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Original Articles

Geometrically constrained stabilization of wave equations with Wentzell boundary conditions

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Pages 1427-1452 | Received 10 Jun 2011, Accepted 30 Nov 2011, Published online: 17 Jan 2012

References

  • Gulliver , R , Lasiecka , I , Littman , W and Triggiani , R . 2004 . “ The case for differential geometry in the control of single and coupled PDEs: The Structural Acoustic Chamber ” . In in Geometric Methods in Inverse problems and PDE control , Vol. 137 , 73 – 181 . Berlin : Springer .
  • Littman , W and Markus , L . 1988 . Stabilization of hybrid system of elasticity by feedback boundary damping . Ann. Mat. Pura Appl. , 152 : 281 – 330 .
  • Heminna , A . 2000 . Stabilisation frontière de problèmes de Ventcel . ESAIM Control Optim. Calc. Var. , 5 : 591 – 622 .
  • Gal , C , Ruiz Goldstein , G and Goldstein , J . 2003 . Oscillatory boundary conditions for acoustic wave equations . J. Evol. Eqns. , 3 : 623 – 635 .
  • Barbu , V , Guo , Y , Rammaha , M and Toundykov , D . 2012 . Convex integrals on sobolev spaces . J. Convex Anal. , 19 ( 2 ) (to appear)
  • Barbu , V . 1993 . Analysis and Control of Nonlinear Infinite-dimensional Systems , Vol. 190 , Boston , MA : Academic Press Inc .
  • Coclite , GM , Goldstein , GR and Goldstein , JA . Stability estimates for nonlinear hyperbolic problems with nonlinear Wentzell boundary conditions Preprint
  • Cavalcanti , MM , Cavalcanti , VND , Fukuoka , R and Toundykov , D . 2009 . Stabilization of the damped wave equation with Cauchy–Ventcel boundary conditions . J. Evol. Eqns. , 9 ( 1 ) : 143 – 169 .
  • Cavalcanti , MM , Toundykov , D and Lasiecka , I . 2012 . Wave equation with damping affecting only a subset of static Wentzell boundary is uniformly stable . Trans. Amer. Math. Soc. , (to appear)
  • Lasiecka , I , Triggiani , R and Zhang , X . “ Nonconservative wave equations with unobserved Neumann B.C.: Global uniqueness and observability in one shot ” . In in Differential Geometric Methods in the Control of Partial Differential Equations (Boulder, CO, 1999); Contemporary Mathematics, AMS, Providence, Vol. 268, 2000, pp. 227–325
  • do Carmo , MP . 1992 . “ Riemannian geometry ” . In Mathematics: Theory & Applications , Boston , MA : Birkhäuser Boston Inc. . (Translated from the second Portuguese edition by Francis Flaherty)
  • Jost , J . 1995 . “ Riemannian Geometry and Geometric Analysis ” . In Universitext , Berlin : Springer-Verlag .
  • Lasiecka , I and Tataru , D . 1993 . Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping . Differ. Integral Eqns. , 6 ( 3 ) : 507 – 533 .
  • Lasiecka , I and Toundykov , D . 2006 . Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms . Nonlinear Anal. , 64 ( 8 ) : 1757 – 1797 .
  • Stroock , DW . 2000 . “ An Introduction to the Analysis of Paths on a Riemannian Manifold ” . In Mathematical Surveys and Monographs , Vol. 74 , Providence, RI : AMS .
  • Cavalcanti , MM , Domingos Cavalcanti , VN , Fukuoka , R and Toundykov , D . 2011 . Boundary and local stabilization of the wave equation on compact surfaces Preprint, 2011
  • Bucci , F and Toundykov , D . 2010 . Finite-dimensional attractors for systems of composite wave/plate equations with localized damping . Nonlinearity , 23 ( 9 ) : 2271 – 2306 .
  • Coclite , GM , Goldstein , GR and Goldstein , JA . 2008 . Stability estimates for parabolic problems with Wentzell boundary conditions . J. Differ. Eqns. , 245 ( 9 ) : 2595 – 2626 .
  • Coclite , GM , Goldstein , GR and Goldstein , JA . 2009 . Stability of parabolic problems with nonlinear Wentzell boundary conditions . J. Differ. Eqns. , 246 ( 6 ) : 2434 – 2447 .
  • Cavalcanti , MM , Khemmoudj , A and Medjden , M . 2007 . Uniform stabilization of the damped Cauchy–Ventcel problem with variable coefficients and dynamic boundary conditions . J. Math. Anal. Appl. , 328 ( 2 ) : 900 – 930 .
  • Daoulatli , M , Lasiecka , I and Toundykov , D . 2009 . Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions . Disc. Contin. Dyn. Syst. , 2 ( 1 ) : 67 – 94 .
  • Heminna , A . 2001 . Contrôlabilité exacte d'un problème avec conditions de Ventcel évolutives pour le système linéaire de l'élasticité . Rev. Mat. Comput. , 14 ( 1 ) : 231 – 270 .
  • Isakov , V and Yamamoto , M . 2000 . “ Carleman estimate with the Neumann boundary condition and its applications to the observability inequality and inverse hyperbolic problems ” . In in Differential Geometric Methods in the Control of Partial Differential Equations (Boulder, CO 1999), Vol. 268, pp. 191–225, Contemporary Mathematics , Providence , RI : AMS .
  • Khemmoudj , A and Medjden , M . 2004 . Exponential decay for the semilinear Cauchy–Ventcel problem with localized damping . Bol. Soc. Parana. Mat. , 22 ( 3 ) : 97 – 116 .
  • Kanoune , A and Mehidi , N . 2008 . Stabilization and control of subcritical semilinear wave equation in bounded domain with Cauchy–Ventcel boundary conditions . Appl. Math. Mech. (English Ed.) , 29 ( 6 ) : 787 – 800 .
  • Koch , H and Zuazua , E . 2006 . “ A hybrid system of PDE's arising in multi-structure interaction: Coupling of wave equations in n and n−1 space dimensions ” . In in Recent Trends in Partial Differential Equations , Vol. 409, Contemporary Mathematics , 55 – 77 . Providence , RI : AMS .
  • Komornik , V . 1994 . Exact Controllability and Stabilization Masson , , Paris
  • Lagnese , JE . 1989 . “ Boundary Stabilization of Thin Plates, ” . Vol. 10 , Philadelphia , PA : SIAM .
  • Lax , PD , Morawetz , CS and Phillips , RS . 1962 . The exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle . Bull. Amer. Math. Soc. , 68 : 593 – 595 .
  • Lagnese , J and Lions , JL . 1988 . Modeling Analysis and Control of Thin Plates Masson, Paris
  • Lemrabet , K . 1987 . Le problème de Ventcel pour le système de l'élasticité dans un domaine de R 3 . C.R. Acad. Sci. Paris Sér. I Math. , 304 ( 6 ) : 151 – 154 .
  • Lasiecka , I . 2002 . “ Mathematical Control Theory of Coupled PDEs, ” . Vol. 75 , Philadelphia , PA : SIAM .
  • Nicaise , S . 2003 . Stability and controllability of an abstract evolution equation of hyperbolic type and concrete applications . Rend. Mat. Appl. (7) , 23 ( 1 ) : 83 – 116 .
  • Quinn , JP and Russell , DL . 1977 . Asymptotic stability and energy decay rates for solutions of hyperbolic equations with boundary damping . Proc. Roy. Soc. Edinburgh Sect. A , 77 ( 1–2 ) : 97 – 127 .
  • Samarski , A and Andréev , V . 1976 . Razrostnye Metody Dlya Ellipticheskikh Uravnenij (Difference Methods for Elliptic Equations) Nauka, Moscow
  • Strauss , W . 1975 . Dispersal of waves vanishing on the boundary of an exterior domain . Comm. Pure Appl. Math. , 28 : 265 – 278 .
  • Tataru. , D . Private Communication
  • Toundykov , D . 2007 . Optimal decay rates for solutions of a nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponent source terms under mixed boundary conditions . Nonlinear Anal. , 67 ( 2 ) : 512 – 544 .
  • Vancostenoble , J and Martinez , P . 2000 . Optimality of energy estimates for the wave equation with nonlinear boundary velocity feedbacks . SIAM J. Control Optim. , 39 ( 3 ) : 776 – 797 . electronic
  • Warma , M . 2006 . The Robin and Wentzell–Robin Laplacians on Lipschitz domains . Semigroup Forum , 73 ( 1 ) : 10 – 30 .
  • Warma , M . 2010 . Analyticity on L 1 of the heat semigroup with Wentzell boundary conditions . Arch. Math. (Basel) , 94 ( 1 ) : 85 – 89 .
  • Ventcel' , AD . 1959 . On boundary conditions for multi-dimensional diffusion processes . Theor. Probability Appl. , 4 : 164 – 177 .

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