Studies On Dynamics And Optimal Control Problem For Some Haptotaxis Models

This thesis studies the dynamical properties and optimal control problem for several kinds of haptotaxis models,including the global existence,uniqueness,boundedness,and asymptotic behavior of solutions to the associated Neumann initial-boundary value problem,the existence of optimal control,the first-order nec-essary optimality condition and numerical simulations.The thesis consists of the following six chapters.Chapter 1 mainly focuses on introducing the background and development of the optimal control theory and haptotaxis or chemotaxis-haptotaxis cancer invasion model.In addition,we briefly state the main results of this thesis.Chapter 2 mainly studies the optimal problem for a haptotaxis cancer invasion model,where the dosages of radiations and chemotherapeutic drugs are regarded as the control functions.Our primary goal is to characterize an optimal control which balances the therapeutic benefits with its side effects.Firstly,we obtain the well-posedness of solution to the state system by applying the Leray-Schauder fixed point theorem and developing adapted a priori estimates,and prove the existence of op-timal pair by using the technique of minimizing sequence.Secondly,we prove the Lipschitz continuity of the control-to-state mapping,and thereafter use the method of convex perturbation to derive the first-order necessary optimality condition.Finally,we present the optimal control strategies and validate the theoretical results by some numerical simulations.Chapter 3 considers a quasilinear chemotaxis-haptotaxis cancer invasion model with general logistic source and nonlinear signal production.When the exponents of cell diffusion,chemotactic and haptotactic sensitivities,logistic dampening and nonlinear signal production satisfy certain explicit conditions,we prove the glob-al existence,uniqueness and boundedness of classical solution in any dimensional bounded domain by means of the maximal Sobolev regularity,L~p-L~qestimates of the Neumann heat semigroup,Moser-Alikakos iteration and so forth.Moreover,the asymptotic behavior of this solution is investigated by utilizing the adapted Moser-Alikakos iteration technique and constructing the appropriate Lyapunov functional.Chapter 4 deals with the optimal problem for a haptotaxis cancer invasion mod-el with two cancer cell species,where the dosages of two different chemotherapeutic drugs are regarded as the control functions.Our main goal is to characterize an opti-mal control which balances the therapeutic benefits with its side effects.Firstly,we prove the well-posedness of solution to the sate system by using the Banach fixed point theorem,L~pestimate and Moser-Alikakos iteration technique.Subsequently,the existence of optimal pair is proved by employing the technique of minimizing se-quence.Furthermore,we prove the Lipschitz continuity of the control-to-state map-ping,and establish the first-order necessary optimality condition by employing the method of convex perturbation.Finally,we numerically present the optimal control and optimal regimen of therapeutic agents administrated,and validate some clinical findings.Chapter 5 discusses a two species cancer invasion haptotaxis model with tissue remodeling.Relying on some a priori estimates and iterative technique,we estab-lish the global existence,uniqueness and boundedness of classical solution in two-dimensional bounded domain for arbitrary logistic damping and in three-dimensional bounded domain for large logistic damping.Chapter 6 summarizes the main results of this thesis,and gives the prospect of future researches.