Theory and applications of impulse extension equations

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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.