# Finite rank, relatively bounded perturbations of semigroups generators

- 64 Downloads
- 10 Citations

## 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 A_{F}=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 A_{F} and lack thereof, the present Part II focuses on the following topics: (i) spectrum assignment of A_{F}, 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 A_{F}—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 {λ_{n}} of A, the given vector a ∈ H and a given suitable sequence {ε_{n}} of nonzero complex numbers, which guarantee the existence of a suitable vector b ∈ H such that A_{F} possesses the following two desirable properties: (i) the eigenvalues of A_{F} are precisely equal to λ_{n}+ε_{n}; (ii) the corresponding eigenvectors of A_{F} form a Riesz basis (a fortiori, A_{F} is spectral). While finitely many ε_{n}′s can be preassigned arbitrarily, it must be however that ε_{n} → 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## Preview

Unable to display preview. Download preview PDF.

### References

- [B-B.1]P. Butzer - H.Berens,
*Semigroups of Operators and Approximations*, Springer-Verlag, 1967.Google Scholar - [C.1]G. Chen,
*Control and Stabilization for the wave equation*, SIAM J. Control,**17**(1979), pp. 66–81.Google Scholar - [C-H.1]R. Courant -D. Hilbert,
*Methods of Mathematical Physics, I*, Interscience Pubs., John Wiley, New York, 1953.Google Scholar - [C-L.1E. A. Coddington -N. Levison,
*Theory of Ordinary Differential Equations*, McGraw Hill, New York, 1955.Google Scholar - [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 - [D-S.1]N. Dunford -J. Schwartz,
*Linear operators*, I (1958), II (1963) and III (1971), Interscience Pubs., John Wiley, New York.Google Scholar - [K.1]
- [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 - [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 - [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 - [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 - [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 - [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 - [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 - [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 - [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 - [S.1]J. Schwartz,
*Perturbations of spectral operators and applications I, Bounded Perturbations*, Pacific J. Mathematics,**4**(1954), pp. 415–458.Google Scholar - [S.2]Sun Shun-Hua,
*On spectrum distribution of completely controllable linear systems*, SIAM J. Control & Optimization,**19**(1981), pp. 730–743.Google Scholar - [S.3]S.Shun Hua,
*Boundary stabilization of hyperbolic systems with no dissipative conditions*.Google Scholar - [S.4]
- [T.1]H. Triebel,
*Interpolation Theory, Function Spaces, Differential Operators*, VEB Dentsher Verlag, Berlin, 1978.Google Scholar - [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 - [T.3]R. Triggiani,
*Boundary feedback stabilizability of parabolic equations*, Applied Math O Optim.,**6**(1980), pp. 201–220.Google Scholar - [T.5]R. Triggiani,
*Well posedness and regularity of boundary feedback parabolic systems*, J. Diff. Equats.,**36**(1980), pp. 347–362.Google Scholar - [T-L.1]A. E. Taylor -D. Lay,
*Introduction to Functional Analysis*, 2nd Edition, John Wiley, New York, 1980.Google Scholar