# American Institute of Mathematical Sciences

December  2020, 9(4): 995-1008. doi: 10.3934/eect.2020046

## Stability analysis and optimal control of a stationary Stokes hemivariational inequality

 1 College of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China 2 Key Lab of Intelligent Analysis and Decision on Complex Systems, Chongqing University of Posts and Telecommunications, Chongqing 400065, China 3 Department of Mathematics, University of Iowa, Iowa City, IA 52242-1410, USA

* Corresponding author: Weimin Han

Dedicated to Professor Meir Shillor on the occasion of his 70th birthday

Received  October 2019 Published  March 2020

Fund Project: The first author is supported by the National Natural Science Foundation of China (No. 11771350), Basic and Advanced Research Project of CQ CSTC (Nos. cstc2016jcyjA0163 and cstc2018jcyjAX0605)

In this paper, we provide stability analysis for a stationary Stokes hemivariational inequality where along the tangential direction of the slip boundary, an inclusion relation involving the generalized subdifferential of a superpotential is specified. With viscous incompressible fluid flows as application background, stability is analyzed for solutions with respect to perturbations in the superpotential and the density of external forces. We also present a result on the existence of a solution to an optimal control problem for the stationary Stokes hemivariational inequality.

Citation: Changjie Fang, Weimin Han. Stability analysis and optimal control of a stationary Stokes hemivariational inequality. Evolution Equations & Control Theory, 2020, 9 (4) : 995-1008. doi: 10.3934/eect.2020046
##### References:
 [1] C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free-Boundary Problems, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984.  Google Scholar [2] V. Barbu, Optimal Control of Variational Inequalities, Research Notes in Mathematics, 100. Pitman, Boston, MA, 1984.  Google Scholar [3] H. Brézis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier, 18 (1968), 115-175.  doi: 10.5802/aif.280.  Google Scholar [4] F. H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc., 205 (1975), 247-262.  doi: 10.1090/S0002-9947-1975-0367131-6.  Google Scholar [5] F. H. Clarke, Optimization and Nonsmooth Analysis, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.  Google Scholar [6] G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Grundlehren der Mathematischen Wissenschaften, 219. Springer-Verlag, Berlin-New York, 1976.  Google Scholar [7] C. Fang, K. Czuprynski, W. Han, X.-L. Cheng and X. Dai, Finite element method for a stationary Stokes hemivariational inequality with slip boundary condition, IMA Journal of Numerical Analysis, (2019). doi: 10.1093/imanum/drz032.  Google Scholar [8] C. Fang and W. Han, Well-posedness and optimal control of a hemivariational inequality for nonstationary Stokes fluid flow, Discrete Contin. Dyn. Syst., 36 (2016), 5369-5386.  doi: 10.3934/dcds.2016036.  Google Scholar [9] G. Fichera, Problemi elastostatici con vincoli unilaterali. II. Problema di Signorini con ambique condizioni al contorno, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia, 7 (1963/64), 91-140.   Google Scholar [10] A. Friedman, Optimal control for variational inequalities, SIAM J. Control Optim., 24 (1986), 439-451.  doi: 10.1137/0324025.  Google Scholar [11] H. Fujita, Flow Problems with Unilateral Boundary Conditions, College de France, Lecons, 1993. Google Scholar [12] H. Fujita, A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions, RIMS Kôkyûroku, 888 (1994), 199–216.  Google Scholar [13] H. Fujita and H. Kawarada, Variational inequalities for the Stokes equation with boundary conditions of friction type, Recent Developments in Domain Decomposition Methods and Flow Problems, GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 11 (1998), 15-33.   Google Scholar [14] H. Fujita, H. Kawarada and A. Sasamoto, Analytical and numerical approaches to stationary flow problems with leak and slip boundary conditions, Lecture Notes Numer. Appl. Anal., Kinokuniya, Tokyo, 14 (1995), 17-31.   Google Scholar [15] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer Series in Computational Physics. Springer-Verlag, New York, 1984. doi: 10.1007/978-3-662-12613-4.  Google Scholar [16] R. Glowinski, J.-L. Lions and R. Trémolières, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981. Google Scholar [17] W. Han, S. Migórski and M. Sofonea, A class of variational-hemivariational inequalities with applications to frictional contact problems, SIAM J. Math. Anal., 46 (2014), 3891-3912.  doi: 10.1137/140963248.  Google Scholar [18] W. Han and M. Sofonea, Numerical analysis of hemivariational inequalities in contact mechanics, Acta Numerica, 28 (2019), 175-286.  doi: 10.1017/S0962492919000023.  Google Scholar [19] W. Han and Y. Li, Stability analysis of stationary variational and hemivariational inequalities with applications, Nonlinear Anal. Real World Appl., 50 (2019), 171-191.  doi: 10.1016/j.nonrwa.2019.04.009.  Google Scholar [20] J. Haslinger, I. Hlaváček and J. Nečas, Numerical methods for unilateral problems in solid mechanics, Handbook of Numerical Analysis, Handb. Numer. Anal., North-Holland, Amsterdam, 4 (1996), 313-485.   Google Scholar [21] J. Haslinger, M. Miettinen and P. D. Panagiotopoulos, Finite Element Method for Hemivariational Inequalities: Theory, Methods and Applications, Nonconvex Optimization and its Applications, 35. Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-1-4757-5233-5.  Google Scholar [22] J. Haslinger and P. D. Panagiotopoulos, Optimal control of systems governed by hemivariational inequalities. Existence and approximation results, Nonlinear Anal., 24 (1995), 105-119.  doi: 10.1016/0362-546X(93)E0022-U.  Google Scholar [23] I. Hlaváček, J. Haslinger, J. Nečas and J. Lovíšek, Solution of Variational Inequalities in Mechanics, Applied Mathematical Sciences, 66. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1048-1.  Google Scholar [24] L. S. Hou and S. S. Ravindran, Computations of boundary optimal control problems for an electrically conducting fluid, Journal of Computational Physics, 128 (1996), 319-330.  doi: 10.1006/jcph.1996.0213.  Google Scholar [25] N. Kikuchi and J. T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM Studies in Applied Mathematics, 8. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1988. doi: 10.1137/1.9781611970845.  Google Scholar [26] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Pure and Applied Mathematics, 88. Academic Press, Inc., New York-London, 1980.   Google Scholar [27] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Die Grundlehren der Mathematischen Wissenschaften, Band 170 Springer-Verlag, New York-Berlin, 1971.  Google Scholar [28] J.-L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math., 20 (1967), 493-519.  doi: 10.1002/cpa.3160200302.  Google Scholar [29] S. Migórski, A note on optimal control problem for a hemivariational inequality modeling fluid flow, Discrete and Continuous Dynam. Systems - Supplement, (2013), 545–554. doi: 10.3934/proc.2013.2013.545.  Google Scholar [30] S. Migórski and A. Ochal, Optimal control of parabolic hemivariational inequalities, J. Global Optim., 17 (2000), 285-300.  doi: 10.1023/A:1026555014562.  Google Scholar [31] 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. doi: 10.1007/978-1-4614-4232-5.  Google Scholar [32] Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Monographs and Textbooks in Pure and Applied Mathematics, 188. Marcel Dekker, Inc., New York, 1995.  Google Scholar [33] P. D. Panagiotopoulos, Nonconvex energy functions. Hemivariational inequalities and substationary principles, Acta Mech., 42 (1983), 111-130.  doi: 10.1007/BF01170410.  Google Scholar [34] P. D. Panagiotopoulos, Hemivariational Inequalities: Applications in Mechanics and Engineering, SpringerVerlag, Berlin, 1993. doi: 10.1007/978-3-642-51677-1.  Google Scholar [35] F. Saidi, Non-Newtonian Stokes flow with frictional boundary conditions, Math. Model. Anal., 12 (2007), 483-495.  doi: 10.3846/1392-6292.2007.12.483-495.  Google Scholar [36] A. Signorini, Sopra a une questioni di elastostatica, Attidella Società Italiana per il Progresso delle Scienze, (1933). Google Scholar [37] M. Sofonea and S. Migórski, Variational-Hemivariational Inequalities with Applications, CRC Press, Boca Raton, FL, 2018.   Google Scholar [38] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Studies in Mathematics and its Applications, Vol. 2. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.  Google Scholar [39] D. Tiba, Optimal Control of Nonsmooth Distributed Parameter Systems, Lecture Notes in Mathematics, 1459. Springer-Verlag, Berlin, 1990. doi: 10.1007/BFb0085564.  Google Scholar [40] F. Tröltzsch, Optimal Control of Partial Differential Equations, American Mathematical Society, Providence, Rhode Island, 2010. Google Scholar [41] Y.-B. Xiao and M. Sofonea, On the optimal control of variational-hemivariational inequalities, J. Math. Anal. Appl., 475 (2019), 364-384.  doi: 10.1016/j.jmaa.2019.02.046.  Google Scholar

show all references

##### References:
 [1] C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free-Boundary Problems, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984.  Google Scholar [2] V. Barbu, Optimal Control of Variational Inequalities, Research Notes in Mathematics, 100. Pitman, Boston, MA, 1984.  Google Scholar [3] H. Brézis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier, 18 (1968), 115-175.  doi: 10.5802/aif.280.  Google Scholar [4] F. H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc., 205 (1975), 247-262.  doi: 10.1090/S0002-9947-1975-0367131-6.  Google Scholar [5] F. H. Clarke, Optimization and Nonsmooth Analysis, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.  Google Scholar [6] G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Grundlehren der Mathematischen Wissenschaften, 219. Springer-Verlag, Berlin-New York, 1976.  Google Scholar [7] C. Fang, K. Czuprynski, W. Han, X.-L. Cheng and X. Dai, Finite element method for a stationary Stokes hemivariational inequality with slip boundary condition, IMA Journal of Numerical Analysis, (2019). doi: 10.1093/imanum/drz032.  Google Scholar [8] C. Fang and W. Han, Well-posedness and optimal control of a hemivariational inequality for nonstationary Stokes fluid flow, Discrete Contin. Dyn. Syst., 36 (2016), 5369-5386.  doi: 10.3934/dcds.2016036.  Google Scholar [9] G. Fichera, Problemi elastostatici con vincoli unilaterali. II. Problema di Signorini con ambique condizioni al contorno, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia, 7 (1963/64), 91-140.   Google Scholar [10] A. Friedman, Optimal control for variational inequalities, SIAM J. Control Optim., 24 (1986), 439-451.  doi: 10.1137/0324025.  Google Scholar [11] H. Fujita, Flow Problems with Unilateral Boundary Conditions, College de France, Lecons, 1993. Google Scholar [12] H. Fujita, A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions, RIMS Kôkyûroku, 888 (1994), 199–216.  Google Scholar [13] H. Fujita and H. Kawarada, Variational inequalities for the Stokes equation with boundary conditions of friction type, Recent Developments in Domain Decomposition Methods and Flow Problems, GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 11 (1998), 15-33.   Google Scholar [14] H. Fujita, H. Kawarada and A. Sasamoto, Analytical and numerical approaches to stationary flow problems with leak and slip boundary conditions, Lecture Notes Numer. Appl. Anal., Kinokuniya, Tokyo, 14 (1995), 17-31.   Google Scholar [15] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer Series in Computational Physics. Springer-Verlag, New York, 1984. doi: 10.1007/978-3-662-12613-4.  Google Scholar [16] R. Glowinski, J.-L. Lions and R. Trémolières, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981. Google Scholar [17] W. Han, S. Migórski and M. Sofonea, A class of variational-hemivariational inequalities with applications to frictional contact problems, SIAM J. Math. Anal., 46 (2014), 3891-3912.  doi: 10.1137/140963248.  Google Scholar [18] W. Han and M. Sofonea, Numerical analysis of hemivariational inequalities in contact mechanics, Acta Numerica, 28 (2019), 175-286.  doi: 10.1017/S0962492919000023.  Google Scholar [19] W. Han and Y. Li, Stability analysis of stationary variational and hemivariational inequalities with applications, Nonlinear Anal. Real World Appl., 50 (2019), 171-191.  doi: 10.1016/j.nonrwa.2019.04.009.  Google Scholar [20] J. Haslinger, I. Hlaváček and J. Nečas, Numerical methods for unilateral problems in solid mechanics, Handbook of Numerical Analysis, Handb. Numer. Anal., North-Holland, Amsterdam, 4 (1996), 313-485.   Google Scholar [21] J. Haslinger, M. Miettinen and P. D. Panagiotopoulos, Finite Element Method for Hemivariational Inequalities: Theory, Methods and Applications, Nonconvex Optimization and its Applications, 35. Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-1-4757-5233-5.  Google Scholar [22] J. Haslinger and P. D. Panagiotopoulos, Optimal control of systems governed by hemivariational inequalities. Existence and approximation results, Nonlinear Anal., 24 (1995), 105-119.  doi: 10.1016/0362-546X(93)E0022-U.  Google Scholar [23] I. Hlaváček, J. Haslinger, J. Nečas and J. Lovíšek, Solution of Variational Inequalities in Mechanics, Applied Mathematical Sciences, 66. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1048-1.  Google Scholar [24] L. S. Hou and S. S. Ravindran, Computations of boundary optimal control problems for an electrically conducting fluid, Journal of Computational Physics, 128 (1996), 319-330.  doi: 10.1006/jcph.1996.0213.  Google Scholar [25] N. Kikuchi and J. T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM Studies in Applied Mathematics, 8. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1988. doi: 10.1137/1.9781611970845.  Google Scholar [26] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Pure and Applied Mathematics, 88. Academic Press, Inc., New York-London, 1980.   Google Scholar [27] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Die Grundlehren der Mathematischen Wissenschaften, Band 170 Springer-Verlag, New York-Berlin, 1971.  Google Scholar [28] J.-L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math., 20 (1967), 493-519.  doi: 10.1002/cpa.3160200302.  Google Scholar [29] S. Migórski, A note on optimal control problem for a hemivariational inequality modeling fluid flow, Discrete and Continuous Dynam. Systems - Supplement, (2013), 545–554. doi: 10.3934/proc.2013.2013.545.  Google Scholar [30] S. Migórski and A. Ochal, Optimal control of parabolic hemivariational inequalities, J. Global Optim., 17 (2000), 285-300.  doi: 10.1023/A:1026555014562.  Google Scholar [31] 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. doi: 10.1007/978-1-4614-4232-5.  Google Scholar [32] Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Monographs and Textbooks in Pure and Applied Mathematics, 188. Marcel Dekker, Inc., New York, 1995.  Google Scholar [33] P. D. Panagiotopoulos, Nonconvex energy functions. Hemivariational inequalities and substationary principles, Acta Mech., 42 (1983), 111-130.  doi: 10.1007/BF01170410.  Google Scholar [34] P. D. Panagiotopoulos, Hemivariational Inequalities: Applications in Mechanics and Engineering, SpringerVerlag, Berlin, 1993. doi: 10.1007/978-3-642-51677-1.  Google Scholar [35] F. Saidi, Non-Newtonian Stokes flow with frictional boundary conditions, Math. Model. Anal., 12 (2007), 483-495.  doi: 10.3846/1392-6292.2007.12.483-495.  Google Scholar [36] A. Signorini, Sopra a une questioni di elastostatica, Attidella Società Italiana per il Progresso delle Scienze, (1933). Google Scholar [37] M. Sofonea and S. Migórski, Variational-Hemivariational Inequalities with Applications, CRC Press, Boca Raton, FL, 2018.   Google Scholar [38] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Studies in Mathematics and its Applications, Vol. 2. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.  Google Scholar [39] D. Tiba, Optimal Control of Nonsmooth Distributed Parameter Systems, Lecture Notes in Mathematics, 1459. Springer-Verlag, Berlin, 1990. doi: 10.1007/BFb0085564.  Google Scholar [40] F. Tröltzsch, Optimal Control of Partial Differential Equations, American Mathematical Society, Providence, Rhode Island, 2010. Google Scholar [41] Y.-B. Xiao and M. Sofonea, On the optimal control of variational-hemivariational inequalities, J. Math. Anal. Appl., 475 (2019), 364-384.  doi: 10.1016/j.jmaa.2019.02.046.  Google Scholar
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