Sensitivity analysis for active control of the Helmholtz equation

The results in \cite{O2} (see \cite{O1} for the quasistatics regime) consider the Helmholtz equation with fixed frequency $k$ and, in particular imply that, for $k$ outside a discrete set of resonant frequencies and given a source region $D_a\subset \mathbb{R}^{d}$ ($d=\overline{2,3}$) and $u_0$, a solution of the homogeneous scalar Helmholtz equation in a set containing the control region $D_c\subset \mathbb{R}^{d}$, there exists an infinite class of boundary data on $\partial D_a$ so that the radiating solution to the corresponding exterior scalar Helmholtz problem in $\mathbb{R}^{d} \setminus D_a$ will closely approximate $u_0$ in $D_c$. Moreover, it will have vanishingly small values beyond a certain large enough "far-field" radius $R$. In this paper we study the minimal energy solution of the above problem (e.g. the solution obtained by using Tikhonov regularization with the Morozov discrepancy principle) and perform a detailed sensitivity analysis. In this regard we discuss the stability of the the minimal energy solution with respect to measurement errors as well as the feasibility of the active scheme (power budget and accuracy) depending on: the mutual distances between the antenna, control region and far field radius $R$, value of regularization parameter, frequency, location of the source.