Stabilization of a fluid structure interaction with nonlinear damping

Lasiecka, I.; Lu, Y.

Artykuł - szczegóły

Tytuł artykułu

Stabilization of a fluid structure interaction with nonlinear damping

Autorzy

Lasiecka I. , Lu Y.

Treść / Zawartość

Pełne teksty:

Pobierz

Identyfikatory

Warianty tytułu

Języki publikacji

Abstrakty

Asymptotic stability of finite energy solutions to a fluid-structure interaction with a static interface in a bounded domain Ω ∈ Rⁿ, n = 2 is considered. It is shown that the undamped model subject to ”partial flatness” geometric condition on the in- terface produces solutions whose energy converges strongly to zero; while with a stress type feedback control applied on the interface of the structure, the model produces solutions whose energy is exponentially stable. An addition of a static damping on the interface produces solutions whose full norm in the phase space is exponentially stable. Without a static damping an interesting phenomenon occurs that steady state solutions (equilibria) might generate genuinely growing in time solutions. This is purely nonlinear phenomenon captured by newly developed techniques amenable to handle instability of steady state solutions arising from nonlinearity.

Słowa kluczowe

fluid structure interaction interface Navier-Stokes equation system of elasticity feedback boundary control strong stability uniform stability optimal control passive damping active damping dynamic and static damping

Wydawca

Systems Research Institute, Polish Academy of Sciences

Czasopismo

Control and Cybernetics

Rocznik

2013

Tom

Vol. 42, no. 1

Strony

155--181

Opis fizyczny

Bibliogr. 36 poz.

Twórcy

autor

Lasiecka I.

il2v@virginia.edu

Department of Mathematics, University of Virginia Charlottesville, VA 22904, USA
Systems Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw, Poland

autor

Lu Y.

ylu@vsu.edu

Department of Mathematics and Computer Science, Virginia State University Petersburg, VA 23806, USA

Bibliografia

1. Arend, W. and Batty, C.J.K. (1988) Tauberian theorems and stability of one-parameter semigroups. Tran. Amer. Math. Soc. 306(8), 837-852.
2. Avalos, G. and Triggiani, R. (2007) The coupled PDE system arising in fluid/structure interaction. Part I: explicit semigroup generator and its spectral properties. Cont. Math. 440, 15-54.
3. Avalos, G. and Triggiani, R. (2008) Uniform Stabilization of A coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface. Dis. & Cont. Dyn. Sys. 22 (4), 817-833.
4. Avalos, G. and Triggiani, R. (2009a) Boundary feedback stabilization of a coupled parabolic-hyperbolic Stoke-Lam´e PDE system. J. Evol. Eqns. 9, 341-370.
5. Avalos, G. and Triggiani, R. (2009b) Coupled parabolic-hyperbolic Stoke-Lam´e PDE system: limit behavior of the resolvent operator on the imaginary axis. Applicable Analysis, 88(9), 1357-1396.
6. Ball, J.M. (1977) Stongly continuous semigroups, weak solutions, and the variation of constants formula. Proc. Am. Math. Soc. 63, 370-373.
7. Ball, J.M. (1978) On the asymptotic behavior of generalized processes, with application to nonlinear evolution equations.J. Diff. Equ. 27, 224-265.
8. Ball, J. M. and Slemrod, M. (1979) Feedback stabilization of distributed semilinear control systems, Appl. Math. Optim. 5, 169-179.
9. Barbu, V., Grujic, Z., Lasiecka, I. and Tuffaha, A. (2007) Existence of the Energy-Level Weak Solutions for a Nonlinear Fluid-Structure Interaction Model. Cont. Math. 440, 55-82.
10. Barbu, V., Grujic, Z., Lasiecka, I. and Tuffaha, A. (2008) Smoothness of Weak Solutions to a Nonlinear Fluid-structure Interaction Model. Indiana University Mathematics Journal 57(2), 1173-1207.
11. Bociu, L. and Zolesio, J.P. (2010) Linearization of a coupled system of nonlinear elasticity and viscuous fluid. Modern Theory of PDE’s. ISAACS Proceedings.
12. Brezis, H. (1978) Asymptotic behavior of some evolutionary systems. Nonlinear Evol. Equ., 141-154 (Academic Press).
13. Caputo, R. and Hammer, D. (2002) Effects of microvillus deformability on leukocyte adhesion explored using adhesive dynamics simulations. Biophysics 92, 2183-2192.
14. Chueshov, I. and Lasiecka, I. (2008) Long tome behavior of second order evolutions with nonlinear damping. Memoires of American Mathematical Society 195(912).
15. Chueshov, I. and Lasiecka, I. (2010) Von Karman Evolutions. Springer Verlag.
16. Coutand, D. and Shokller, S. (2005) Motion of an elastic solid inside an incompressible viscous fluid. Arch. Ration. Mech. Anal. 176, 25-102.
17. Du, Q., Gunzburger, M.D., Hou, L.S. and Lee, J. (2003) Analysis of a linear fluid-structure interaction problem. Dis. Con. Dyn. Sys. 9, 633-650.
18. Fernandez, M.A. and Moubachir, M. (2003) An exact Block-Newton algorithm for solving fluid-structure interaction problems. C.R. Acad. Sci Paris, Ser. I 336, 681-686.
19. Haraux, A. (2006) Decay rate of the range component of solutions to some semilinear evolution equations. NoDea 13, 435-445.
20. Khismatullin, D. and Truskey, G. (2005) hree dimensional numerical simulation of receptor-medicated leukocyte adhesion to surfaces. Effects of cell deformability and viscoelasticity. Physics of Fluids 17, 031505.
21. Komornik, V. (1998) Exact Controllability by Multipliers Method, Masson.
22. Kukavica, I., Tuffaha, A. and Ziane, M. (2009) Strong solutions to nonlinear fluid structure interactions, J.Diff. Equ.. 247, 1452-1478, and Advances in Differential Equations 15(3-4), 231-254, 2010.
23. Kukavica, I., Tuffaha, A. and Ziane, M. (2011) Strong solutions to a Navier-Stokes-Lamsystem on a domain with a non-flat boundary. Nonlinearity 24(1), 159-176,
24. Lagnese, J. (1983) Decay of solutions of wave equations in a bounded region with boundary dissipation. J. Diff. Eqns. 50, 163-182.
25. LaSalle, J.P. (1976) Stability theory and invariance principles,Dynamical Systems Vol. 1, L. Cerasir, J.K. Hale, and J.P. LaSalle, eds.,Academic Press, New York, 1976, pp. 211-222.
26. Lasiecka, I. (2002) Control Theory of Coupled PDE’s, CBMS-SIAM Lecture Notes, SIAM.
27. Lasiecka, I., Lions, J.L. and Triggiani, R. (1986) Nonhomogeneous boundary value problems for second order hyperbolic operators. J. Math. Pures Appl., 65, 149-152.
28. Lasiecka, I. and Lu, Y. (2011) Asymptotic stability of finite energy in Navier Stokes-elastic wave interaction. Semigroup Forum 82, 61-82.
29. Lasiecka, I. and Lu, Y. (2012) Interface feedback control stabilization of a nonlinear fluid-structure interaction. Nonlinear Analysis 75, 1449-1460.
30. Lasiecka, I. and Seidman, T. (2003) Strong stability of elastic control systems with dissipative saturating feedback. Systems and Control Letters 48, 243-252.
31. Lasiecka, I. and Tataru, D., (1993) Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Diff. & Inte. Eqns. 6(3), 507-533.
32. Lions, J.L. (1988) Controllability Exact and Stabilisation de Systèmes Distribués, Masson, Paris.
33. Moubachir, M. and Zolesio, J. (2006) Moving Shape Analysis and Control: Applications to Fluid Structure Interactions. Chapman & Hall/CRC, 2006.
34. Slemrod, M. (1989) Weak asymptotic decay via a ”relaxed invariance principle” for a wave equation with nonlinear, non-monotone damping, Proceedings of the Royal Society of Edinburgh 113A, 87-97.
35. Temam, R. 1977) Navier-Stokes Equations., Studies in Math. and its Applications, North Holland, Amsterdam.
36. Walker, J.A. (1980) Dynamical Systems and Evolution Equations. Plenum Press, New York.

Typ dokumentu

Bibliografia

Identyfikator YADDA

bwmeta1.element.baztech-e3501ce1-c441-4c68-8799-f24d62a6e9b8