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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

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 and 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.

[2]

A. Bensoussan, J. L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures," North-Holland Company, Amsterdam, 1978.

[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.

[4]

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

[5]

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

[6]

X. Li and J. Yong, "Optimal Control Theory for Infinite Dimensional Systems," Birkhäuser, Boston, 1995.

[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.

[8]

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

[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).

[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).

[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.

[12]

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

show all references

References:
[1]

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

[2]

A. Bensoussan, J. L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures," North-Holland Company, Amsterdam, 1978.

[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.

[4]

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

[5]

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

[6]

X. Li and J. Yong, "Optimal Control Theory for Infinite Dimensional Systems," Birkhäuser, Boston, 1995.

[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.

[8]

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

[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).

[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).

[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.

[12]

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

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