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. 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.