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Optimal design of stratified media. (English) Zbl 1030.49016

Summary: An optimal control problem governed by a linear elliptic equation with a control acting as a coefficient matrix is considered. This matrix is symmetric, uniformly positive definite, can be large but may vary only in a single prescribed direction. The optimal control (i.e., design of the stratified medium) may not exist as usual because of oscillations of the coefficient matrix but also because of concentration effects due to lack of an upper bound on the control variable. Using a homogenization theory for possibly stiff stratified media, a relaxation theory is developed and, in a special case, differentiability of the extended cost functional is proved and then a pointwise maximum principle necessary for an optimal composite is derived.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49J20 Existence theories for optimal control problems involving partial differential equations
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35J20 Variational methods for second-order elliptic equations
49K20 Optimality conditions for problems involving partial differential equations
80M40 Homogenization for problems in thermodynamics and heat transfer
80M50 Optimization problems in thermodynamics and heat transfer
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