The Existence And Uniqueness Of Classical Solutions And The Optimal Control Analysis For A Class Of Nonlinear Mixed Forest Evolution Systems

The paper is intended to study a class of nonlinear mixed forest evolution models and proved the existence and uniqueness of classical solutions of them in a determined condition by characteristic-line and Schauder Fixed-point theorem. Then the existences of the optimal control about effective afforestation update rate and the cutting rate are proved by some theorems of compact space. The necessary conditions for the optimality are gotten at last.In the first chapter, some common models of quantitative distribution structures are introduced and also illuminate and analyze some researches on these forest models at home and abroad. Besides, it is shown by some basic knowledge that this paper will be used. In the second chapter, the existence and uniqueness of solution is discussed in the nonlinear age-structured mixed forest evolution system by characteristic-line and Schauder Fixed-point theorem while effective afforestation update rateα_i (t )and the cutting rateμ_i ( x , t)are anything.In the third chapter, while the control variable isα_i (t ), the target is the net receipt that is the difference of economic price and investment economic value to achieve maximum in an operating cycle in the nonlinear age-structured mixed forest evolution system. The existence of optimal solution and the optimal necessary condition of optimal control problem are obtained in the nonlinear age-structured mixed forest evolution system about the effective afforestation update rate. While the control variable isμ_i ( x , t), the target is the net receipt. The existence of optimal solution and the optimal necessary condition of optimal control problem are obtained in the nonlinear age-structured mixed forest evolution system about the cutting rate.In the fourth chapter, discussing the nonlinear age-structured mixed forest evolution system whose boundary condition is changed, the existence and uniqueness of solution of it is proved in determined conditions by characteristic-line and Schauder Fixed-point theorem. Then we get the necessary conditions for the optimality.This subject has significant meaning, whether in theory or in practice. In theory it gives the solutions of some partial differential equations exist under certain conditions. In practice according to the theory of forest knowledge forest can be optimized renovation, which economic forest can create better benefits and shelter forest can make trees much more.