## 1 Introduction
Unlike previous works that only considered single or multiple users and a single objective, a hybrid IRS-aided integrated sensing and communication network with multiple users and objectives was investigated in [39], aimed at maximizing the worst-case target illumination power; an approach was proposed to design precoding and IRS coefficients. In fact, an IRS needs to be installed on the surface of the object. However, when there is no installation plane and/or emergency communication is needed, such as disaster relief, mounting an IRS on an UAV is an effective way to solve this challenge. In this way, not only can signal enhancement be achieved, but the position of the IRS can also be flexibly adjusted, and signal coverage may be extended. Given the benefits of UAV and hybrid IRS in terms of enhancing the performance of a wireless network, it is a reasonable choice to combine them with a conventional DM network to balance the performance and power consumption. So far, to the best of the authors' knowledge, hybrid IRS and UAV-aided DM systems have not been investigated yet. In this paper, we employ a hybrid IRS to further enhance the performance of passive IRS-aided DM networks. The main contributions of this work are summarized as follows:  
1. To balance performance, cost, and power consumption well, a hybrid IRS and UAV-aided DM system model is proposed. Aiming at maximizing the achievable rate, an optimization problem of maximizing the signal-to-noise ratio (SNR) is established, and the maximal SNR-fractional programming (FP) (Max-SNR-FP) method is proposed to jointly optimize the transmit beamforming vector and hybrid IRS PSM by solving one and giving another. In this scheme, the beamforming vector and passive IRS PSM are derived via the SCA algorithm, and the active IRS PSM is computed with the FP method. 2. Given the high computational complexity of the Max-SNR-FP scheme, a low-complexity alternating iteration method named maximum SNR-equal amplitude reflecting (EAR) (Max-SNR-EAR) is subsequently proposed. In this method, by utilizing the maximal signal-to-leakage-noise ratio (SLNR) criterion, the beamforming vector is obtained. Then, the phases of passive and active IRS phase shift matrices are computed on the basis of the criteria of phase alignment, while the amplitude of the active IRS PSM is obtained via the EAR criterion. 3. Given that the passive and active IRS phase shift matrices of the proposed Max-SNR-FP and Max-SNR-EAR methods were designed separately, to investigate the effect of jointly optimizing them on performance improvement, low-complexity alternating optimization algorithm Max-SNR-MM is proposed to maximize the achievable rate. The majorization-minimization (MM) criterion was employed to optimize the hybrid IRS phase-shift matrix. The simulation results clearly show that the achievable rates harvested with the three proposed methods were higher than those without IRS, random-phase IRS, and passive IRS. In addition, when the number of hybrid IRS phase shift elements tended towards a large scale, the difference in achievable rates among these three proposed methods was trivial. The remainder of this paper is organized as follows. In Section 2, we describe the system model of the hybrid IRS and UAV-aided DM network. Section 3 presents the Max-SNR-FP scheme. The Max-SNR-EAR scheme is described in Section 4. The Max-SNR-MM scheme is outlined in Section 5. We present the numerical simulation results in Section 6. In Section 7, the conclusions are drawn. **Notations:** in this article, vectors and matrices are shown in boldface lowercase and uppercase letters, respectively. \((\cdot)^{*}\), \((\cdot)^{T}\), \((\cdot)^{H}\), \(\mathbf{Tr}(\cdot)\), \(\Re\{\cdot\}\), \(\lambda_{\max}\{\cdot\}\), \(\text{diag}\{\cdot\}\), and \(\text{blkdiag}\{\cdot\}\) stand for the conjugate, transpose, conjugate transpose, trace, real part, maximal eigenvalue of the matrix, diagonal, and block diagonal matrix operations, respectively. \(|\cdot|\) refers to the scalar's absolute value or the matrix's determinant.