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Properties of difference inclusions with computable reachable set

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Fitzsimmons_Maxwell.pdf (870.27 KB)

Date

2023-12-13

Authors

Fitzsimmons, Maxwell

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University of Waterloo

Abstract

Dynamical systems have important applications in science and engineering. For example, if a dynamical system describes the motion of a drone, it is important to know if the drone can reach a desired location; the dual problem of safety is also important: if an area is unsafe, it is important to know that the drone cannot reach the unsafe area. These types of problems fall under the area of reachability analysis and are important problems to solve whenever something is moving. Computer algorithms have been used to solve these reachability problems. These algorithms are primarily viewed from a numerical simulation perspective, where guarantees about the dynamical system are made only in the short-term (i.e. on a finite time horizon). Yet, the important properties of dynamical systems often arise from their long-term (asymptotic) behaviours. Furthermore, in sensitive applications it may be important to determine if the system is provably safe or unsafe instead of approximately safe or unsafe. The theory of computation (or computability theory) can be used to investigate whether computer algorithms can determine weather a dynamical system is provably safe. Computability theory, broadly speaking, is a field of computer science that studies what kind of problems a computer can solve (or cannot solve). For difference inclusions, a characterization of when the reachable set is computable was found by Pieter Collins. Difference inclusions, are one way of modelling discrete time dynamical systems with control. This thesis is an investigation into this characterization. Broadly, it is argued that this characterization is far to restrictive on the dynamical system to be of general practical use. For example, a continuous function f which maps the real line to itself, has a computable reachable set if and only if there is a metric d on the real line (which is equivalent to the standard metric) for which f is a contraction map with respect to d.

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Keywords

difference inclusion, reachable set, computability, asymptotic behaviour, multifunction, set valued analysis

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http://hdl.handle.net/10012/20160

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Applied Mathematics