Asymptotic analysis of a boundary optimal control problem on a general branched structure

Abstract

We consider an optimal control problem posed on a domain with a highly oscillating smooth boundary where the controls are applied on the oscillating part of the boundary. There are many results on domains with oscillating boundaries where the oscillations are pillar-type (non-smooth) while the literature on smooth oscillating boundary is very few. In this article, we use appropriate scaling on the controls acting on the oscillating boundary leading to different limit control problems, namely, boundary optimal control and interior optimal control problem. In the last part of the article, we visualize the domains as a branched structure, and we introduce unfolding operators to get contributions from each level at every branch. KEYWORDS asymptotic analysis, homogenization, optimal control, oscillating boundary domain, unfolding operator MSC CLASSIFICATION 80M35; 80M40; 49J20; 35B27 Math Meth Appl Sci. 2019;42:6407-6434.wileyonlinelibrary.com/journal/mma

Unfolding operator on Heisenberg Group and applications in Homogenization

“…Since, ξ 0 and ∇ H u are in (L 2 (Ω × Y )) 2 , we see that u 1 ∈ L 2 (Ω; H 1 #,H (Y )/R). Hence, we have the second convergence.…”

“…For the second part, we will use the test function of the form 2 with div H,y ψ = 0. Let us consider the following,…”

“…We are under investigation of the applicability of the introduced unfolding operator in particular to optimal control problems. At this stage we would like to recall that the unfolding operator can be used to characterize the optimal control in homogenization problem (see [2,12,14]).…”

“…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 and exact controllability for problems with imperfect interface

“…Later on, in [49] the authors deal with the case of a Neumann boundary value problem in the same framework. In [2]- [5], [12]- [16], [30]- [32], [55]- [58] optimal control and exact controllability problems in domains with highly oscillating boundary are studied. Moreover we refer to [42] and [9,10,11] for the exact controllability of hyperbolic problems with oscillating coefficients in fixed and in perforated domains respectively, to [36,37] and [34,35,54] for the optimal control and the exact controllability, respectively, of hyperbolic problems in composites with imperfect interface.…”