A Stochastic Model to Account for System Uncertainties Applied to an Optimal Harvesting Problem

Abstract In the modeling of stochastic nonlinear systems it is often necessary to consider the existence of uncertainties about some parameters of the system or in the distribution implied by the stochastic shocks, and in this setting, the control policy can be obtained by the rules of robust control theory. To present a different way to handle control systems in this scenario, this paper introduces a theoretical framework for continuous-time stochastic nonlinear systems in which the Control Variations Increase the state Uncertainty (CVIU systems). This type of system can be applied in several areas of science and engineering, due to its ability in dealing with complex stochastic systems, for which the dynamics are not completely known. In particular, the paper applies this model to a problem of fishery management in an unknown environment, and establishes a harvesting rule for it based on the CVIU approach. Using dynamic programming and tools from nonsmooth analysis, the control solution indicates the existence of a region in the state space in which the optimal harvesting is of no variation, yielding periods of inaction. This behavior obtained from optimality is somehow expected from the cautionary nature of controlling underdetermined systems.