Optimal Control Problem Of Non-Linear Population System With Age-Dependent

The survival environment of human has been deteriorated, therefor,some biologicalpopulations become extinct. In this condition,people have realized that the sustainabledevelopment is extremely significant in control theory. So that the problem of optimalcontrol for ecological population dynamics is a fully active field. We can work up amathematic model to describe the optimal control problem of ecological populations andchoose suitable control variables. It is a highly efective tools for solving the optimalcontrol problems. At the same time, the optimal control theory is invaluable for ecologicalenvironment. Since1980s,the research team of the biomathematics have become moreand more.In recent twenty years, scholar’s research can provide reasonable, scientific andefective management methods for related department.We discuss the optimal control of three classes of non-linear population system withage-dependent. These systems’ fertility and mortality continuous rely on age-dependentand weighted size. The paper solve the system solution on characteristic line, andby Gronwall Lemma and Banach-fixed-point Theorem, the existence, uniqueness,non-negativeness, boundedness and continuous dependence on initial data of solution of thesystem are proved. Then, we can obtain the optimal control problem of existence andnecessary condition of optimality by using Mazur’s Theorem, the conception of normalcone means and conjugate equation theory.This paper prepare to divide into three chapters:(Ⅰ) Optimal control of a competing system with age dependent and weighted size.In the second chapter, using Banach Fixed Point Theorem,Gronwall Lemma andBellman Lemma,the well-posedness of the solution of the system are proved firstly. Thenby Mazur’s Theorem, we demonstrate existence of solution of the optimal control problemand by means of normal cone, we obtain the first order necessary conditions of optimalityfor problem.(Ⅱ) Optimal control of a predator-prey system with age dependent and weightedsize.In the third chapter, we analyze the system of predator-prey and obtain the solutionof this system. Then we prove the well-posedeness of the solution, the existence of theoptimal solution and necessary condition optimality.(Ⅲ) Optimal control of three competing species system with age dependent andweighted size.In the forth chapter, we introduce the optimal control of three competing species with age and weighted size. Similar, the above conclusions are obtained. This biologicalmodel is more realistic.