# American Institute of Mathematical Sciences

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

## Optimal control in a free boundary fluid-elasticity interaction

 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.
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

show all references

##### 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
 [1] Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241 [2] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348 [3] Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019 [4] Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020462 [5] Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $L^2-$norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 [6] Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046 [7] Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347 [8] Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $p$ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442 [9] Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032 [10] Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461 [11] Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467 [12] José Madrid, João P. G. Ramos. On optimal autocorrelation inequalities on the real line. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020271 [13] Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074 [14] Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453 [15] Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020 [16] Tommi Brander, Joonas Ilmavirta, Petteri Piiroinen, Teemu Tyni. Optimal recovery of a radiating source with multiple frequencies along one line. Inverse Problems & Imaging, 2020, 14 (6) : 967-983. doi: 10.3934/ipi.2020044 [17] Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444 [18] Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248 [19] Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267 [20] Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

Impact Factor: