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Xu, Gen Qi; Yung, Siu Pang

The expansion of a semigroup and a Riesz basis criterion. (English) Zbl 1131.47042

J. Differ. Equations 210, No. 1, 1-24 (2005).
Let \(\mathcal{A}\) be the generator for a strongly continuous semigroup \(T(t)\) on a Hilbert space \(X\). Suppose that the singular set for \(\mathcal{A}\) can be split into two parts \(\sigma(\mathcal{A})=\sigma_1(\mathcal{A})\cup\sigma_2(\mathcal{A})\), where \(\sigma_2(\mathcal{A})\) consists all isolated eigenvalues.
Under the assumption that the isolated eigenvalues are contained in a vertical strip, that the multiplicity of the eigenvalues are uniformly bounded and that they satisfy \(\inf_{k\neq l}| \lambda_k-\lambda_l| >0\), the authors prove their main theorem, which says that there exist two \(T(t)\)-invariant subspaces \(X_1\), \(X_2\) such that \(\sigma(\left.\mathcal{A}\right| _{X_1})=\sigma_1(\mathcal{A})\), \(\sigma(\left.\mathcal{A}\right| _{X_2})=\sigma_2(\mathcal{A})\) and \(X_1\oplus X_2\subset X\) (the topological direct sum).
In addition, if the Riesz projectors for the eigenvalues satisfy a certain inequality, then \(X=X_1\oplus X_2\). As an application, the authors consider a heat exchanger system with boundary feedback. It is shown that in a certain Hilbert space, the corresponding semigroup fulfils the conditions of the main theorem.
Reviewer: Stefan Jakobsson (Göteborg)

Cited in 2 Reviews
Cited in 59 Documents

MSC:

47D06 One-parameter semigroups and linear evolution equations
35F15 Boundary value problems for linear first-order PDEs
47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces

Keywords:

semigroup expansion; Riesz basis; heat exchanger equation
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References:

[1] Adamjan, V.; Pivovarchik, V.; Tretter, C., On a class of non-self-adjoint quadratic matrix operator pencils Setarising in elasticity theory, J. Operator Theory, 47, 2, 325-341 (2002) · Zbl 1019.47017
[2] Avdonin, S. A.; Ivanov, S. A., Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems (1995), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0866.93001
[3] Avdonin, S. A.; Ivanov, S. A., Riesz bases of exponentials and divided differences (Russian), Algebra i Anal., 13, 3, 1-17 (2001), transl. St. Petersburg Math. J. 13(3) (2002) 339-351 · Zbl 0999.42018
[4] Avdonin, S. A.; Ivanov, S. A.; Russell, D. L., Exponential bases in Sobolev spaces, Control and Observation Problems, (Hoffmann, K.-H.; Leugering, G.; Troltzsch, F., Optimal Control of Partial Differential Equations (Chemnitz, 1998), International Series of Numerical Mathematics, vol. 133 (1999), Birkhäuser: Birkhäuser Basel), 33-42 · Zbl 0935.35014
[5] Avdonin, S. A.; Ivanov, S. A.; Russell, D. L., Exponential bases in Sobolev spaces, Control and observation problems for the wave equation, Proc. Roy. Soc. Edinburgh Section A, 130, 5, 947-970 (2000) · Zbl 0966.93057
[6] Avdonin, S. A.; Moran, W., Ingham-type inequalities and Riesz bases of divided of differences, Mathematical methods of optimization and control large-scale systems (Ekaterinburg, 2000), Internat. J. Appl. Math. Comput. Sci., 11, 4, 803-820 (2001) · Zbl 1031.93098
[7] Avdonin, S. A.; Tucsnark, M., Simultaneous controllability in sharp time for two elastic strings, ESAIM Control Optim. Calculus of Variations, 6, 259-273 (2001), (electronic) · Zbl 0995.93036
[8] Conrad, F.; Morgül, Ö., On the stabilization of a flexible beam with a tip mass, SIAM J. Control Optim., 36, 1962-1986 (1998) · Zbl 0927.93031
[9] Curtain, R. F.; Zwart, H. J., An Introduction to Infinite Dimensional Linear System Theory (1995), Springer: Springer New York · Zbl 0839.93001
[10] Dunford, N.; Schwartz, J. T., Linear Operators, Part III, Spectral Operators (1971), Wiley-Interscience: Wiley-Interscience New York · Zbl 0243.47001
[11] Gohberg, I.; Goldberg, S.; Kaashoek, M. A., (Classes of Linear Operators, vol. 1 (1990), Birkhäuser, Verlag: Birkhäuser, Verlag Basel, Boston, Berlin), 329-336 · Zbl 0745.47002
[12] Gohberg, I. C.; Krein, M. G., Introduction to the theory of linear nonselfadjoint operators, AMS Transl. Math. Monographs, 18 (1969) · Zbl 0181.13504
[13] Guo, B. Z., On the boundary control for a hybrid system with variable coefficients, J. Optim. Theory Appl., 114, 2, 373-395 (2002) · Zbl 1061.93055
[14] Guo, B. Z., Riesz basis property and exponential stability of controlled Euler-Bernoulli beam equation with variable coefficients, SIAM J. Control Optim., 40, 6, 1905-1923 (2002) · Zbl 1015.93025
[15] Kaashoek, M. A.; Verduyn Lunel, S. M., An integrability condition on the resolvent for hyperbolicity of the semigroup, J. Differential Equations, 112, 2, 374-406 (1994), (doi:10.1006/jdeq.1994.1109) · Zbl 0834.47036
[16] Kunimatsu, N.; Sano, H., Stability analysis of heat-exchanger equations with boundary feedbacks, IMA J. Math. Control Inform., 15, 317-330 (1998) · Zbl 0917.93034
[17] Larsen, E. W.; Zweifel, P. F., On the spectrum of the linear transport operator, J. Math. Phys., 15, 1987-1997 (1974)
[18] Y.I. Lyubarskii, K. Seip, A splitting problem for unconditional bases of complex exponentials. Entire functions in modern analysis (Tel-Aviv, 1997), Israel Mathematical Conference Proceedings, vol. 15, Bar-Ilan University, Ramat Gan, 2001, pp. 233-242.; Y.I. Lyubarskii, K. Seip, A splitting problem for unconditional bases of complex exponentials. Entire functions in modern analysis (Tel-Aviv, 1997), Israel Mathematical Conference Proceedings, vol. 15, Bar-Ilan University, Ramat Gan, 2001, pp. 233-242. · Zbl 1015.42022
[19] Lyubarskii, Y. I.; Seip, K., Weighted Paley-Wiener spaces, J. Amer. Math. Soc., 15, 4, 979-1006 (2002), (electronic) · Zbl 1017.46018
[20] Naimark, M. A., (Linear Differential Operators, vol. I (1967), Ungar: Ungar New York) · Zbl 0219.34001
[21] Pazy, A., Semigroup of Linear Operators and Applications to Partial Differential Equations (1983), Springer: Springer New York · Zbl 0516.47023
[22] Rao, B. P., Optimal energy decay rate in a damped Rayleigh beam, optimization methods, (Cox, S.; Lasiecka, I., Partial Differential Equations, Contemporary Mathematics, vol. 209 (1997)), 221-229
[23] Shkalikov, A. A., Boundary value problems for ordinary differential equations with a parameter in the boundary conditions, Engl. transl. J. Sov. Math., 33, 6, 1311-1342 (1986) · Zbl 0609.34019
[24] Shubov, M. A., The Riesz basis property of the system of root vectors for the equation of a nonhomogeneous damped stringtransformation operators method, Methods Appl. Anal., 6, 4, 571-591 (1999) · Zbl 0986.74039
[25] Shubov, M. A., Asymptotic and spectral analysis of the spatially nonhomogeneous Timoshenko beam model, Math. Nachr., 241, 125-162 (2002) · Zbl 1037.47032
[26] Tretter, C., On \(\lambda \)-Nonlinear Boundary Eigenvalue Problem, (Mathematical Research, vol. 71 (1993), Akademie Verlag: Akademie Verlag Berlin) · Zbl 0974.34079
[27] Verduyn Lunel, S. M.; Yakubovich, D. V., A functional model approach to linear neutral functional differential equations, Integral Equations Operator Theory, 27, 347-378 (1997) · Zbl 0891.34077
[28] Xu, G. Q.; Wang, S. H., The completeness of systems of generalized eigenfunctions of generators of Riesz semigroups (Chinese), Acta Math. Sinica, 39, 2, 263-267 (1996) · Zbl 0867.47015
[29] Xu, G. Q.; Feng, D. X., The Riesz basis property of a Timoshenko beam with boundary feedback and application, IMA J. Appl. Math., 67, 4, 357-370 (2002) · Zbl 1136.74343
[30] Young, R. M., An Introduction to Nonharmonic Fourier Series (1980), Academic Press: Academic Press New York · Zbl 0493.42001
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