Spectral operators generated by damped hyperbolic equations

Shubov, Marianna A.

doi:10.1007/bf01294159

Cited by 32 publications

(19 citation statements)

References 17 publications

(15 reference statements)

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“…This approach has been used in dealing with the beam equations with low order perturbation of variable coefficients (see [6], [7], and [8]). A similar adoption can also be found in the study of the string equations with variable coefficients, for which we refer to [9], [10], and [11] as well as the references therein.…”

Section: Introductionmentioning

confidence: 80%

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Riesz Basis Property and Exponential Stability of Controlled Euler--Bernoulli Beam Equations with Variable Coefficients

Guo¹

2002

SIAM J. Control Optim.

107

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Abstract. This paper studies the basis property and the stability of a distributed system described by a nonuniform Euler-Bernoulli beam equation under linear boundary feedback control. It is shown that there is a sequence of generalized eigenfunctions of the system, which forms a Riesz basis for the state Hilbert space. The asymptotic distribution of eigenvalues, the spectrumdetermined growth condition, and the exponential stability are concluded. The results are applied to a nonuniform beam equation with viscous damping of variable coefficient as a generalization of existing results for the uniform beam.

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Section: Introductionmentioning

confidence: 80%

“…The result greatly simplifies the procedure in establishing the Riesz basis property for systems described by discrete operators in a Hilbert space since the result eliminates the requirement of estimation of low eigenfunctions, which is rather difficult by other methods found in all previous papers [10], [11], [12].…”

Section: Introductionmentioning

confidence: 99%

Riesz Basis Property and Exponential Stability of Controlled Euler--Bernoulli Beam Equations with Variable Coefficients

Guo¹

2002

SIAM J. Control Optim.

107

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“…One of the main results of the above series of works (see [19,20,22,23]) is the fact that the operators A h defined in (1.11) and (1.16) provide a class of nontrivial examples of spectral operators. To explain this statement, we recall the definition of a spectral operator.…”

Section: 2mentioning

confidence: 99%

Nonselfadjoint operators generated by the equation of a nonhomogeneous damped string

Shubov

1997

Trans. Amer. Math. Soc.

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Abstract. We consider a one-dimensional wave equation, which governs the vibrations of a damped string with spatially nonhomogeneous density and damping coefficients. We introduce a family of boundary conditions depending on a complex parameter h. Corresponding to different values of h, the problem describes either vibrations of a finite string or propagation of elastic waves on an infinite string. Our main object of interest is the family of nonselfadjoint operators A h in the energy space of two-component initial data. These operators are the generators of the dynamical semigroups corresponding to the above boundary-value problems. We show that the operators A h are dissipative, simple, maximal operators, which differ from each other by rank-one perturbations. We also prove that the operator A 1 (h = 1) coincides with the generator of the Lax-Phillips semigroup, which plays an important role in the aforementioned scattering problem. The results of this work are applied in our two forthcoming papers both to the proof of the Riesz basis property of the eigenvectors and associated vectors of the operators A h and to establishing the exact and approximate controllability of the system governed by the damped wave equation.

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“…In addition, we would like to emphasize that the restriction considered in our paper [12] was not unique, i.e. in many of our previous works [12,[20][21][22][23]27,28] devoted to asymptotic and spectral analysis on non-selfadjoint operators, generated by di erent problems on the vibrations of exible structures, we had to impose similar connections between the boundary parameters and the mechanical and structural parameters of the corresponding models (strings, rods, beams, etc). Moreover, in many papers of di erent authors [16,17,[29][30][31][32], the same type of a restriction on the boundary parameter is imposed.…”

Section: Introduction and Statement Of Problemmentioning

confidence: 96%

Asymptotic and spectral analysis of non‐selfadjoint operators generated by a filament model with a critical value of a boundary parameter

Shubov¹

2002

Math Methods in App Sciences

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SUMMARYWe consider a class of non-selfadjoint operators generated by the equation and the boundary conditions, which govern small vibrations of an ideal ÿlament with non-conservative boundary conditions at one end and a heavy load at the other end. The ÿlament has a non-constant density and is subject to a viscous damping with a non-constant damping coe cient. The boundary conditions contain two arbitrary complex parameters. In our previous paper (Mathematical Methods in the Applied Sciences 2001; 24(15):1139-1169), we have derived the asymptotic approximations for the eigenvalues and eigenfunctions of the aforementioned non-selfadjoint operators when the boundary parameters were arbitrary complex numbers except for one speciÿc value of one of the parameters. We call this value the critical value of the boundary parameter. It has been shown (in Mathematical Methods in the Applied Sciences 2001; 24(15):1139-1169) that the entire set of the eigenvalues is located in a strip parallel to the real axis. The latter property is crucial for the proof of the fact that the set of the root vectors of the operator forms a Riesz basis in the state space of the system. In the present paper, we derive the asymptotics of the spectrum exactly in the case of the critical value of the boundary parameter. We show that in this case, the asymptotics of the eigenvalues is totally di erent, i.e. both the imaginary and real parts of eigenvalues tend to ∞ as the number of an eigenvalue increases. We will show in our next paper, that as an indirect consequence of such a behaviour of the eigenvalues, the set of the root vectors of the corresponding operator is not uniformly minimal (let alone the Riesz basis property).

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Spectral operators generated by damped hyperbolic equations

Cited by 32 publications

References 17 publications

Riesz Basis Property and Exponential Stability of Controlled Euler--Bernoulli Beam Equations with Variable Coefficients

Riesz Basis Property and Exponential Stability of Controlled Euler--Bernoulli Beam Equations with Variable Coefficients

Nonselfadjoint operators generated by the equation of a nonhomogeneous damped string

Asymptotic and spectral analysis of non‐selfadjoint operators generated by a filament model with a critical value of a boundary parameter

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