Abstract
This article examines a linear-quadratic elliptic optimal control problem in which the cost functional and the state equation involve a highly oscillatory periodic coefficient Aε. The small parameter ε > 0 denotes the periodicity length. We propose a high-order effective control problem with constant coefficients that provides an approximation of the original one with error O(εM), where M ∈ ℕ is as large as one likes. Our analysis relies on a Bloch wave expansion of the optimal solution and is performed in two steps. In the first step, we expand the lowest Bloch eigenvalue in a Taylor series to obtain a high-order effective optimal control problem. In the second step, the original and the effective problem are rewritten in terms of the Bloch and the Fourier transform, respectively. This allows for a direct comparison of the optimal control problems via the corresponding variational inequalities, leading to our main theoretical result on the high-oder approximation.
Highlights
Many modern key technologies call for mathematical modeling through partial differential equations (PDEs) with macro- and microstructures
In Kogut and Leugering [24], the concept of variational S-homogenization was proposed and analyzed for the limiting behavior of an optimal control problem governed by a linear elliptic equation with control and state constraints
We propose the use of a spectral method involving the so-called Bloch waves, which allows for a variety of L2-conditions in the admissible set
Summary
Many modern key technologies call for mathematical modeling through partial differential equations (PDEs) with macro- and microstructures. In Kogut and Leugering [24], the concept of variational S-homogenization was proposed and analyzed for the limiting behavior of an optimal control problem governed by a linear elliptic equation with control and state constraints. While in the setting of [22] this strategy is suitable, it seems difficult to apply the classical two-scale approach to optimal control problems with control or state constraints since adding correctors to the effective control function in general will destroy its admissibility To circumvent this difficulty, we propose the use of a spectral method involving the so-called Bloch waves, which allows for a variety of L2-conditions in the admissible set. This paper is aimed at the high-order asymptotic behavior of a linear-quadratic elliptic optimal control problem with ε-periodic coefficients and ε-dependent admissible sets. The final section is devoted to the analysis of an alternative effective problem
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