The Optimal Control Policies And Algorithm Design For Some Periodic Ecosystems With Impulsive Effects

The sustainable development has recently become a general consensus of all the countries. At the same time the sustainable development and the utilization of resources, as an important part of the general consensus, are receiving more and more attentions. It has been a focus studying how to optimize the management of the renewable biological resources, and much research was focused on optimizing the development of the resources modelled by population models.The exploiture to biological populations can be classified as continuous or im-pulsive. At present, the research on optimizing impulsive exploiture focused mainly on how to optimize the harvest yield without considering the cost factors of ex-ploiture. In the real development process, the cost is indispensable, so it is of important practical significance considering the cost when choosing a optimization object.Generally speaking, finding a solution to an optimal control problem is difficult. In many cases, the analytical expression of the solution is unlikely obtained and in these cases an approximate numerical solution is necessary. In designing algorithms for optimal control problems, there are some better results obtained, but there is little research on the numerical algorithms for the impulsive optimal control problem. The research on this class of problems is necessary from the point of both the theoretical and the application view.In this dissertation, we study some impulsive optimal control problems of some population systems. Under the sustainable development condition, we study the periodic optimal control of an ecological system with impulsive harvest, and design a numerical algorithm for finding the optimal numerical solution of the general pe-riodic impulsive control system. We further design a universal numerical simulation algorithm for some impulsive population system. The researches enrich the system control theory of impulsive differential systems. Because impulsive optimal control has various practical applications, these results can be applied to some practical systems, and have practical values.The following problems are discussed and results are obtained in this thesis: (1) We consider the optimal harvesting control of a biological species system whose growth is governed by the Gompertz equation with periodic coefficients. The species is harvested at fixed moments for economic profit. The purpose is to control the harvesting effort so as to maximize the profit which is the difference between economic revenue and cost, assuming that the cost in the payoff functional depends linearly on the control variable. We show the existence of an optimal impulsive har-vest control, and by the maximum principle for optimal impulse control problem, we characterize the optimal control in terms of an optimality system which is expressed by a numerical equations. Two special cases are considered, and the precise optimal controls are achieved for the cases.(2) We design an algorithm for a class of impulsive harvest control problem with general form. The control system is described by a periodic differential system with impulsive harvest, and the objective function is choose as harvesting yield, and the maximum harvesting yield is needed to seek. By using the steepest descent method, we design the numerical algorithm to find the optimal solution and verify the correctness of the algorithm by the simulation results for an example.(3) In the final part of this thesis, we mainly study numerical simulation al-gorithms for impulsive differential systems, an algorithm of assignment functions is established. According to the characteristics of impulsive differential systems, and using Runge-Kutta algorithm for ordinary differential equations as base, combining with the method to determine the impulsive time, we design a new numerical al-gorithm to carry out numerical simulation for impulsive differential systems. The new numerical algorithms can be applied to typical impulsive differential systems to compute their numerical solution, to plot final state diagram and time series dia-gram. Two typical examples are discussed and the numerical simulations are carried out, some meaningful results are obtained.