一类具非凸控制约束的不适定最优控制问题

A classe of non-well-posed optimal control problem will be studied in this project under the non-convex case. Specifically, the state equation considered is a semilinear non-well-posed parabolic type equation.The control domain and the object functional maybe non-convex. Our main purpose is to obtain the necessary condition of an optimal pair..　　In the non-well-posed system,the state does not depend continuously on the control. Hence, in the non-well-posed optimal control problem,to discuss the necessary condition for an optimal pair is essentially to find the necessary condition for the extreme point of a discontinuous functional. This is the largest difficulty of the non-well-posed optimal control problem.In addition,the restriction of non-convexity makes us cannot apply the classical penalizition method.We will propose a new idea to overcome the difficulty caused by the non-convexity. That is, we will transformate the original problem to a well-posed problem with the point-point mixed constraint. Then we will use the Ekeland''s variational principle and the homogenized spike variational method. Finally, we can obtain the necessary conditions for an optimal pair by a process of passing to the limit..　　The non-well-posed equation has extensive practical backgrounds, which can describe the bifurcation phenomena, enzymatic reactions phenomena and some phenomenen in plasma physics and so on. Therefore the relevant results can better enrich the content of the optimal control theory and can provide theoretical guidance for some practical problems.

本项目拟在非凸条件下研究一类不适定最优控制问题。具体地说，考虑的状态方程为半线性不适定抛物型方程，控制区域以及指标泛函都可能非凸。我们的主要目标是得到最优对的必要条件。.　　在不适定系统中，状态关于控制是不连续的。因此在不适定最优控制问题中，讨论最优对的必要条件，本质上是寻求一个不连续泛函的极值点的必要条件。这是不适定最优控制问题的最大难点。此外，非凸条件的限制使我们不能应用传统的惩罚方法。我们拟提出一个新想法来克服非凸带来的困难，即将原问题转化为一个有点点混合约束的适定问题，然后利用爱克兰变分原理和齐次针状变分方法，最后再结合取极限的过程，得到原问题最优对的必要条件。.　　不适定方程有着广泛的实际背景，比如可以描述分歧现象、酶反应现象、等离子体物理学中的一些现象等等。因此相关结果可以更好地丰富最优控制理论的内容，并且可为某些实际问题提供理论指导。

不适定方程有着广泛的实际背景，比如可以描述分歧现象、酶反应现象、等离子体物理学中的一些现象等等，由不适定方程所支配的最优控制问题被简称为不适定问题。本项目在非凸条件下研究一类不适定问题，目标是得到最优对的必要条件。这其中的主要工作是克服非凸条件带来的困难，进而得到惩罚问题最优控制的存在性。关于这个问题，我们进行了深入研究。通过分析松弛控制是否是古典控制的方法，得到了最优控制的存在性结果。该成果已经发表，在最优控制理论中具有单独的意义。