American Institute of Mathematical Sciences

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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

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