Active manipulation of fields modeled by the Helmholtz equation

Abstract

In this paper we extend the results proposed in \cite{Onofrei-S} and study the problem of active control in the context of a scalar Helmholtz equation. Given a source region $D_a$ and $\{v_0,v_1,\ldots,v_n\}$, a set of solutions of the homogeneous scalar Helmholtz equation in $n$ mutually disjoint ``control" regions $\{D_0,D_1,\ldots,D_n\}$ of $\RR^2$ or $\RR^3$, respectively, the main objective of this paper is to characterize the necessary boundary data on $\partial D_a$ so that the solution to the corresponding exterior scalar Helmholtz problem will closely approximate $v_i$ in $D_i$, respectively, for each $i\in\{0,\ldots,n\}$. Building up on the previous ideas presented in \cite{Onofrei-S} we show the existence of a class of solutions to the problem, provide numerical support of the results in 2D and 3D and discuss the existence of a minimal energy solution and its stability.