Strong stability of nonlinear semigroups with weak dissipation and non-compact resolvent–applications to structural acoustics

References

• J. Lagnese, Stabilization of Thin Plates, SIAM, Philadelphia, 1989
   Google Scholar
• Lasiecka , I . 2002 . Mathematical Control Theory of Coupled PDEs , CBMS–NSF Regional Conference Series Philadelphia : SIAM .
   Google Scholar
• Komornik , V . 1998 . Exact Controllability, Multipliers Method , Paris : Masson .
   Google Scholar
• Haraux , A . 1987 . “ Semilinear Hyperbolic Problems in Bounded Domains ” . In Mathematical Reports , Vol. 3 , New York : Harwood Gordon Breach .
   Google Scholar
• Chueshov , I and Lasiecka , I . 2008 . Long-time behavior of second order evolution equations with nonlinear damping . Mem. Amer. Math. Soc. , 917
   Google Scholar
• Chueshov , I , Eller , M and Lasiecka , I . 2002 . On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation . Commun. Partial Differ. Equ. , 27 : 1901 – 1951 .
   Web of Science ®Google Scholar
• Bucci , F , Chueshov , I and Lasiecka , I . 2007 . Global attractor for a composite system of nonlinear wave and plate equations . Commun. Pure Appl. Anal. , 6 : 113 – 140 .
   Google Scholar
• Triggiani , R . 1989 . Wave equation on a bounded domain with boundary dissipation: An operator approach . J. Math. Anal. Appl. , 137 : 438 – 461 .
   Web of Science ®Google Scholar
• Walker , JA . 1980 . Dynamical Systems and Evolution Equations , New York : Plenum Press .
   Google Scholar
• Nagy-Sz , B and Foias , C . 1970 . Harmonic Analysis of Operators on Hilber Space , Amsterdam : North-Holland .
   Google Scholar
• Pazy , A . 1983 . Semigroups of Linear Operators and Applications to Partial Differential Equations , New York, Berlin : Springer-Verlag .
   Google Scholar
• Slemorod , M . 1976 . Stabilization of boundary control systems . J. Differ. Equ. , 22 : 402 – 415 .
   Google Scholar
• Dafermos , C . 1968 . On the existence and asymptotic stability of solutions to the equations of linear thermoelasticity . Arch. Ration. Mech. Anal. , 29 : 241 – 271 .
   Web of Science ®Google Scholar
• Ball , JM . 1978 . On the asymptotic behavior of generalized processes, with applications to nonlinear evolution equations . J. Differ. Equ. , 27 : 224 – 265 .
   Web of Science ®Google Scholar
• Dafermos , C . 1976 . “ Contraction Semigroups and Trend to Equilibrium in Continuum Mechanics ” . In Applications of Methods of Functional Analysis to Problems in Mechanics, Lecture Notes in Mathematics , Vol. 503 , 295 – 306 . Berlin, New York : Springer-Verlag .
   Google Scholar
• Dafermos , C and Slemrod , M . 1973 . Asymptotic behavior of nonlinear contraction semigroups . J. Funct. Anal. , 13 : 97 – 106 .
   Google Scholar
• Arendt , W and Batty , CJK . 1988 . Tauberian theorems and stability of one-parameter semigroups . Trans. Amer. Math. Soc. , 306 ( 8 ) : 837 – 852 .
   Web of Science ®Google Scholar
• Lyubich , YI and Phong , VQ . 1988 . Asymptotic stability of linear differential equations in Banach spaces . Studia Math. , LXXXXVII : 37 – 42 .
   Google Scholar
• G.M. Sklyar, On the maximal asymptotica for linear equations in Banach spaces, 2009
   Google Scholar
• Avalos , G and Lasiecka , I . 1998 . The strong stability of a semigroup arising from a coupled hyperbolic/parabolic system . Semigroup Forum , 57 : 278 – 292 .
   Web of Science ®Google Scholar
• Avalos , G . 2007 . The strong stability and instability of a fluid-structure semigroup . Appl. Math. Optim. , 55 ( 2 ) : 163 – 184 .
   Web of Science ®Google Scholar
• Avalos , G . 2008 . Strong stability of semigroups via a generator resolvent criterion . Discrete and Contin. Dyn. Syst. Ser. S , 1 ( 2 ) : 207 – 218 .
   Web of Science ®Google Scholar
• H.T. Banks, W. Fang, R.J. Silcos and R.C. Smith, Approximation methods for control of acoustic/structure models with piezoceramic actuators, NASA Contract Rep. 189578
   Google Scholar
• Lasiecka , I and Tataru , D . 1993 . Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping . Differ. Integral Equ. , 6 ( 3 ) : 507 – 533 .
   Google Scholar
• Avalos , G . 2004 . The null controllability of thermoelastic plates and singularity of the associated minimal energy function . J. Math. Anal. Appl. , 294 : 34 – 61 .
   Web of Science ®Google Scholar
• Avalos , G and Lasiecka , I . 2003 . Optimal blow up rates for the minimal energy null control of the strongly damped abstract wave equation . Ann. Sc. Norm. Super. , II : 601 – 616 .
   Google Scholar
• Lasiecka , I and Triggiani , R . 2000 . Control Theory for Partial Differential Equations , Vol. I , Cambridge : Cambridge University Press .
   Google Scholar
• Grisvard , P . 1985 . Elliptic Problems in Nonsmooth Domains , Boston, MA : Pitman .
   Google Scholar
• Barbu , V . 1975 . Nonlinear Semigroups and Differential Equations in B-spaces , Leyden : Nordhoff .
   Google Scholar
• Lasiecka , I . 1989 . Stabilization of Wave and Plate-like Equations with Nonlinear Dissipation on the Boundary . J. Differ. Equ. , 79 ( 2 ) : 340 – 381 .
   Web of Science ®Google Scholar
• LaSalle , JP . 1976 . “ Stability theory and invariance principles ” . In Dynamical Systems , Edited by: Cerasir , L , Hale , JK and LaSalle , JP . Vol. 1 , 211 – 222 . New York : Academic Press .
   Google Scholar