press, pp. 119–150.Google Scholar
  17. [S.1]
    J. Schwartz,Perturbations of spectral operators and applications I, Bounded Perturbations, Pacific J. Mathematics,4 (1954), pp. 415–458.Google Scholar
  18. [S.2]
    Sun Shun-Hua,On spectrum distribution of completely controllable linear systems, SIAM J. Control & Optimization,19 (1981), pp. 730–743.Google Scholar
  19. [S.3]
    S.Shun Hua,Boundary stabilization of hyperbolic systems with no dissipative conditions.Google Scholar
  20. [S.4]
    M.Schechter,Principles of Functional Analysis, Academic Press, 1971.Google Scholar
  21. [T.1]
    H. Triebel,Interpolation Theory, Function Spaces, Differential Operators, VEB Dentsher Verlag, Berlin, 1978.Google Scholar
  22. [T.2]
    R. Triggiani,On the stabilizability problem in Banach space, J. Math. Anal. & Applic,52 (1975), pp. 383–403; Ibid,56 (1976).Google Scholar
  23. [T.3]
    R. Triggiani,Boundary feedback stabilizability of parabolic equations, Applied Math O Optim.,6 (1980), pp. 201–220.Google Scholar
  24. [T.5]
    R. Triggiani,Well posedness and regularity of boundary feedback parabolic systems, J. Diff. Equats.,36 (1980), pp. 347–362.Google Scholar
  25. [T-L.1]
    A. E. Taylor -D. Lay,Introduction to Functional Analysis, 2nd Edition, John Wiley, New York, 1980.Google Scholar

Copyright information

© Nicola Zanichelli Editore 1986

Authors and Affiliations

  • I. Lasiecka
    • 1
  • R. Triggiani
    • 1
  1. 1.Department of Mathematics201 Walker Hall University of FloridaGainesvilleUSA

Personalised recommendations