# Optimality conditions for semilinear parabolic equations with controls in leading term*

ESAIM: Control, Optimisation and Calculus of Variations (2011)

- Volume: 17, Issue: 4, page 975-994
- ISSN: 1292-8119

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topLou, Hongwei. "Optimality conditions for semilinear parabolic equations with controls in leading term*." ESAIM: Control, Optimisation and Calculus of Variations 17.4 (2011): 975-994. <http://eudml.org/doc/221936>.

@article{Lou2011,

abstract = {
An optimal control problem for
semilinear 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 the
method of homogenizing spike variation. Results for problems with
state constraints are also stated.
},

author = {Lou, Hongwei},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Optimal control; necessary conditions; parabolic equation; homogenized spike variation; semilinear parabolic partial differential equations; necessary conditions for optimal controls; method of homogenizing spike variation},

language = {eng},

month = {11},

number = {4},

pages = {975-994},

publisher = {EDP Sciences},

title = {Optimality conditions for semilinear parabolic equations with controls in leading term*},

url = {http://eudml.org/doc/221936},

volume = {17},

year = {2011},

}

TY - JOUR

AU - Lou, Hongwei

TI - Optimality conditions for semilinear parabolic equations with controls in leading term*

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2011/11//

PB - EDP Sciences

VL - 17

IS - 4

SP - 975

EP - 994

AB -
An optimal control problem for
semilinear 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 the
method of homogenizing spike variation. Results for problems with
state constraints are also stated.

LA - eng

KW - Optimal control; necessary conditions; parabolic equation; homogenized spike variation; semilinear parabolic partial differential equations; necessary conditions for optimal controls; method of homogenizing spike variation

UR - http://eudml.org/doc/221936

ER -

## References

top- G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal.23 (1992) 1482–1518.
- A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. North-Holland Company, Amsterdam (1978).
- C. Calvo-Jurado and J. Casado-Diaz, 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.
- J. Casado-Diaz, J. Couce-Calvo and J.D. Martin-Gómez, Optimality conditions for nonconvex multistate control problems in the coefficients. SIAM J. Control Optim.43 (2004) 216–239.
- E. Casas, Optimal Control in coefficients of elliptic equations with state constraints. Appl. Math. Optim.26 (1992) 21–37.
- I. Ciuperca, M. El Alaoui Talibi and M. Jai, On the optimal control of coefficients in elliptic problems, Application to the optimization of the head slider. ESAIM: COCV11 (2005) 102–121.
- H. Gao and X. Li, Necessary conditions for optimal control of elliptic systems. J. Australian Math. Soc. Ser. B41 (2000) 542–567.
- A. Holmbom, Homogenization of parabolic equations an alternative approach and some corrector-type results. Appl. Math.42 (1997) 321–343.
- O.A. Ladyženskaja, V.A. Solonnikov and N.N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, Transl. Math. Monographs23. American Mathematical Society, Providence (1968).
- X. Li, and J. Yong, Optimal Control Theory for Infinite Dimensional Systems. Birkhäuser, Boston (1995).
- H. Lou and J. Yong, Optimality Conditions for Semilinear Elliptic Equations with Leading Term Containing Controls. SIAM J. Control Optim.48 (2009) 2366–2387.
- 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 Applications31, L. Cherkaev and R.V. Kohn Eds., Birkaüser, Boston (1998) 139–174.
- U. Raitums and W.H. Schmidt, On necessary optimal conditions for optimal control problems governed by elliptic systems. Optimization54 (2005) 149–160.
- S.Y. Serovajsky, Sequential extension in the problem of control in coefficients for elliptic-type equations. J. Inverse Ill-Posed Probl.11 (2003) 523–536.
- R.K. Tagiyev, 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 Mathematics125, P. Krée Ed., Pitman, London (1985) 168–187.

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