The Optimal Inputting Rate Control For Two Classes Of Age-dependent Competing System Of N Species

The value of biological resources has long been recognized, and more and more people's attention focus on it. As a renewable resource, biological resources not only have extremely high economic value, but also have immeasurable value for ecological environment. The present situation is that biological resources are reducing rapidly and the ecological environment also has been deteriorated sharply. Therefore, to achieve the sustainable development of biological resources, it must be rationally used. The optimal control problem of population dynamics system studies population's development and control by mathematical methods, and it is an active field in control theory. At home and abroad many academicians try their best to research the optimal control problem of population dynamics system and obtain a lot of achievements. Taking into account the reality that there exist multiple populations in the same environment, in the paper the optimal inputting rate control problem for age-dependent competing system of n species is researched. According to the contents, this paper is divided into three chapters:In the first chapter, it introduces the background and significance, the model of population dynamics system, the present study of population dynamics system at home and abroad and the preparative theories.In the second chapter, the optimal inputting rate control problem for a class of age-dependent competing system of n species is discussed. By means of Ekeland's variational principle, Gronwall Lemma, Bellman Lemma, Fatou Lemma, the Banach fixed point theorem, conjugated system and so on, the existence and uniqueness of the optimal control are proven, furthermore the necessary condition of the optimal control problem is obtained by the concep-tion of normal cone.In the third chapter, it is about the optimal inputting rate control problem for a class of age-dependent competing system of n species with diffusion. The model of the conjugated system is given and analyzed, moreover the necessary condition of optimal control problem is obtained by the conception of normal cone.