Exact controllability of the damped wave equation

SIAM Journal on Control and Optimization

Volumn 35, Issue 5, 1997, Pages 1773-1789

  Shubov, Marianna A ^a   Martin, Clyde F ^a   Dauer, Jerald P ^b   Belinskiy, Boris P ^b  

^a TEXAS TECH UNIVERSITY   (United States)

^b UNIVERSITY OF TENNESSEE AT CHATTANOOGA   (United States)

Author keywords

Damping; Distributed parameter control; Eigenvectors and root vectors; Exact and approximate controllability; Hyperbolic equations; Nonharmonic exponential basis; Nonselfadjoint operators; Riesz basis; Stability

Indexed keywords

APPROXIMATION THEORY; BOUNDARY CONDITIONS; CONTROLLABILITY; DAMPING; EIGENVALUES AND EIGENFUNCTIONS; FUNCTIONS; MATHEMATICAL MODELS; MATHEMATICAL OPERATORS; SYSTEM STABILITY;

DAMPED WAVE EQUATION;

DISTRIBUTED PARAMETER CONTROL SYSTEMS;

EID: 0031233798     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012996291616     Document Type: Article

References (31)

1. D.L. RUSSELL, On boundary-value controllability of linear symmetric hyperbolic system, in Mathematical Theory of Control, A.V. Balakrishnan and L.W. Neustadt, eds., Academic Press, New York, (1967), pp. 312-321.
2. D.L. RUSSELL, Nonharmonic Fourier series in the control theory of distributed parameter systems, J. Math. Anal. Appl., 18 (1967), pp. 542-559.
3. D.L. RUSSELL, Boundary value control of the higher dimensional wave equation, SIAM J. Control, 9 (1971), pp. 29-42, pp. 401-419.
4. D.L. RUSSELL, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations, Stud. Appl. Math., LII (1973), pp. 189-211.
5. K. GRAHAM AND D.L. RUSSELL, Boundary value control of the wave equation in a spherical region, SIAM J. Control, 13 (1975), pp. 174-196.
6. M.A. SHUBOV, Asymptotics of resonances and geometry of resonance states in the problem of scattering of acoustical waves by a spherically symmetric inhomogeneity of the density, Differential Integral Equations, 8 (1995), pp. 1073-1115.
7. M.A. SHUBOV, Asymptotics of resonances and eigenvalues for nonhomogeneous damped string, Asymptotic Anal., 13 (1996), pp. 31-78.
8. M.A. SHUBOV, Basis property of eigenfunctions of nonselfadjoint operator pencils generated by the equation of nonhomogeneous damped string, Integral Equations Operator Theory, 25 (1996), pp. 289-328.
9. M.A. SHUBOV, Transformation Operators and Basis Property of Root Vectors for Damped Wave Equation, Texas Tech University, Lubbock, TX, 1996, preprint.
10. J.-L. LIONS, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York, 1971.
11. R. DAUTRAY AND J.-L. LIONS, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 5, Springer-Verlag, New York, 1992.
12. P.D. LAX AND R.S. PHILLIPS, Scattering Theory, rev. ed., Academic Press, New York, 1989.
13. A.S. MARCUS, Introduction to the Spectral Theory of Polynomial Pencils, Transl. Math. Monographs, 71, AMS, Providence, RI, 1988.
14. N.K. NIKOL'SKII AND B.S. PAVLOV, Eigenvector bases of completely nonunitary contractions and the characteristic function, Math. USSR-Izv., 4 (1970), pp. 90-133.
15. N.K. NIKOL'SKII, Treatise on the Shift Operator, Springer-Verlag, Berlin, 1986.
16. S.A. IVANOV AND B.S. PAVLOV, Carleson series of resonances and the Regge problem, Math. USSR-Izv., 12(1), (1978), pp. 26-55.
17. B. YA. LEVIN, Distribution of Zeroes on Entire Functions, Transl. Math. Monographs 5, AMS, Providence, RI, 1964.
18. A.M. SEDLETSKII, On convergence of nonharmonic Fourier series in systems of exponentials, cosines and sines, Soviet Math. Dokl., 38 (1989), pp. 179-183.
19. I. TS. GOHBERG AND M.G. KREIN, Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monographs 18, AMS, Providence, RI, 1969.
20. R.M. YOUNG, An Introduction to Nonharmonic Fourier Series, Academic Press, New York, 1980.
21. A. PAZY, Semigroup of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sciences 44, Springer-Verlag, New York, 1983.
22. J.-L. LIONS AND E. MAGENES, Problèmes aux Limites non Homogènes et Applications, Vol. I, II, Dunod, Paris, 1968, Vol. III, 1970.
23. S. COX AND E. ZUAZUA, Estimations sur le taux de decroissance exponentielle de l'energie dans des équations des ondes dissipatives linéaires, C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), pp. 249-254.
24. S. COX AND E. ZUAZUA, The rate at which the energy decays in a damped string, Comm. Partial Differential Equations, 19 (1994), pp. 213-243.
25. S. COX AND E. ZUAZUA, The rate at which the energy decays in the string damped at one end, Indiana Univ. Math. J., 44 (1995), pp. 545-573.
26. M.A. PEKKER (M.A. SHUBOV), Resonances in the scattering of acoustic waves by a spherical inhomogeneity of the density, Soviet Math. Dokl., 15 (1974), pp. 1131-1134.
27. M.A. PEKKER (M.A. SHUBOV), Resonances in the scattering of acoustic waves by a spherical inhomogeneity of the density, Amer. Math. Soc. Transl. (2), 115 (1980), pp. 143-163.
28. S.V. HRUŠČEV, N.K. NIKOĹSKII, AND B.S. PAVLOV, Unconditional Bases of Exponentials and of Reproducing Kernels, Lecture Notes in Math. 864, Springer-Verlag, New York, 1981, pp. 214-335.
29. J.-L. LIONS, Contrôlabilité exacte, perturbations et stabilisation de systemes distribues, T.1: Contrôlabilité exacte, Rech. Math. Appl. 8, Masson, Paris, 1988; T.2, Perturbations, Rech. Math. Appl. 9, Masson, Paris, 1988.
30. J.-L. LIONS, Contrôlabilité exacte, perturbations et stabilisation de systemes distribues, T.1: Contrôlabilité exacte, Rech. Math. Appl. 8, Masson, Paris, 1988; T.2, Perturbations, Rech. Math. Appl. 9, Masson, Paris, 1988.
31. C. BARDOS, G. LEBEAU, AND J. RAUCH, Contrôle et stabilisation dans les problèmes hyperboliques, Appendix II, in J.-L. Lions, Contrôlabilité exacte et stabilisation de systèmes distribués, Vols. 1,2, Masson, 1990.