Maxwell Fitzsimmons| Applied Math, University of Waterloo
Theory of Computation for Analysis and Control of Discrete-Time Dynamical Systems
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). Although computability theory is well studied by computer scientists, it is still in its infancy when it comes to studying dynamical systems. In my thesis research, I will develop computability theory for reachability analysis and control of discrete-time dynamical systems.