In this paper an indirect spectral method of an optimal control problem governed by a space-fractional diffusion equation with variable diffusivity coefficient is studied. First-order optimality conditions of the proposed model are derived and the regularity of the solutions is analyzed. Indirect spectral methods via weighted Jacobi polynomials are built up based on the “first optimize, then discretize” strategy, and a priori error estimates of the discrete optimal control problem in weighted norms are derived. As proposed indirect spectral discrete schemes are designed to accommodate the impact of the variable coefficients, which are complicated and not formulated under the variational framework, conventional error estimate techniques of the discrete space-fractional optimal control problems do not apply. Novel treatments of the discrete variational inequality are developed to resolve the aforementioned issues and to support the error estimates. Numerical examples are presented to verify the theoretical findings.

本文研究了由变扩散系数的空间分数扩散方程控制的最优控制问题的间接谱方法。推导了所提出模型的一阶最优性条件，并分析了解的规律性。基于“先优化，后离散”的策略，通过加权雅可比多项式建立间接谱方法，并推导出加权范数下离散最优控制问题的先验误差估计。由于所提出的间接谱离散方案旨在适应复杂且未在变分框架下制定的可变系数的影响，因此离散空间分数最优控制问题的常规误差估计技术不适用。离散变分不等式的新处理被开发来解决上述问题并支持误差估计。给出了数值例子来验证理论发现。