Abstract
In this paper we study an optimal control problem for the mixed Dirichlet–Neumann boundary value problem for the strongly non-linear elliptic equation with p-Laplace operator and \(L^1\)-nonlinearity in their right-hand side. A density of surface traction u acting on a part of boundary of open domain is taken as a boundary control. The optimal control problem is to minimize the discrepancy between a given distribution \(y_d\in L^2(\varOmega )\) and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for any admissible control. After defining a suitable functional class in which we look for solutions and assuming that this problem admits at least one feasible solution, we prove the existence of optimal pairs. In order to handle the strong non-linearity in the right-hand side of elliptic equation, we involve a special two-parametric fictitious optimization problem. We derive existence of optimal solutions to the regularized optimization problems at each \(({\varepsilon },k)\)-level of approximation and discuss the asymptotic behaviour of the optimal solutions to regularized problems as the parameters \({\varepsilon }\) and k tend to zero and infinity, respectively.
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References
Boccardo, L., Murat, F.: Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal. Theory Methods Appl. 19, 581–597 (1992)
Brezis, H., Cazenave, Th, Martel, Y., Ramiandrisoa, A.: Blow-up for \(u_t- {\varDelta }u=g(u)\) revisited. Adv. P.D.E. 1, 73–90 (1996)
Brezis, H., Vázquez, J.L.: Blow-up solutions of some nonlinear elliptic problems. Revista Matemática de la Universidad Compluense de Madrid 10(2), 443–469 (1997)
Casas, E., Fernandez, L.A.: Optimal control of quasilinear elliptic equations with non differentiable coefficients at the origin. Rev. Matematica Univ. Compl. Madr. 4(2–3), 227–250 (1991)
Casas, E., Fernandez, L.A.: Distributed controls of systems governed by a general class of quasilinear elliptic system. J. Differ. Equ. 104, 20–47 (1993)
Casas, E., Kavian, O., Puel, J.P.: Optimal control of an ill-posed elliptic semilinear equation with an exponential nonlinearity. ESAIM Control Optim. Calc. Var. 3, 361–380 (1998)
Casas, E., Kogut, P.I., Leugerin, G.: Approximation of optimal control problems in the coefficient for the \(p\)-Laplace equation I. Convergence result. SIAM J. Control Optim. 54(3), 1406–1422 (2016)
Chandrasekhar, S.: An Introduction to the Study of Stellar Structures. Dover Publishing Inc, New York (1985)
D’Apice, C., De Maio, U., Kogut, P.I.: Gap Phenomenon in the homogenization of parabolic optimal control problems. IMA J. Math. Control Inf. 25, 461–489 (2008)
D’Apice, C., De Maio, U., Kogut, P.I.: Thermistor problem: multi-dimensional modelling, optimization, and approximation. In: Proceedings of the 32nd European Conference on Modelling and Simulations, 22–25 May 2018, Wilhelmshaven, Germany, pp. 348–356 (2018)
D’Apice, C., De Maio, U., Kogut, P.I.: On optimal control of quasi-linear elliptic equation with variable \(p(x)\)-Laplacian. In: Proceedings of the 2018 International Conference of Applied and Engineering Mathematics, 4–6 July 2018, London, U.K., pp. 1–6 (2018)
D’Apice, C., De Maio, U., Kogut, P.I., Manzo, R.: On the solvability of an optimal control problem in coeffcients for ill-posed elliptic boundary value problems. Electron. J. Differ. Equ. 2014(166), 1–23 (2014)
Durante, T., Kupenko, O.P., Manzo, R.: On attainability of optimal controls in coefficients for system of Hammerstein type with anisotropic p-Laplacian. Ricerche di Matematica 66(2), 259–292 (2017)
Ferreira, R., De Pablo, A., Vazquez, J.L.: Classification of blow-up with nonlinear diffusion and localized reaction. J. Differ. Equ. 231, 195–211 (2006)
Franck-Kamenetskii, D.A.: Diffusion and Heat Transfer in Chemical Kinetics, 2nd edn. Plenum Press, New York (1969)
Fujita, H.: On the blowing up of the solutions to the Cauchy problem for \(u_t={\varDelta } u+u^{1+\alpha }\). J. Fac. Sci. Univ. Tokyo Sect. IA Math 13, 109–124 (1996)
Gallouët, T., Mignot, F., Puel, J.P.: Quelques résultats sur le problème \(-{\varDelta } u=\lambda e^u\). C. R. Acad. Sci. Paris Série I 307, 289–292 (1988)
Gelfand, I.M.: Some problems in the theory of quasi-linear equations. Am. Math. Soc. Transl. Ser. 2 29, 289–292 (1963)
Crandall, M.G., Rabinowitz, P.H.: Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems. Arch. Ration. Mech. Anal. 58, 207–218 (1975)
Joseph, D.D., Lundgren, T.S.: Quasilinear Dirichlet problems driven by positive sources. Arch. Ration. Mech. Anal. 49, 241–269 (1973)
Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)
Kogut, P.I., Leugering, G.: Optimal and approximate boundary control of an elastic body with quasistatic evolution of damage. Math. Methods Appl. Sci. 38(13), 2739–2760 (2015)
Kogut, P.I., Kupenko, O.P.: On optimal control problem for an ill-posed strongly nonlinear elliptic equation with \(p\)-Laplace operator and \(L^1\)-type of nonlinearity. Discrete Contin. Dyn. Syst. Ser. B 24(3), 1273–1295 (2019)
Kogut, P.I., Kupenko, O.P.: On approximation of an optimal control problem for ill-posed strongly nonlinear elliptic equation with \(p\)-Laplace operator. In: Advances in Dynamical Systems and Control. Springer (to appear)
Kogut, P.I., Manzo, R., Putchenko, A.O.: On approximate solutions to the Neumann elliptic boundary value problem with non-linearity of exponential type. Bound. Value Probl. 2016(1), 1–32 (2016)
Kogut, P.I., Putchenko, A.O.: On approximate solutions to one class of non-linear Dirichlet elliptic boundary value problems. Visnyk DNU. Ser. Math. Model. Dnipropetrovsk DNU 24(8), 27–55 (2016)
Kupenko, O.P., Manzo, R.: Approximation of an optimal control problem in the coefficients for variational inequality with anisotropic p-Laplacian. Nonlinear Differ. Equ. Appl. NoDEA 23, 35 (2016). https://doi.org/10.1007/s00030-016-0387-9(2016)
Kupenko, O.P., Manzo, R.: On optimal controls in coefficients for ill-posed non-linear elliptic Dirichlet boundary value problems. Discrete Contin. Dyn. Syst. Ser. B 23(4), 1363–1393 (2018)
Lions, J.-L.: Some Methods of Solving Non-Linear Boundary Value Problems. Dunod-Gauthier-Villars, Paris (1969)
Lions, J.-L., Magenes, E.: Problèmes aux Limites non Homogènes et Applications, vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunon, Paris (1968)
Mignot, F., Pue, J.P.: Sur une classe de problèmes non linéaires avec nonlinéarité positive, croissante, convexe. Commun. PDE 5(8), 791–836 (1980)
Peral, I.: Multiplicity of Solutions for the \(p\)-Laplacian, Second School of Nonlinear Functional Analysis and Applications to Differential Equations, Miramare–Trieste (1997)
Pinsky, R.G.: Existence and nonexistence of global solutions \(u_t={\varDelta } u+a(x) u^p\) in \({\mathbb{R}}^d\). J. Differ. Equ. 133, 152–177 (1997)
Roubček, T.: Relaxation in Optimization Theory and Variational Calculus, Series in Nonlinear Analysis and Applications 4. Walter de Gruyter, Berlin (1997)
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The author wishes to thank Prof P.I. Kogut for the useful discussions and suggestions.
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Manzo, R. On Neumann boundary control problem for ill-posed strongly nonlinear elliptic equation with p-Laplace operator and \(L^1\)-type of nonlinearity. Ricerche mat 68, 769–802 (2019). https://doi.org/10.1007/s11587-019-00439-x
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DOI: https://doi.org/10.1007/s11587-019-00439-x
Keywords
- Approximation approach
- Existence result
- Optimal control
- p-Laplace operator
- Elliptic equation
- Fictitious control