Impulsive hybrid systems with delays and applications

在过去的二十年里，脉冲混合系统已经成为连接应用数学、控制工程和计算机科学的一门新学科。它们为复杂反应系统的数学建模提供了一个框架，在这些系统中，物理过程与人为的自动化环境相互作用。这类系统是由微分方程组(或差分方程组)控制的动力系统，由于模型特性的突然变化，这些系统经历矢量场切换和/或状态跳跃。它们产生于广泛的应用，如人工智能、通信、空间技术、基于互联网的医疗系统和无人驾驶汽车。所涉及的微分方程组可以是一组常微分方程组、时滞微分方程组、积分微分方程组、偏微分方程组、随机微分方程组，甚至是上述方程的混合体。相应的差分方程式也可以有多种形式。在提出的研究方案中，我将研究脉冲混合系统，它包含状态依赖脉冲、状态依赖切换、状态依赖时滞、随机扰动，甚至是这些特征中的两个或两个以上的组合。我的研究计划的长期目标是推进脉冲混合系统的理论，开发新的思想和新的方法来处理依赖状态的脉冲/切换和颤振问题，并使其有可能应用于现实世界的问题。(I)建立具有状态依赖脉冲和/或分布时滞和/或随机扰动的脉冲混合系统的稳定性和鲁棒稳定性理论；(Ii)发展具有状态依赖脉冲和时滞脉冲的脉冲时滞微分方程的分支理论；以及(Iii)将所获得的理论和方法应用于实际问题，如无人自主车辆的编队控制和网络上的地方病模型。这一研究项目为加拿大提供了一个走在尖端技术前沿的绝佳机会，并为加拿大的大学和工业培养高素质人才提供了一种手段。预计拟议的研究将产生许多高质量的科学出版物、几名训练有素的研究生和本科生研究人员，以及一系列有效的解决方法和相关软件，使更多的非专家能够解决各自学科内的实际问题。这些学生将获得足够的分析和计算技能来进行独立研究，指导和教授学生，并在学术界或某些工业部门找到职业生涯，如机器人、人工智能和公共卫生。这一成果将极大地提高加拿大在脉冲混合系统领域的研究声誉。

Impulsive hybrid systems have emerged into a new discipline over the past two decades that bridges applied mathematics, control engineering, and computer science. They provide a framework for mathematical modeling of complex reactive systems where physical processes interact with man-made automated environments. Such systems are dynamical systems governed by differential (or difference) equations, which undergo vector field switching and/or state jumps, due to sudden changes in model characteristics. They arise from a wide variety of applications, such as artificial intelligence, communications, space technology, Internet-based health system, and unmanned autonomous vehicles. The differential equations involved may be a set of ordinary differential equations, delay differential equations, integro-differential equations, partial differential equations, stochastic differential equations, or even a mixture of the equations mentioned above. The corresponding difference equations can have a variety of forms as well. In the proposed research program, I will study impulsive hybrid systems that contain state-dependent impulses, state-dependent switching, state-dependent delays, random disturbance, even a combination of two or more of these features. The long-term goal of my research program is to advance the theory of impulsive hybrid systems and develop new ideas and novel methods to deal with state-dependent impulses/switching, and chattering problems, and make possible applications to real world problems. The objectives over the proposed period are to : (i) establish stability and robust stability theory for impulsive hybrid systems with state-dependent impulses and/or distributed delays and/or stochastic disturbances; (ii) develop bifurcation theory for impulsive delay differential equations with state-dependent impulses and those with delayed impulses; and (iii) apply the obtained theory and methods to practical problems, such as formation control of unmanned autonomous vehicles, and endemic models over networks. This research program represents an excellent opportunity for Canada to be at the fore-front of the cutting-edge technology and provide a means to train highly qualified personnel for Canadian universities and industry. It is envisaged that the proposed research will result in many high-quality scientific publications, several well-trained graduate students and undergraduate researchers, and a range of effective solution methods with associated software that will enable a wider range of non-experts to solve practical problems within their own disciplines. These students will obtain adequate analytical and computational skills to conduct independent research, mentor and teach students, and land a career in academia or in certain sectors of industry, such as robotics, artificial intelligence, and public health. The outcomes will greatly enhance Canada's research reputation in the field of impulsive hybrid systems.