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Optimal control of a linear unsteady fluid-structure interaction problem. (English) Zbl 1346.35033
Summary: In this paper, we consider optimal control problems governed by linear unsteady fluid-structure interaction problems. Based on a novel symmetric monolithic formulation, we derive optimality systems and provide regularity results for optimal solutions. The proposed formulation allows for natural application of gradient-based optimization algorithms and for space-time finite element discretizations.

MSC:
35B65 Smoothness and regularity of solutions to PDEs
35M33 Initial-boundary value problems for mixed-type systems of PDEs
49K20 Optimality conditions for problems involving partial differential equations
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
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[1] Bertoglio, C; Barber, D; Gaddum, N; Valverde, I; Rutten, M; Beerbaum, P; Moireau, P; Hose, R; Gerbeau, J, Identification of artery wall stiffness: in vitro validation and in vivo results of data assimilation procedure applied to a 3D fluid-structure interaction model, J. Biomech., 47, 1027-1034, (2014)
[2] Bertoglio, C; Chapelle, D; Fernández, M; Gerbeau, JF; Moireau, P, State observers of a vascular fluid-structure interaction model through measurements in the solid, Comput. Methods Appl. mech. Eng., 256, 149-168, (2013) · Zbl 1352.74103
[3] Bertoglio, C; Moireau, P; Gerbeau, JF, Sequential parameter estimation for fluid-structure problems: application to hemodynamics, Int. J. Numer. Meth. Biomed Eng., 28, 434-455, (2012)
[4] D’Elia, M., Mirabella, L., Passerini, T., Perego, M., Piccinelli, M.,Vergara, C.,Veneziani, A.: Applications of variational data assimilation in computational hemodynamics. In: Modeling of Physiological Flows. MS&A—Modeling, Simulation and Applications, vol. 5, pp. 363-394. Springer (2012)
[5] Lassila, T; Manzoni, A; Quarteroni, A; Rozza, G, A reduced computational and geometric framework for inverse problems in hemodynamics, Int. J. Numer. Methods Biomed. Eng., 29, 741-776, (2013)
[6] Razzaq, M; Tsotskas, C; Turek, S; Kipouros, T; Savill, M; Hron, J, Multi-objective optimization of a fluid structure interaction benchmarking, CMES Comput. Model. Eng. Sci., 90, 303-337, (2013) · Zbl 1356.74062
[7] Degroote, J; Hojjat, M; Stavropoulou, E; Wüchner, R; Bletzinger, KU, Partitioned solution of an unsteady adjoint for strongly coupled fluid-structure interactions and application to parameter identification of a one-dimensional problem, Struct. Multidiscip. Optim., 47, 77-94, (2013) · Zbl 1274.74103
[8] Martin, V; Clément, F; Decoene, A; Gerbeau, JF, Parameter identification for a one-dimensional blood flow model, ESAIM Proc., 14, 174-200, (2005) · Zbl 1070.92015
[9] Richter, T; Wick, T, Optimal control and parameter estimation for stationary fluid-structure interaction problems, SIAM J. Sci. Comput., 35, b1085-b1104, (2013) · Zbl 1282.35287
[10] Bertagna, L., D’Elia, M., Perego, M., Veneziani, A.: Data assimilation in cardiovascular fluid-structure interaction problems: an introduction. In: Fluid-Structure Interaction and Biomedical Applications. Advances in Mathematical Fluid Mechanics, pp. 395-481. Springer (2014) · Zbl 1426.76761
[11] Perego, M; Veneziani, A; Vergara, C, A variational approach for estimating the compliance of the cardiovascular tissue: an inverse fluid-structure interaction problem, SIAM J. Sci. Comput., 33, 1181-1211, (2011) · Zbl 1227.92010
[12] Moubachir, M., Zolesio, J.: Moving Shape Analysis and Control: Applications to Fluid Structure Interaction. Chapmann & Hall/CRC, Boca Raton, FL (2006) · Zbl 1117.49003
[13] Bucci, F; Lasiecka, I, Optimal boundary control with critical penalization for a PDE model of fluid-solid interaction, Calc. Var., 37, 217-235, (2010) · Zbl 1198.35278
[14] Lasiecka, I; Tuffaha, A, Optimal feedback synthesis for Bolza control problem arising in linearized fluid structure interaction, Int. Ser. Numer. Math., 158, 171-190, (2009) · Zbl 1206.35054
[15] Lasiecka, I; Tuffaha, A, Riccati theory and singular estimates for a Bolza control problem arising in linearized fluid-structure interaction, Syst. Control Lett., 58, 499-509, (2009) · Zbl 1166.49036
[16] Manzoni, A; Quarteroni, A; Rozza, G, Shape optimization for viscous flows by reduced basis methods and free-form deformation, Int. J. Numer. Methods Fluids, 70, 646-670, (2012)
[17] Becker, R; Meidner, D; Vexler, B, Efficient numerical solution of parabolic optimization problems by finite element methods, Optim. Methods Softw., 22, 813-833, (2007) · Zbl 1135.35317
[18] Meidner, D; Vexler, B, Adaptive space-time finite element methods for parabolic optimization problems, SIAM J. Control Optim., 46, 116-142, (2007) · Zbl 1149.65051
[19] Meidner, D; Vexler, B, A priori error estimates for space-time finite element discretization of parabolic optimal control problems part I: problems without control constraints, SIAM J. Control Optim., 47, 1150-1177, (2008) · Zbl 1161.49026
[20] Meidner, D; Vexler, B, A priori error estimates for space-time finite element discretization of parabolic optimal control problems part II: problems with control constraints, SIAM J. Control Optim., 47, 1301-1329, (2008) · Zbl 1161.49035
[21] Meidner, D; Vexler, B, A priori error analysis of the Petrov-Galerkin Crank-Nicolson scheme for parabolic optimal control problems, SIAM J. Control Optim., 49, 2183-2211, (2011) · Zbl 1234.49030
[22] Razzaq, M.: Finite element simulation techniques for incompressible fluid-structure interaction with applications to bio-engineering and optimization. Ph.D. thesis, Technische Universität Dortmund (2011) · Zbl 1234.49030
[23] Wick, T.: Adaptive finite element simulation of fluid-structure interaction with application to heart-valve dynamics. Ph.D. thesis, University of Heidelberg (2011) · Zbl 1278.76005
[24] Du, Q; Gunzburger, MD; Hou, L; Lee, J, Analysis of a linear fluid-structure interaction problem, Discrete Contin. Dyn. Syst., 9, 633-650, (2003) · Zbl 1039.35076
[25] Du, Q; Gunzburger, MD; Hou, L; Lee, J, Semidiscrete finite element approximation of a linear fluid-structure interaction problem, SIAM J. Numer. Anal., 42, 1-29, (2004) · Zbl 1159.74343
[26] Avalos, G; Lasiecka, I; Triggiani, R, Higher regularity of a coupled parabolic-hyperbolic fluid-structure interactive system, Georgian Math. J., 15, 403-437, (2008) · Zbl 1157.35085
[27] Avalos, G., Triggiani, R.: The coupled PDE system arising in fluid/structure interaction. I. Explicit semigroup generator and its spectral properties. In: Fluids and waves, Contemp. Math., vol. 440, pp. 15-54. Amer. Math. Soc., Providence, RI (2007) · Zbl 1297.74035
[28] Avalos, G; Triggiani, R, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction, Discrete Contin. Dyn. Syst. Ser. S, 2, 417-447, (2009) · Zbl 1179.35010
[29] Avalos, G; Triggiani, R, Fluid-structure interaction with and without internal dissipation of the structure: a contrast study in stability, Evol. Equ. Control Theory, 2, 563-598, (2013) · Zbl 1277.35045
[30] Avalos, G; Triggiani, R, Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface, Discrete Contin. Dyn. Syst., 22, 817-833, (2008) · Zbl 1158.35320
[31] Avalos, G; Triggiani, R, Boundary feedback stabilization of a coupled parabolic-hyperbolic Stokes-Lamé PDE system, J. Evol. Equ., 9, 341-370, (2009) · Zbl 1239.74022
[32] Avalos, G., Lasiecka, I., Triggiani, R.: Beyond lack of compactness and lack of stability of a coupled parabolic-hyperbolic fluid-structure system. In: Optimal control of coupled systems of partial differential equations, Internat. Ser. Numer. Math., vol. 158, pp. 1-33. Birkhäuser Verlag, Basel (2009) · Zbl 1205.35182
[33] Ignatova, M; Kukavica, I; Lasiecka, I; Tuffaha, A, On well-posedness and small data global existence for an interface damped free boundary fluid-structure model, Nonlinearity, 27, 467-499, (2014) · Zbl 1286.35272
[34] Zhang, X; Zuazua, E, Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction, Arch. Rational Mech. Anal., 184, 49-120, (2007) · Zbl 1178.74075
[35] Barbu, V; Grujic, Z; Lasiecka, I; Tuffaha, A, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, Recent Trends Appl. Anal., 440, 5582, (2007) · Zbl 1297.35234
[36] Barbu, V; Grujic, Z; Lasiecka, I; Tuffaha, A, Smoothness of weak solutions to a nonlinear fluid-structure interaction model, Indiana Univ. Math. J., 57, 1173-1207, (2008) · Zbl 1147.74016
[37] Kukavica, I; Tuffaha, A; Ziane, M, Strong solutions to a Navier-Stokes-Lamé system on a domain with non-flat boundary, Nonlinearity, 24, 159-176, (2011) · Zbl 1372.76029
[38] Kukavica, I; Tuffaha, A; Ziane, M, Strong solutions to a nonlinear fluid structure interaction system, J. Differ. Equ., 247, 1452-1478, (2009) · Zbl 1180.35415
[39] Dunne, T., Rannacher, R., Richter, T.: Numerical simulation of fluid-structure interaction based on monolithic variational formulations. In: G.P. Galdi, R. Rannacher (eds.) Contemporary Challenges in Mathematical Fluid Mechanics. World Scientific (2009)
[40] Richter, T; Wick, T, Finite elements for fluid-structure interaction in ALE and fully Eulerian coordinates, Comput. Methods Appl. Mech. Eng., 199, 2633-2642, (2010) · Zbl 1231.74436
[41] Johnson, C, Discontinuous Galerkin finite element methods for second order hyperbolic problems, Comput. Method Appl. Mech., 107, 117-129, (1993) · Zbl 0787.65070
[42] Grisvards, P.: Elliptic Problems in Nonsmooth Domains. Pitman Advanced Pub. Program (1985) · Zbl 1239.74022
[43] Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications. Die Grundlehren der mathematischen Wissenschaften. Springer, Berlin (1972)
[44] Temam, R.: Navier-Stokes equations. Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam (1984) · Zbl 0568.35002
[45] Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Springer, New York (1994) · Zbl 0949.35004
[46] Tröltzsch, F.: Optimale Steuerung partieller Differentialgleichungen. Vieweg+Teubner, Wiesbaden (2009) · Zbl 1320.49001
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