论文标题
HJB和Fokker-Planck方程，用于基于随机冲动控制的河流环境管理，并具有离散和随机观察
HJB and Fokker-Planck equations for river environmental management based on stochastic impulse control with discrete and random observation
论文作者
论文摘要
我们根据在大坝下游的河流中管理沉积物存储和滋扰底栖藻类种群动态的基于跳跃随机微分方程（SDE）的新两变量环境恢复问题。控制动力学是通过脉冲沉积物补充和随机观察/干预措施来进行的，以避免沉积物消耗和厚藻类的生长。我们考虑了SDE的一个成本效益的管理问题，以实现其解决方案减少解决汉密尔顿 - 雅各比 - 贝尔曼（HJB）方程的目标。我们还考虑了控制受控动力学概率密度函数的Fokker-Planck（FP）方程。 HJB方程有一个不连续的解，而FP方程式沿边界具有狄拉克的三角洲。我们表明，在简化的情况下，值函数（优化的目标函数）受HJB方程的控制，进一步是阈值类型控制是最佳的。我们证明了简单的数值方案可以处理这些方程。最后，我们通过数值分析最佳对照和产生的概率密度函数。
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.
