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。虽然脉冲微分方程对非脉冲微分方程的近似的粗糙度很重要，但我们选择从更广泛的角度来解决这个“近似”问题。特别地，如果我们将脉冲微分方程视为物理过程的极限情况(例如，当扰动时间趋于零时)，我们能确信脉冲周期解的存在保证了对于足够短的脉冲，在原始物理过程中存在周期解吗？我们的目标是开发一个过程，通过这个过程，我们可以从脉冲微分方程构造一个常微分方程，该方程带有脉冲条件的“结构”，但允许脉冲持续有限的、非零的时间量。我们应该具备的条件是：1.这种新的、“延长的”或“延长的”冲动持续的时间应该足够短，以便在前一个冲动结束之前不会出现新的冲动。2.在没有系统演化的情况下，冲量“延伸”对系统的作用应与原始冲量相同。3.脉冲扩展应具有与原始脉冲相等的周期条件。我们称这类新的微分方程为脉冲扩展方程。人们可以用两种方式之一来解释它：一种是“近似”建立它的脉冲微分方程的微分方程，另一种是脉冲微分方程通过假设脉冲在系统上瞬时作用而努力模仿的更现实的系统模型。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。我们还将把脉冲延拓方程与Fillipv系统联系起来，Fillipv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。我在脉冲微分方程及其应用方面的大量工作对我这个项目很有帮助。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. Although the degree of coarseness of an approximation of a nonimpulsive differential equation by an impulsive one is important, we choose to tackle this question of “approximation” from a broader angle. In particular, if we view an impulsive differential equation as a limiting case of a physical process (eg as the perturbation time approaches zero), can we be confident that the existence of an impulsive periodic solution guarantees that a periodic solution exists in the original physical process, for sufficiently short impulses? Our goal is to develop a process by which we can construct an ordinary differential equation from the impulsive differential equation that carries with it the “structure” of the impulse condition, but allows the impulse to last a finite, nonzero amount of time. The conditions that we should have are: 1. The amount of time this new, “stretched” or “extended” impulse lasts should be short enough that new impulses do not occur before the previous one has finished. 2. The impulse “extension” should have the same effect on the system as the original impulse in the absence of system evolution. 3. The impulse extension should have a periodicity condition equivalent to the original impulse. We call this new class of differential equations "impulse extension equations". One can interpret it in one of two ways: as a differential equation which “approximates” the impulsive differential equation it is built from, or as a more realistic model of a system that the impulsive differential equation strives to emulate by the assumption that impulses are acting instantaneously on the system. 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 Fillipov 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. My considerable work on impulsive differential equations and their applications stands me in good stead for this project. 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.