Studies On Well-posedness And Optimal Control Of Several Fractional Biological Model

This paper mainly considers the well-posedness and optimal control problems of several types of fractional-order biological systems,including the well-posedness,blow up and asymptotic behavior,the existence of optimal control and the first order necessary conditions and so on.It is divided into five chapters.Chapter 1 firstly elaborates the background of fractional biological model and introduces the research status at home and abroad.Then some preliminaries required in this paper are reviewed.At last,we also give the main results of this article.Chapter 2 considers the global existence of a generalized Keller-Segel system with Logistic sources.Based on the known local solutions,we first establish the L~?criterion for the existence of global solutions.The L~pboundedness can be obtained through difference inequalities and L~1uniformly boundedness of the solutions.Hence the global existence of the solutions is obtained by the maximum principle.In order to reduce the limitation of the parameter?,we also consider the weak solutions of the system.Based on the previous L~pestimate,by use of the Aubin-Lions compactness lemma,the global existence of weak solutions can be obtained.Chapter 3 studies the well-posedness problem of a class of generalized parabolic elliptic chemotaxis models with Logistic sources.Compared with the fractional order chemotaxis model in Chapter 2,this model also has a nonlinear secretion term.This makes it necessary to discuss and analyze the nonlinear secretion term firstly.Similar to the method in Chapter 2,through L~1boundedness,it is possible to more accurate-ly calculate and obtain the L~puniformly bounded estimate of the solutions,thereby establishing a uniformly bounded global solution.Therefore,on the basis of above,we continue to discuss the asymptotic behavior of the solutions.By the definition and related properties of the fractional Laplace on the n dimensional torus,we construct the upper and lower homogeneous solutions and conclude that the solution converges to 1 with exponential decay at infinite time.Finally,for the critical case?=1 in one-dimensional torus,we also discuss the stability of the homogeneous solutions and accurately give the range of the relevant parameters in the system.Chapter 4 studies the weak solution and blow-up criterion of a class of time-space fractional order attract-repulsive chemotaxis systems.By use of the mollified tech-niques and the compactness theorem of time fractional differential equations,we first establish the global weak solution of the system:when the repulsion is greater than the attraction,the existence of the global weak solution does not require other restrictions on the initial mass,and when the attraction is greater than the repulsion Time,the ini-tial mass must be small enough.Therefore,in the case where attraction is greater than repulsion,we consider whether the system has a blow up solution.We use the method of truncated moments to consider our problem in the two-dimensional space.By con-structing a smooth concave function and giving the bilateral estimates of the concave function,we can establish the fractional differential inequality for the local moment of the solution.Then the blow up criterion can be obtained.Our results are news.Chapter 5 studies the dynamic analysis and optimal control of the fractional singu-lar Leslie-Gower prey-predator model.Firstly,we discuss the stability of the singular model.In particular,singularity induced bifurcation phenomenon will cause an im-pulse phenomenon of the ecosystem,which may lead to the collapse of the proposed model system.Therefore,we eliminate this negative effect by introducing feedback control.In order to reduce the energy consumption in the control process,we also con-sider the related optimal control problem and obtain the first-order necessary conditions through the variational method and singular system theory.Finally,relevant numerical simulation are given to confirm the validity of our methods and theories.Chapter 6 gives a summary of the contents of this doctoral thesis and draws the future research prospects.