Quadratic optimization in the problems of active control of sound Original Research Article

We analyze the problem of suppressing the unwanted component of a time-harmonic acoustic field (noise) on a predetermined region of interest. The suppression is rendered by active means, i.e., by introducing the additional acoustic sources called controls that generate the appropriate anti-sound. Previously, we have obtained general solutions for active controls in both continuous and discrete formulation of the problem. We have also obtained optimal solutions that minimize the overall absolute acoustic source strength of active control sources, which is equivalent to minimization in the sense of L1L1. By contrast, in the current paper we formulate and study optimization problems that involve quadratic functions of merit. Specifically, we minimize the L2L2 norm of the control sources, and we consider both the unconstrained and constrained minimization. The unconstrained L2L2 minimization is an easy problem to address numerically. On the other hand, the constrained approach allows one to analyze sophisticated geometries. In a special case, we can compare our finite-difference optimal solutions to the continuous optimal solutions obtained previously using a semi-analytic technique. We also show that the optima obtained in the sense of L2L2 differ drastically from those obtained in the sense of L1L1.