## 2 System Model and Problem Formulation  
The system under consideration is an RIS-aided batteryless IoT network consisting of several key components: an RF Source, an IoT device, an IoT receiver, an RIS, an RIS controller, and an IoT server, as shown in Figure 1. Each component plays a critical role in enabling the network's communication and energy harvesting functionalities. The RF source (S) emits RF signals, which serve two primary purposes: communication and energy harvesting. In this network, the RF source provides the necessary signals that the batteryless IoT devices capture to power their operations. These signals are also used for data transmission between the RF source, the IoT device (D), and the IoT receiver (R). The RF source could be a wireless access point, base station, or any other ambient or dedicated RF-emitting device, which broadcasts unmodulated continuous signals over a designated frequency band. The IoT device relies on harvested energy from the RF signals for its operation. This device captures the RF signals to extract the necessary power for its operations and, subsequently, transmits or backscatters data. Due to its energy constraints, efficient energy harvesting is critical to ensure the sustained functionality of the IoT device. In addition to energy harvesting, the IoT device also transmits sensed data, such as environmental or monitoring information, to the IoT receiver either directly or via the RIS-aided link. The IoT receiver plays the role of collecting the data transmitted from the IoTdevice [33]. It is equipped to handle incoming signals from both direct and RIS-aided links, ensuring reliable communication. Given the potential for interference caused by multiple signal reflections, the IoT receiver must utilize advanced signal processing techniques to suppress interference and decode the received signals efficiently. The IoT receiver also provides feedback to an IoT server. The RIS (I) is a key component of the system, designed to optimize the wireless communication environment. The RIS consists of multiple passive reflective elements that can be reconfigured to alter the phase of the incoming signals. By adjusting these phase shifts, the RIS can reflect the RF signals to improve the signal quality and increase the energy harvesting efficiency of the IoT device. Additionally, the RIS enhances communication between the IoT device and the IoT receiver by strategically reflecting signals away from unwanted paths to improve the SINR. The RIS controller is responsible for dynamically adjusting the phase shifts of the RIS elements. It determines the optimal configuration based on real-time network conditions and objectives, such as the channel state information (CSI) of the involved links in Figure 1. Although CSI estimation in batteryless IoT networks remains a challenging task, recent research has started to yield promising advancements, with a number of key studies providing insights into potential solutions, as discussed in [34]. The IoT server acts as the central management and data processing unit of the system. It collects the data received by the IoT receiver, processes them, and stores them for further analysis. The server also plays a critical role in coordinating the activities of the RIS controller, sending commands to optimize RIS configurations based on the network's performance metrics [35]. Additionally, the IoT server handles the network's security, managing user authentication and ensuring that the IoT devices and communication links are secure. For simplicity and without loss of generality, we assume that all components of the RIS-enabled IoT network, including the RF source, IoT receiver, and IoT device, are equipped with a single antenna. Additionally, we consider an RIS consisting of \(N\) passive reflective elements. The direct channels between S-D, S-R, and D-R are represented as \(h_{SD}\), \(h_{SR}\), and \(h_{DR}\), respectively.