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November  2015, 20(9): 2967-2992. doi: 10.3934/dcdsb.2015.20.2967

Shape stability of optimal control problems in coefficients for coupled system of Hammerstein type

Olha P. Kupenko 1, and  Rosanna Manzo 2,

1. 

Dnipropetrovsk Mining University, Department of System Analysis and Control, Karl Marks av., 19, 49005 Dnipropetrovsk, Ukraine
2. 

Università degli Studi di Salerno, Dipartimento di Ingegneria dell'Informazione, Ingegneria Elettrica e Matematica Applicata, Via Giovanni Paolo II, 132, 84084 Fisciano (SA), Italy

Received  April 2014 Revised  October 2014 Published  September 2015

In this paper we consider an optimal control problem (OCP) for the coupled system of a nonlinear monotone Dirichlet problem with matrix-valued $L^\infty(\Omega;\mathbb{R}^{N\times N} )$-controls in coefficients and a nonlinear equation of Hammerstein type. Since problems of this type have no solutions in general, we make a special assumption on the coefficients of the state equation and introduce the class of so-called solenoidal admissible controls. Using the direct method in calculus of variations, we prove the existence of an optimal control. We also study the stability of the optimal control problem with respect to the domain perturbation. In particular, we derive the sufficient conditions of the Mosco-stability for the given class of OCPs.
Keywords: control in coefficients, Nonlinear monotone Dirichlet problem, domain perturbation., equation of Hammerstein type.
Mathematics Subject Classification: Primary: 49J20, 35J57; Secondary: 49J45, 93C7.
Citation: Olha P. Kupenko, Rosanna Manzo. Shape stability of optimal control problems in coefficients for coupled system of Hammerstein type. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2967-2992. doi: 10.3934/dcdsb.2015.20.2967
References:
[1]

D. E. Akbarov, V. S. Melnik and V. V. Jasinskiy, Coupled Systems Control Methods, Viriy, Kyiv, 1998 (in Russian). Google Scholar

[2]

G. Allaire, Shape Optimization by the Homogenization Method, Applied Mathematical Sciences, vol. 146, Springer, New York, 2002. doi: 10.1007/978-1-4684-9286-6.  Google Scholar

[3]

T. Bagby, Quasi topologies and rational approximation, J. Func. Anal., 10 (1972), 259-268. doi: 10.1016/0022-1236(72)90025-0.  Google Scholar

[4]

D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems, Birkhäuser, Boston: in Progress in Nonlinear Differential Equations and their Applications, Vol. 65, 2005.  Google Scholar

[5]

D. Bucur and P. Trebeschi, Shape optimization problems governed by nonlinear state equations, Proc. Roy. Soc. Edinburgh, Ser. A, 128 (1998), 943-963. doi: 10.1017/S0308210500030006.  Google Scholar

[6]

D. Bucur and J. P. Zolésio, $N$-Dimensional Shape Optimization under Capacitary Constraints, J. Differential Equations, 123 (1995), 504-522. doi: 10.1006/jdeq.1995.1171.  Google Scholar

[7]

G. Buttazzo and G. Dal Maso, Shape optimization for Dirichlet problems. Relaxed SIS and optimally conditions, Appl. Math. Optim., 23 (1991), 17-49. doi: 10.1007/BF01442391.  Google Scholar

[8]

C. Calvo-Jurado and J. Casado-Diaz, Results on existence of solution for an optimal design problem, Extracta Mathematicae, 18 (2003), 263-271.  Google Scholar

[9]

G. Dal Maso and F. Murat, Asymptotic behaviour and correctors for Dirichlet problem in perforated domains with homogeneous monotone operators, Ann. Scuola Norm. Sup. Pisa Cl.Sci., 24 (1997), 239-290.  Google Scholar

[10]

G. Dal Maso, F. Ebobisse and M. Ponsiglione, A stability result for nonlinear Neumann problems under boundary variations, J. Math. Pures Appl., 82 (2003), 503-532. doi: 10.1016/S0021-7824(03)00014-X.  Google Scholar

[11]

E. N. Dancer, The effect of domains shape on the number of positive solutions of certain nonlinear equations, J. Diff. Equations, 87 (1990), 316-339. doi: 10.1016/0022-0396(90)90005-A.  Google Scholar

[12]

D. Daners, Domain perturbation for linear and nonlinear parabolic equations, J. Diff. Equations, 129 (1996), 358-402. doi: 10.1006/jdeq.1996.0122.  Google Scholar

[13]

C. D'Apice, U. De Maio and O. P. Kogut, On shape stability of Dirichlet optimal control problems in coefficients for nonlinear elliptic equations, Advances in Differential Equations, 15 (2010), 689-720.  Google Scholar

[14]

C. D'Apice, U. De Maio and O. P. Kogut, Optimal control problems in coefficients for degenerate equations of monotone type: shape stability and attainability problems, SIAM Journal on Control and Optimization, 50 (2012), 1174-1199. doi: 10.1137/100815761.  Google Scholar

[15]

C. D'Apice, U. De Maio and P. I. Kogut, Suboptimal boundary control for elliptic equations in critically perforated domains, Ann. Inst. H. Poincaré Anal. Non Line'aire, 25 (2008), 1073-1101. doi: 10.1016/j.anihpc.2007.07.001.  Google Scholar

[16]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992.  Google Scholar

[17]

K. J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, Cambridge, 1986.  Google Scholar

[18]

H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Academie-Varlar, Berlin, 1974.  Google Scholar

[19]

J. Haslinger and P. Neittaanmäki, Finite Element Approximation of Optimal Shape. Material and Topology Design, John Wiley and Sons, Chichester, 1996.  Google Scholar

[20]

J. Heinonen, T. Kilpelainen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Unabridged republication of the 1993 original. Dover Publications, Inc., Mineola, NY, 2006.  Google Scholar

[21]

V. I. Ivanenko and V. S. Mel'nik, Varational Metods in Control Problems for Systems with Distributed Parameters, Naukova Dumka, Kiev, 1988 (in Russian).  Google Scholar

[22]

O. P. Kogut, Qualitative Analysis of one Class of Optimization Problems for Nonlinear Elliptic Operators, PhD thesis at Gluskov Institute of Cyberentics NAS Kiev, 2010. Google Scholar

[23]

P. I. Kogut and G. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains, Series: Systems and Control, Birkhäuser Verlag, 2011. doi: 10.1007/978-0-8176-8149-4.  Google Scholar

[24]

O. P. Kupenko, Optimal control problems in coefficients for degenerate variational inequalities of monotone type.I Existence of solutions, Journal of Computational & Applied Mathematics, 106 (2011), 88-104. Google Scholar

[25]

I. Lasiecka, NSF-CMBS Lecture Notes: Mathematical Control Theory of Coupled Systems of Partial Differential Equations, SIAM, 2002. Google Scholar

[26]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer Verlag, New York, 1971.  Google Scholar

[27]

K. A. Lurie, Applied Optimal Control Theory of Distributed Systems, Plenum Press, NewYork, 1993. doi: 10.1007/978-1-4757-9262-1.  Google Scholar

[28]

U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. Math., 3 (1969), 510-585. doi: 10.1016/0001-8708(69)90009-7.  Google Scholar

[29]

F. Murat, Un contre-exemple pour le probleme du controle dans les coefficients, C. R. Acad. Sci. Paris Ser. A-B, 273 (1971), A708-A711.  Google Scholar

[30]

F. Murat and L. Tartar, H-convergence. Topics in the mathematical modelling of composite materials, Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 31 (1997), 21-43.  Google Scholar

[31]

O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-87722-3.  Google Scholar

[32]

U. Ë. Raytum, Optimal Control Problems for Elliptic Equations, Zinatne, Riga, 1989 (in Russian).  Google Scholar

[33]

J. Sokolowski and J. P. Zolesio, Introduction to Shape Optimization, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58106-9.  Google Scholar

[34]

D. Tiba, Lectures on the Control of Elliptic Systems, in: Lecture Notes, 32, Department of Mathematics, University of Jyväskylä, Finland, 1995. Google Scholar

[35]

M. M. Vainberg and I. M. Lavrentieff, Nonlinear equations of hammerstein type with potential and monotone operators in banach spaces, Matematicheskij Sbornik, no. 3, 87 (1972), 324-337 (in Russian).  Google Scholar

[36]

M. Z. Zgurovski and V. S. Mel'nik, Nonlinear Analysis and Control of Physical Processes and Fields, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-642-18770-4.  Google Scholar

[37]

M. Z. Zgurovski, V. S. Mel'nik and A. N. Novikov, Applied Methods for Analysis and Control of Nonlinear Processes and Fields, Naukova Dumka, Kiev, 2004 (in Russian). Google Scholar

[38]

W. P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

show all references

References:
[1]

D. E. Akbarov, V. S. Melnik and V. V. Jasinskiy, Coupled Systems Control Methods, Viriy, Kyiv, 1998 (in Russian). Google Scholar

[2]

G. Allaire, Shape Optimization by the Homogenization Method, Applied Mathematical Sciences, vol. 146, Springer, New York, 2002. doi: 10.1007/978-1-4684-9286-6.  Google Scholar

[3]

T. Bagby, Quasi topologies and rational approximation, J. Func. Anal., 10 (1972), 259-268. doi: 10.1016/0022-1236(72)90025-0.  Google Scholar

[4]

D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems, Birkhäuser, Boston: in Progress in Nonlinear Differential Equations and their Applications, Vol. 65, 2005.  Google Scholar

[5]

D. Bucur and P. Trebeschi, Shape optimization problems governed by nonlinear state equations, Proc. Roy. Soc. Edinburgh, Ser. A, 128 (1998), 943-963. doi: 10.1017/S0308210500030006.  Google Scholar

[6]

D. Bucur and J. P. Zolésio, $N$-Dimensional Shape Optimization under Capacitary Constraints, J. Differential Equations, 123 (1995), 504-522. doi: 10.1006/jdeq.1995.1171.  Google Scholar

[7]

G. Buttazzo and G. Dal Maso, Shape optimization for Dirichlet problems. Relaxed SIS and optimally conditions, Appl. Math. Optim., 23 (1991), 17-49. doi: 10.1007/BF01442391.  Google Scholar

[8]

C. Calvo-Jurado and J. Casado-Diaz, Results on existence of solution for an optimal design problem, Extracta Mathematicae, 18 (2003), 263-271.  Google Scholar

[9]

G. Dal Maso and F. Murat, Asymptotic behaviour and correctors for Dirichlet problem in perforated domains with homogeneous monotone operators, Ann. Scuola Norm. Sup. Pisa Cl.Sci., 24 (1997), 239-290.  Google Scholar

[10]

G. Dal Maso, F. Ebobisse and M. Ponsiglione, A stability result for nonlinear Neumann problems under boundary variations, J. Math. Pures Appl., 82 (2003), 503-532. doi: 10.1016/S0021-7824(03)00014-X.  Google Scholar

[11]

E. N. Dancer, The effect of domains shape on the number of positive solutions of certain nonlinear equations, J. Diff. Equations, 87 (1990), 316-339. doi: 10.1016/0022-0396(90)90005-A.  Google Scholar

[12]

D. Daners, Domain perturbation for linear and nonlinear parabolic equations, J. Diff. Equations, 129 (1996), 358-402. doi: 10.1006/jdeq.1996.0122.  Google Scholar

[13]

C. D'Apice, U. De Maio and O. P. Kogut, On shape stability of Dirichlet optimal control problems in coefficients for nonlinear elliptic equations, Advances in Differential Equations, 15 (2010), 689-720.  Google Scholar

[14]

C. D'Apice, U. De Maio and O. P. Kogut, Optimal control problems in coefficients for degenerate equations of monotone type: shape stability and attainability problems, SIAM Journal on Control and Optimization, 50 (2012), 1174-1199. doi: 10.1137/100815761.  Google Scholar

[15]

C. D'Apice, U. De Maio and P. I. Kogut, Suboptimal boundary control for elliptic equations in critically perforated domains, Ann. Inst. H. Poincaré Anal. Non Line'aire, 25 (2008), 1073-1101. doi: 10.1016/j.anihpc.2007.07.001.  Google Scholar

[16]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992.  Google Scholar

[17]

K. J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, Cambridge, 1986.  Google Scholar

[18]

H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Academie-Varlar, Berlin, 1974.  Google Scholar

[19]

J. Haslinger and P. Neittaanmäki, Finite Element Approximation of Optimal Shape. Material and Topology Design, John Wiley and Sons, Chichester, 1996.  Google Scholar

[20]

J. Heinonen, T. Kilpelainen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Unabridged republication of the 1993 original. Dover Publications, Inc., Mineola, NY, 2006.  Google Scholar

[21]

V. I. Ivanenko and V. S. Mel'nik, Varational Metods in Control Problems for Systems with Distributed Parameters, Naukova Dumka, Kiev, 1988 (in Russian).  Google Scholar

[22]

O. P. Kogut, Qualitative Analysis of one Class of Optimization Problems for Nonlinear Elliptic Operators, PhD thesis at Gluskov Institute of Cyberentics NAS Kiev, 2010. Google Scholar

[23]

P. I. Kogut and G. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains, Series: Systems and Control, Birkhäuser Verlag, 2011. doi: 10.1007/978-0-8176-8149-4.  Google Scholar

[24]

O. P. Kupenko, Optimal control problems in coefficients for degenerate variational inequalities of monotone type.I Existence of solutions, Journal of Computational & Applied Mathematics, 106 (2011), 88-104. Google Scholar

[25]

I. Lasiecka, NSF-CMBS Lecture Notes: Mathematical Control Theory of Coupled Systems of Partial Differential Equations, SIAM, 2002. Google Scholar

[26]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer Verlag, New York, 1971.  Google Scholar

[27]

K. A. Lurie, Applied Optimal Control Theory of Distributed Systems, Plenum Press, NewYork, 1993. doi: 10.1007/978-1-4757-9262-1.  Google Scholar

[28]

U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. Math., 3 (1969), 510-585. doi: 10.1016/0001-8708(69)90009-7.  Google Scholar

[29]

F. Murat, Un contre-exemple pour le probleme du controle dans les coefficients, C. R. Acad. Sci. Paris Ser. A-B, 273 (1971), A708-A711.  Google Scholar

[30]

F. Murat and L. Tartar, H-convergence. Topics in the mathematical modelling of composite materials, Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 31 (1997), 21-43.  Google Scholar

[31]

O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-87722-3.  Google Scholar

[32]

U. Ë. Raytum, Optimal Control Problems for Elliptic Equations, Zinatne, Riga, 1989 (in Russian).  Google Scholar

[33]

J. Sokolowski and J. P. Zolesio, Introduction to Shape Optimization, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58106-9.  Google Scholar

[34]

D. Tiba, Lectures on the Control of Elliptic Systems, in: Lecture Notes, 32, Department of Mathematics, University of Jyväskylä, Finland, 1995. Google Scholar

[35]

M. M. Vainberg and I. M. Lavrentieff, Nonlinear equations of hammerstein type with potential and monotone operators in banach spaces, Matematicheskij Sbornik, no. 3, 87 (1972), 324-337 (in Russian).  Google Scholar

[36]

M. Z. Zgurovski and V. S. Mel'nik, Nonlinear Analysis and Control of Physical Processes and Fields, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-642-18770-4.  Google Scholar

[37]

M. Z. Zgurovski, V. S. Mel'nik and A. N. Novikov, Applied Methods for Analysis and Control of Nonlinear Processes and Fields, Naukova Dumka, Kiev, 2004 (in Russian). Google Scholar

[38]

W. P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

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