Semi-linear optimal control problem on a smooth oscillating domain

Abstract

We demonstrate the asymptotic analysis of a semi-linear optimal control problem posed on a smooth oscillating boundary domain in the present paper. We have considered a more general oscillating domain than the usual “pillar-type” domains. Consideration of such general domains will be useful in more realistic applications like circular domain with rugose boundary. We study the asymptotic behavior of the problem under consideration using a new generalized periodic unfolding operator. Further, we are studying the homogenization of a non-linear optimal control problem and such non-linear problems are limited in the literature despite the fact that they have enormous real-life applications. Among several other technical difficulties, the absence of a sufficient criteria for the optimal control is one of the most attention-grabbing issues in the current setting. We also obtain corrector results in this paper.

Oscillating PDE in a rough domain with a curved interface: Homogenization of an Optimal Control Problem

“…Let ψ(x, y ) ∈ (C ∞ c (Ω u )) 2 . Let T ε (∇ x u ε ) D in (L 2 (Ω u )) 2 . A simple integration by parts gives us the following,…”

“…The unfolding operator which we develop is quiet new and we derive various properties together with the convergences enjoyed by the newly introduced unfolding operators. In the last 10 years or so the unfolding operators have been used extensively by various authors including the present authors and their collaborators (see [1,2,19,35,36]). Thus, we deal with, at least, three important aspects in this paper, namely (i) Consideration of two different oscillating matrices A ε and B ε , respectively for the equation and cost functional.…”

“…The authors have used unfolding operator corresponding to the pillar-type oscillations to characterize the optimal control for the first time. In [1], unfolding operator for general periodic oscillating domain with flat base has been introduced and authors have investigated asymptotic behavior of a semilinear PDE with principle part as the Laplace operator and a corresponding interior optimal control problem have been studied in [2] by the same authors. In [3], unfolding operators for locally periodic oscillating domain has been defined, again the base of the oscillations is flat.…”

Unfolding operator on Heisenberg Group and applications in Homogenization

“…Observe that definition of two-scale convergence reduced to weakly convergence in L 2 (Ω × Y ) and it is easy to apply as it is technicality less demanding. There are some advantages of using this method; for example, while doing optimal control problems in periodic setup, the optimal control is easily characterized by the unfolding of the adjoint state, which helps to analyze asymptotic behavior see [2,3,12,13,14]. This method reduces the definition of two-scale convergence in L p (Ω) to weak convergence of the unfolding sequence in L p (Ω × Y ) for 1 < p < ∞.…”

Homogenization of a nonlinear monotone problem in a locally periodic domain via unfolding method

“…All these above works are of pillar type periodic oscillations except a few. There are some works on non uniform pillar type, that is the thickness or the cross section of the pillar changes when the height changes, see [25,1,3,36,31] .…”