# American Institute of Mathematical Sciences

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

 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.

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. Google Scholar [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. Google Scholar [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. Google Scholar [4] S. Chandrasekhar, An Introduction to the Study of Stellar Structures, Dover Publications, Inc., New York, N. Y., 1957. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [8] D. A. Franck-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, Second edition, Plenum Press, 1969.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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). Google Scholar [13] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [21] J.-L. Lions, Some Methods of Solving Non-Linear Boundary Value Problems, Dunod-Gauthier-Villars, Paris 1969.Google Scholar [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. Google Scholar [23] I. Peral, Multiplicity of Solutions for the p-Laplacian, Second School of Nonlinear Functional Analysis and Applications to Differential Equations, Miramare–Trieste, 1997.Google Scholar [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. Google Scholar [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. Google Scholar

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##### 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. Google Scholar [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. Google Scholar [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. Google Scholar [4] S. Chandrasekhar, An Introduction to the Study of Stellar Structures, Dover Publications, Inc., New York, N. Y., 1957. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [8] D. A. Franck-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, Second edition, Plenum Press, 1969.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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). Google Scholar [13] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [21] J.-L. Lions, Some Methods of Solving Non-Linear Boundary Value Problems, Dunod-Gauthier-Villars, Paris 1969.Google Scholar [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. Google Scholar [23] I. Peral, Multiplicity of Solutions for the p-Laplacian, Second School of Nonlinear Functional Analysis and Applications to Differential Equations, Miramare–Trieste, 1997.Google Scholar [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. Google Scholar [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. Google Scholar
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