# American Institute of Mathematical Sciences

June  2018, 38(6): 3055-3083. doi: 10.3934/dcds.2018133

## Interface stabilization of a parabolic-hyperbolic pde system with delay in the interaction

 1 Department of Mathematics and Computer Science, University of the Philippines Baguio, Governor Pack Road, 2600 Baguio, Philippines 2 Institute for Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria 3 RICAM Institute, Austrian Academy of Sciences, Altenbergerstrasse 69, 4040 Linz, Austria

* Corresponding author

Received  October 2017 Revised  January 2018 Published  April 2018

Fund Project: This work was supported in part by the ERC advanced grant 668998 (OCLOC) under the EUs H2020 research program

A coupled parabolic-hyperbolic system of partial differential equations modeling the interaction of a structure submerged in a fluid is studied. The system being considered incorporates delays in the interaction on the interface between the fluid and the solid. We study the stability properties of the interaction model under suitable assumptions between the competing strengths of the delays and the feedback controls.

Citation: Gilbert Peralta, Karl Kunisch. Interface stabilization of a parabolic-hyperbolic pde system with delay in the interaction. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3055-3083. doi: 10.3934/dcds.2018133
##### References:
 [1] W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 360 (1988), 837-852. doi: 10.1090/S0002-9947-1988-0933321-3. Google Scholar [2] G. Avalos and F. Bucci, Rational rates of uniform decay for strong solutions to a fluid-structure PDE system, J. Differential Equations, 258 (2015), 4398-4423. doi: 10.1016/j.jde.2015.01.037. Google Scholar [3] G. Avalos and R. Triggiani, The coupled PDE system arising in fluid-structure interaction, Part Ⅰ: Explicit semigroup generator and its spectral properties, Contemporary Mathematics, 440 (2007), 15-54. Google Scholar [4] G. Avalos and R. Triggiani, Semigroup wellposedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction, Discr. Cont. Dynam. Sys., 2 (2009), 417-447. doi: 10.3934/dcdss.2009.2.417. Google Scholar [5] G. Avalos and R. Triggiani, Rational decay rates for a PDE heat-structure interaction: A frequency domain approach, Evol. Equ. Control Theory, 2 (2013), 233-253. doi: 10.3934/eect.2013.2.233. Google Scholar [6] G. Avalos and R. Triggiani, Fluid structure interaction with and without internal dissipation of the structure: A contrast study in stability, Evol. Equ. Control Theory, 2 (2013), 563-598. doi: 10.3934/eect.2013.2.563. Google Scholar [7] G. Avalos, I. Lasiecka and R. Trigianni, Higher regularity of a coupled parabolic-hyperbolic fluid structure interactive system, Georgian Math. J., 15 (2008), 403-437. Google Scholar [8] G. Avalos, I. Lasiecka and R. Trigianni, Heat-wave interaction in 2-3 dimensions: Optimal rational decay rate, Journal of Mathematical Analysis and its Applications, 437 (2016), 782-815. doi: 10.1016/j.jmaa.2015.12.051. Google Scholar [9] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, Contemporary Mathematics, 440 (2007), 55-82. Google Scholar [10] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Smoothness of weak solutions to a nonlinear fluid-structure interaction model, Indiana U. Math. J., 57 (2008), 1173-1207. doi: 10.1512/iumj.2008.57.3284. Google Scholar [11] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Mathematische Annalen, 347 (2010), 455-478. doi: 10.1007/s00208-009-0439-0. Google Scholar [12] R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedback, SIAM J. Control Optim., 26 (1988), 697-713. doi: 10.1137/0326040. Google Scholar [13] R. Datko, J. Lagnese and P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156. doi: 10.1137/0324007. Google Scholar [14] W. Desch, E. Fašangová, J. Milota and G. Propst, Stabilization through viscoelastic boundary damping: A semigroup approach, Semigroup Forum, 80 (2010), 405-415. doi: 10.1007/s00233-009-9197-2. Google Scholar [15] Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem, Discr. and Contin. Dynam. Systems, 9 (2003), 633-650. doi: 10.3934/dcds.2003.9.633. Google Scholar [16] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, 2nd ed., Springer, Berlin, 2000. Google Scholar [17] L. C. Evans, Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1993.Google Scholar [18] M. Kirane and B. Said-Houari, Existence and asymptotic stability of a viscoelastic wave equation with delay, Z. Angew. Math. Phys., 62 (2011), 1065-1082. doi: 10.1007/s00033-011-0145-0. Google Scholar [19] Lasiecka and Y. Lu, Asymptotic stability of finite energy in Navier Stokes-elastic wave interaction, Semigroup Forum, 82 (2011), 61-82. doi: 10.1007/s00233-010-9281-7. Google Scholar [20] I. Lasiecka and Y. Lu, Interface feedback control stabilization of a nonlinear fluid-structure interaction, Nonlinear Analysis, 75 (2012), 1449-1460. doi: 10.1016/j.na.2011.04.018. Google Scholar [21] I. Lasiecka and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pure Appl., 65 (1986), 149-192. Google Scholar [22] I. Lasiecka and R. Triggiani, Exponential uniform energy decay rates of the wave equation in a bounded region with $L^2(0,T;L^2(Γ))$-boundary feedback in the Dirichlet B.C., J. Diff. Eqns., 66 (1987), 340-390. doi: 10.1016/0022-0396(87)90025-8. Google Scholar [23] I. Lasiecka and R. Triggiani, Exact controllability for the wave equation with Neumann boundary control, Appl. Math. Optimiz., 19 (1989), 243-290. doi: 10.1007/BF01448201. Google Scholar [24] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories; Vol. II: Abstract Hyperbolic Equations, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2000. Google Scholar [25] J. L. Lions, Contrôlabilité Exacte, Stabilisation et Perturbations des Systémes Distribués, Vol. 1, Masson, Paris, 1988. Google Scholar [26] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, New York, (1972) Google Scholar [27] J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. 2, Springer, Heidelberg, 1972. Google Scholar [28] L. Lu, Numerical stability of θ-methods for systems of differential equations with several delay terms, J. Comput. Apple. Math., 34 (1991), 291-304. doi: 10.1016/0377-0427(91)90090-7. Google Scholar [29] Y. Lu, Stabilization of a fluid structure interaction with nonlinear damping, Control and Cybernetics, 42 (2013), 155-181. Google Scholar [30] Y. Lu, Uniform decay rates for the energy in nonlinear fluid structure interaction with monotone viscous damping, Palest. J. Math., 2 (2013), 215-232. Google Scholar [31] Y. I. Lyubich and V. Q. Phong, Asymptotic stability of linear differential equations in Banach spaces, Studia Matematica, 88 (1988), 37-42. doi: 10.4064/sm-88-1-37-42. Google Scholar [32] S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585. doi: 10.1137/060648891. Google Scholar [33] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. Google Scholar [34] G. Peralta, A fluid-structure interaction model with damping and delay in the structure, Z. Angew. Math. Phys., 67 (2016), Art. 10, 20 pp. Google Scholar [35] G. Peralta and G. Propst, Stability and boundary controllability of a linearized model of flow in an elastic tube, ESAIM: Control, Optimisation and Calculus of Variations, 21 (2015), 583-601. doi: 10.1051/cocv/2014039. Google Scholar [36] G. Peralta and G. Propst, Well-posedness and regularity of linear hyperbolic systems with dynamic boundary conditions, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 146 (2016), 1047-1080. doi: 10.1017/S0308210515000827. Google Scholar [37] G. Peralta and Y. Ueda, Stability conditions for a system of delay differential equations and its application, in preparation.Google Scholar [38] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser-Verlag, Basel, 2009. Google Scholar

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##### References:
 [1] W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 360 (1988), 837-852. doi: 10.1090/S0002-9947-1988-0933321-3. Google Scholar [2] G. Avalos and F. Bucci, Rational rates of uniform decay for strong solutions to a fluid-structure PDE system, J. Differential Equations, 258 (2015), 4398-4423. doi: 10.1016/j.jde.2015.01.037. Google Scholar [3] G. Avalos and R. Triggiani, The coupled PDE system arising in fluid-structure interaction, Part Ⅰ: Explicit semigroup generator and its spectral properties, Contemporary Mathematics, 440 (2007), 15-54. Google Scholar [4] G. Avalos and R. Triggiani, Semigroup wellposedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction, Discr. Cont. Dynam. Sys., 2 (2009), 417-447. doi: 10.3934/dcdss.2009.2.417. Google Scholar [5] G. Avalos and R. Triggiani, Rational decay rates for a PDE heat-structure interaction: A frequency domain approach, Evol. Equ. Control Theory, 2 (2013), 233-253. doi: 10.3934/eect.2013.2.233. Google Scholar [6] G. Avalos and R. Triggiani, Fluid structure interaction with and without internal dissipation of the structure: A contrast study in stability, Evol. Equ. Control Theory, 2 (2013), 563-598. doi: 10.3934/eect.2013.2.563. Google Scholar [7] G. Avalos, I. Lasiecka and R. Trigianni, Higher regularity of a coupled parabolic-hyperbolic fluid structure interactive system, Georgian Math. J., 15 (2008), 403-437. Google Scholar [8] G. Avalos, I. Lasiecka and R. Trigianni, Heat-wave interaction in 2-3 dimensions: Optimal rational decay rate, Journal of Mathematical Analysis and its Applications, 437 (2016), 782-815. doi: 10.1016/j.jmaa.2015.12.051. Google Scholar [9] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, Contemporary Mathematics, 440 (2007), 55-82. Google Scholar [10] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Smoothness of weak solutions to a nonlinear fluid-structure interaction model, Indiana U. Math. J., 57 (2008), 1173-1207. doi: 10.1512/iumj.2008.57.3284. Google Scholar [11] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Mathematische Annalen, 347 (2010), 455-478. doi: 10.1007/s00208-009-0439-0. Google Scholar [12] R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedback, SIAM J. Control Optim., 26 (1988), 697-713. doi: 10.1137/0326040. Google Scholar [13] R. Datko, J. Lagnese and P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156. doi: 10.1137/0324007. Google Scholar [14] W. Desch, E. Fašangová, J. Milota and G. Propst, Stabilization through viscoelastic boundary damping: A semigroup approach, Semigroup Forum, 80 (2010), 405-415. doi: 10.1007/s00233-009-9197-2. Google Scholar [15] Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem, Discr. and Contin. Dynam. Systems, 9 (2003), 633-650. doi: 10.3934/dcds.2003.9.633. Google Scholar [16] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, 2nd ed., Springer, Berlin, 2000. Google Scholar [17] L. C. Evans, Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1993.Google Scholar [18] M. Kirane and B. Said-Houari, Existence and asymptotic stability of a viscoelastic wave equation with delay, Z. Angew. Math. Phys., 62 (2011), 1065-1082. doi: 10.1007/s00033-011-0145-0. Google Scholar [19] Lasiecka and Y. Lu, Asymptotic stability of finite energy in Navier Stokes-elastic wave interaction, Semigroup Forum, 82 (2011), 61-82. doi: 10.1007/s00233-010-9281-7. Google Scholar [20] I. Lasiecka and Y. Lu, Interface feedback control stabilization of a nonlinear fluid-structure interaction, Nonlinear Analysis, 75 (2012), 1449-1460. doi: 10.1016/j.na.2011.04.018. Google Scholar [21] I. Lasiecka and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pure Appl., 65 (1986), 149-192. Google Scholar [22] I. Lasiecka and R. Triggiani, Exponential uniform energy decay rates of the wave equation in a bounded region with $L^2(0,T;L^2(Γ))$-boundary feedback in the Dirichlet B.C., J. Diff. Eqns., 66 (1987), 340-390. doi: 10.1016/0022-0396(87)90025-8. Google Scholar [23] I. Lasiecka and R. Triggiani, Exact controllability for the wave equation with Neumann boundary control, Appl. Math. Optimiz., 19 (1989), 243-290. doi: 10.1007/BF01448201. Google Scholar [24] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories; Vol. II: Abstract Hyperbolic Equations, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2000. Google Scholar [25] J. L. Lions, Contrôlabilité Exacte, Stabilisation et Perturbations des Systémes Distribués, Vol. 1, Masson, Paris, 1988. Google Scholar [26] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, New York, (1972) Google Scholar [27] J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. 2, Springer, Heidelberg, 1972. Google Scholar [28] L. Lu, Numerical stability of θ-methods for systems of differential equations with several delay terms, J. Comput. Apple. Math., 34 (1991), 291-304. doi: 10.1016/0377-0427(91)90090-7. Google Scholar [29] Y. Lu, Stabilization of a fluid structure interaction with nonlinear damping, Control and Cybernetics, 42 (2013), 155-181. Google Scholar [30] Y. Lu, Uniform decay rates for the energy in nonlinear fluid structure interaction with monotone viscous damping, Palest. J. Math., 2 (2013), 215-232. Google Scholar [31] Y. I. Lyubich and V. Q. Phong, Asymptotic stability of linear differential equations in Banach spaces, Studia Matematica, 88 (1988), 37-42. doi: 10.4064/sm-88-1-37-42. Google Scholar [32] S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585. doi: 10.1137/060648891. Google Scholar [33] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. Google Scholar [34] G. Peralta, A fluid-structure interaction model with damping and delay in the structure, Z. Angew. Math. Phys., 67 (2016), Art. 10, 20 pp. Google Scholar [35] G. Peralta and G. Propst, Stability and boundary controllability of a linearized model of flow in an elastic tube, ESAIM: Control, Optimisation and Calculus of Variations, 21 (2015), 583-601. doi: 10.1051/cocv/2014039. Google Scholar [36] G. Peralta and G. Propst, Well-posedness and regularity of linear hyperbolic systems with dynamic boundary conditions, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 146 (2016), 1047-1080. doi: 10.1017/S0308210515000827. Google Scholar [37] G. Peralta and Y. Ueda, Stability conditions for a system of delay differential equations and its application, in preparation.Google Scholar [38] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser-Verlag, Basel, 2009. Google Scholar
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