Positive Solutions And Time Optimal Control Problem For A Competitive Reaction-Diffusion Equations

In this thesis,we concern the controllability of parabolic systems of equations with one internal control.Such kind of model represents a large number of types in partial differential equations of parabolic type,and such kind of problem arises among the recently popular research in the theory of controllability.We obtain the positive solutions of the competitive reaction-diffusion equations and controllability.Firstly,we establish the existence of positive solutions for the competitive reactiondiffusion equations by the method of upper and lower solutions and energy estimates.With the positive trajectory,we study the controllability and time optimal control problems of the competitive reaction-diffusion equations.Secondly,we study the null controllability of the one-control reaction-diffusion system under the idea of that exact null controllability of the linear system is equivalent to the observability estimate of the conjugate system,which is one of the essential steps in the study of controllabilty of nonlinear systems.Thirdly,we prove the locally exact controllability of the nonlinear reaction-diffusion equations.This is accomplished by linearizing the nonlinear system and adopting the method which combines the null controllability of the linear system and the Kakutani's fixed point theorem.Finally,we obtain an application of the locally exact controllability in the existence of time optimal control.By calculating the explicit estimates related to time T in Carlman inequality and observability estimates and their consequent results,we show the existence of time optimal control.The controllability problems of competitive reaction-diffusion equations with onecontrol and other similar types of parabolic equations are not yet fully covered.There are many scientific problems worthy of discussion.The research of this paper may supplement the research in this research field.