## 1 Introduction  
For the purposes of public safety, environmental protection, scientific research, etc., people need to observe, measure and collect information in certain areas over a long time, and then make decisions based on the results of these observations, measurements and collection. This is generally called a persistent monitoring problem [1, 2, 3]. Monitoring in person or by hand is usually constrained by weather, geography, working hours and labor costs, and intelligent equipment can greatly overcome the above deficiencies of human based monitoring. Unmanned aerial vehicles (UAV) are examples of one of these typical intelligent monitoring devices. Because they are free of human intervention and offer stable flight, a wide range of motion, and low cost, UAVs are often used to perform persistent monitoring tasks [4], target detection and tracking [5], and border patrols [6]. This research mainly studies the UAV path planning problem when they are used in persistent monitoring. With the emergence of various complex environments and complex tasks, a single UAV will find it hard to meet the requirements of increasingly complex inspection operations. Consequently, there has been extensive research on multi-UAV cooperation. Compared with single-UAV operation, multi-UAV cooperation has demonstrated greater advantages. For example, multi-UAV cooperation [7, 8] can obtain more comprehensive and wide information and can realize multi-angle monitoring of the target area. However, such problems as cooperation strategy, inconsistent monitoring frequency, unsynchronized monitoringinformation, and unsafe monitoring strategies still exists for multi-UAV cooperation. The task decisions of multi-UAV persistent monitoring have become popular issues in the application field of UAVs. The multi-UAV persistent monitoring problem can be divided into two levels. One level is the monitoring frequency constraint, and the other is the persistent monitoring security, i.e., monitoring privacy preservation. The above two levels correspond to the two so-called modes of UAV persistent monitoring. One is the regular monitoring mode, that is, the route planned for the UAV to minimize the time delay between each adjacent visit of the task nodes and to improve their monitor frequency as much as possible. The other is the adversarial monitoring mode, which is to plan uncertain, unpredictable and non-periodic monitoring paths for UAVs in order to prevent any intelligent intruders from detecting the monitoring regularity [9]. If the monitoring frequency constraint is considered as the only criterion, the monitoring path is usually a certain periodic path. Once an intelligent intrusion appears in the monitoring environment, the privacy of the UAV monitoring intention cannot be protected, and the monitoring task is easily destroyed by intelligent intruders. On the other hand, when only the security of persistent monitoring is considered, it may be difficult to satisfy the monitoring frequency requirements of each node due to excessive consideration of path privacy security. Therefore, it is of great theoretical significance and practical value to study the joint optimization problem of monitoring frequency and privacy protection. [PERSON] [10] reviewed the multi-robot cooperative patrol algorithms that has been studied in recent years and pointed out that a distributed, non-deterministic and cooperative strategy represents the future trend. [PERSON] [11] studied the persistent monitoring problem of a single robot. The optimization goal is to minimize the revisit duration of the given monitoring tasks. Two approximate algorithms with complexity \(O(\log\rho_{G})\) and \(O(\log n)\) were proposed, respectively. [PERSON] [12] studied the patrol problem in a closed area and proposed the patrol frequency optimization criterion for the first time, and each point in the area should be repeatedly visited by multiple robots.