Theory and applications of impulse extension equations

脉冲微分方程在生物和物理问题上有着广泛的应用，如传染病模型、控制理论和种群动力学。该理论建立在这样的假设之上，即通常很自然地假设系统中的足够短的扰动是瞬时发生的，因为与过程的持续时间相比，这些扰动的长度可以忽略不计。尽管如此，人们很自然地会问这样一个问题：“假设过程足够短是瞬间发生的，总是安全的吗？”在实践中，当人们试图找到脉冲周期轨道的全局最大值的估计时，这个问题就会出现，这在现实世界中有很多应用。我打算研究具有非平凡齐次分量的固定脉冲的情形及其相应的脉冲扩展方程，并将结果推广到具有不固定脉冲和自治脉冲的方程。在此之后，我们将致力于为非线性系统开发类似的技术。我们还将把脉冲扩展方程与Filippov系统联系起来，Filippov系统是导数不连续的动力系统。Filippov系统在科学和工程中有许多应用，包括收获阈值、油井钻探和液气反应，因此将微分方程式推广到微分包裹体。通过了解短脉冲行为的脉冲近似的性质和局限性，我已经做好了充分的准备来发展和应用脉冲扩展方程的理论。在我看来，这将是一个与脉冲微分方程一起使用的工具，在处理结果精度很重要的生物、物理或其他真实世界模型时，这将在应用环境中有用。

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.