微分包含与不连续微分方程的理论及应用研究

On the basis of existing research, some theoretical and applied problems of differential inclusions are further investigated . In particular, we establish and develop the basic theories and research methods of the solutions to functional differential inclusions, such as the Lyapunov method of stability for functional differential inclusions, the asymptotic and exponential stability in Lagrange sense, etc. And then, by employing the topological degree theory in set-valued functions, some new fixed point theorems of the set-valued analysis, non-smooth critical point theory and variational methods, comparison principle and other modern mathematical theory and methods, we thoroughly study some classes of differential equations with discontinuous right-hand sides. Especially, considerable efforts will be devoted to investigate the basic theory and large-time dynamics of solutions for discontinuous functional differential equations, such as the local or global existence and uniqueness of initial value problems, the continuation of solutions, the continuous dependence of solutions on initial value or on systematic parameters, the convergence of solutions in finite time (especially for periodic solutions), the existence of (positive) equilibrium point, the (positive) periodic solution, the almost periodic solution, the sliding motion, the homoclinic solutions and heteroclinic solutions, as well as the bifurcations and chaos caused by the parameters changes. At last, to provide reliable theoretical basis for practitioners, we perform a thorough analysis of the dynamic behaviors for some discontinuous differential equations which come from the actual field such as the input-to-state stability and the integral input-to-state stability of discontinuous non-autonomous control system with time-delays.

在已有研究成果的基础上，进一步探讨微分包含若干理论和应用问题，特别是建立和发展泛函微分包含解的基本理论和研究方法，如泛函微分包含稳定性研究的Lyapunov方法，Lagrange意义下的渐近与指数稳定性等。在此基础上，辅之以发展和应用集值函数的拓扑度理论，集值分析中一些新的不动点定理，非光滑临界点理论和变分方法，比较原理等现代数学理论与方法，深入研究一些类型的不连续微分方程，特别是不连续泛函微分方程解的基本理论和大时间状态，包括解的局部(整体)存在性和唯一性、延拓性、对初值及系统参数的连续依赖性、有限时间收敛性（特别是对周期解），（正）平衡点、（正）周期解、概周期解、滑模解、同宿解和异宿解的存在性，以及由参数变化所引起的分岔和混沌等。深入分析一些实际领域中不连续微分方程模型解的动力学性质，如不连续非自治时滞控制系统的输入-状态稳定性与积分输入-状态稳定性等，为实际工作者提供可靠的理论依据。

本项目进一步探讨了微分包含若干理论和应用问题，特别是建立和发展了泛函微分包含解的基本理论和研究方法，如泛函微分包含稳定性研究的Lyapunov方法，Lagrange意义下的渐近与指数稳定性等。在此基础上，通过发展和应用集值函数的拓扑度理论，集值分析中一些新的不动点定理，非光滑临界点理论和变分方法，比较原理等现代数学理论与方法，深入研究了几类不连续微分方程，特别是不连续泛函微分方程解的基本理论和大时间状态，包括解的局部(整体)存在性和唯一性、延拓性、对初值及系统参数的连续依赖性、有限时间收敛性（特别是对周期解），（正）平衡点、（正）周期解、概周期解、滑模解、同宿解和异宿解的存在性，以及由参数变化所引起的分岔和混沌等。此外，本项目还深入分析了一些实际领域中不连续微分方程模型解的动力学性质，如不连续非自治时滞控制系统的输入-状态稳定性与积分输入-状态稳定性等，这将为实际工作者提供可靠的理论依据。