图像修复问题中，局部细节特征的辨识修复和全局特征的保护是至关重要的。基于分数阶偏微分方程的模型具有丰富的演化行为能力，在图像修复中能较好地理解图像细节并兼具一定的锐化作用，但也易出现不能准确辨识较大尺度特征和过度锐化等问题。为此提出以图像整体特征的总变差能量为目标函数，空间分数阶向量值Cahn-Hilliard方程为约束的最优控制模型，以达到局部细节修复和整体特征保持的均衡效果。通过L₂梯度流、H^-1梯度流和凸分裂设计非凸约束条件的数值计算格式，再结合分裂Bregman方法优化目标函数，并引入灰度级动态调整策略，保持灰度辨识能力的同时，进一步提升计算效率。数值实验表明，新模型修复结果的峰值信噪比（peak signal to noise ratio, PSNR）相较其他方法提升0.371 8 ~ 9.935 2 dB，结构相似指数（structural similarity, SSIM）表现出较强的竞争力，且在碎片破损的图像上更具效用；相较传统的分数阶方程模型，计算时间减少49.50% ~ 52.91%。

In image inpainting, it is crucial that the identification and inpainting of local detail features and the preservation of global features. The models based on fractional-order partial differential equations were characterized by rich evolutionary behaviors, which allow image details to be effectively understood and a certain sharpening effect to be exhibited in image inpainting. However, issues such as inaccurate identification of large-scale features and over-sharpening are prone to be encountered. An optimal control model was proposed and the objective function was defined by the total variation energy of image global features and the constraint was formulated by a spatial fractional-order vector-valued Cahn-Hilliard equation, aiming to achieve a balanced effect between local detail restoration and preservation of global features. L₂ gradient flow, H^-1 gradient flow, and convex splitting were applied to design a numerical scheme for non-convex constraint conditions. And then the split bregman method was used to optimize the objective function with a dynamic grayscale adjustment strategy was introduced to maintain grayscale discrimination capability while enhancing computational efficiency. The numerical experiments demonstrate that the new model achieves an improvement on peak signal to noise ratio(PSNR) ranging from 0.371 8 dB to 9.935 2 dB compared to other methods, exhibiting strong competitiveness in terms of structural similarity(SSIM) and greater effectiveness on images with fragmental damages. Moreover, compared to traditional fractional-order equation models, the computational time is reduced by a factor of 49.50% to 52.91%.