Optimal Inputting Rate Control For Two Classes Of Age-dependent Competitive Population System

In modern society, people have more and more realized the importance of the sustainable development. Base on sustainable development, we should not only satisfy humanity's material want but also protect Ecological balance in the exploitation Biological resources, so that our offsprings can gain better survival and development. At present, many academicians at home and abroad try their best to research the problems of the optimal control for population dynamics. By the principles of biology and ecology, we set up more appropriate capability target, select more appropriate control variable, research the problems of optimal control and exploitation for age-dependent competing population dynamics. It has practical meaning and applied value. According to the contents, this paper is divided into three chapters.In the first chapter, we give the preface. It mainly introduces the background, the present studying of population dynamics at home and abroad and the preparative theory.In the second chapter, the problem of the optimal inputting rate control for a class of competitive population dynamics system with age-structure is discussed. By means of characteristic line, Gronwall Lemma, Bellman Lemma, adjoint system and so on, the Lipschitz continuity properties of the system in terms of the control variable is certificated, and the necessary condition of the optimal control is obtained.The third chapter is about the optimal control for a class of age-dependent competitive population dynamics with diffusion, the continuity properties of the system solution in terms of the control variable is certificated, the Normal cone theory is applied and the necessary conditions of optimal control problem is obtained.The fourth chapter is the summary-of paper.