Optimal Control In Non-Linear Population Diffusion System With Age-Dependence

This doctoral dissertation discusses the non-linear problems of the fertilityrate controls in linear age-dependent time-varying population system (P₀) andthe existence and uniqueness of the generalized solution and the optimal controlproblems for non-linear age-dependent population diffusion system (P). This is aquestion with pratical background which one is concerned with.Chapter 1 is a introduction which summarizes the state of the internal andexternal research and the latest development in the field of population systems,then draws forth the problems to study and narrates briefly the main results ob-tained.Chapter 2 discusses the existence and uniqueness of the generalized solutionfor the non-linear age-dependent population diffusion system (P) and obtains theexistence and uniqueness theorem, which provides theoretical basis for the opti-mal control problems for the system.Chapter 3 is concerned with the fertility rate control problems for a linearage-dependent time-varying population system (P₀). When the cost functionJ₁ (β) takes the observation of the final state, it proves the existence of the opti-mal fertility rate control and obtains the necessary conditions for a controlβ^* tobe optimal and optimality systems; when the cost function J₂(β) takes the dis-tributed observation, it proves the existence of the optimal fertility rate controlβ^* and obtains first-order necessary conditions for a controlβ^* to be optimal andthe optimal feedback formula of expression forβ^*.Chapter 4 considers the optimal control problems for the non-linear age-de- pendent time-varying population diffusion system (P), where the variable v(r,t,x), which reflects the outside enviroment influence of population, is the control.When the cost function J₃ (v) contains the distributed observation, it proves theexistence of the optimal control u of system (P) and establishes the necessaryconditions for a control u to be optimal and the feedback formula of expression foru; When the cost function J₄(v) takes the observation of the final state, itproves the existence of the optimal control u and obtains the necessary conditionsfor a control u to be optimal and optimality systems.The theoretical results obtained in this doctoral dissertation have not beenfound in previous documents. These results provide theoretical foundation for thepratical research of population problems and therefore are of theoretical meaningand pratical value.