Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T17:26:18.057Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimality conditions for semilinear parabolic equations withcontrols in leading term*

Published online by Cambridge University Press:  23 August 2010

Hongwei Lou
Hongwei Lou*
Affiliation:
School of Mathematical Sciences, and LMNS, Fudan University, Shanghai 200433, P.R. China. [email protected]
Get access

Abstract

An optimal control problem forsemilinear parabolic partial differential equations is considered.The control variable appears in the leading term of the equation.Necessary conditions for optimal controls are established by themethod of homogenizing spike variation. Results for problems withstate constraints are also stated.

Keywords

Optimal controlnecessary conditionsparabolic equationhomogenized spike variation
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 4 , October 2011 , pp. 975 - 994
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Allaire, G., Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992) 1482–1518. CrossRef
A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. North-Holland Company, Amsterdam (1978).
Calvo-Jurado, C. and Casado-Diaz, J., Homogenization of Dirichlet parabolic problems for coefficients and open sets simultaneously variable and applications to optimal design. J. Comput. Appl. Math. 192 (2006) 20–29. CrossRef
Casado-Diaz, J., Couce-Calvo, J. and Martin-Gómez, J.D., Optimality conditions for nonconvex multistate control problems in the coefficients. SIAM J. Control Optim. 43 (2004) 216–239. CrossRef
Casas, E., Optimal Control in coefficients of elliptic equations with state constraints. Appl. Math. Optim. 26 (1992) 21–37. CrossRef
Ciuperca, I., El Alaoui Talibi, M. and Jai, M., On the optimal control of coefficients in elliptic problems, Application to the optimization of the head slider. ESAIM: COCV 11 (2005) 102–121. CrossRef
Gao, H. and Necessary, X. Li conditions for optimal control of elliptic systems. J. Australian Math. Soc. Ser. B 41 (2000) 542–567. CrossRef
Holmbom, A., Homogenization of parabolic equations an alternative approach and some corrector-type results. Appl. Math. 42 (1997) 321–343. CrossRef
O.A. Ladyženskaja, V.A. Solonnikov and N.N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, Transl. Math. Monographs 23. American Mathematical Society, Providence (1968).
X. Li, and J. Yong, Optimal Control Theory for Infinite Dimensional Systems. Birkhäuser, Boston (1995).
Lou, H. and Yong, J., Optimality Conditions for Semilinear Elliptic Equations with Leading Term Containing Controls. SIAM J. Control Optim. 48 (2009) 2366–2387. CrossRef
F. Murat and L. Tartar, Calculus of variations and homogenization, in Topics in the Mathematical Modelling of Composite Materials, Progress in Nonlinear Diffrential Equations and their Applications 31, L. Cherkaev and R.V. Kohn Eds., Birkaüser, Boston (1998) 139–174.
Raitums, U. and Schmidt, W.H., On necessary optimal conditions for optimal control problems governed by elliptic systems. Optimization 54 (2005) 149–160. CrossRef
Serovajsky, S.Y., Sequential extension in the problem of control in coefficients for elliptic-type equations. J. Inverse Ill-Posed Probl. 11 (2003) 523–536. CrossRef
Tagiyev, R.K., Optimal control by the coefficients of a parabolic equation. Trans. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. Math. Mech. 24 (2004) 247–256.
L. Tartar, Estimations fines de coefficients homogénéisés, Ennio de Giorgi Colloquium, in Pitman Research Notes in Mathematics 125, P. Krée Ed., Pitman, London (1985) 168–187.