## 1 Introduction  
High-precision large reflectors are garnering widespread research attention. Many studies on improving reflector surface accuracies have been conducted. [PERSON] et al. [1] designed and analyzed a large and high-precision inflatable membrane reflector. [PERSON] et al. [2] developed a design concept for high-precision shell reflectors with a structural optimization and a shape adjustment method. A pseudo-geometric method for the generation of the surface mesh geometry of deployable reflectors was proposed by [PERSON] et al. [3]. [PERSON] et al. developed a technique for placing nodes off the working surface of a mesh reflector [4] and applied an additive structure, named as a self-standing-truss with hardpoints [5, 6], to improve the surface accuracy of the reflector. [PERSON] et al. [7] designed and analyzed a high-precision offset-stiffened spring-back reflector with carbon fiber-reinforced composite (CFRC) materials. Active shape control is an effective approach to improve reflector surface accuracies. Smart materials and smart structures such as PVDF (poly vinylidene fluoride), PZT (piezoelectric lead zirconate titanate), MFC (macro fiber composite), and SMA (shape memory alloy material) actuators/sensors are widely used in active shape and vibration controls. [PERSON] et al. [8] investigated the feasibility of an active shape correction method for a deployable space antenna reflector. [PERSON] [9] presented a high-precision adaptive control architecture with PVDF actuators for large membrane and thin shell reflectors, in which both an analytical model and an experimental system were developed. [PERSON] [10] introduced an active shape adjustment method that used active cables to obtain a desired shape. [PERSON] et al. [11] developed an active shape control technology for the reflecting surfaces using large spatially distributed actuator arrays. [PERSON] [12] presented an approach to shape control problems with a constrained number of discrete actuators. [PERSON] et al. [13] used SMA actuators to improve the surface accuracy of a circular membrane structure. In their study, an analytical model and a solution method were developed, and the finite element method was then used to validate analytical model under typical thermal loads. [PERSON] et al. [13] proposed a novel method to reduce the global and local surface error of a CFRC reflector using PZT and MFC actuators. [PERSON] et al. [14] investigated the feasibility of using an active gore/seam cable-based control system to reduce the global root mean square errors arising from thermal loads. CFRC materials are widely used for high-precision reflectors owing to their high strength, high modulus, and low Coefficient Thermal Expansion (CTE). For improving the control efficiency, the arrangement of the actuators is essential to actively control the problems of space structures, especially for those with a weight budget and constrained energy supply. Many studies have investigated the geometry sizes, locations, and applied voltages of actuators for beam and plate structures. The previously studied methods for actuator arrangement optimization include gradient-based algorithms, group elimination methods, and random search algorithms. [PERSON] and [PERSON] [15] optimized the sizes, thicknesses, and locations of the actuators for a rectangle cantilever composite beam to achieve maximum nodal displacements, minimum masses and minimum voltages for shape control task. [PERSON] et al. [16] used a one-dimensional optimization algorithm to determine the locations of actuators for a laminated beam structure. [PERSON] et al. [17] optimized the lengths and locations of the actuators to maximize the weighted controllability. [PERSON] et al. [18] optimized the lengths and locations of the actuators for a simply supported beam. [PERSON] et al. [19] used a gradient method to optimize the lengths and locations of the actuators in a three-beam structure to achieve the minimum mechanical energy and maximum observability.