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 and Continuous Dynamical Systems, 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.

[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.

[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. 

[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.

[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.

[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.

[7]

G. AvalosI. Lasiecka and R. Trigianni, Higher regularity of a coupled parabolic-hyperbolic fluid structure interactive system, Georgian Math. J., 15 (2008), 403-437. 

[8]

G. AvalosI. 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.

[9]

V. BarbuZ. GrujićI. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, Contemporary Mathematics, 440 (2007), 55-82. 

[10]

V. BarbuZ. GrujićI. 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.

[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.

[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.

[13]

R. DatkoJ. 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.

[14]

W. DeschE. FašangováJ. 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.

[15]

Q. DuM. D. GunzburgerL. 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.

[16]

K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, 2nd ed., Springer, Berlin, 2000.

[17]

L. C. Evans, Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1993.

[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.

[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.

[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.

[21]

I. Lasiecka and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pure Appl., 65 (1986), 149-192. 

[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.

[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.

[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.

[25]

J. L. Lions, Contrôlabilité Exacte, Stabilisation et Perturbations des Systémes Distribués, Vol. 1, Masson, Paris, 1988.

[26]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, New York, (1972)

[27]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. 2, Springer, Heidelberg, 1972.

[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.

[29]

Y. Lu, Stabilization of a fluid structure interaction with nonlinear damping, Control and Cybernetics, 42 (2013), 155-181. 

[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. 

[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.

[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.

[33]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.

[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.

[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.

[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.

[37]

G. Peralta and Y. Ueda, Stability conditions for a system of delay differential equations and its application, in preparation.

[38]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser-Verlag, Basel, 2009.

show all references

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.

[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.

[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. 

[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.

[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.

[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.

[7]

G. AvalosI. Lasiecka and R. Trigianni, Higher regularity of a coupled parabolic-hyperbolic fluid structure interactive system, Georgian Math. J., 15 (2008), 403-437. 

[8]

G. AvalosI. 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.

[9]

V. BarbuZ. GrujićI. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, Contemporary Mathematics, 440 (2007), 55-82. 

[10]

V. BarbuZ. GrujićI. 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.

[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.

[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.

[13]

R. DatkoJ. 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.

[14]

W. DeschE. FašangováJ. 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.

[15]

Q. DuM. D. GunzburgerL. 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.

[16]

K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, 2nd ed., Springer, Berlin, 2000.

[17]

L. C. Evans, Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1993.

[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.

[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.

[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.

[21]

I. Lasiecka and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pure Appl., 65 (1986), 149-192. 

[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.

[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.

[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.

[25]

J. L. Lions, Contrôlabilité Exacte, Stabilisation et Perturbations des Systémes Distribués, Vol. 1, Masson, Paris, 1988.

[26]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, New York, (1972)

[27]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. 2, Springer, Heidelberg, 1972.

[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.

[29]

Y. Lu, Stabilization of a fluid structure interaction with nonlinear damping, Control and Cybernetics, 42 (2013), 155-181. 

[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. 

[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.

[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.

[33]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.

[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.

[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.

[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.

[37]

G. Peralta and Y. Ueda, Stability conditions for a system of delay differential equations and its application, in preparation.

[38]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser-Verlag, Basel, 2009.

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