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2013, 2013(special): 545-554. doi: 10.3934/proc.2013.2013.545

A note on optimal control problem for a hemivariational inequality modeling fluid flow

1. 

Jagiellonian University, Faculty of Mathematics and Computer Science, Institute of Computer Science, ul. Łojasiewicza 6, 30348 Krakow

Received  October 2012 Published  November 2013

We consider a class of distributed parameter optimal control problems for the boundary value problem for the stationary Navier--Stokes equation with a subdifferential boundary condition in a bounded domain. The weak formulation of the boundary value problem is a hemivariational inequality associated with a nonconvex nonsmooth locally Lipschitz superpotential. We establish the existence of solutions to the optimal control problem. We also address an open problem of potential identification in the hemivariational inequality.
Citation: Stanisław Migórski. A note on optimal control problem for a hemivariational inequality modeling fluid flow. Conference Publications, 2013, 2013 (special) : 545-554. doi: 10.3934/proc.2013.2013.545
References:
[1]

E. B. Bykhovski and N. V. Smirnov, On the orthogonal decomposition of the space of vector-valued square summable functions and the operators of vector analysis (in Russian), Trudy Mat. Inst. im. V. A. Steklova AN SSSR 59 (1960), 6-36.

[2]

A. Yu. Chebotarev, Subdifferential boundary value problems for stationary Navier-Stokes equations, Differensialnye Uravneniya [Differential Equations] 28 (1992), 1443-1450.

[3]

A. Yu. Chebotarev, Stationary variational inequalities in the model of inhomogeneous incompressible fluids, Sibirsk. Math. Zh. [Siberian Math. J.] 38 (1997), 1184-1193.

[4]

F. H. Clarke, "Optimization and Nonsmooth Analysis", Wiley, Interscience, New York, 1983.

[5]

Z. Denkowski, S. Migórski and N.S. Papageorgiou, "An Introduction to Nonlinear Analysis: Theory", Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003.

[6]

Z. Denkowski, S. Migórski and N.S. Papageorgiou, "An Introduction to Nonlinear Analysis: Applications", Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003.

[7]

D. S. Konovalova, Subdifferential boundary value problems for evolution Navier-Stokes equations, Differensialnye Uravneniya [Differential Equations] 36 (2000), 792-798.

[8]

S. Migórski, Evolution hemivariational inequalities in infinite dimension and their control, Nonlinear Analysis Theory Methods and Applications 47 (2001), 101-112.

[9]

S. Migórski, Hemivariational inequality for a frictional contact problem in elasto-piezoelectricity, Discrete Continuous Dynam. Syst. Ser. B 6 (2006), 1339-1356.

[10]

S. Migórski, A. Ochal, Optimal control of parabolic hemivariational inequalities, Journal of Global Optimization 17 (2000), 285-300.

[11]

S. Migórski and A. Ochal, Hemivariational inequalites for stationary Navier-Stokes equations, J. Math. Anal. Appl. 306 (2005), 197-217.

[12]

S. Migórski and A. Ochal, Navier-Stokes models modeled by evolution hemivariational inequalities, Discrete and Continuous Dynamical Systems (Suppl.) (2007) 731-740.

[13]

S. Migórski, A. Ochal and M. Sofonea, "Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems", Advances in Mechanics and Mathematics 26, Springer, New York, 2013.

[14]

Z. Naniewicz and P. D. Panagiotopoulos, "Mathematical Theory of Hemivariational Inequalities and Applications", Marcel Dekker, Inc., New York, Basel, Hong Kong, 1995.

[15]

P. D. Panagiotopoulos, "Hemivariational Inequalities, Applications in Mechanics and Engineering", Springer-Verlag, Berlin, 1993.

[16]

R. Temam, "Navier-Stokes Equations", North-Holland, Amsterdam, 1979.

[17]

E. Zeidler, "Nonlinear Functional Analysis and Applications II A/B", Springer, New York, 1990.

show all references

References:
[1]

E. B. Bykhovski and N. V. Smirnov, On the orthogonal decomposition of the space of vector-valued square summable functions and the operators of vector analysis (in Russian), Trudy Mat. Inst. im. V. A. Steklova AN SSSR 59 (1960), 6-36.

[2]

A. Yu. Chebotarev, Subdifferential boundary value problems for stationary Navier-Stokes equations, Differensialnye Uravneniya [Differential Equations] 28 (1992), 1443-1450.

[3]

A. Yu. Chebotarev, Stationary variational inequalities in the model of inhomogeneous incompressible fluids, Sibirsk. Math. Zh. [Siberian Math. J.] 38 (1997), 1184-1193.

[4]

F. H. Clarke, "Optimization and Nonsmooth Analysis", Wiley, Interscience, New York, 1983.

[5]

Z. Denkowski, S. Migórski and N.S. Papageorgiou, "An Introduction to Nonlinear Analysis: Theory", Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003.

[6]

Z. Denkowski, S. Migórski and N.S. Papageorgiou, "An Introduction to Nonlinear Analysis: Applications", Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003.

[7]

D. S. Konovalova, Subdifferential boundary value problems for evolution Navier-Stokes equations, Differensialnye Uravneniya [Differential Equations] 36 (2000), 792-798.

[8]

S. Migórski, Evolution hemivariational inequalities in infinite dimension and their control, Nonlinear Analysis Theory Methods and Applications 47 (2001), 101-112.

[9]

S. Migórski, Hemivariational inequality for a frictional contact problem in elasto-piezoelectricity, Discrete Continuous Dynam. Syst. Ser. B 6 (2006), 1339-1356.

[10]

S. Migórski, A. Ochal, Optimal control of parabolic hemivariational inequalities, Journal of Global Optimization 17 (2000), 285-300.

[11]

S. Migórski and A. Ochal, Hemivariational inequalites for stationary Navier-Stokes equations, J. Math. Anal. Appl. 306 (2005), 197-217.

[12]

S. Migórski and A. Ochal, Navier-Stokes models modeled by evolution hemivariational inequalities, Discrete and Continuous Dynamical Systems (Suppl.) (2007) 731-740.

[13]

S. Migórski, A. Ochal and M. Sofonea, "Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems", Advances in Mechanics and Mathematics 26, Springer, New York, 2013.

[14]

Z. Naniewicz and P. D. Panagiotopoulos, "Mathematical Theory of Hemivariational Inequalities and Applications", Marcel Dekker, Inc., New York, Basel, Hong Kong, 1995.

[15]

P. D. Panagiotopoulos, "Hemivariational Inequalities, Applications in Mechanics and Engineering", Springer-Verlag, Berlin, 1993.

[16]

R. Temam, "Navier-Stokes Equations", North-Holland, Amsterdam, 1979.

[17]

E. Zeidler, "Nonlinear Functional Analysis and Applications II A/B", Springer, New York, 1990.

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