Optimal Impulsive Control Of Some Resources Management Models

With the rapid development of market economy, many mathematicians, ecologists andeconomists are interested in the reasonable utilization of renewable resources. As a renewableresource, the optimal management of fishery resource is more important. For the situation ofharvesting a large of fish population in a short time, the optimal control theory with impulsiveeffect is a powerful tool to deal with these problems. Based on these, the main contents of thispaper are as follows:Firstly, on the basis of single population model with Logistic growth, the correspondingmodel with impulsive effect is proposed. According to the principle of economics, areasonable objective function is given to get the exploitation model with impulsive effect.Using the optimal impulsive control principle, the control is exerted to get the optimalimpulsive harvesting time and the corresponding harvesting level which make fishman getmaximum economic benefit. They provide the theoretical foundation for the study on fisheryresource.Secondly, on the basis of single population model with Logistic growth, thecorresponding model with continuous harvest and impulsive harvest is proposed. Accordingto the principle of economics, a reasonable objective function is given to get the exploitationmodel. Using the optimal impulsive control principle, the switch time of optimal continuousharvesting and the corresponding harvesting level are obtained. Further, the optimal impulsiveharvesting time and the corresponding harvesting level are obtained also. They provide thebackground for the applications of optimal impulsive control principle.Thirdly, on the basis of a prey-predator model, people obtain the economic benefit byharvesting predator population. According to the principle of economics, a reasonableobjective functional is given to get the exploitation model. Using the geometric theory ofdifferential equations, the sufficient conditions of the existence of periodic solution to themodel are got. Using the optimal impulsive control principle, the optimal impulsiveharvesting time and the corresponding harvest level which make fishermen get maximumeconomic benefit are obtained. Further, the optimal impulsive harvest time with periodicityand the corresponding harvest level are obtained.