A Class Of Two-dimensional Purebamboo Development System With The Free Boundary Problems

In this paper, a class of two-dimensional pure bamboo development system model of the free boundary has been studied,and in the area of "diameter-age of bamboo", according to the actual demand,We need introduce a special kind of curve to simplified the model,then the existence systematic solution could be proved by the methods of characteristic line, the second type of Volterra’s integral and fixed point theorem of Schauder. However, the uniqueness of classical solution of the problem mainly use free boundary to conduct zero-padding which can extend to the space fixed boundary. At the same time, we conduct the optimal control for the cutting edge.The paper contains four parts.Chapter1is devoted to the introduction of the formation and development of the two-dimensional pure bamboo system, then give the research background and the related knowledge of this paper.In chapter2,we build a free boundary problem developing in model of two-dimensional pure bamboo system,and study the existence and uniqueness of solution. The main results is as follows:The model has the form as below.Theorem2.1Let P0∈O1(Ω),β(.)∈O1([0,T]),k(.)∈O1([0,T]) and M>0,such that‖g(x,y,t)‖O1(QT)≤M,‖φ(x,y)‖O1(QT)≤M Then system (2.1.1) has a unique and nonnegative solution Ph (x,y,t)∈O1,1(QT). If there exists a constant m0>0and for any po(x,y)>0in Q,then ph(x,y,t)>0,p(l,y,t)=0,p(x,l,t)=0in QT t∈[0,T],N(t)≥m0Chapter3is devoted to a class of two-dimensional affected by environment of pure bamboo about free boundary model,and discuss the existence and uniqueness of the solution.The main results is as follows:The model has the form as below.The model is based on chapter2, the same conclusion with chapter2can be obtained.In chapter4,we we conduct the optimal control for the cutting edge. The main results is as follows:Theorem4.2Let ph(x,y,t) is the state of dominate system about (4.1.2); U is admissible control set; J[h] is performance index function,then there is a h*(y,t)∈U such that J｜h*｜=maxJ[h].