中文摘要：

该文考虑抑制剂作用下肿瘤生长的模型.假设肿瘤是球对称的,其表面为运动边界,用函数r=R（t）表示.既然多细胞肿瘤扁球体（MTS）通常作为肿瘤生长的体外模型,在实验室能够被观察和控制,因此研究如下反问题：根据观察到的MTS动态增长（即给定R（t））,来确定抑制剂的参数.运用极大值原理,作者证明了该抛物反问题解的唯一性.进一步,用最优控制框架来重构模型中的抑制剂参数,证明了最优控制问题解的存在性,并推导了最优控制满足的最优性必要条件.

英文摘要：

In this paper,the authors consider a model of tumor growth in the presence of inhibitors.The tumor is assumed to be spherically symmetric and its surface is a moving boundary denoted by a function r = R（t）.Since multicellular tumor spheroids（MTS） are routinely used as in vitro models of cancer growth and they can be observed and controlled in the laboratory,the authors study the following inverse problem：Given observed dynamics of MTS growth（i.e.,given R（t））,the authors determine the inhibitor＇s parameter. The authors first prove the uniqueness of solution to the inverse parabolic problem by the maximum principle. Then the authors develop an optimal control framework for studying the reconstruction of the inhibitor＇s parameter. The authors prove the existence of solution to the optimal control problem, and the authors derive the necessary optimality conditions which have to be satisfied by each optimal control.