In this paper we present and analyze a weighted residual a posteriori error estimate for an optimal control problem. The problem involves a nondifferentiable cost functional, a state equation with an integral fractional Laplacian, and control constraints. We employ subdifferentiation in the context of nondifferentiable convex analysis to obtain first-order optimality conditions. Piecewise linear polynomials are utilized to approximate the solutions of the state and adjoint equations. The control variable is discretized using the variational discretization method. Upper and lower bounds for the a posteriori error estimate of the finite element approximation of the optimal control problem are derived. In the region where 3/2 < alpha < 2, the residuals do not satisfy the L2(Omega) regularity. To address this issue, an additional weight is included in the weighted residual estimator, which is based on a power of the distance from the mesh skeleton. Furthermore, we propose an h-adaptive algorithm driven by the posterior view error estimator, utilizing the Dorfler labeling criterion. The convergence analysis results show that the approximation sequence generated by the adaptive algorithm converges at the optimal algebraic rate. Finally, numerical experiments are conducted to validate the theoretical results.

翻译：本文针对一类最优控制问题，提出并分析了加权残差后验误差估计。该问题涉及不可微代价泛函、含积分分数阶拉普拉斯算子的状态方程以及控制约束。我们利用不可微凸分析中的次微分方法获得一阶最优性条件。采用分段线性多项式逼近状态方程和伴随方程的解，并通过变分离散化方法对控制变量进行离散。推导了该最优控制问题有限元逼近的后验误差估计的上界和下界。在3/2 < α < 2的区域内，残差不满足L2(Ω)正则性。为解决此问题，我们在加权残差估计子中引入基于网格骨架距离幂次的额外权重。进一步，我们提出基于后验视角误差估计子的h自适应算法，并采用Dörfler标记准则。收敛性分析表明，自适应算法生成的逼近序列以最优代数速率收敛。最后通过数值实验验证了理论结果。