具有混合约束的非光滑双层最优控制理论和算法

Bilevel optimal control models have been widely used in diverse fields such as supply chain management, logistics engineering, etc., especially in the conflicts and coordination issues of hierarchical dynamical systems. It is very challenging to solve bilevel optimal control problems. This is because the bilevel optimal control problems related to intersect different branches of mathematics including the optimal control theory, nonsmooth analysis theory, and bilevel programming theory. Theoretical studies of bilevel optimal control problems is just getting started, and even many fundamental problems in this research field have not yet been solved satisfactorily. In this study, we are going to investigate a class of nonsmooth nonconvex bilevel optimal control problems with mixed constraints. We plan to investigate the fundamental properties and necessary optimality conditions for these problems, propose algorithms for solving such problems, and conduct numerical simulations. In this project, the first step is to find the necessary optimality conditions and the optimization algorithms for bilevel optimal control problem in which the lower level problem is a finite dimensional parametric optimization problem. The general nonsmooth nonconvex bilevel optimal control problems will then be studied. The outcomes of this study will fill in the gaps in the theories of bilevel optimal control problems and promote the development of application of the bilevel optimal control models in solving practical problems of management and engineering fields. It is therefore that our research is of important theoretical significance and great application prospects.

双层最优控制模型在供应链管理、物流工程等问题的建模过程中有着广泛的应用，特别是在建模分等级的动态系统的冲突和合作问题。由于双层最优控制问题涉及到数学不同分支的相互交叉，包括最优控制理论、非光滑分析理论和双层规划理论等，难度较大，要解决这类问题很具有挑战性。国际上对双层最优控制问题的理论研究才刚起步，乃至许多基础性的问题都尚未得到令人满意的解答。本项目拟针对一类具有混合约束的非光滑非凸双层最优控制展开研究，探讨此类问题的基本性质和最优性必要条件等理论问题，并提出求解这类问题的优化算法，进行数值试验。本项目首先解决下层问题是有限维情形的双层最优控制问题的最优性必要条件和优化算法，再研究一般的非光滑非凸的双层最优控制问题。上述研究不仅可以填补双层最优控制理论的某些空白，又可推动管理、工程等领域运用双层最优控制去建模解决实际问题，因此具有重要的理论意义和应用前景。

许多实际控制问题需要在一些特殊的混合状态-控制约束（如微分变分不等式约束）下寻求最优控制策略，其中双层最优控制问题作为具有广泛应用背景的一类特殊最优控制问题，其研究还未达到对关键限制因素的理论突破。本项目研究了非光滑非自治动力系统的最优控制问题的最优性必要条件。针对上述问题，我们提出一个新的约束规格：Weak basic constraint qualification(WBCQ)+Calmness。我们得到了在WBCQ+Calmness约束规格条件下的最优性必要条件。我们将该结论应用到隐控制系统最优控制问题中，隐控制系统最大的挑战是约束映射的Jacobin矩阵是奇异的，我们发现WBCQ+Calmness约束规格允许约束映射的Jacobin矩阵是奇异的，因此对隐控制系统采用该约束规格是很合适的。利用上述结果我们针对下层是有限维参数优化情形的双层最优控制问题展开研究，得到了此类问题的基本性质和最优性必要条件等理论成果。..本项目还研究了一类生化反应系统的辨识和动态优化问题。我们给出了一个测量系统的浓度鲁棒性的定量指标。基于提出的鲁棒性指标，构造了一个minimax动态优化问题用于辨识动力学参数以及未知的代谢机制。利用Monte-Carlo方法来近似评估鲁棒性指标，并得到了收敛性结果。构造了一种求解动态优化问题的算法，数值结果表明所提出的鲁棒性指标可以正确地测量该系统的浓度鲁棒性。

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