We design and compute a class of optimal control problems for reaction-diffusion systems. They form mean field control problems related to multi-density reaction-diffusion systems. To solve proposed optimal control problems numerically, we first apply high-order finite element methods to discretize the space-time domain and then solve the optimal control problem using augmented Lagrangian methods (ALG2). Numerical examples, including generalized optimal transport and mean field control problems between Gaussian distributions and image densities, demonstrate the effectiveness of the proposed modeling and computational methods for mean field control problems involving reaction-diffusion equations/systems.

翻译：我们设计并计算了一类反应扩散系统的最优控制问题。这些问题构成了与多密度反应扩散系统相关的平均场控制问题。为了对所提出的最优控制问题进行数值求解，我们首先采用高阶有限元方法对时空域进行离散，然后利用增广拉格朗日方法（ALG2）求解该最优控制问题。数值算例（包括高斯分布与图像密度之间的广义最优输运和平均场控制问题）验证了所提出的建模与计算方法在涉及反应扩散方程/系统的平均场控制问题中的有效性。