Theory and applications of impulse extension equations

脉冲微分方程在生物和物理问题中有许多应用，如传染病建模，控制理论和种群动力学。该理论是建立在这样一个假设之上的，即假设系统中足够短的扰动是瞬时发生的，这是很自然的，因为它们的长度与过程的持续时间相比是可以忽略不计的。尽管如此，人们还是很自然地提出这样一个问题：“假设足够短的过程是瞬间发生的，这总是安全的吗？“这个问题出现在实践中，当人们试图找到一个估计的全球最大的脉冲周期轨道，这有许多现实世界的应用。我打算探讨的情况下，固定的脉冲，有一个非平凡的齐次分量和相应的脉冲扩展方程，以及扩展的结果与不固定的和自主的脉冲方程。在此之后，我们将致力于为非线性系统开发类似的技术。我们还将脉冲扩展方程联系到Filippov系统，Filippov系统是导数中具有不连续性的动力系统。Filippov系统在科学和工程中有着广泛的应用，包括收获阈值、油井钻探和液气反应等。通过理解脉冲近似的性质和局限性，以短突发行为，我准备好发展和应用脉冲扩展方程的理论。我认为，这将是一个工具，可以与脉冲微分方程一起使用，这将是有用的，在应用背景下，当处理生物，物理或其他真实世界的模型，其中精度的结果是重要的。

Impulsive differential equations have a host of applications to both biological and physical problems, such as infectious disease modelling, control theory and population dynamics. The theory is founded upon the assumption that it is often natural to assume that sufficiently short perturbations in the system occur instantaneously, since their length is negligible in comparison with the duration of the process. Still, it is natural to ask the question: "Is it always safe to assume that sufficiently short processes occur instantaneously?" This question comes up in practice when one attempts to find an estimate on the global maximum of an impulsive periodic orbit, which has many real-world applications. I intend to explore the case of fixed impulses that have a nontrivial homogeneous component and their corresponding impulse extension equations, as well as extend the results to equations with unfixed and autonomous impulses. Following this, we will work at developing similar techniques for nonlinear systems. We will also link impulse extension equations to Filippov systems, which are dynamical systems with discontinuities in the derivatives. Filippov systems have many applications in science and engineering, including harvesting thresholds, oilwell drilling and liquid-gas reactions, for which the differential equation is extended to a differential inclusion. By understanding the nature and limitations of impulsive approximations to short-burst behaviour, I am well-poised to develop and apply the theory of impulse extension equations. It is my view that this will be a tool to be employed alongside impulsive differential equations, which will be useful in an applied context when dealing with biological, physical or other real-word models where precision of the results are important.