非瞬时脉冲周期系统及其可控性研究

Non-instantaneous impulsive action characterizes the perturbation depending on system state and starts at an arbitrary fixed point and remains active on a finite time interval. Non-instantaneous impulsive periodic systems describe that period phenomena mingles with non-instantaneous impulse phenomena, which can be applied to pharmacotherapy, fishery resources sustainable development and etc. The current contributions are concentrated on autonomous systems. There are few works on other type systems, especially for fractional order periodic systems, and some essential problems are still not solved. This project will investigate integer order and fractional order non-instantaneous impulsive periodic systems. Based on theory analysis results, we study controllability problems. Firstly, we study fundamental solutions operator for integer order non-instantaneous linear impulsive periodic systems and fundamental property, Lyapunov characteristic exponents and asymptotical stability of solutions, adjoint linear system and Perron theorem, further, we discuss existence of periodic solutions, subharmonic solutions and attractor for the global attractor operator. Secondly, we study non-existence of periodic solutions and discuss global existence and stability of solutions and stable manifolds, and asymptotical periodic solutions via almost periodic solutions. Finally, we study completed controllability for integer order non-instantaneous impulsive periodic systems and approximate controllability of fractional order non-instantaneous impulsive periodic systems, which will provide some necessary theory for the research on optimal controls and pulse-width sampler controller.

非瞬时脉冲是指干扰过程依赖于状态且持续作用一段时间。非瞬时脉冲周期系统适合描述自然界中周期现象和非瞬时脉冲现象相互交织和影响的过程，在药物动力学、渔业资源等方面有着广泛的应用。现有成果主要集中在自治系统，对其它系统研究的还很少，特别是对分数阶周期系统，一些关键问题还没有突破。本课题拟对整数阶和分数阶非瞬时脉冲周期系统展开定性分析，并讨论相应的可控性问题。首先，研究整数阶非瞬时线性周期系统对应的基本解算子及其性质，解的Lyapunov指数与渐近稳定性，共轭线性系统，讨论非线性周期系统周期解、次谐波解及全局吸引子的存在性。其次，研究分数阶非瞬时线性脉冲系统周期解不存在性，讨论非线性系统有界解的全局存在性与稳定性及中心稳定域，渐近周期解和概周期解的存在性。最后，研究整数阶非瞬时脉冲周期系统完全可控性和分数阶非瞬时脉冲周期系统逼近可控性，为后续研究最优控制问题和调宽采样控制系统奠定必要的理论基础。

本课题针对整数阶和分数阶非瞬时脉冲周期系统展开定性分析，并讨论相应的可控性问题。首先，研究了整数阶非瞬时线性周期系统对应的基本解算子及其性质，解的Lyapunov指数与渐近稳定性，共轭线性系统，讨论非线性周期系统周期解的存在性与稳定性。其次，研究了非线性系统有界解的全局存在性与稳定性及中心稳定域，渐近周期解和概周期解的存在性。最后，研究整数阶非瞬时脉冲系统完全可控性和分数阶非瞬时脉冲系统逼近可控性。在此基础上，在迭代学习控制、时滞系统、多智能体系统、洋流方程等方面做了一些探索。