# American Institute of Mathematical Sciences

June 2018, 23(4): 1363-1393. doi: 10.3934/dcdsb.2018155

## On optimal controls in coefficients for ill-posed non-Linear elliptic Dirichlet boundary value problems

 1 National Mining University, Department of System Analysis and Control, Yavornitskii av., 19, 49005 Dnipro, Ukraine 2 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 3 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  July 2016 Revised  January 2018 Published  April 2018

We consider an optimal control problem associated to Dirichlet boundary value problem for non-linear elliptic equation on a bounded domain $Ω$. We take the coefficient $u(x)∈ L^∞(Ω)\cap BV(Ω)$ in the main part of the non-linear differential operator as a control and in the linear part of differential operator we consider coefficients to be unbounded skew-symmetric matrix $A_{skew}∈ L^q(Ω;\mathbb{S}^N_{skew})$. We show that, in spite of unboundedness of the non-linear differential operator, the considered Dirichlet problem admits at least one weak solution and the corresponding OCP is well-possed and solvable. At the same time, optimal solutions to such problem can inherit a singular character of the matrices $A^{skew}$. We indicate two types of optimal solutions to the above problem and show that one of them can be attained by optimal solutions of regularized problems for coercive elliptic equations with bounded coefficients, using the two-parametric regularization of the initial OCP.

Citation: Olha P. Kupenko, Rosanna Manzo. On optimal controls in coefficients for ill-posed non-Linear elliptic Dirichlet boundary value problems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1363-1393. doi: 10.3934/dcdsb.2018155
##### References:
 [1] A. B. Al'shin, M. O. Korpusov and A. G. Sveshnikov, Blow-up in Nonlinear Sobolev Type Equations, De Gruyter, Berlin, 2011. [2] G. Buttazzo and P. I. Kogut, Weak optimal controls in coefficients for linear elliptic problems, Revista Matematica Complutense, 24 (2011), 83-94. doi: 10.1007/s13163-010-0030-y. [3] E. Casas, Optimal control in the coefficients of elliptic equations with state constraints, Appl. Math. Optim., 26 (1992), 21-37. doi: 10.1007/BF01218394. [4] E. Casas and L. A. Fernandez, Optimal control of quasilinear elliptic equations with non differentiable coefficients at the origin, Rev. Matematica Univ. Compl. Madrid, 4 (1991), 227-250. [5] 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. [6] M. Chicco and M. Venturino, Dirichlet problem for a divergence form elliptic equation with unbounded coefficients in an unbounded domain, Annali di Matematica Pura ed Applicata, 178 (2000), 325-338. doi: 10.1007/BF02505902. [7] R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72 (1993), 247-286. [8] 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. [9] P. Drabek and Y. Milota, Methods of Nonlinear Analysis. Applications to Differential Equations, Birkhäuser, Berlin, 2007. [10] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992. [11] H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Academie-Varlar, Berlin, 1974. [12] S. A. Gorbonos, On approximation of solutions to one class of the optimal control problems for parabolic equation with unbounded coefficients, Journal of Automation and Information Sciences, 46 (2014), 12-28. doi: 10.1615/JAutomatInfScien.v46.i9.20. [13] T. Horsin and P. I. Kogut, Optimal L2-control problem in coefficients for a linear elliptic equation. I. Existence result, Mathematical Control and Related Fields, 5 (2015), 73-96. doi: 10.3934/mcrf.2015.5.73. [14] T. Horsin and P. I. Kogut, On unbounded optimal controls in coefficients for ill-posed elliptic Dirichlet boundary value problems, Asymptotic Analysis, 98 (2016), 155-188. doi: 10.3233/ASY-161365. [15] A. Ioffe and V. Tikhomirov, Extremal Problems, North-Holland, Amsterdam, 1979. [16] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980. [17] P. I. Kogut, On Approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients, Discrete and Continuous Dynamical Systems -Series A, 34 (2014), 2105-2133. [18] P. I. Kogut, On some properties of unbounded bilinear forms associated with skew-symmetric L2(Omega)-matrices, Bulletin of Dnipro National University, Series: Modelling, 5 (2013), 84-97. [19] P. I. Kogut and G. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains: Approximation and Asymptotic Analysis, Birkhäuser, Boston, 2011. [20] O. P. Kupenko, Optimal control problems in coefficients for degenerate variational inequalities of monotone type. Ⅰ. Existence of optimal solutions, J. Computational and Appl. Mathematics, 106 (2011), 88-104. [21] O. P. Kupenko, Optimal control problems in coefficients for degenerate variational inequalities of monotone type. Ⅱ. Attainability problem, J. Computational and Appl. Mathematics, 107 (2012), 15-34. [22] O. P. Kupenko and R. Manzo, Approximation of an optimal control problem in coefficient for variational inequality with anisotropic p-Laplacian Nonlinear Differential Equations and Applications, 23 (2016), Art. 35, 18 pp. doi: 10.1007/s00030-016-0387-9. [23] J. -L. Lions, Some Methods of Solving Non-Linear Boundary Value Problems, Dunod-Gauthier-Villars, Paris, 1969. [24] J. -L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York-Berlin, 1971. [25] K. A. Lurie, Optimum control of conductivity of a fluid moving in a channel in a magnetic field, J. Appl. Math. Mech., 28 (1964), 316-327. doi: 10.1016/0021-8928(64)90165-0. [26] V. G. Mazya and I. E. Verbitsky, Form boundedness of the general second-order differential operator, Comm. Pure Appl. Math., 59 (2006), 1286-1329. doi: 10.1002/cpa.20122. [27] S. E. Pastukhova, Degenerate equations of monotone type: Lavrentiev phenomenon and attainability problems, Sbornik: Mathematics, 198 (2007), 1465-1494. [28] T. Phan, Regularity gradient estimates for weak solutions of singular quasi-linear parabolic equations, J. Differential Equations, 263 (2017), 8329-8361. doi: 10.1016/j.jde.2017.08.043. [29] O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer-Verlag, Berlin, 1984. [30] F. Punzo and A. Tesei, Uniqueness of solutions to degenerate elliptic problems with unbounded coefficients, Ann. I.H. Poincaré, 26 (2009), 2001-2024. doi: 10.1016/j.anihpc.2009.04.005. [31] T. Radice, Regularity result for nondivergence elliptic equations with unbounded coefficients, Diff. Integral Equa., 23 (2010), 989-1000. [32] T. Radice and G. Zecca, Existence and uniqueness for nonlinear elliptic equations with unbounded coefficients, Ricerche Mat., 63 (2014), 355-367. doi: 10.1007/s11587-014-0202-z. [33] T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2013. [34] M. V. Safonov, Non-divergence elliptic equations of second order with unbounded drift, Nonlinear Partial Diff. Equa. and Related Topics, 229 (2010), 211-232. [35] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, American Mathematical Society, 1997. [36] V. V. Zhikov, Weighted Sobolev spaces, Sbornik: Mathematics, 189 (1998), 27-58. [37] V. V. Zhikov, Remarks on the uniqueness of a solutionof the Dirichlet problem for second-order elliptic equations with lower-order terms, Functional Analysis and Its Applications, 38 (2004), 173-183.

show all references

##### References:
 [1] A. B. Al'shin, M. O. Korpusov and A. G. Sveshnikov, Blow-up in Nonlinear Sobolev Type Equations, De Gruyter, Berlin, 2011. [2] G. Buttazzo and P. I. Kogut, Weak optimal controls in coefficients for linear elliptic problems, Revista Matematica Complutense, 24 (2011), 83-94. doi: 10.1007/s13163-010-0030-y. [3] E. Casas, Optimal control in the coefficients of elliptic equations with state constraints, Appl. Math. Optim., 26 (1992), 21-37. doi: 10.1007/BF01218394. [4] E. Casas and L. A. Fernandez, Optimal control of quasilinear elliptic equations with non differentiable coefficients at the origin, Rev. Matematica Univ. Compl. Madrid, 4 (1991), 227-250. [5] 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. [6] M. Chicco and M. Venturino, Dirichlet problem for a divergence form elliptic equation with unbounded coefficients in an unbounded domain, Annali di Matematica Pura ed Applicata, 178 (2000), 325-338. doi: 10.1007/BF02505902. [7] R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72 (1993), 247-286. [8] 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. [9] P. Drabek and Y. Milota, Methods of Nonlinear Analysis. Applications to Differential Equations, Birkhäuser, Berlin, 2007. [10] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992. [11] H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Academie-Varlar, Berlin, 1974. [12] S. A. Gorbonos, On approximation of solutions to one class of the optimal control problems for parabolic equation with unbounded coefficients, Journal of Automation and Information Sciences, 46 (2014), 12-28. doi: 10.1615/JAutomatInfScien.v46.i9.20. [13] T. Horsin and P. I. Kogut, Optimal L2-control problem in coefficients for a linear elliptic equation. I. Existence result, Mathematical Control and Related Fields, 5 (2015), 73-96. doi: 10.3934/mcrf.2015.5.73. [14] T. Horsin and P. I. Kogut, On unbounded optimal controls in coefficients for ill-posed elliptic Dirichlet boundary value problems, Asymptotic Analysis, 98 (2016), 155-188. doi: 10.3233/ASY-161365. [15] A. Ioffe and V. Tikhomirov, Extremal Problems, North-Holland, Amsterdam, 1979. [16] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980. [17] P. I. Kogut, On Approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients, Discrete and Continuous Dynamical Systems -Series A, 34 (2014), 2105-2133. [18] P. I. Kogut, On some properties of unbounded bilinear forms associated with skew-symmetric L2(Omega)-matrices, Bulletin of Dnipro National University, Series: Modelling, 5 (2013), 84-97. [19] P. I. Kogut and G. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains: Approximation and Asymptotic Analysis, Birkhäuser, Boston, 2011. [20] O. P. Kupenko, Optimal control problems in coefficients for degenerate variational inequalities of monotone type. Ⅰ. Existence of optimal solutions, J. Computational and Appl. Mathematics, 106 (2011), 88-104. [21] O. P. Kupenko, Optimal control problems in coefficients for degenerate variational inequalities of monotone type. Ⅱ. Attainability problem, J. Computational and Appl. Mathematics, 107 (2012), 15-34. [22] O. P. Kupenko and R. Manzo, Approximation of an optimal control problem in coefficient for variational inequality with anisotropic p-Laplacian Nonlinear Differential Equations and Applications, 23 (2016), Art. 35, 18 pp. doi: 10.1007/s00030-016-0387-9. [23] J. -L. Lions, Some Methods of Solving Non-Linear Boundary Value Problems, Dunod-Gauthier-Villars, Paris, 1969. [24] J. -L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York-Berlin, 1971. [25] K. A. Lurie, Optimum control of conductivity of a fluid moving in a channel in a magnetic field, J. Appl. Math. Mech., 28 (1964), 316-327. doi: 10.1016/0021-8928(64)90165-0. [26] V. G. Mazya and I. E. Verbitsky, Form boundedness of the general second-order differential operator, Comm. Pure Appl. Math., 59 (2006), 1286-1329. doi: 10.1002/cpa.20122. [27] S. E. Pastukhova, Degenerate equations of monotone type: Lavrentiev phenomenon and attainability problems, Sbornik: Mathematics, 198 (2007), 1465-1494. [28] T. Phan, Regularity gradient estimates for weak solutions of singular quasi-linear parabolic equations, J. Differential Equations, 263 (2017), 8329-8361. doi: 10.1016/j.jde.2017.08.043. [29] O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer-Verlag, Berlin, 1984. [30] F. Punzo and A. Tesei, Uniqueness of solutions to degenerate elliptic problems with unbounded coefficients, Ann. I.H. Poincaré, 26 (2009), 2001-2024. doi: 10.1016/j.anihpc.2009.04.005. [31] T. Radice, Regularity result for nondivergence elliptic equations with unbounded coefficients, Diff. Integral Equa., 23 (2010), 989-1000. [32] T. Radice and G. Zecca, Existence and uniqueness for nonlinear elliptic equations with unbounded coefficients, Ricerche Mat., 63 (2014), 355-367. doi: 10.1007/s11587-014-0202-z. [33] T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2013. [34] M. V. Safonov, Non-divergence elliptic equations of second order with unbounded drift, Nonlinear Partial Diff. Equa. and Related Topics, 229 (2010), 211-232. [35] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, American Mathematical Society, 1997. [36] V. V. Zhikov, Weighted Sobolev spaces, Sbornik: Mathematics, 189 (1998), 27-58. [37] V. V. Zhikov, Remarks on the uniqueness of a solutionof the Dirichlet problem for second-order elliptic equations with lower-order terms, Functional Analysis and Its Applications, 38 (2004), 173-183.
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