An optimal control problem in a tubular thin domain with rough boundary

Abstract

In this paper we analyze the asymptotic behavior of a control problem set by a convection-reaction-diffusion equation with mixed boundary conditions and defined in a tubular thin domain with rough boundary. The control term acts on a subset of the rough boundary where a Robin-type boundary condition and a catalyzed reaction mechanism are set. The reaction mechanism depends on a parameter α∈R. Such parameter establishes different regimes which also depend on the profile and geometry of the tube defined by a periodic function g:R2↦R. We see that, if ∂2g is not null (that is, when g really depends on the second variable), then three regimes with respect to α are established: α<2, α=2 (the critical value) and α>2. On the other hand, if ∂2g≡0, similar regimes are obtained but now with a different critical value. Indeed, if ∂2g≡0, then the critical value is achieved at α=1. For each one of these six regimes, we obtain the asymptotic behavior of the control problem when the cylindrical thin domain degenerates to the unit interval. We show that the problem is asymptotically controllable just when α assumes the critical values. Our analysis is mainly performed using the periodic unfolding method adapted to cylindrical coordinates in R3.