Optimal control in a free boundary fluid-elasticity interaction

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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

Abstract
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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. 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), 1389. 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. 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, 216 (2011), 93. 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, 1 (2011), 184. Google Scholar
[7]	L. Bociu and J.-P. Zolésio, Sensitivity analysis for a free boundary fluid-elasticity interaction,, Evol. Equ. Control Theory, 2 (2013), 55. Google Scholar
[8]	G. Coppola and K. Liu, Study of compliance mismatch within a stented artery,, Proceedings of the COMSOL Conference, (2008). Google Scholar
[9]	R. Correa and A. Seeger, Directional derivative of a minimax function,, Nonlinear Analysis, 9 (1985), 13. Google Scholar
[10]	D. Coutand and S. Shkoller, Motion of an elastic solid inside an incompressible viscous fluid,, Arch. Rational Mech. Anal., 176 (2005), 303. Google Scholar
[11]	D. Coutand and S. Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations,, Arch. Rational Mech. Anal., 179 (2006), 25. Google Scholar
[12]	R. Dziri and J.-P. Zolésio, Drag reduction for non-cylindrical Navier-Stokes flows,, Optimization Methods and Software, 26 (2011), 4. 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), 2191. 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), 852. Google Scholar
[15]	C. Grandmont, Existence for a three-dimensional steady state fluid-structure interaction problem,, J. Math. Fluid. Mech., 4 (2002), 76. 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. 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), 167. 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). 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), 1. Google Scholar
[20]	I. Kukavica and A. Tuffaha, Solutions to a free boundary problem of fluid-structure interaction,, Discrete Contin. Dyn. Syst., (2012). 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), 1817. 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), 971. Google Scholar
[23]	I. Lasiecka and A. Tuffaha, A Bolza optimal synthesis problem for singular estimate control system,, Cont. Cybernetics, 38 (2009), 1429. 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, (2008), 203. 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), 65. 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), 78. 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), 1922. Google Scholar
[29]	R. Wood, N. Radhika and W. Gu-Yeon, Flight of the robobees,, Scientific American, 308 (2013). Google Scholar
[30]	J. Yong, Existence theory of optimal controls for distributed parameter systems,, Kodai Math. J., 15 (1992), 193. Google Scholar

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References:

[1]	F. Abergel and R. Temam, On some control problems in fluid mechanics,, Theoret. Comput. Fluid Dynamics, 1 (1990), 303. 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), 1389. 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. 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, 216 (2011), 93. 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, 1 (2011), 184. Google Scholar
[7]	L. Bociu and J.-P. Zolésio, Sensitivity analysis for a free boundary fluid-elasticity interaction,, Evol. Equ. Control Theory, 2 (2013), 55. Google Scholar
[8]	G. Coppola and K. Liu, Study of compliance mismatch within a stented artery,, Proceedings of the COMSOL Conference, (2008). Google Scholar
[9]	R. Correa and A. Seeger, Directional derivative of a minimax function,, Nonlinear Analysis, 9 (1985), 13. Google Scholar
[10]	D. Coutand and S. Shkoller, Motion of an elastic solid inside an incompressible viscous fluid,, Arch. Rational Mech. Anal., 176 (2005), 303. Google Scholar
[11]	D. Coutand and S. Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations,, Arch. Rational Mech. Anal., 179 (2006), 25. Google Scholar
[12]	R. Dziri and J.-P. Zolésio, Drag reduction for non-cylindrical Navier-Stokes flows,, Optimization Methods and Software, 26 (2011), 4. 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), 2191. 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), 852. Google Scholar
[15]	C. Grandmont, Existence for a three-dimensional steady state fluid-structure interaction problem,, J. Math. Fluid. Mech., 4 (2002), 76. 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. 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), 167. 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). 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), 1. Google Scholar
[20]	I. Kukavica and A. Tuffaha, Solutions to a free boundary problem of fluid-structure interaction,, Discrete Contin. Dyn. Syst., (2012). 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), 1817. 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), 971. Google Scholar
[23]	I. Lasiecka and A. Tuffaha, A Bolza optimal synthesis problem for singular estimate control system,, Cont. Cybernetics, 38 (2009), 1429. 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, (2008), 203. 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), 65. 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), 78. 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), 1922. Google Scholar
[29]	R. Wood, N. Radhika and W. Gu-Yeon, Flight of the robobees,, Scientific American, 308 (2013). Google Scholar
[30]	J. Yong, Existence theory of optimal controls for distributed parameter systems,, Kodai Math. J., 15 (1992), 193. Google Scholar

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