American Institute of Mathematical Sciences

Advanced
2015, 2015(special): 122-131. doi: 10.3934/proc.2015.0122

Optimal control in a free boundary fluid-elasticity interaction

Lorena Bociu 1, , Lucas Castle 2, , Kristina Martin 2, and  Daniel Toundykov 3,

1. 

Department of Mathematics, NC State University, Raleigh, NC 27695
2. 

Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8501, United States, United States
3. 

Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588

Received  September 2014 Revised  January 2015 Published  November 2015

We establish existence of an optimal control for the problem of minimizing flow turbulence in the case of a nonlinear fluid-structure interaction model in the framework of the known local well-posedness theory. If the initial configuration is regular, in an appropriate sense, then a class of sufficiently smooth control inputs contains an element that minimizes, within the control class, the vorticity of the fluid flow around a moving and deforming elastic solid.
Keywords: fluid-structure, Optimal control, nonlinear elasticity, Navier-Stokes, moving boundary..
Mathematics Subject Classification: Primary: 49J20, 74F10; Secondary: 74B20, 76D05, 35R3.
Citation: Lorena Bociu, Lucas Castle, Kristina Martin, Daniel Toundykov. Optimal control in a free boundary fluid-elasticity interaction. Conference Publications, 2015, 2015 (special) : 122-131. doi: 10.3934/proc.2015.0122
References:
[1]

F. Abergel and R. Temam, On some control problems in fluid mechanics, Theoret. Comput. Fluid Dynamics, 1 (1990), 303-325. Google Scholar

[2]

P. Acquistapace, F. Bucci and I. Lasiecka, Optimal boundary control and Riccati theory for abstract dynamics motivated by hybrid systems of PDEs, Adv. Differential Equations, 10 2005, 12, 1389-1436.  Google Scholar

[3]

F. Bucci and I. Lasiecka, Optimal boundary control with critical penalization for a PDE model of fluid-solid interactions, Calc. Var. Partial Differential Equations, 37 2010, 1-2, 217-235.  Google Scholar

[4]

L. Bociu, D. Toundykov, and J.-P. and Zolésio, Well-posedness analysis for the total linearization of a fluid-elasticity interaction, Preprint, 2015. Google Scholar

[5]

L. Bociu and J.-P. Zolésio, Linearization of a coupled system of nonlinear elasticity and viscous fluid, Modern aspects of the theory of partial differential equations, 93-120, Oper. Theory Adv. Appl., 216 (2011), Birkhäuser/Springer Basel AG, Basel.  Google Scholar

[6]

L. Bociu and J.-P. Zolésio, Existence for the linearization of a steady state fluid - nonlinear elasticity interaction, Discrete and Continuous Dynamical Systems, Supplement, 1 (2011), 184-197.  Google Scholar

[7]

L. Bociu and J.-P. Zolésio, Sensitivity analysis for a free boundary fluid-elasticity interaction, Evol. Equ. Control Theory, 2 (2013), 1, 55-79.  Google Scholar

[8]

G. Coppola and K. Liu, Study of compliance mismatch within a stented artery, Proceedings of the COMSOL Conference, (2008), Boston. Google Scholar

[9]

R. Correa and A. Seeger, Directional derivative of a minimax function, Nonlinear Analysis, Theory, Methods and Applications, 9 (1985), 1, 13-22.  Google Scholar

[10]

D. Coutand and S. Shkoller, Motion of an elastic solid inside an incompressible viscous fluid, Arch. Rational Mech. Anal., 176 (2005), 1, 303-352.  Google Scholar

[11]

D. Coutand and S. Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations, Arch. Rational Mech. Anal., 179 (2006), 3, 25-102.  Google Scholar

[12]

R. Dziri and J.-P. Zolésio, Drag reduction for non-cylindrical Navier-Stokes flows, Optimization Methods and Software, 26 (2011), 4-5, 575-600.  Google Scholar

[13]

A. V. Fursikov, M. D. Gunzburger and L. S. Hou, Optimal boundary control for the evolutionary Navier-Stokes system: the three-dimensional case, SIAM J. Control Optim., 43 (2005), 6, 2191-2232.  Google Scholar

[14]

A. V. Fursikov, M. D. Gunzburger and L. S. Hou, Boundary value problems and optimal boundary control for the Navier-Stokes system: the two-dimensional case, SIAM J. Control Optim., 36 (1998), 3, 852-894.  Google Scholar

[15]

C. Grandmont, Existence for a three-dimensional steady state fluid-structure interaction problem, J. Math. Fluid. Mech., 4 (2002), 76-94.  Google Scholar

[16]

M. Gunzburger and H. Kim, Existence of an optimal solution of a shape control problem for the stationary Navier-Stokes equations, SIAM J. Control Optim., 36 (1998), 895-909.  Google Scholar

[17]

M. Gunzburger, L. Hou and T. P. Svobodny, Boundary velocity control of incompressible flow with an application to viscous drag reduction, SIAM J. Control Optim., 30 (1992), 1, 167-181.  Google Scholar

[18]

M. Ignatova, I. Kukavica, I. Lasiecka and A. Tuffaha, On well-posedness and small data global existence for a damped free boundary fluid-structure model, J. Math. Phys., 53 (2012), 115624.  Google Scholar

[19]

H. Kim, Penalized approach and analysis of an optimal shape control problem for the stationary Navier-Stokes equations, J. Korean Math. Soc., 38 (2001), no. 1, 1-23.  Google Scholar

[20]

I. Kukavica and A. Tuffaha, Solutions to a free boundary problem of fluid-structure interaction, Discrete Contin. Dyn. Syst., (2012), no. 4, 1355 - 1389.  Google Scholar

[21]

I. Kukavica and A. Tuffaha., Regularity of solutions to a free boundary problem of fluid-structure interaction. Indiana Univ. Math. J., 61 (2012), no. 5, 1817-1859.  Google Scholar

[22]

I. Lasiecka, R. Triggiani and J. Zhang, The fluid-structure interaction model with both control and disturbance at the interface: a game theory problem via an abstract approach, Applicable Analysis: An International Journal, 90 (2011), 6, 971-1009.  Google Scholar

[23]

I. Lasiecka and A. Tuffaha, A Bolza optimal synthesis problem for singular estimate control system, Cont. Cybernetics, 38 (2009), 4, 1429-1460.  Google Scholar

[24]

I. Lasiecka and A. Tuffaha, Boundary feedback control in fluid-structure interactions, Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, (2008), 203-208. Google Scholar

[25]

M. Moubachir and J.-P. Zolésio, Moving Shape Analysis and Control: Applications to Fluid Structure Interactions, Chapman and Hall/CRC Pure and Applied Mathematics, (2006).  Google Scholar

[26]

J. Simon, Compact sets in the space $L_p(0,T;B)$, Annali di Mat. Pura et Applicate,146 (1987), 4, 65-96.  Google Scholar

[27]

M. Sarma, Analysis of blood flow through stenosed vessel under effect of magnetic field, International Journal for Basic Sciences and Social Sciences, 1 (2012), 3, 78-88. Google Scholar

[28]

J. Tambača, M. Kosor, S. Čanić and D. Paniagua, M.D., Mathematical modeling of vascular stents, SIAM J. Appl. Math., 70 (2010), 6, 1922-1952.  Google Scholar

[29]

R. Wood, N. Radhika and W. Gu-Yeon, Flight of the robobees, Scientific American, 308 (2013), 60 - 65. Google Scholar

[30]

J. Yong, Existence theory of optimal controls for distributed parameter systems, Kodai Math. J., 15 (1992), 193-220.  Google Scholar

show all references

References:
[1]

F. Abergel and R. Temam, On some control problems in fluid mechanics, Theoret. Comput. Fluid Dynamics, 1 (1990), 303-325. Google Scholar

[2]

P. Acquistapace, F. Bucci and I. Lasiecka, Optimal boundary control and Riccati theory for abstract dynamics motivated by hybrid systems of PDEs, Adv. Differential Equations, 10 2005, 12, 1389-1436.  Google Scholar

[3]

F. Bucci and I. Lasiecka, Optimal boundary control with critical penalization for a PDE model of fluid-solid interactions, Calc. Var. Partial Differential Equations, 37 2010, 1-2, 217-235.  Google Scholar

[4]

L. Bociu, D. Toundykov, and J.-P. and Zolésio, Well-posedness analysis for the total linearization of a fluid-elasticity interaction, Preprint, 2015. Google Scholar

[5]

L. Bociu and J.-P. Zolésio, Linearization of a coupled system of nonlinear elasticity and viscous fluid, Modern aspects of the theory of partial differential equations, 93-120, Oper. Theory Adv. Appl., 216 (2011), Birkhäuser/Springer Basel AG, Basel.  Google Scholar

[6]

L. Bociu and J.-P. Zolésio, Existence for the linearization of a steady state fluid - nonlinear elasticity interaction, Discrete and Continuous Dynamical Systems, Supplement, 1 (2011), 184-197.  Google Scholar

[7]

L. Bociu and J.-P. Zolésio, Sensitivity analysis for a free boundary fluid-elasticity interaction, Evol. Equ. Control Theory, 2 (2013), 1, 55-79.  Google Scholar

[8]

G. Coppola and K. Liu, Study of compliance mismatch within a stented artery, Proceedings of the COMSOL Conference, (2008), Boston. Google Scholar

[9]

R. Correa and A. Seeger, Directional derivative of a minimax function, Nonlinear Analysis, Theory, Methods and Applications, 9 (1985), 1, 13-22.  Google Scholar

[10]

D. Coutand and S. Shkoller, Motion of an elastic solid inside an incompressible viscous fluid, Arch. Rational Mech. Anal., 176 (2005), 1, 303-352.  Google Scholar

[11]

D. Coutand and S. Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations, Arch. Rational Mech. Anal., 179 (2006), 3, 25-102.  Google Scholar

[12]

R. Dziri and J.-P. Zolésio, Drag reduction for non-cylindrical Navier-Stokes flows, Optimization Methods and Software, 26 (2011), 4-5, 575-600.  Google Scholar

[13]

A. V. Fursikov, M. D. Gunzburger and L. S. Hou, Optimal boundary control for the evolutionary Navier-Stokes system: the three-dimensional case, SIAM J. Control Optim., 43 (2005), 6, 2191-2232.  Google Scholar

[14]

A. V. Fursikov, M. D. Gunzburger and L. S. Hou, Boundary value problems and optimal boundary control for the Navier-Stokes system: the two-dimensional case, SIAM J. Control Optim., 36 (1998), 3, 852-894.  Google Scholar

[15]

C. Grandmont, Existence for a three-dimensional steady state fluid-structure interaction problem, J. Math. Fluid. Mech., 4 (2002), 76-94.  Google Scholar

[16]

M. Gunzburger and H. Kim, Existence of an optimal solution of a shape control problem for the stationary Navier-Stokes equations, SIAM J. Control Optim., 36 (1998), 895-909.  Google Scholar

[17]

M. Gunzburger, L. Hou and T. P. Svobodny, Boundary velocity control of incompressible flow with an application to viscous drag reduction, SIAM J. Control Optim., 30 (1992), 1, 167-181.  Google Scholar

[18]

M. Ignatova, I. Kukavica, I. Lasiecka and A. Tuffaha, On well-posedness and small data global existence for a damped free boundary fluid-structure model, J. Math. Phys., 53 (2012), 115624.  Google Scholar

[19]

H. Kim, Penalized approach and analysis of an optimal shape control problem for the stationary Navier-Stokes equations, J. Korean Math. Soc., 38 (2001), no. 1, 1-23.  Google Scholar

[20]

I. Kukavica and A. Tuffaha, Solutions to a free boundary problem of fluid-structure interaction, Discrete Contin. Dyn. Syst., (2012), no. 4, 1355 - 1389.  Google Scholar

[21]

I. Kukavica and A. Tuffaha., Regularity of solutions to a free boundary problem of fluid-structure interaction. Indiana Univ. Math. J., 61 (2012), no. 5, 1817-1859.  Google Scholar

[22]

I. Lasiecka, R. Triggiani and J. Zhang, The fluid-structure interaction model with both control and disturbance at the interface: a game theory problem via an abstract approach, Applicable Analysis: An International Journal, 90 (2011), 6, 971-1009.  Google Scholar

[23]

I. Lasiecka and A. Tuffaha, A Bolza optimal synthesis problem for singular estimate control system, Cont. Cybernetics, 38 (2009), 4, 1429-1460.  Google Scholar

[24]

I. Lasiecka and A. Tuffaha, Boundary feedback control in fluid-structure interactions, Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, (2008), 203-208. Google Scholar

[25]

M. Moubachir and J.-P. Zolésio, Moving Shape Analysis and Control: Applications to Fluid Structure Interactions, Chapman and Hall/CRC Pure and Applied Mathematics, (2006).  Google Scholar

[26]

J. Simon, Compact sets in the space $L_p(0,T;B)$, Annali di Mat. Pura et Applicate,146 (1987), 4, 65-96.  Google Scholar

[27]

M. Sarma, Analysis of blood flow through stenosed vessel under effect of magnetic field, International Journal for Basic Sciences and Social Sciences, 1 (2012), 3, 78-88. Google Scholar

[28]

J. Tambača, M. Kosor, S. Čanić and D. Paniagua, M.D., Mathematical modeling of vascular stents, SIAM J. Appl. Math., 70 (2010), 6, 1922-1952.  Google Scholar

[29]

R. Wood, N. Radhika and W. Gu-Yeon, Flight of the robobees, Scientific American, 308 (2013), 60 - 65. Google Scholar

[30]

J. Yong, Existence theory of optimal controls for distributed parameter systems, Kodai Math. J., 15 (1992), 193-220.  Google Scholar

[1]

Henry Jacobs, Joris Vankerschaver. Fluid-structure interaction in the Lagrange-Poincaré formalism: The Navier-Stokes and inviscid regimes. Journal of Geometric Mechanics, 2014, 6 (1) : 39-66. doi: 10.3934/jgm.2014.6.39
[2]

Igor Kukavica, Amjad Tuffaha. Solutions to a fluid-structure interaction free boundary problem. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1355-1389. doi: 10.3934/dcds.2012.32.1355
[3]

Andro Mikelić, Giovanna Guidoboni, Sunčica Čanić. Fluid-structure interaction in a pre-stressed tube with thick elastic walls I: the stationary Stokes problem. Networks & Heterogeneous Media, 2007, 2 (3) : 397-423. doi: 10.3934/nhm.2007.2.397
[4]

George Avalos, Roberto Triggiani. Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface. Discrete & Continuous Dynamical Systems, 2008, 22 (4) : 817-833. doi: 10.3934/dcds.2008.22.817
[5]

Pavel Eichler, Radek Fučík, Robert Straka. Computational study of immersed boundary - lattice Boltzmann method for fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 819-833. doi: 10.3934/dcdss.2020349
[6]

Daniele Boffi, Lucia Gastaldi. Discrete models for fluid-structure interactions: The finite element Immersed Boundary Method. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 89-107. doi: 10.3934/dcdss.2016.9.89
[7]

Mehdi Badra, Takéo Takahashi. Feedback boundary stabilization of 2d fluid-structure interaction systems. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2315-2373. doi: 10.3934/dcds.2017102
[8]

Qiang Du, M. D. Gunzburger, L. S. Hou, J. Lee. Analysis of a linear fluid-structure interaction problem. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 633-650. doi: 10.3934/dcds.2003.9.633
[9]

Pavel I. Plotnikov, Jan Sokolowski. Optimal shape control of airfoil in compressible gas flow governed by Navier-Stokes equations. Evolution Equations & Control Theory, 2013, 2 (3) : 495-516. doi: 10.3934/eect.2013.2.495
[10]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110
[11]

A. V. Fursikov. Stabilization for the 3D Navier-Stokes system by feedback boundary control. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 289-314. doi: 10.3934/dcds.2004.10.289
[12]

Jie Liao, Xiao-Ping Wang. Stability of an efficient Navier-Stokes solver with Navier boundary condition. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 153-171. doi: 10.3934/dcdsb.2012.17.153
[13]

George Avalos, Roberto Triggiani. Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 417-447. doi: 10.3934/dcdss.2009.2.417
[14]

Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control & Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61
[15]

Yoshikazu Giga. A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1277-1289. doi: 10.3934/dcdss.2013.6.1277
[16]

Shuguang Shao, Shu Wang, Wen-Qing Xu, Bin Han. Global existence for the 2D Navier-Stokes flow in the exterior of a moving or rotating obstacle. Kinetic & Related Models, 2016, 9 (4) : 767-776. doi: 10.3934/krm.2016015
[17]

Mehdi Badra. Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1169-1208. doi: 10.3934/dcds.2012.32.1169
[18]

Enrique Fernández-Cara. Motivation, analysis and control of the variable density Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2012, 5 (6) : 1021-1090. doi: 10.3934/dcdss.2012.5.1021
[19]

Hongjie Dong, Kunrui Wang. Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5289-5323. doi: 10.3934/dcds.2020228
[20]

Jing Wang, Lining Tong. Stability of boundary layers for the inflow compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2595-2613. doi: 10.3934/dcdsb.2012.17.2595

 Impact Factor: 

Tools

Metrics

  • PDF downloads (70)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy