A simplified stochastic optimization model for logistic dynamics with control-dependent carrying capacity

A simplified stochastic control model for optimization of logistic dynamics with the control-dependent carrying capacity, which is motivated by a recent algae population management problem in the river environment, is presented. Solving the optimization problem reduces to finding a solution to a non-local first-order differential equation called tt-Jacobi-Bellman (HJB) equation. It is shown that the HJB equation has a unique viscosity solution and that the solution can be approximated with a finite difference scheme. Asymptotic estimates of the solution and the optimal control are derived and compared with numerical solutions. Finally, parameter dependence of the optimal control is examined numerically with implications to river environmental management.