Establishment of a long-term biological population management policy requires balancing its cost and utility/disutility. We approach this issue from the viewpoint of stochastic control as an efficient candidate for its modelling and analysis. A remarkable point of the present model is making good use of an unbounded performance index, which naturally penalises too much population decrease. The goal of the present optimal control problem is approached through solving a Hamilton–Jacobi–Bellman equation having a solution blowing up at a boundary. We thus positively use a singular mathematical model. Solution behaviour of the HJB equation is analysed from the viewpoint of viscosity solutions with the help of an asymptotic expansion technique, which can handle the blow up under regularity assumptions. Numerical framework for analysing the optimal control problem is presented and examined as well, to qualitatively determine the profile of the solution, optimal human intervention, and probability density function.

建立长期的生物种群管理政策需要平衡其成本和效用/效用。我们从随机控制的角度来处理这个问题，作为其建模和分析的有效候选者。本模型的一个显着点是很好地利用了无界性能指数，这自然会惩罚过多的人口减少。当前最优控制问题的目标是通过求解具有边界处解法的 Hamilton-Jacobi-Bellman 方程来实现的。因此，我们积极使用奇异数学模型。借助渐近展开技术从粘度解的角度分析了 HJB 方程的解行为，该技术可以在规则性假设下处理爆炸。