## 1 Introduction
These methods are based on the gradients of objective functions with respect to optimized variables. However, for discrete optimizing problems or complicated non-convex problems, the gradient-based methods are not suitable because the gradients of objective functions may be difficult to obtain and the solutions derived are the local optimum. The random search algorithm is another type of optimization method for shape control problems. [PERSON] and [PERSON] [20] optimized a truss topology based on a multipoint approximate function and a Genetic Algorithm (GA). An optimized actuator was positioned for an adaptive truss using a two-level multipoint approximation method [21]. [PERSON] and [PERSON] [22] used a Simulated Annealing (SA) algorithm to track the shape of a smart plate and optimized the distribution of the voltage channels as well as the control voltages. [PERSON] et al. [23] investigated multidisciplinary design optimization with the GA for vibration controls of smart laminated composite structures. Random search algorithms do not require gradients of the objective function. Therefore, variables are allowed to be discrete and the global optimum can be obtained. However, the accuracies of solutions from a random search algorithm may be lower than that from a gradient-based method due to its uncertain searching trajectory. For large adaptive space mirrors and reflectors, researchers have investigated the arrangement of the actuators. [PERSON] [24] used a linear quadratic Gaussian controller, combining with an efficient weight-selection technique to optimize the configurations of actuators and sensors. [PERSON] and [PERSON] [25] used a modified Simulated Annealing Algorithm to optimize the locations of the actuators of a honeycomb composite panel. [PERSON] and [PERSON] [26] used the GA to optimize the locations of the actuators for a primary mirror to achieve the minimum surface error caused by thermal distortion. [PERSON] et al. [27] used the elimination method to determine the optimal grouping of the PVDF actuators for a membrane reflector. [PERSON] et al. [28] studied the optimal design of actuator locations in a honeycomb mirror by using the Simulated Annealing Algorithm. We have previously proposed a novel analysis method to reduce the global and local surface error of a CFRC reflector analytically with PZT (Piezo Direct, Burlingame, CA, USA) and MFC (MFC Manufacturing, Memphis, TN, USA) actuators employed [14]. For further research, in this article, the optimal placement of PZT actuators and the shape control experiment with PZT actuators are studied. Firstly, a finite element model with independent degrees of freedom of applied voltages is set up, and based on the finite element formulation, the optimal arrangement of the actuators is determined by a numerical method using the proposed modified GA. Both single and multi-objective optimizations are studied in optimizing the placement of the actuators. For the single-objective optimization, a hybrid method that combines the GA with the gradient projection method is proposed. The proposed method has higher convergence accuracy than the classical GA due to combining the random and deterministic methods. Numerical examples are provided to verify the effectiveness of this method. For the multi-objective optimization, the Pareto optimal solution of the shape control problem is derived. Unlike traditional approaches to converting a multi-objective optimization problem to a single-objective problems by enforcing the weighted sum of the objectives, here, a modified multi-objective Genetic Algorithm is proposed to derive the Pareto optimal solution.