We formulate a new two-variable river environmental restoration problem based on jump stochastic differential equations (SDEs) governing the sediment storage and nuisance benthic algae population dynamics in a dam-downstream river. Controlling the dynamics is carried out through impulsive sediment replenishment with discrete and random observation/intervention to avoid sediment depletion and thick algae growth. We consider a cost-efficient management problem of the SDEs to achieve the objectives whose resolution reduces to solving a Hamilton-Jacobi-Bellman (HJB) equation. We also consider a Fokker-Planck (FP) equation governing the probability density function of the controlled dynamics. The HJB equation has a discontinuous solution, while the FP equation has a Dirac's delta along boundaries. We show that the value function, the optimized objective function, is governed by the HJB equation in the simplified case and further that a threshold-type control is optimal. We demonstrate that simple numerical schemes can handle these equations. Finally, we numerically analyze the optimal controls and the resulting probability density functions.

我们基于跳跃随机微分方程（SDEs），制定了一个新的两变量河流环境恢复问题，该方程控制着大坝下游河流中的沉积物存储和讨厌的底栖藻类种群动态。通过离散和随机观察/干预的脉冲沉积物补给来控制动力学，以避免沉积物枯竭和浓密的藻类生长。我们考虑了SDE的一种经济高效的管理问题，以实现其分辨率降低为解决Hamilton-Jacobi-Bellman（HJB）方程的目标。我们还考虑了一个Fokker-Planck（FP）方程，该方程控制着受控动力学的概率密度函数。HJB方程具有不连续解，而FP方程沿边界具有狄拉克增量。我们证明了价值函数，在简化的情况下，优化的目标函数由HJB方程控制，此外，阈值类型的控制是最佳的。我们证明了简单的数值方案可以处理这些方程式。最后，我们对最优控制和由此产生的概率密度函数进行数值分析。