Sensitivity analysis for active control of the Helmholtz equation

In previous works we considered the Helmholtz equation with fixed frequency k outside a discrete set of resonant frequencies, where it is implied that, given a source region _Da⊂R^d(d=2,3¯) and u₀, a solution of the homogeneous scalar Helmholtz equation in a set containing the control region D_c⊂R^d , there exists an infinite class of boundary data on ∂D_a so that the radiating solution to the corresponding exterior scalar Helmholtz problem in R^d\_Da will closely approximate u₀ 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 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 the regularization parameter; frequency; location of the source.