Abstract
We formulate our results on the spectral analysis for a class of nonselfadjoint operators in a Hilbert space and on the applications of this analysis to the control theory of linear distributed parameter systems. The operators, we consider, are the dynamics generators for systems governed by 3-dimensional wave equation which has spacially nonhomogeneous spherically symmetric coefficients and contains a first order damping term. We consider this equation with a one-parameter family of linear first order boundary conditions on a sphere. These conditions contain a damping term as well. Our main object of interest is the class of operators in the energy space of 2-component initial data which generate the dynamics of the above systems. Our first main result is the fact that these operators are spectral in the sense of N. Dunford. This result is obtained as a corollary of two groups of results: (i) asymptotic representations for the complex eigenvalues and eigenfunctions, and (ii) the fact that the systems of eigenvectors and associated vectors form Riesz bases in the energy space. We also present an explicit solution of the controllability problem for the distributed parameter systems governed by the aforementioned equation using the spectral decomposition method.
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References
Agranovich, M. S., Convergence of series in root vectors of operators that are nearly selfadjoint. (Russian)-Trudy Moskov. Mat. Obshch., 41, pp. 163–180, (1980).
Agranovich, M. S., Elliptic operators on closed manifolds. (Russian)-Current Problems in Mathematics, Fundomental directions, Vol. 63, pp. 5-169.
Aubin, T., Non-linear Analysis on Manifolds, Monge-Ampere Equations, Springer, 1982.
Bardos, C., Lebeau, G., and Rauch, J., Controle et Stabilisation dans les Problemes Hyperboliques, Appendix II in J.-L. Lions, controllabilite exacte et Stabilisation de Systems Distributes, Vols. 1,2, Masson, 1990.
Bardos, G., Lebeau, G., and Rauch, J., Sharp sufficient conditions for the observations, control and stabilization of waves from the boundary, SIAMI Cont. Opt., 30 (1992), 1024–1065.
Cox, S. and Zuazua, E., The rate at which the energy decays in the string damped at one end, Indiana Univ. Math. J., 44 (1995), 545–573.
Dunford, N. and Schwartz, J. T., Linear Operators, Part III, Spectral Operators, Wiley Interscience, 1971.
Gallot, S., Hulin, D., and Lafontaine, J., Riemannian Geometry, Springer-Verlag, 1987.
Gohberg, I. Ts. and Krein, M. G., Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space, AMS Translations, Vol. 18, (1969).
Hruščev, S. V., The Regge problem for strings, unconditionally convergent eigenfunction expansions, and unconditional basis of exponentials in L 2 (−T, T), J. Oper. Theory 14 (1985), 57–85.
Lions, J.-L., Exact controllability, stabilizability and perturbations for distributed systems, SIAM Rev. 30 (1988), 1–68.
Marcus, A. S., Introduction to the Spectral Theory of Polynomial Pencils, Translation of Mathematical Monographs, Vol. 71, AMS, Providence, RI, (1988).
Pekker, M. A. (M. A. Shubov), Resonances in the scattering of acoustic waves by a spherical inhomogeneity of the density, Soviet Math. Dokl., 15 (1974), 1131–1134.
Pekker, M. A. (M. A. Shubov), Resonances in the scattering of acoustic waves by a spherical inhomogeneity of the density, Amer. Math. Soc. Transi. (2), 115 (1980), 143–163.
Ramm, A. S., Spectral properties of some nonselfadjoint operators and some applications, Spectral Theory of Differential Operators (Birmingham, Ala., 1981), pp. 349-354, North Holland Math. Stud., 55.
Russell, D. L., Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Review, Vol. 20, No. 4 (1978), 639–839.
Russell, D. L., Nonharmonic Fourier series in the control theory of distributed parameter systems, Journal of Mathematical Analysis and Applications, 18, (1967), 542–560.
Shubova, M. A., Construction of resonances states for the three-dimensional Schrödinger operator, Problems of Mathematical Physics 9, Leningrad University, (1979) 145-180 (Russian), Editor and Reviewer: M. S. Birman.
Shubova, M. A., The resonance selection principle for the three-dimensional Schrödinger operator, Proc. Steklov Inst. Math., 159, (1984), 181–195.
Shubova, M. A., The serial structure of resonances of the three-dimensional Schrödinger operator, Proc. Steklov Inst. Math., 159, (1984), 197–217.
Shubov, Marianna A., Asymptotics of spectrum and eigenfunctions for nonselfadjoint operators generated by radial nonhomogeneous damped wave equation, to appear in Asymptotic Analysis.
Shubov, Marianna A., Riesz basis property of root vectors of nonselfadjoint operators generated by radial damped wave equations. Exact controllability results. Preprint TTU, (1997).
Shubov, Marianna A., Riesz basis property of root vectors of dynamics generator for 3-dimensional damped wave equation. Preprint TTU, (1997).
Shubov, Marianna A., Basis Property of Eigenfunctions of Nonselfadjoint Operator Pencils Generated by the Equation of Nonhomogeneous Damped String, Integral Equations and Operator Theory, 25, (1996), pp. 289–328.
Shubov, Marianna A., Asymptotics of Resonances and Eigenvalues for Nonhomogeneous Damped String, Asymptotic Analysis, 13, (1996), pp. 31–78.
Shubov, Marianna A., Nonselfadjoint Operators Generated by the Equation of Nonhomogeneous Damped String, to appear in Transactions of American Math. Society.
Shubov, Marianna A., Transformation operators for class of damped hyperbolic equations, Preprint TTU, (1997).
Shubov, Marianna A., Spectral operators generated by damped hyperbolic equations, Int. Eqs. and Oper. Theory, 28, (1997), pp. 358–372.
Shubov, Marianna A., Asymptotics of resonances and geometry of resonance states in the problem of scattering of acoustic waves by a spherically symmetric inhomogeneity of the density, Differ, and Integ. Eq. V.8, No. 5 (1995), 1073–1115.
Shubov, Marianna A., Martin, C. F. Dauer, J. P., and Belinskiy, B. P., Exact controllability of damped wave equation, to appear in SIAM J. on Control and Optimization.
Shubov, Marianna A., Exact boundary and distributed controllability of nonhomogeneous damped string. Preprint TTU, (1996).
Shubov, Marianna A., Spectral decomposition method for controlled damped string. Reduction of control time. To appear in Applicable Analysis.
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Shubov, M.A. (1998). Spectral Operators Generated by 3-Dimensional Damped Wave Equation and Applications to Control Theory. In: Ramm, A.G. (eds) Spectral and Scattering Theory. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-1552-8_11
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DOI: https://doi.org/10.1007/978-1-4899-1552-8_11
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