一类非凸不适定问题最优对的存在性及不存在性

A class of nonconvex non-well-posed optimal control problem will be studied in this project. The state equation considered is a semilinear non-well-posed elliptic type equation. The control domain and the object functional maybe all nonconvex. The purpose is to obtain the existence and nonexistence result of an optimal pair. . In the theory of optimal control, the minimizing sequence method is usually used to discuss the existence problem. However, this method is mostly applicable to the case where the state equation is well-posed and the condition of convexity is satisfied. As far as the problem of this project is concerned, firstly the state equation is non-well-posed and the state does not depend continuously on the control, thus one cannot obtain the estimate inequality of a state with respect to a control using the theory of PDEs directly. Secondly, the control domain and the object functional maybe all nonconvex, these constraint conditions lead to the minimizing sequence method is not applicable to this problem. This is the largest difficult point of the problem which will be studied in this project. To overcome this difficulty, we will use the relaxed control method. The conclusion that when an optimal relaxed control is a Dirac measure, it is certainly a classical optimal control is mainly used. . The non-well-posed equation has extensive practical backgrounds, which can describe the bifurcation phenomena, enzymatic reactions phenomena 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.

本项目拟研究一类非凸不适定最优控制问题，考虑的状态方程为半线性不适定椭圆型方程，控制区域以及指标泛函都可能非凸，目标是得到最优对的存在性及不存在性结果。 . 在最优控制理论中，讨论存在性问题，通常采用极小化序列方法，但该方法大多适用于状态方程适定且凸性条件满足的情形。就本项目中的问题而言，首先，状态方程不适定，状态关于控制没有连续依赖性，无法直接利用方程的理论得到状态关于控制的估计不等式；其次，控制区域以及指标泛函都可能非凸，该约束条件决定了极小化序列方法对于本问题的不适用性。这是本项目拟研究问题的最大难点。为克服这个困难，我们拟采用松弛控制方法，主要是利用当最优松弛控制是一个Dirac测度时，则它一定是古典最优控制这一结论。 . 不适定方程有广泛的实际背景，比如可以描述分歧现象、酶反应现象等。因此相关结果可以更好地丰富最优控制理论的内容，并可为某些实际问题提供理论指导。

不适定偏微分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。本项目主要研究由半线性不适定椭圆型偏微分方程所支配的最优控制问题，得到以下四个研究成果。一、在非凸条件下研究了半线性不适定椭圆方程所支配的最优控制问题，得到了最优对的必要条件。为克服非凸带来的困难，我们提出了一个新的想法，即将原问题转化成一个带有点点混合约束条件的适定问题，然后利用爱克兰变分原理证明适定问题存在近似解，最后在近似解的必要条件中取极限并将约束条件还原，就得到了原问题最优对所满足的Pontryagin’s最大值原理。二、研究了一类只有两个解的半线性椭圆方程所支配的最优控制问题，分别在不同条件下得到了最优对的存在性和不存在性结果。主要的工具是松弛控制。具体地，首先考虑原问题所对应的松弛问题，然后给出最优松弛对的Pontryagin’s最大值原理。最后利用最大值条件证明某个最优松弛控制几乎处处是一个Dirac测度，那么它一定是古典最优控制，故原问题的最优对存在，否则不存在。三、同样利用松弛控制，考虑了一类带有非线性Neumann边界条件的半线性椭圆方程所支配的最优控制问题，得到了最优控制的存在性。四、研究了一类多解常微分方程的时间最优控制问题。利用方程解的性质，得到了最优控制的存在性及Pongtryagin’s最大值原理。其中三个研究成果已经发表。