基于Lojasiewicz不等式的广义梯度系统大时间行为及应用

Large-time behavior analysis of a dynamic system described by ordinary differential equations and partial differential equations is an important and complex problem. At present, to our best knowledge, there is no general method for this problem. A gradient system is a special dynamic system in mathematics, physics, mechanics and engineering calculation etc. and has been widely applied. Additionally, one of valid tools for the large-time behavior of a gradient system is ?ojasiewicz inequality and its extended forms..Concerned on different generalizations of the ?ojasiewicz inequality in finite and infinite dimensional spaces, this project focuses on the analysis of large-time behavior of generalized gradient system- - - - a generalized version of the usual gradient system, based on the structure of functional spaces and analytic algebra geometry.In theory, we investigate the relation between trajectory length of generalized gradient systems and geometric algebraic property of function. Then we discuss the internal relations between large-time behavior and space structure for first-order and second-order generalized gradient systems to provide a general functional analysis framework for large-time behavior of the corresponding partial differential equations. In practice, we search for a quick and effective algorithm based on dynamic neural network. Then, we give a new method to analyze the synchronous behavior of Cuker-Samle model and Kuramot model under complex conditions. The research of this project is the enrichment and development of gradient system theory, and will provide new tools for optimization and multiagent systems.

常微分方程与偏微分方程所描述的动力系统大时间行为分析是一重要且复杂的问题，尚无统一的处理方法。梯度系统作为特殊的动力系统在数学、物理、力学以及工程计算等领域有广泛应用，其大时间行为分析的有效工具之一是?ojasiewicz不等式及其推广。.本项目立足于梯度系统的推广-广义梯度系统的大时间行为分析，以泛函空间结构和解析代数几何为工具，围绕?ojasiewicz不等式在有限维空间与无穷维空间的各种推广，理论上，揭示广义梯度系统的轨道长度与函数的几何代数性质之间的关系；探索无穷维空间一阶、二阶广义梯度系统大时间行为与空间结构的内在联系，以期为相应偏微分方程的大时间行为提供统一泛函分析框架；应用上，寻找求解优化问题的快速、高效动态神经网络算法以及分析复杂情况下Cuker-Samle模型与Kuramoto模型同步的新方法。本项目的研究是梯度系统理论的丰富和发展，将为优化和多智体系统提供新的研究工具。

本项目以Lojasiewicz梯度不等式为工具，立足于一阶、二阶广义梯度系统大时间行为的动力学特性分析，采用微分方程、泛函空间与非线性理论相结合的方法，对该系统的收敛性、收敛率、吸引域估计、稳定性做了理论研究。以此为基础，应用于具体系统，如 Kirchhoff系统的边界控制稳定性、非光滑优化算法的收敛性、群体运动的 Cucker-Smael 模型的群体行为，Kuramoto 模型的同步与锁相、电力系统暂态稳定的吸引域估计等，揭示了这些系统长时间行为的机理。项目取得的成果不仅丰富了梯度系统的相关理论，而且在电力系统稳定性分析、飞行器和机器人编队、优化算法的设计等方面具有指导意义。