多解p-Laplacian型椭圆方程的最优控制

Optimal control for multisolution elliptic equations of p-Laplacian type will be studied in this project. The control domain and the object functional may be nonconvex and our goal is to obtain the maximum principle for optimal pairs. As a representative of degenerate quasilinear equations, multisolution p-Laplacian equations have a wide range of application background. Therefore, on the research of optimal control problems for them has important theoretical and practical significance. In theory, the combination of multiplicity, degeneracy of state equations and nonconvex conditions is the main difficulty that we are going to overcome. First, the state equation admits multiple solutions and then a state does not depend continuously on a control, thus the necessary conditions of optimal pairs can not be obtained by the classical variational method directly. We need to consider the corresponding penalization problem. Second, the state equation is degenerate and then the corresponding nondegenerate perturbation problem must be considered. Moreover, a solution of the penalization problem or the nondegenerate perturbation problem may not exist under nonconvex conditions. To overcome the difficulties mentioned above, we will use a skill of the combination of the relaxation method and the penalization method. The relaxation method is essentially a convexification technique, while the penalization method is an effective way to deal with the case of a state equation admits multiple solutions.

本项目拟研究多解p-Laplacian型椭圆方程的最优控制问题，控制区域和指标泛函可能非凸，目标是得到最优对的最大值原理。多解p-Laplacian方程作为退化拟线性方程的代表，具有广泛的应用背景，故对其最优控制问题的研究具有重要的理论和实际意义。从理论上讲，状态方程的多解性、退化性和非凸条件相结合是我们拟克服的主要困难。首先，状态方程多解，那么状态关于控制不连续，故不能直接应用古典变分法得到最优对的必要条件，必须要考虑其惩罚问题。其次，状态方程退化，则要考虑其非退化扰动问题。再者，非凸条件下惩罚问题或非退化扰动问题的解可能不存在。为克服上述困难，我们拟采取松弛方法与惩罚方法相结合的技巧。松弛方法本质上是一种凸化技术，而惩罚方法是处理状态方程多解的有效途径。