An optimal control model for cloud seeding in a deterministic and stochastic environment

To promote artificial rain in India and other such developing countries, in this article, we have proposed and analyzed a nonlinear mathematical model for cloud seeding by considering that aerosols are introduced proportional to the density of water vapors present in the atmosphere. The model is analyzed using Lyapunov's stability theory of differential equations. To reduce the cost of cloud seeding, an optimal control strategy is designed by incorporating four control parameters. We have shown the existence and uniqueness of solution of this optimal control problem, using Pontryagins Maximum Principle. To minimize the cost in making artificial rain, the optimal control problem provides the strategy for the rate of introduction of aerosols in the atmosphere. To capture the effects of environmental noise on control strategies, the model in deterministic framework is converted into stochastic framework. In this regard, the Hamilton-Jacobi-Bellman equation for stochastic control cost function has been formed. The stochastic analysis implies that the control strategy is effective in reducing the cost and increasing rainfall.