## 1 Introduction
\right]\mathbf{q}_{p} \tag{4}\]  
where \(\mathbf{Z}_{sel\_uni}\) is the matrix (dimension \(N_{e1}\times NL\)) of transfer functions from the \(NL\) uniformly placed secondary loudspeakers to target area 1, \(\mathbf{Z}_{so2\_uni}\) is the matrix (dimension \(N_{e2}\times NL\)) of transfer functions from the \(NL\) uniformly placed secondary loudspeakers to target area 2, and \(\mathbf{q}_{s}\) is the strength vector (dimension \(NL\times 1\)) of the secondary loudspeakers. The RLS algorithm [14; 20] using uniformly placed secondary loudspeakers aims to minimize the sound pressure in the target area under the premise that the secondary loudspeaker power is constrained \(\|\mathbf{q}_{s}\|_{2}^{2}\leq P_{\max}\). \(P_{\max}\) is the maximum allowable power of the secondary loudspeakers, and the cost function of the RLS algorithm is given by:  
\[J=\left\|\left[ \begin{array}{c}\mathbf{Z}_{se1\_uni}\\ \mathbf{Z}_{se2\_uni}\end{array} \right]\mathbf{q}_{s}+\left[ \begin{array}{c} \mathbf{Z}_{pe1\_sca}\\ \mathbf{Z}_{pe2\_sca}\end{array} \right]\mathbf{q}_{p}\right\|_{2}^{2}+\lambda \|\mathbf{q}_{s}\|_{2}^{2} \tag{5}\]  
where \(\lambda\) is the regularization parameter. By setting the partial derivative of Equation (5) with respect to \(\mathbf{q}_{s}\) to zero, the solution to Equation (5) can be written as:  
\[\mathbf{q}_{s}=-(\mathbf{Z}_{left}^{H}\mathbf{Z}_{left}+\lambda\mathbf{I})^{- 1}\mathbf{Z}_{left}\mathbf{Z}_{right}\mathbf{q}_{p} \tag{6}\]  
where \(\mathbf{Z}_{left}=\left[ \begin{array}{cc}\mathbf{Z}_{so1\_uni}&\mathbf{Z}_{ so2\_uni}\end{array} \right]^{T}\),\(\mathbf{Z}_{right}=\left[ \begin{array}{cc}\mathbf{Z}_{pe1\_sca}&\mathbf{Z}_{pe2\_ sca}\end{array} \right]^{T}\) and \(\mathbf{I}\) is an identity matrix. After calculating the secondary loudspeaker strength vector \(\mathbf{q}_{s}\) with the initialized regularization parameter \(\lambda=10^{-4}\), if the power constraint condition \(\|\mathbf{q}_{s}\|_{2}^{2}\leq P_{\max}\) is not satisfied, \(\lambda\) is increased until the condition is satisfied [18; 26]. Proposed CLASSO-RLS Method for Multi-Zone Active Noise Control of the Scattered Sound from an Infinite Rigid Cylinder  
As shown in Figure 1b, one plane wave is incident, \(NS\) candidate secondary loudspeakers are uniformly placed at a distance of \(r_{s}\) from the cylinder, and \(N_{c}\) constraint points are uniformly placed between the two target areas. To promote sparsity of the strength vector of the secondary loudspeakers and suppress the sound amplification of the constraint points, the CLASSO-RLS algorithm adds the \(l_{1}\)-norm \(\|\mathbf{q}_{s}\|_{1}\) and the noise reduction constraint of the constraint points to the cost function of the RLS algorithm. The cost function of the CLASSO-RLS algorithm can be expressed as:  
\[ \begin{split}\min\left\|\left[\begin{array}{c}\mathbf{Z}_{ se1}\\ \mathbf{Z}_{se2}\end{array}\right]\mathbf{q}_{s}+\left[\begin{array}{c} \mathbf{Z}_{pe1\_sca}\\ \mathbf{Z}_{pe2\_sca}\end{array}\right]\mathbf{q}_{p}\right\|_{2}^{2}+\gamma_ {CLASSO}\|\mathbf{q}_{s}\|_{1}\\ s.t.NR_{c\_7n}\geq NR_{c0}\\ s.t.\|\mathbf{q}_{s}\|_{2}^{2}\leq P_{\max}\end{split}  \tag{7}\]  
where \(\mathbf{Z}_{se1}\) is the matrix (dimension \(N_{e1}\times NS\)) of transfer functions from the \(NS\) candidate secondary loudspeakers to target area 1, \(\mathbf{Z}_{so2}\) is the matrix (dimension \(N_{e2}\times NS\)) of transfer functions from the \(NS\) candidate secondary loudspeakers to target area 2, \(\mathbf{q}_{s}\) is the strength vector (dimension \(NS\times 1\)) of \(NS\) candidate secondary loudspeakers, \(NR_{c\_7n}\) denotes the noise reduction of the constraint points, and \(NR_{c0}\) is the maximum noise reduction of the constraint points under a certain condition, which is defined in detail below. The sparse penalty parameter \(\gamma_{CLASSO}\) regulates the number of non-zero entries of \(\mathbf{q}_{s}\)[14]; the larger sparse penalty parameter \(\gamma_{CLASSO}\) results in fewer non-zero entries.