On optimal control problem for an ill-posed strongly nonlinear elliptic equation with p-Laplace operator and L^1-type of nonlinearity

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March 2019, 24(3): 1273-1295. doi: 10.3934/dcdsb.2019016

On optimal control problem for an ill-posed strongly nonlinear elliptic equation with $p$-Laplace operator and $L^1$-type of nonlinearity

Peter I. Kogut ^1, and Olha P. Kupenko ^2,3,

1.	Oles Honchar Dnipro National University, Department of Differential Equations, Gagarin av., 72, 49010 Dnipro, Ukraine
2.	Dnipro University of Technology, Department of System Analysis and Control, Yavornitskii av., 19, 49005 Dnipro, Ukraine
3.	Institute for Applied System Analysis, National Academy of Sciences and Ministry of Education and Science of Ukraine, Peremogy av., 37/35, IASA, 03056 Kyiv, Ukraine

To the memory of our big Friend and Teacher V. S. Mel'nik

Received December 2017 Revised March 2018 Published January 2019

We study an optimal control problem for one class of non-linear elliptic equations with $p$-Laplace operator and $L^1$-nonlinearity. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for any given control. After defining a suitable functional class in which we look for solutions, we reformulate the original problem and prove the existence of optimal pairs. In order to ensure the validity of such reformulation, we provide its substantiation using a special family of fictitious optimal control problems. The idea to involve the fictitious optimization problems was mainly inspired by the brilliant book of V.S. Mel'nik and V.I. Ivanenko "Variational Methods in Control Problems for the Systems with Distributed Parameters", Kyiv, 1998.

Keywords: Existence result, optimal control, $p$-Laplace operator, elliptic equation, fictitious control.

Mathematics Subject Classification: Primary: 49J20, 49K20; Secondary: 58J37.

Citation: Peter I. Kogut, Olha P. Kupenko. On optimal control problem for an ill-posed strongly nonlinear elliptic equation with $p$-Laplace operator and $L^1$-type of nonlinearity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1273-1295. doi: 10.3934/dcdsb.2019016

References:

[1]	L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal., Theory, Methods, Appl., 19 (1992), 581-597. doi: 10.1016/0362-546X(92)90023-8.
[2]	E. Casas, O. Kavian and J. P. Puel, Optimal control of an ill-posed elliptic semilinear equation with an exponential nonlinearity, ESAIM: Control, Optimization and Calculus of Variations, 3 (1998), 361-380. doi: 10.1051/cocv:1998116.
[3]	E. Casas, P. I. Kogut and G. Leugering, Approximation of optimal control problems in the coefficient for the $p$-Laplace equation. I. Convergence result, SIAM Journal on Control and Optimization, 54 (2016), 1406-1422. doi: 10.1137/15M1028108.
[4]	S. Chandrasekhar, An Introduction to the Study of Stellar Structures, Dover Publications, Inc., New York, N. Y., 1957.
[5]	M. G. Crandall and P. H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rational Mech. Anal., 58 (1975), 207-218. doi: 10.1007/BF00280741.
[6]	J. Dolbeault and R. Stańczy, Non-existence and uniqueness results for supercritical semilinear elliptic equations, Annales Henri Poincaré, 10 (2010), 1311-1333. doi: 10.1007/s00023-009-0016-9.
[7]	R. Ferreira, A. De Pablo and J. L. Vazquez, Classification of blow-up with nonlinear diffusion and localized reaction, J. Differential Equations, 231 (2006), 195-211. doi: 10.1016/j.jde.2006.04.017.
[8]	D. A. Franck-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, Second edition, Plenum Press, 1969.
[9]	H. Fujita, On the blowing up of the solutions to the Cauchy problem for $u_t = Δ u+u^{1+α}$, J. Fac. Sci. Univ. Tokyo Sect. IA, Math., 13 (1996), 109-124.
[10]	T. Gallouët, F. Mignot and J. P. Puel, Quelques résultats sur le problème $-Δ u = λ e^u$, C. R. Acad. Sci. Paris, Série I, 307 (1988), 289–292.
[11]	I. M. Gelfand, Some problems in the theory of quasi-linear equations, Amer. Math. Soc. Transl., Ser. 2, 29 (1963), 295–381. doi: 10.1090/trans2/029/12.
[12]	V. I. Ivanenko and V. S. Mel'nik, Variational Methods in Control Problems for the Systems with Distributed Parameters, Naukova Dumka, Kyiv, 1988 (in Russian).
[13]	D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.
[14]	P. I. Kogut and G. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains. Approximation and Asymptotic Analysis, Series: Systems and Control, Birkhäuser, Boston, 2011. doi: 10.1007/978-0-8176-8149-4.
[15]	P. I. Kogut, R. Manzo and A. O. Putchenko, On approximate solutions to the Neumann elliptic boundary value problem with non-linearity of exponential type, Boundary Value Problems, 2016 (2016), 1-32. doi: 10.1186/s13661-016-0717-1.
[16]	P. I. Kogut and A. O. Putchenko, On approximate solutions to one class of non-linear Dirichlet elliptic boundary value problems, Visnyk DNU. Series: Mathematical Modelling, Dnipropetrovsk: DNU, 24 (2016), 27-25.
[17]	P. I. Kogut and V. S. Mel'nik, On one class of extremum problems for nonlinear operator systems, Cybern. Syst. Anal., 34 (1998), 894-904.
[18]	P. I. Kogut and V. S. Mel'nik, On weak compactness of bounded sets in Banach and locally convex spaces, Ukrainian Mathematical Journal, 52 (2001), 837-846. doi: 10.1007/BF02591778.
[19]	P. I. Kogut and O. P. Kupenko, On attainability of optimal solutions for linear elliptic equations with unbounded coefficients, Visnyk DNU. Series: Mathematical Modelling, Dnipropetrovsk: DNU, 20 (2012), 63-82.
[20]	O. P. Kupenko and R. Manzo, On optimal controls in coefficients for ill-posed non-linear elliptic Dirichlet boundary value problems, Discrete and Continuous Dynamic Systems. Series B, 23 (2018), 1363-1393. doi: 10.3934/dcdsb.2018155.
[21]	J.-L. Lions, Some Methods of Solving Non-Linear Boundary Value Problems, Dunod-Gauthier-Villars, Paris 1969.
[22]	F. Mignot and J. P. Puel, Sur une classe de problémes non linéaires avec nonlinéarité positive, croissante, convexe, Comm. in PDE, 5 (1980), 791-836. doi: 10.1080/03605308008820155.
[23]	I. Peral, Multiplicity of Solutions for the p-Laplacian, Second School of Nonlinear Functional Analysis and Applications to Differential Equations, Miramare–Trieste, 1997.
[24]	R. G. Pinsky, Existence and Nonexistence of global solutions $u_t=Δ u+a(x) u^p$ in $\mathbb{R}^d$, J. of Differential Equations, 133 (1997), 152-177. doi: 10.1006/jdeq.1996.3196.
[25]	D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math., 21 (1972), 979-1000. doi: 10.1512/iumj.1972.21.21079.

show all references

References:

[1]	L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal., Theory, Methods, Appl., 19 (1992), 581-597. doi: 10.1016/0362-546X(92)90023-8.
[2]	E. Casas, O. Kavian and J. P. Puel, Optimal control of an ill-posed elliptic semilinear equation with an exponential nonlinearity, ESAIM: Control, Optimization and Calculus of Variations, 3 (1998), 361-380. doi: 10.1051/cocv:1998116.
[3]	E. Casas, P. I. Kogut and G. Leugering, Approximation of optimal control problems in the coefficient for the $p$-Laplace equation. I. Convergence result, SIAM Journal on Control and Optimization, 54 (2016), 1406-1422. doi: 10.1137/15M1028108.
[4]	S. Chandrasekhar, An Introduction to the Study of Stellar Structures, Dover Publications, Inc., New York, N. Y., 1957.
[5]	M. G. Crandall and P. H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rational Mech. Anal., 58 (1975), 207-218. doi: 10.1007/BF00280741.
[6]	J. Dolbeault and R. Stańczy, Non-existence and uniqueness results for supercritical semilinear elliptic equations, Annales Henri Poincaré, 10 (2010), 1311-1333. doi: 10.1007/s00023-009-0016-9.
[7]	R. Ferreira, A. De Pablo and J. L. Vazquez, Classification of blow-up with nonlinear diffusion and localized reaction, J. Differential Equations, 231 (2006), 195-211. doi: 10.1016/j.jde.2006.04.017.
[8]	D. A. Franck-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, Second edition, Plenum Press, 1969.
[9]	H. Fujita, On the blowing up of the solutions to the Cauchy problem for $u_t = Δ u+u^{1+α}$, J. Fac. Sci. Univ. Tokyo Sect. IA, Math., 13 (1996), 109-124.
[10]	T. Gallouët, F. Mignot and J. P. Puel, Quelques résultats sur le problème $-Δ u = λ e^u$, C. R. Acad. Sci. Paris, Série I, 307 (1988), 289–292.
[11]	I. M. Gelfand, Some problems in the theory of quasi-linear equations, Amer. Math. Soc. Transl., Ser. 2, 29 (1963), 295–381. doi: 10.1090/trans2/029/12.
[12]	V. I. Ivanenko and V. S. Mel'nik, Variational Methods in Control Problems for the Systems with Distributed Parameters, Naukova Dumka, Kyiv, 1988 (in Russian).
[13]	D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.
[14]	P. I. Kogut and G. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains. Approximation and Asymptotic Analysis, Series: Systems and Control, Birkhäuser, Boston, 2011. doi: 10.1007/978-0-8176-8149-4.
[15]	P. I. Kogut, R. Manzo and A. O. Putchenko, On approximate solutions to the Neumann elliptic boundary value problem with non-linearity of exponential type, Boundary Value Problems, 2016 (2016), 1-32. doi: 10.1186/s13661-016-0717-1.
[16]	P. I. Kogut and A. O. Putchenko, On approximate solutions to one class of non-linear Dirichlet elliptic boundary value problems, Visnyk DNU. Series: Mathematical Modelling, Dnipropetrovsk: DNU, 24 (2016), 27-25.
[17]	P. I. Kogut and V. S. Mel'nik, On one class of extremum problems for nonlinear operator systems, Cybern. Syst. Anal., 34 (1998), 894-904.
[18]	P. I. Kogut and V. S. Mel'nik, On weak compactness of bounded sets in Banach and locally convex spaces, Ukrainian Mathematical Journal, 52 (2001), 837-846. doi: 10.1007/BF02591778.
[19]	P. I. Kogut and O. P. Kupenko, On attainability of optimal solutions for linear elliptic equations with unbounded coefficients, Visnyk DNU. Series: Mathematical Modelling, Dnipropetrovsk: DNU, 20 (2012), 63-82.
[20]	O. P. Kupenko and R. Manzo, On optimal controls in coefficients for ill-posed non-linear elliptic Dirichlet boundary value problems, Discrete and Continuous Dynamic Systems. Series B, 23 (2018), 1363-1393. doi: 10.3934/dcdsb.2018155.
[21]	J.-L. Lions, Some Methods of Solving Non-Linear Boundary Value Problems, Dunod-Gauthier-Villars, Paris 1969.
[22]	F. Mignot and J. P. Puel, Sur une classe de problémes non linéaires avec nonlinéarité positive, croissante, convexe, Comm. in PDE, 5 (1980), 791-836. doi: 10.1080/03605308008820155.
[23]	I. Peral, Multiplicity of Solutions for the p-Laplacian, Second School of Nonlinear Functional Analysis and Applications to Differential Equations, Miramare–Trieste, 1997.
[24]	R. G. Pinsky, Existence and Nonexistence of global solutions $u_t=Δ u+a(x) u^p$ in $\mathbb{R}^d$, J. of Differential Equations, 133 (1997), 152-177. doi: 10.1006/jdeq.1996.3196.
[25]	D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math., 21 (1972), 979-1000. doi: 10.1512/iumj.1972.21.21079.

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