Study On Optimal Control Of Several Chemotactic Systems

In different environments,by controlling the concentration of related chemicals,enzyme activity,ambient temperature and other conditions,the biological concentration can be kept within a reasonable range,which has important applications in scientific research.Moreover,these mathematical models have important reference value in real life,such as the control of cancer cell concentration,the limitation of biological population and so on.The development of optimal control theory began in the early 1950s.In response to the great demand of simulating real life,more and more mathematicians began to pay attention to the optimal control of partial differential equations.Therefore,we study the control of several chemotaxis models.The first chapter is the introduction,which first introduces the biological background of the research problem and the research progress at home and abroad.Next,it introduces the use methods and innovations of this paper.In Chapter 2,the following optimal control problem of chemotactic-fluid equation is considered where ?(?)R2 is a bounded region with smooth boundary (?)?,? and ? are normal numbers.Firstly,we overcome the computational difficulties caused by the coupling of u and other parabolic equations,prove the global existence of the solution by using the Leray-Schauder fixed point theorem,and deduce the existence of the global optimal solution.In addition,based on some prior estimates,we study the differentiability of the control to the state operator.Next,In order to overcome the difficulties of u ·?n,u ·?c and u·?u in pushing to the adjoint system.With the help of special treatment of fluid velocity u,we prove the regularity of the adjoint system solution and the first-order necessary conditions of the local optimal solution.In Chapter 3,optimal control of chemotaxis-haptotaxis model system is discussed where(?)R2 is a bounded region with smooth boundary (?)?,?,?,r,? are normal numbers.Since the first equation in(5)contains the second-order spatial derivative of w,and the third equation is an ordinary differential equation that cannot improve the spatial regularity of w,vwis included in the w equation,which strengthens the coupling of the spatial regularity of v and w,so we use the expression formula about w different from that used to prove the classical solution,At the same time,in order to reduce the regularity requirements for w and v when the strong solution of the system exists,In addition,we quote the conclusion on the regularity of heat equation with Neumann boundary to prove the global existence and uniqueness of the strong solution,and use the minimization sequence to obtain the existence of the optimal solution.Through complex operations to solve the difficulties caused by ordinary differential equations,the differentiability of the control to the state operator is obtained.Then,using the Lagrange multiplier theorem,and combining with Lp-estimates of the linear parabolic equations,the regularity estimation of the solution of the adjoint problem is obtained.Finally,the first-order necessary conditions for the local solution are derived.In Chapter 4,we consider the following chemotaxis-growth system where ?(?) R2 is a bounded region with smooth boundary (?)?,and ? is a normal number.Introduce f as f(u)=1-?u,p,is a normal number.First,we use the the special treatment of ?u?L2 and the properties of ln(u+1)function solve the difficulties caused by f(u)=1-?u and ordinary differential equations,and obtain the existence and uniqueness of the system solution in two-dimensional space.Then the existence of the optimal solution is obtained by using the minimization sequence.In addition,based on some prior estimates,we prove the Lipschiz continuity of the control to state mapping,and obtain the differentiability of the control to state operator by using the convex perturbation method.Next,we obtain the regularity of the adjoint system solution by using the Lp-estimates of the linear parabolic equations,and establish the optimality system of the problem(6).