In this paper, we investigate the optimal control of a semi-linear fractional PDEs involving the spectral diffusion operator, or the realization of the integral fractional Laplace operator with the zero Dirichlet exterior condition, both of order s with \(s\in (0,1)\). The state equation contains a non-smooth nonlinearity, and the objective functional is convex in the control variable but contains non-smooth terms. As the mappings involved may not be Gâteaux differentiable, we use a regularization technique to regularize these nonlinear terms, aiming to obtain Gâteaux differentiable mappings. By employing this regularization technique, we are able to derive the first-order optimality condition for the regularized control problem by using the associated adjoint system. Furthermore, we conduct a limit analysis on the regularized term resulting in an optimality system for the non-smooth problem of C-stationary type. Subsequently, we establish a primal optimality condition, specifically B-stationarity. Under the assumption of “constraint qualification”, we derive the strong stationarity conditions for the non-smooth optimization problem with control constraints and establish the equivalence between B-stationarity and strong stationarity conditions.

在本文中，我们研究了涉及光谱扩散算子的半线性分数阶偏微分方程的最优控制，或具有零狄利克雷外部条件的积分分数阶拉普拉斯算子的实现，两者都是 s 阶的 \（s\in （0,1）\）。状态方程包含非平滑非线性，目标泛函在控制变量中是凸的，但包含非平滑项。由于所涉及的映射可能不是 Gâteaux 可微的，我们使用正则化技术来正则化这些非线性项，旨在获得 Gâteaux 可微分映射。通过采用这种正则化技术，我们能够通过使用相关的伴随系统推导出正则化控制问题的一阶最优条件。此外，我们对正则化项进行了极限分析，从而得到了 C 平稳型非平滑问题的最优系统。随后，我们建立了一个原始最优性条件，特别是 B 平稳性。在 “约束限定” 的假设下，我们推导出了具有控制约束的非平滑优化问题的强平稳性条件，并建立了 B 平稳性和强平稳性条件之间的等效性。