一类新Willis环脑动脉瘤系统的构造及最优控制

本文基于6-氨基乙酸减小血流速度及缓冲血流波动的性质,在6-氨基乙酸的作用下,构建了一个新的脑动脉瘤数学模型.该模型很好地体现了6-氨基乙酸、血流速度及血流速度变化率三者之间的相互作用关系.鉴于脑动脉瘤的医疗费用颇高及破裂后的死亡率较高,从而有必要从数学的角度研究该模型的最优控制,以致在一定条件下花费最小且医疗效果最佳.本文首先证明了该模型最优控制的存在性;其次通过构造Lagrangian函数及运用最大值原理,证明了最优控制的唯一性.从理论上得到Willis环脑动脉瘤内血流波动最小的条件,这为预防脑动脉瘤的破裂提供了理论依据..

Modeling and Optimal Control for A New System of Aneurysm of Circle of Willis

Based on the properties of aminocaproic acid to slow down fluctuation of blood flow in aneurysm,in this paper,we propose a new mathematic model of aneurysm of circle of Willis, which shows the relationship among aminocaproic acid、blood flow and the rate of blood flow in aneurysm.Because of the high cost and mortality of aneurysm,for mathematical resons,it＇s necessary to study the optimal control of the new model for smaller cost and better cure.First ,we prove the existence of an optimal control.Second,we construct a Largrangian function and use Pontryagin＇s Maximum Principle to prove the uniqueness of the optimal control of the new aneurysm Model.Theoretically,this paper denotes that,if the normal blood flow is ensured,the fluctuation of blood flow in aneurysm will be smallest under certain conditions,so the aneurysm won＇t rupture easily,and the research of optimal control actively help to prevent the rupture of aneurysm of circle of Willis.