Geometric properties of root vectors for equation of nonhomogeneous damped string: transformation operators method

References

• Agranovich , Z. S. and Marchenko , V. A. 1963 . “ The Inverse Problem in Scattering Theory ” . In Gordon and Breach New York, London
   Google Scholar
• Aktosun , T. , Klaus , M. and Van der Mee , C. 1998 . Wave scattering in one dimension with absorption . J. Math. Physics , 39 ( 4 ) : 1957 – 1992 .
   Google Scholar
• Aktosun , T. , Klaus , M. and Van der Mee , C. 1998 . Inverse scattering in one dimensional non— conservative media . Integral Eqs. Oper. Theory , 30 ( 3 ) : 279 – 316 .
   Google Scholar
• Belinskiy , B. P. , Dauer , J. P. , Martin , C. F. and Shubov , M. A. 1998 . On controllability of elastic string with viscous damping . Numer. Funct. Anal. Optim , 19 ( 3, 4 ) : 227 – 255 .
   Google Scholar
• Chadan , K. and Sabatier , P. C. 1977 . The Inverse Problem in Quantum Scattering Theory , New York : Springer—Verlag .
   Google Scholar
• Dunford , N. and Schwartz , J. T. 1971 . Linear Operators, Part III: Spectral Operators New York-London-Toronto
   Google Scholar
• Foias , C. and —Nagy , B. Sz. 1970 . Harmonic Analysis of Operators in Hilbert Space , Amsterdam : North Holland . Rev Ed
   Google Scholar
• Gohberg , I. Ts. and Krein , M. G. 1969 . “ Introduction to Linear Nonselfadjoint Operators ” . In Transl. of Math. Monogr , Vol. 18 , Providence, RI : AMS .
   Google Scholar
• Hruscev , S. V. , Nikol'skii , N. K. and Pavlov , B. S. 1981 . Unconditional bases of exponentials and reproducing kernels . Lecture Notes in Math , 864 : 214 – 335 .
   Google Scholar
• Jaulent , M. and Jean , C. 1976 . The inverse problem for the one—dimensional Schrödinger equation with an energy—dependent potential I and II . Ann. Inst Henri Poincare , A25 : 105 – 118 . ll9–l37
   Google Scholar
• Levitan , B. M. 1973 . Theory of Generalized Translation Operators , Moscow : “Nauka” .
   Google Scholar
• Marchenko , V. A. 1986 . Sturm—Liouville Operators and Applications , Basel—Boston—Stuttgart : Birkhäuser Verlag .
   Google Scholar
• Newton , R. G. 1982 . Scattering Theory of Waves and Particles , New York : Springer—Verlag .
   Google Scholar
• Nikol'skii , N. K. 1986 . Treatise on the Shift Operator , Berlin : Springer—Verlag .
   Google Scholar
• Russell , D. L. 1967 . Nonharmonic Fourier series in the control theory of distributed parameter system . J. Math. Anal. Appl , 18 : 542 – 559 .
   Web of Science ®Google Scholar
• Russell , D. L. 1978 . Controllability, stabilization theory for linear partial differential equations. Recent progress and open questions . SIAM J. Contr. Opt , 20 : 639 – 739 .
   Google Scholar
• Shubov Marianna A. Transformation operators for class of damped hyperbolic equations to appear in Asymptotic Analysis
   Google Scholar
• Shubov , Marianna A. 1999 . The Riesz basis property of the system of root vectors of non—homogenous damped string. Transformation operators method . Methods Anal. Appl , 6 ( 4 ) : 571 – 592 .
   Google Scholar
• Shubov , Marianna A. 1998 . Exact boundary and distributed controllability of radial damped wave equation . J. Math. Pures Appl , 77 ( 4 ) : 415 – 437 .
   Google Scholar
• Shubov , Marianna A. 1998 . Spectral decomposition method for the damped string. Reduction of control time . Applicable Analysis , 68 ( 3–4 ) : 241 – 259 .
   Google Scholar
• Shubov , Marianna A. , Martin , C. F. , Dauer , J. P. and Belinskiy , B. P. 1997 . Unique controllability of damped wave equation . SIAM J. Cont. Optim , 35 ( 5 ) : 1773 – 1789 .
   Web of Science ®Google Scholar
• Shubov , Marianna A. and Martin , C. F. 1999 . Controllability of nonhomogeneous damped string. Reduction of control time . J. Math. Anal. Appl , 240 ( 5 ) : 16 – 36 .
   Google Scholar
• Shubov , Marianna A. 1997 . Nonselfadjoint operators generated by equation of nonhomogeneous damped string . Trans. of Amer. Math. Soc , 349 ( 11 ) : 4481 – 4499 .
   Google Scholar
• Shubov , Marianna A. 1996 . Basis property of eigenfunctions of nonselfadjoint operator pencils generated by the equation of nonhomogeneous damped string . Int. Eg. and Oper. Theory , 25 ( 11 ) : 289 – 328 .
   Google Scholar
• Shubov , Marianna A. 1997 . Spectral operators generated by damped hyperbolic equations . Integr. Eq. and Oper. Theory , 28 ( 11 ) : 358 – 372 .
   Google Scholar
• Shubov , Marianna A. 1996 . Asymptotics of resonances and eigenvalues for nonhomogeneous damped string . Asymp. Anal , 13 ( 11 ) : 31 – 78 .
   Google Scholar
• Shubov , Marianna. A. 1998 . Asymptotics of spectrum and eigenfunctions for nonselfad joint operators generated by radial nonhomogeneous damped wave equations . Asymp. Anal , 16 ( 11 ) : 245 – 272 .
   Google Scholar
• Shubov , Marianna A. 1998 . “ Spectral operators generated by 3–dimensional damped wave equation and their application to control theory ” . In Spectral and Scattering theory , Edited by: Ramm , A. G. 177 – 188 . New York : Plenum Press .
   Google Scholar
• Shubov , Marianna A. 1995 . Asyrnptotics of resonances and geometry of resonance states in the problem of scattering of acoustical waves by a spherically symmetric inhomogeneity of the density . Dif. Int. Eg , 8 ( 5 ) : 1073 – 1115 .
   Google Scholar
• Shubov , Marianna A. 1990 . Certain class of unconditional bases in Hilbert space and its applications to functional model and scattering theory . Integ. Eq. and Oper. Theory , 13 ( 5 ) : 750 – 770 .
   Google Scholar
• Young , R. M. 1980 . An Introduction to Nonharmonic Fourier Series , New York : Academic Press .
   Google Scholar