# American Institute of Mathematical Sciences

• Previous Article
Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain
• MCRF Home
• This Issue
• Next Article
Numerical methods for dividend optimization using regime-switching jump-diffusion models
March  2011, 1(1): 41-59. doi: 10.3934/mcrf.2011.1.41

## Cesari-type conditions for semilinear elliptic equation with leading term containing controls

 1 School of Mathematical Sciences, Fudan University, Shanghai 200433, China 2 School of Mathematical Sciences and LMNS, Fudan University, Shanghai 200433

Received  November 2010 Revised  February 2011 Published  March 2011

An optimal control problem governed by semilinear elliptic partial differential equation is considered. The equation is in divergence form with the leading term containing controls. By studying the $G$-closure of the leading term, an existence result is established under a Cesari-type condition.
Citation: Bo Li, Hongwei Lou. Cesari-type conditions for semilinear elliptic equation with leading term containing controls. Mathematical Control & Related Fields, 2011, 1 (1) : 41-59. doi: 10.3934/mcrf.2011.1.41
##### References:
 [1] G. Allaire, "Shape Optimization by the Homogenization Method,", Springer, (2002). Google Scholar [2] A. Bensoussan, J. L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures,", North-Holland Company, (1978). Google Scholar [3] E. Cabib and G. Dal Maso, On a class of optimum problems in structural design,, J. Optim. Theory Appl., 56 (1988), 39. doi: 10.1007/BF00938526. Google Scholar [4] L. Cesari, "Optimization Theory and Applications, Problems with Ordinary Equations,", Applications of Mathematics \textbf{17}, 17 (1983). Google Scholar [5] A. F. Filippov, On certain questions in the theory of optimal control,, SAIM J. Control Optim., 1 (1962), 76. Google Scholar [6] X. Li and J. Yong, "Optimal Control Theory for Infinite Dimensional Systems,", Birkh\, (1995). Google Scholar [7] N. G. Meyers, An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations,, Ann. Sc. Norm. Sup. Pisa, 17 (1963), 189. Google Scholar [8] G. Milton and R. V. Kohn, Variational bounds on the effective moduli of anisotropic composites,, J. Mech. Phys. Solids, 36 (1988), 597. Google Scholar [9] F. Murat and L. Tartar, H-convergence,, in:, (1997), 21. Google Scholar [10] F. Murat and L. Tartar, Calculus of variations and homogenization,, in:, (1997), 139. Google Scholar [11] L. Tartar, Estimations fines des coefficitents homogénéisés,, Ennio de Giorgi colloquium, 125 (1985), 168. doi: i:10.1016/0022-5096(88)90001-4. Google Scholar [12] J. Warga, "Optimal Control of Differential and Functional Equations,", Academic Press, (1972). Google Scholar

show all references

##### References:
 [1] G. Allaire, "Shape Optimization by the Homogenization Method,", Springer, (2002). Google Scholar [2] A. Bensoussan, J. L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures,", North-Holland Company, (1978). Google Scholar [3] E. Cabib and G. Dal Maso, On a class of optimum problems in structural design,, J. Optim. Theory Appl., 56 (1988), 39. doi: 10.1007/BF00938526. Google Scholar [4] L. Cesari, "Optimization Theory and Applications, Problems with Ordinary Equations,", Applications of Mathematics \textbf{17}, 17 (1983). Google Scholar [5] A. F. Filippov, On certain questions in the theory of optimal control,, SAIM J. Control Optim., 1 (1962), 76. Google Scholar [6] X. Li and J. Yong, "Optimal Control Theory for Infinite Dimensional Systems,", Birkh\, (1995). Google Scholar [7] N. G. Meyers, An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations,, Ann. Sc. Norm. Sup. Pisa, 17 (1963), 189. Google Scholar [8] G. Milton and R. V. Kohn, Variational bounds on the effective moduli of anisotropic composites,, J. Mech. Phys. Solids, 36 (1988), 597. Google Scholar [9] F. Murat and L. Tartar, H-convergence,, in:, (1997), 21. Google Scholar [10] F. Murat and L. Tartar, Calculus of variations and homogenization,, in:, (1997), 139. Google Scholar [11] L. Tartar, Estimations fines des coefficitents homogénéisés,, Ennio de Giorgi colloquium, 125 (1985), 168. doi: i:10.1016/0022-5096(88)90001-4. Google Scholar [12] J. Warga, "Optimal Control of Differential and Functional Equations,", Academic Press, (1972). Google Scholar
 [1] Omid S. Fard, Javad Soolaki, Delfim F. M. Torres. A necessary condition of Pontryagin type for fuzzy fractional optimal control problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 59-76. doi: 10.3934/dcdss.2018004 [2] Muhammad I. Mustafa. On the control of the wave equation by memory-type boundary condition. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1179-1192. doi: 10.3934/dcds.2015.35.1179 [3] Hakima Bessaih, Yalchin Efendiev, Florin Maris. Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition. Networks & Heterogeneous Media, 2015, 10 (2) : 343-367. doi: 10.3934/nhm.2015.10.343 [4] Piero Montecchiari, Paul H. Rabinowitz. A nondegeneracy condition for a semilinear elliptic system and the existence of 1- bump solutions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6995-7012. doi: 10.3934/dcds.2019241 [5] Ana P. Lemos-Paião, Cristiana J. Silva, Delfim F. M. Torres. A sufficient optimality condition for delayed state-linear optimal control problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2293-2313. doi: 10.3934/dcdsb.2019096 [6] Khadijah Sharaf. A perturbation result for a critical elliptic equation with zero Dirichlet boundary condition. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1691-1706. doi: 10.3934/dcds.2017070 [7] Alain Hertzog, Antoine Mondoloni. Existence of a weak solution for a quasilinear wave equation with boundary condition. Communications on Pure & Applied Analysis, 2002, 1 (2) : 191-219. doi: 10.3934/cpaa.2002.1.191 [8] Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233 [9] Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control & Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73 [10] 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 [11] Tsung-Fang Wu. Multiplicity of positive solutions for a semilinear elliptic equation in $R_+^N$ with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1675-1696. doi: 10.3934/cpaa.2010.9.1675 [12] Nicolas Forcadel, Wilfredo Salazar, Mamdouh Zaydan. Specified homogenization of a discrete traffic model leading to an effective junction condition. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2173-2206. doi: 10.3934/cpaa.2018104 [13] 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 [14] Shu Luan. On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions. Mathematical Control & Related Fields, 2017, 7 (3) : 493-506. doi: 10.3934/mcrf.2017018 [15] Imen Benabbas, Djamel Eddine Teniou. Observability of wave equation with Ventcel dynamic condition. Evolution Equations & Control Theory, 2018, 7 (4) : 545-570. doi: 10.3934/eect.2018026 [16] Constantin Christof, Christian Meyer, Stephan Walther, Christian Clason. Optimal control of a non-smooth semilinear elliptic equation. Mathematical Control & Related Fields, 2018, 8 (1) : 247-276. doi: 10.3934/mcrf.2018011 [17] Peter I. Kogut. On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2105-2133. doi: 10.3934/dcds.2014.34.2105 [18] Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 [19] Gökçe Dİlek Küçük, Gabil Yagub, Ercan Çelİk. On the existence and uniqueness of the solution of an optimal control problem for Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 503-512. doi: 10.3934/dcdss.2019033 [20] J. García-Melián, Julio D. Rossi, José Sabina de Lis. A convex-concave elliptic problem with a parameter on the boundary condition. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1095-1124. doi: 10.3934/dcds.2012.32.1095

2018 Impact Factor: 1.292