Exact Controllability of the Damped Wave Equation

Shubov, Marianna A.; Martin, Clyde F.; Dauer, Jerald P.; Belinskiy, Boris P.

doi:10.1137/s0363012996291616

Cited by 35 publications

(17 citation statements)

References 11 publications

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“…To conclude this Introduction, we mention that the above Riesz basis property result allows us to solve the so-called controllability problem for the damped string equation [21,24]. In the late sixties D. Russell [15]- [17] suggested the spectral decomposition method for the solution of this problem in the case of an undamped string: d(x) = 0.…”

Section: 2mentioning

confidence: 97%

“…The present paper, while being a closed piece of work, is, in fact, one in a series of papers by the author [18]- [24]. These works, together with two forthcoming papers, are devoted to the spectral analysis of the above described operators A h (and the generalizations of these operators related to 3-dimensional spherically symmetric nonhomogeneous damped wave equation) and to applications of the spectral results to the control theory of distributed parameter systems.…”

Section: 2mentioning

confidence: 99%

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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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Section: 2mentioning

confidence: 97%

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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“…For the completeness of the eigenvectors of nonsymmetric system and closely related eigenvalue problems for pencils of ordinary differential operations, we can further refer to Cox and Zuazua [3], Rykhlov [12], Shkalikov [13], Shubov [14], Shubov, Martin, Dauer and Belinskiy [15], Vagabov [18].…”

Section: D(a) = {U ∈ {Hmentioning

confidence: 99%

Spectral properties and an inverse eigenvalue problem for non-symmetric systems of ordinary differential operators

Trooshin

Yamamoto

2002

Journal of Inverse and Ill-posed Problems

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We consider a nonsymmetric first-order differential operator (Au)(x) = 0 1 1 0 (du/dx)(x) + P (x)u(x), 0 < x < 1, where P is a 2 × 2 matrix whose components are in L 2 (0, 1). We study an eigenvalue problem for A with boundary conditions at x = 0, 1. We establish an asymptotic form of the eigenvalues and prove that the set of the root vectors forms a Riesz basis in {L 2 (0, 1)} 2 . Moreover we show some characterization of L 2 -coefficients in an inverse eigenvalue problem. The key is a transformation formula.

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“…First, they are useful in stabilization problems, because they imply that the rate of the energy decay in the considered systems is equal to the spectral abscissa of the corresponding dynamical semigroup. Secondly, they allow us to solve the controllability problem for systems governed by the aformentioned equations using the spectral decomposition method [24] in the way similar to the original one suggested by D. Russell [25,26] for equations without damping terms. The spectral method, when it is applicable, can provide a useful information in addition to the general controllability results [27][28][29] due to C: Bardos, G.…”

Section: Definition 1 A)mentioning

confidence: 99%

“…The following statement is important for the solution of the moment problem which appears as a step in the spectral decomposition solution of the control problem [24,30]. Eq.…”

Section: $Hubovmentioning

confidence: 99%

Spectral operators generated by damped hyperbolic equations

Shubov

1997

Integr equ oper theory

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We announce a series of results on the spectral analysis for a class of nonselfadjoint opeators~ which are the dynamics generators for the systems governed by hyperbolic equations containing dissipative terms. Two such equations are considered: the equation of nonhomogeneous damped string and the 3-dimensional damped wave equation with spacially nonhomogeneous spherically symmetric coefficients. Nonselfadjoint boundary conditions are imposed at the ends of a finite interval or on a sphere centered at the origin respectively. Our main result is the fact the aforementioned operators are spectral in the sense of N. Dunford. The result follows from the fact that the systems of root vectors of the above operators form Riesz bases in the corresponding energy spaces. We also give asymptotics of the spectra and state the Riesz basis property results for the nonselfadjoint operator pencils associated with these operators.

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Exact Controllability of the Damped Wave Equation

Cited by 35 publications

References 11 publications

Nonselfadjoint operators generated by the equation of a nonhomogeneous damped string

Nonselfadjoint operators generated by the equation of a nonhomogeneous damped string

Spectral properties and an inverse eigenvalue problem for non-symmetric systems of ordinary differential operators

Spectral operators generated by damped hyperbolic equations

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