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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

Bo Li 1, and  Hongwei Lou 2,

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.
Keywords: existence condition, homogenization, Cesari-type condition., elliptic equation, Optimal control.
Mathematics Subject Classification: Primary: 49J20; Secondary: 35B27, 35B37, 35J2.
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, New York, 2002.  Google Scholar

[2]

A. Bensoussan, J. L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures," North-Holland Company, Amsterdam, 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), 39-65. doi: 10.1007/BF00938526.  Google Scholar

[4]

L. Cesari, "Optimization Theory and Applications, Problems with Ordinary Equations," Applications of Mathematics 17, Springer, New York, 1983.  Google Scholar

[5]

A. F. Filippov, On certain questions in the theory of optimal control, SAIM J. Control Optim., 1 (1962), 76-84.  Google Scholar

[6]

X. Li and J. Yong, "Optimal Control Theory for Infinite Dimensional Systems," Birkhäuser, Boston, 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), 189-206.  Google Scholar

[8]

G. Milton and R. V. Kohn, Variational bounds on the effective moduli of anisotropic composites, J. Mech. Phys. Solids, 36 (1988), 597-629. Google Scholar

[9]

F. Murat and L. Tartar, H-convergence, in: "Topics in the Mathematical Modelling of Composites Materials" (Eds. A. Cherkaev, R.V. Kohn), Boston, Birkhäuser, Boston 1997, 21-44 (French version: F. Murat, H-convergence, Mimeographed notes, Séminaire d'Analyse Fonctionnelle et Numérique de l'Université d'Alger, 1978).  Google Scholar

[10]

F. Murat and L. Tartar, Calculus of variations and homogenization, in: "Topics in the Mathematical Modelling of Composites Materials" (Eds. A. Cherkaev, R. V. Kohn), Birkhäuser," Boston 1997, 139-173 (French version: F. Murat, Calcul des variations et homogénéisation, in: Les méthodes de l'homogénéisation, théorie et applications en physique, Coll. Dir. Etudes et Recherches EDF, Eyrolles, 1985, 319-369).  Google Scholar

[11]

L. Tartar, Estimations fines des coefficitents homogénéisés, Ennio de Giorgi colloquium, P. Krée ed., Pitman Research Notes in Math., 125 (1985), 168-187. doi: i:10.1016/0022-5096(88)90001-4.  Google Scholar

[12]

J. Warga, "Optimal Control of Differential and Functional Equations," Academic Press, New York, 1972.  Google Scholar

show all references

References:
[1]

G. Allaire, "Shape Optimization by the Homogenization Method," Springer, New York, 2002.  Google Scholar

[2]

A. Bensoussan, J. L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures," North-Holland Company, Amsterdam, 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), 39-65. doi: 10.1007/BF00938526.  Google Scholar

[4]

L. Cesari, "Optimization Theory and Applications, Problems with Ordinary Equations," Applications of Mathematics 17, Springer, New York, 1983.  Google Scholar

[5]

A. F. Filippov, On certain questions in the theory of optimal control, SAIM J. Control Optim., 1 (1962), 76-84.  Google Scholar

[6]

X. Li and J. Yong, "Optimal Control Theory for Infinite Dimensional Systems," Birkhäuser, Boston, 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), 189-206.  Google Scholar

[8]

G. Milton and R. V. Kohn, Variational bounds on the effective moduli of anisotropic composites, J. Mech. Phys. Solids, 36 (1988), 597-629. Google Scholar

[9]

F. Murat and L. Tartar, H-convergence, in: "Topics in the Mathematical Modelling of Composites Materials" (Eds. A. Cherkaev, R.V. Kohn), Boston, Birkhäuser, Boston 1997, 21-44 (French version: F. Murat, H-convergence, Mimeographed notes, Séminaire d'Analyse Fonctionnelle et Numérique de l'Université d'Alger, 1978).  Google Scholar

[10]

F. Murat and L. Tartar, Calculus of variations and homogenization, in: "Topics in the Mathematical Modelling of Composites Materials" (Eds. A. Cherkaev, R. V. Kohn), Birkhäuser," Boston 1997, 139-173 (French version: F. Murat, Calcul des variations et homogénéisation, in: Les méthodes de l'homogénéisation, théorie et applications en physique, Coll. Dir. Etudes et Recherches EDF, Eyrolles, 1985, 319-369).  Google Scholar

[11]

L. Tartar, Estimations fines des coefficitents homogénéisés, Ennio de Giorgi colloquium, P. Krée ed., Pitman Research Notes in Math., 125 (1985), 168-187. doi: i:10.1016/0022-5096(88)90001-4.  Google Scholar

[12]

J. Warga, "Optimal Control of Differential and Functional Equations," Academic Press, New York, 1972.  Google Scholar

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