We investigate the application of a posteriori error estimates to a fractional optimal control problem with pointwise control constraints. Specifically, we address a problem in which the state equation is formulated as an integral form of the fractional Laplacian equation, with the control variable embedded within the state equation as a coefficient. We propose two distinct finite element discretization approaches for an optimal control problem. The first approach employs a fully discrete scheme where the control variable is discretized using piecewise constant functions. The second approach, a semi-discrete scheme, does not discretize the control variable. Using the first-order optimality condition, the second-order optimality condition, and a solution regularity analysis for the optimal control problem, we devise a posteriori error estimates. We subsequently demonstrate the reliability and efficiency of the proposed error estimators. Based on the established error estimates framework, an adaptive refinement strategy is developed to help achieve the optimal convergence rate. The effectiveness of the refinement strategy is verified by numerical experiments.

翻译：本文研究带有逐点控制约束的分数阶最优控制问题的后验误差估计应用。具体而言，我们处理一类状态方程以分数阶拉普拉斯方程积分形式表述的问题，其中控制变量作为系数嵌入状态方程。针对该最优控制问题，我们提出两种不同的有限元离散方法。第一种方法采用全离散格式，其中控制变量使用分段常数函数离散化；第二种为半离散格式，不对控制变量进行离散化。基于最优控制问题的一阶最优性条件、二阶最优性条件及解正则性分析，我们推导了后验误差估计，并论证了所提误差估计子的可靠性与有效性。在已建立的误差估计框架基础上，提出自适应细化策略以帮助实现最优收敛速度。数值实验验证了该细化策略的有效性。