关于一类非线性系数反演问题最优解的适定性研究

Well-posedness of an Optimal Solution to the Inverse Problem of a Class of Nonlinear Coefficient

研究一个利用附加条件重构二阶抛物型方程的扩散系数的反问题,它在许多应用科学领域都有重要的意义.基于最优控制理论,原问题被转化为一个优化问题.该问题的主要困难有二:其一,所需反演的未知系数是二阶抛物性方程的二阶项系数,这是一个完全非线性的强不适定问题;其二,附加条件并非通常意义下的终端观测条件,而只是一个平均意义下的观测值.由于控制泛函没有凸性,通常很难得到最优解的唯一性.然而,通过对极小元所满足的必要条件的仔细分析,并结合正问题的先验估计式,发现当终端时刻T取值适当小时,可以证明极小元的局部唯一性和稳定性.

An inverse problem of reconstructing the diffusion coefficient in the second- order parabolic equation is investigated by using additional conditions, which has important application in a large field of applied science. Based on the optimal control framework, the original problem is transformed to an optimization problem. Such problem has two main difficulties. The first is that the unknown coefficient needed is the principle coefficient of parabolic equations ,which is known as a fully nonlinear and severely ill-posed problem. Secondly,the additional condition is an average sense of observation, rather than the common terminal observation condition. It is quite diffi- cult to get the uniqueness of the optimal solution due to the non-convexity of the control functional. However, when the terminal point is relatively small, the local uniqueness and stability of the minimizer can be proved by carefully analyzing the necessary condition and using the priori estimates of the forword problem,which is also the main contribution of this article.