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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

E. CasasP. 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

[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. Google Scholar

[7]

R. CoifmanP.-L. LionsY. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72 (1993), 247-286. Google Scholar

[8]

C. D'ApiceU. 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

[9]

P. Drabek and Y. Milota, Methods of Nonlinear Analysis. Applications to Differential Equations, Birkhäuser, Berlin, 2007. Google Scholar

[10]

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

[11]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

A. Ioffe and V. Tikhomirov, Extremal Problems, North-Holland, Amsterdam, 1979.Google Scholar

[16]

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

P. I. Kogut and G. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains: Approximation and Asymptotic Analysis, Birkhäuser, Boston, 2011. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

J. -L. Lions, Some Methods of Solving Non-Linear Boundary Value Problems, Dunod-Gauthier-Villars, Paris, 1969.Google Scholar

[24]

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

[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. Google Scholar

[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. Google Scholar

[27]

S. E. Pastukhova, Degenerate equations of monotone type: Lavrentiev phenomenon and attainability problems, Sbornik: Mathematics, 198 (2007), 1465-1494. Google Scholar

[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. Google Scholar

[29]

O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer-Verlag, Berlin, 1984. Google Scholar

[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. Google Scholar

[31]

T. Radice, Regularity result for nondivergence elliptic equations with unbounded coefficients, Diff. Integral Equa., 23 (2010), 989-1000. Google Scholar

[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. Google Scholar

[33]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2013. Google Scholar

[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. Google Scholar

[35]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, American Mathematical Society, 1997. Google Scholar

[36]

V. V. Zhikov, Weighted Sobolev spaces, Sbornik: Mathematics, 189 (1998), 27-58. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

E. CasasP. 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

[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. Google Scholar

[7]

R. CoifmanP.-L. LionsY. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72 (1993), 247-286. Google Scholar

[8]

C. D'ApiceU. 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

[9]

P. Drabek and Y. Milota, Methods of Nonlinear Analysis. Applications to Differential Equations, Birkhäuser, Berlin, 2007. Google Scholar

[10]

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

[11]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

A. Ioffe and V. Tikhomirov, Extremal Problems, North-Holland, Amsterdam, 1979.Google Scholar

[16]

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

P. I. Kogut and G. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains: Approximation and Asymptotic Analysis, Birkhäuser, Boston, 2011. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

J. -L. Lions, Some Methods of Solving Non-Linear Boundary Value Problems, Dunod-Gauthier-Villars, Paris, 1969.Google Scholar

[24]

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

[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. Google Scholar

[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. Google Scholar

[27]

S. E. Pastukhova, Degenerate equations of monotone type: Lavrentiev phenomenon and attainability problems, Sbornik: Mathematics, 198 (2007), 1465-1494. Google Scholar

[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. Google Scholar

[29]

O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer-Verlag, Berlin, 1984. Google Scholar

[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. Google Scholar

[31]

T. Radice, Regularity result for nondivergence elliptic equations with unbounded coefficients, Diff. Integral Equa., 23 (2010), 989-1000. Google Scholar

[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. Google Scholar

[33]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2013. Google Scholar

[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. Google Scholar

[35]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, American Mathematical Society, 1997. Google Scholar

[36]

V. V. Zhikov, Weighted Sobolev spaces, Sbornik: Mathematics, 189 (1998), 27-58. Google Scholar

[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. Google Scholar

[1]

Antonio Greco, Giovanni Porru. Optimization problems for the energy integral of p-Laplace equations. Conference Publications, 2013, 2013 (special) : 301-310. doi: 10.3934/proc.2013.2013.301

[2]

Ryuji Kajikiya. Nonradial least energy solutions of the p-Laplace elliptic equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 547-561. doi: 10.3934/dcds.2018024

[3]

Arrigo Cellina. The regularity of solutions to some variational problems, including the p-Laplace equation for 3≤p < 4. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4071-4085. doi: 10.3934/dcds.2018177

[4]

Yangrong Li, Jinyan Yin. Existence, regularity and approximation of global attractors for weakly dissipative p-Laplace equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1939-1957. doi: 10.3934/dcdss.2016079

[5]

Mikhail D. Surnachev, Vasily V. Zhikov. On existence and uniqueness classes for the Cauchy problem for parabolic equations of the p-Laplace type. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1783-1812. doi: 10.3934/cpaa.2013.12.1783

[6]

Harbir Antil, Mahamadi Warma. Optimal control of the coefficient for the regional fractional $p$-Laplace equation: Approximation and convergence. Mathematical Control & Related Fields, 2019, 9 (1) : 1-38. doi: 10.3934/mcrf.2019001

[7]

Joachim Naumann. On the existence of weak solutions of an unsteady p-Laplace thermistor system with strictly monotone electrical conductivities. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 837-852. doi: 10.3934/dcdss.2017042

[8]

Patrizia Pucci, Raffaella Servadei. Nonexistence for $p$--Laplace equations with singular weights. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1421-1438. doi: 10.3934/cpaa.2010.9.1421

[9]

Salvatore A. Marano, Sunra J. N. Mosconi. Some recent results on the Dirichlet problem for $(p, q)$-Laplace equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 279-291. doi: 10.3934/dcdss.2018015

[10]

Vladimir Bobkov, Mieko Tanaka. Remarks on minimizers for (p, q)-Laplace equations with two parameters. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1219-1253. doi: 10.3934/cpaa.2018059

[11]

Zheng Zhou. Layered solutions in $R^2$ for a class of $p$-Laplace equations. Communications on Pure & Applied Analysis, 2010, 9 (3) : 819-837. doi: 10.3934/cpaa.2010.9.819

[12]

Fahd Jarad, Thabet Abdeljawad. Generalized fractional derivatives and Laplace transform. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 709-722. doi: 10.3934/dcdss.2020039

[13]

Eun Bee Choi, Yun-Ho Kim. Existence of nontrivial solutions for equations of $p(x)$-Laplace type without Ambrosetti and Rabinowitz condition. Conference Publications, 2015, 2015 (special) : 276-286. doi: 10.3934/proc.2015.0276

[14]

Amelia Álvarez, José-Luis Bravo, Manuel Fernández. The number of limit cycles for generalized Abel equations with periodic coefficients of definite sign. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1493-1501. doi: 10.3934/cpaa.2009.8.1493

[15]

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

[16]

L. Cherfils, Y. Il'yasov. On the stationary solutions of generalized reaction diffusion equations with $p\& q$-Laplacian. Communications on Pure & Applied Analysis, 2005, 4 (1) : 9-22. doi: 10.3934/cpaa.2005.4.9

[17]

Adam Lipowski, Bogdan Przeradzki, Katarzyna Szymańska-Dębowska. Periodic solutions to differential equations with a generalized p-Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2593-2601. doi: 10.3934/dcdsb.2014.19.2593

[18]

Gabriella Zecca. An optimal control problem for some nonlinear elliptic equations with unbounded coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1393-1409. doi: 10.3934/dcdsb.2019021

[19]

Chao Zhang, Xia Zhang, Shulin Zhou. Gradient estimates for the strong $p(x)$-Laplace equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4109-4129. doi: 10.3934/dcds.2017175

[20]

Pavel Drábek, Martina Langerová. Impulsive control of conservative periodic equations and systems: Variational approach. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3789-3802. doi: 10.3934/dcds.2018164

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (62)
  • HTML views (144)
  • Cited by (0)

Other articles
by authors

[Back to Top]