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

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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

Abstract
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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 and 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).
[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.
[3]	T. Bagby, Quasi topologies and rational approximation, J. Func. Anal., 10 (1972), 259-268. doi: 10.1016/0022-1236(72)90025-0.
[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.
[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.
[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.
[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.
[8]	C. Calvo-Jurado and J. Casado-Diaz, Results on existence of solution for an optimal design problem, Extracta Mathematicae, 18 (2003), 263-271.
[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.
[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.
[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.
[12]	D. Daners, Domain perturbation for linear and nonlinear parabolic equations, J. Diff. Equations, 129 (1996), 358-402. doi: 10.1006/jdeq.1996.0122.
[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.
[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.
[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.
[16]	L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992.
[17]	K. J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, Cambridge, 1986.
[18]	H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Academie-Varlar, Berlin, 1974.
[19]	J. Haslinger and P. Neittaanmäki, Finite Element Approximation of Optimal Shape. Material and Topology Design, John Wiley and Sons, Chichester, 1996.
[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.
[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).
[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.
[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.
[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.
[25]	I. Lasiecka, NSF-CMBS Lecture Notes: Mathematical Control Theory of Coupled Systems of Partial Differential Equations, SIAM, 2002.
[26]	J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer Verlag, New York, 1971.
[27]	K. A. Lurie, Applied Optimal Control Theory of Distributed Systems, Plenum Press, NewYork, 1993. doi: 10.1007/978-1-4757-9262-1.
[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.
[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.
[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.
[31]	O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-87722-3.
[32]	U. Ë. Raytum, Optimal Control Problems for Elliptic Equations, Zinatne, Riga, 1989 (in Russian).
[33]	J. Sokolowski and J. P. Zolesio, Introduction to Shape Optimization, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58106-9.
[34]	D. Tiba, Lectures on the Control of Elliptic Systems, in: Lecture Notes, 32, Department of Mathematics, University of Jyväskylä, Finland, 1995.
[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).
[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.
[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).
[38]	W. P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-1-4612-1015-3.

show all references

References:

[1]	D. E. Akbarov, V. S. Melnik and V. V. Jasinskiy, Coupled Systems Control Methods, Viriy, Kyiv, 1998 (in Russian).
[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.
[3]	T. Bagby, Quasi topologies and rational approximation, J. Func. Anal., 10 (1972), 259-268. doi: 10.1016/0022-1236(72)90025-0.
[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.
[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.
[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.
[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.
[8]	C. Calvo-Jurado and J. Casado-Diaz, Results on existence of solution for an optimal design problem, Extracta Mathematicae, 18 (2003), 263-271.
[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.
[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.
[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.
[12]	D. Daners, Domain perturbation for linear and nonlinear parabolic equations, J. Diff. Equations, 129 (1996), 358-402. doi: 10.1006/jdeq.1996.0122.
[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.
[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.
[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.
[16]	L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992.
[17]	K. J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, Cambridge, 1986.
[18]	H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Academie-Varlar, Berlin, 1974.
[19]	J. Haslinger and P. Neittaanmäki, Finite Element Approximation of Optimal Shape. Material and Topology Design, John Wiley and Sons, Chichester, 1996.
[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.
[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).
[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.
[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.
[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.
[25]	I. Lasiecka, NSF-CMBS Lecture Notes: Mathematical Control Theory of Coupled Systems of Partial Differential Equations, SIAM, 2002.
[26]	J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer Verlag, New York, 1971.
[27]	K. A. Lurie, Applied Optimal Control Theory of Distributed Systems, Plenum Press, NewYork, 1993. doi: 10.1007/978-1-4757-9262-1.
[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.
[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.
[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.
[31]	O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-87722-3.
[32]	U. Ë. Raytum, Optimal Control Problems for Elliptic Equations, Zinatne, Riga, 1989 (in Russian).
[33]	J. Sokolowski and J. P. Zolesio, Introduction to Shape Optimization, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58106-9.
[34]	D. Tiba, Lectures on the Control of Elliptic Systems, in: Lecture Notes, 32, Department of Mathematics, University of Jyväskylä, Finland, 1995.
[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).
[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.
[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).
[38]	W. P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-1-4612-1015-3.

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