Riesz basis of exponential family for a hyperbolic system

• Autorzy: Intissar Abdelkader , Jeribi Aref , Walha Ines

Identyfikatory

DOI

10.1515/jaa-2019-0002

• Języki publikacji: EN

Abstrakty

EN

This paper studies a linear hyperbolic system with boundary conditions thatwas first studied under someweaker conditions in [8, 11]. Problems on the expansion of a semigroup and a criterion for being a Riesz basis are discussed in the present paper. It is shown that the associated linear system is the infinitesimal generator of a C₀-semigroup; its spectrum consists of zeros of a sine-type function, and its exponential system {e^λnt}_n≥1 constitutes a Riesz basis in L²[0, T]. Furthermore, by the spectral analysis method, it is also shown that the linear system has a sequence of eigenvectors, which form a Riesz basis in Hilbert space, and hence the spectrum-determined growth condition is deduced.

Słowa kluczowe

EN

hyperbolic system; eigenvalues; eigenvectors; sine-type function; Riesz basis

PL

układ hiperboliczny; wartości własne; wektory własne; funkcja sinusoidalna; bazy Riesza

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2019
• Tom: Vol. 25, nr 1
• Strony: 13--23
• Opis fizyczny: Bibliogr. 17 poz.

Twórcy

autor

Intissar Abdelkader

• University of Corse, Quartier Grossetti, 20250 Corté, France

autor

Jeribi Aref

• Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, Route Soukra, BP1171, 3000 Sfax, Tunisia

autor

Walha Ines

• Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, Route Soukra, BP1171, 3000 Sfax, Tunisia

Bibliografia

• [1] M.-T. Aimar, A. Intissar and J.-M. Paoli, Densité des vecteurs propres généralisés d’une classe d’opérateurs non auto-adjoints à résolvante compacte, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 4, 393-396.
• [2] M.-T. Aimar, A. Intissar and J.-M. Paoli, Densité des vecteurs propres généralisés d’une classe d’opérateurs compacts non auto-adjoints et applications, Comm. Math. Phys. 156 (1993), no. 1, 169-177.
• [3] S. A. Avdonin and S. A. Ivanov, Families of Exponentials: The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University, Cambridge, 1995.
• [4] N. Ben Ali and A. Jeribi, On the Riesz basis of a family of analytic operators in the sense of Kato and application to the problem of radiation of a vibrating structure in a light fluid, J. Math. Anal. Appl. 320 (2006), no. 1, 78-94.
• [5] H. Brézis, Analysis Functional Theory and Applications, Masson, Paris, 1999.
• [6] S. Charfi, A. Jeribi and I. Walha, Riesz basis property of families of nonharmonic exponentials and application to a problem of a radiation of a vibrating structure in a light fluid, Numer. Funct. Anal. Optim. 32 (2011), no. 4, 370-382.
• [7] N. Dunford and J. T. Schwartz, Linear Operators. Part III: Spectral Operators, John Wiley & Sons, New York, 1971.
• [8] A. Intissar, Analyse functionnelle et theorie spectrale pour les operateurs compacts non auto-adjoints et exercices avec solutions, Cépaduâs, Toulouse, 1997.
• [9] A. Jeribi, Denseness, Bases and Frames in Banach Spaces and Applications, De Gruyter, Berlin, 2018.
• [10] A. Jeribi and A. Intissar, On an Riesz basis of generalized eigenvectors of the nonselfadjoint problem deduced from a perturbation method for sound radiation by a vibrating plate in a light fluid, J. Math. Anal. Appl. 292 (2004), no. 1, 1-16.
• [11] J. Kergomard and V. Debut, Resonance modes in a one-dimensional medium with two purely resistive boundaries: Calulation methods, orthogonality, and completeness, Acoust. Soc. Amer. 119 (2006), no. 3, 1356-1367.
• [12] V. B. Lidskii, Summability of series in the principal vectors of non-selfadjoint operators, Amer. Math. Soc. Transl. Ser. 2 40 (1964), 193-228.
• [13] B. S. Pavlov, Basicity of an exponential systems and Muckenhoupt’s condition, Soviet Math. Dokl. 20 (1979), no. 4, 655-659.
• [14] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer, New York, 1983.
• [15] J.-M. Wang, G.-Q. Xu and S.-P. Yung, Exponential stability of variable coefficients Rayleigh beams under boundary feedback controls: A Riesz basis approach, Systems Control Lett. 51 (2004), no. 1, 33-50.
• [16] G. Q. Xu and S. P. Yung, The expansion of a semigroup and a Riesz basis criterion, J. Differential Equations 210 (2005), no. 1, 1-24.
• [17] R. M. Young, An Introduction to Nonharmonic Fourier Series, Pure Appl. Math. 93, Academic Press, New York, 1980.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).