Nonlinear impulsive and hybrid systems with time delay: theory, methods and applications

近年来，脉冲和混合动力系统的研究在世界范围内引起了越来越大的兴趣。这类系统出现在许多应用中，如通信网络、癌症化疗和糖尿病的药物管理、受限冲击机械、电力电子开关电路、航天器的非平稳和混合控制以及生态系统管理。然而，也有许多具有挑战性的问题。首先，连续动力系统的许多定理对于脉冲动力系统和混合动力系统都是不成立的，特别是当系统含有时滞时。需要开发新的想法，以一种健壮且易于应用的方式处理冲击、跳跃、切换和抖动问题。其次，对其动力学的定性理解需要特定的方法，不能用连续的线性或非线性动态系统的已知技术的简单改编来解决。该研究计划的目标是：(I)发展时滞脉冲和混杂动力系统的基本理论；(Ii)发展一些处理冲击、跳跃、切换和颤振问题的一般方法；(Iii)发展稳定性和鲁棒稳定性判据和计算吸引域的方法；(Iv)发展具有延迟反馈的脉冲和混杂动力系统的镇定控制设计方法；(V)发展复杂网络的脉冲同步算法；(Vi)发展有效的计算算法，用于脉冲和混杂动力系统的仿真和控制；(Vii)将所获得的方法应用于现实世界的问题。在这个项目中，将进行基础研究(哲学、概念和理论研究)和技术开发(系统设计和计算机模拟)。这些成果将提高加拿大在前沿研究方面的声誉，并促进国际合作的机会。它还将为高素质人员的研究培训提供一个极好的机会。

There has been a growing world-wide interest in the study of impulsive and hybrid dynamical systems in recent years. Such systems arise in many applications such as communication networks, drug administration in cancer chemotherapy and diabetes, constrained impact mechanics, switching circuits in power electronics, non-smooth and hybrid control of spacecrafts, and ecosystems management. However, there are many challenging issues. Firstly, many theorems of continuous dynamical systems fail for impulsive and hybrid dynamical systems, especially  when the systems contain time delays. New ideas need to be developed that deal with impacts, jumps, switching, and chattering problems in a robust and easily applicable way. Secondly, the qualitative understanding of their dynamics requires specific methods and cannot be solved with simple adaptations of known techniques for continuous linear or nonlinear dynamical systems. The objectives of the proposed research program are (i) to develop the fundamental theory for impulsive and hybrid dynamical systems with time delay; (ii) to develop some general methods that deal with impacts, jumps, switching, and chattering problems; (iii) to develop stability and robust stability criteria and methods of computing the domain of  attraction; (iv) to develop stabilizing control design methods for impulsive and hybrid dynamical systems with delayed feedback; (v) to develop algorithms for impulsive synchronization of complex networks; (vi) to develop efficient computational algorithms for simulation and control of impulsive and hybrid dynamical systems; (vii) to apply the obtained methods to real world problems. In this program, both fundamental research (philosophy, concepts and theoretical investigation) and technical development (system design and computer simulation) will be conducted. The outcomes will enhance Canada's reputation for leading edge research and facilitate opportunities for international collaboration. It will also provide an excellent opportunity for research training of highly qualified personnel.