A note on optimal control problem  fora hemivariational inequality  modeling fluid flow

Stanisław Migórski

doi:10.3934/proc.2013.2013.545

Article Contents

2013, Volume 2013: 545-554. Doi: 10.3934/proc.2013.2013.545 open access

This issue Previous Article On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients Next Article Existence and multiplicity of solutions in fourth order BVPs with unbounded nonlinearities

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

Stanisław Migórski^1,

1.
Jagiellonian University, Faculty of Mathematics and Computer Science, Institute of Computer Science, ul. Łojasiewicza 6, 30348 Krakow

Received: September 30, 2012

Published: October 2013

Abstract Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35Q30, 76D05, 35J87, 49J20, 49J52.

Citation:

Related Papers

Cited by

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.