Department of Mathematics and Statistics Graduate Theses
http://hdl.handle.net/1974/758
Mon, 26 Aug 2019 14:10:25 GMT2019-08-26T14:10:25ZAccelerated Convergence of Saddle-Point Dynamics
http://hdl.handle.net/1974/26476
Accelerated Convergence of Saddle-Point Dynamics
McCreesh, Michael
In this thesis, a second-order continuous-time saddle-point dynamics is introduced that mimics Nesterov's accelerated gradient flow dynamics. We study the convergence properties of this dynamics using a family of time-varying Lyapunov functions. In particular, we study the convergence rate of the dynamics for classes of strongly convex-strongly concave functions. For a class of quadratic strongly convex-strongly concave functions and under appropriate assumptions, this dynamics achieves global asymptotic convergence; in fact, further conditions lead to an accelerated convergence rate. We also provide conditions for both local asymptotic convergence and local accelerated convergence of general strongly convex-strongly concave functions.
http://hdl.handle.net/1974/26476Filter Stability, Observability and Robustness for Partially Observed Stochastic Dynamical Systems
http://hdl.handle.net/1974/26466
Filter Stability, Observability and Robustness for Partially Observed Stochastic Dynamical Systems
Mcdonald, Curtis
Filter stability refers to the correction of an incorrectly initialized filter for a partially observed stochastic dynamical system with increasing measurements. In this thesis, we study the filter stability problem, develop new methods and results for both controlled and control-free stochastic dynamical systems, and study the implications of filter stability on robustness of optimal solutions for partially observed stochastic control problems. We introduce a definition of non-linear stochastic observability and through this notion of observability, we provide sufficient conditions for when a falsely initialized filter merges with the correctly initialized filter over time. We study stability under different notions such as the weak topology, total variation, and relative entropy. Additionally, we investigate properties of the transition kernel and measurement kernel which result in stability with an exponential rate of merging. We generalize our results to the controlled case, which is an unexplored area in the literature, to our knowledge.
Stability results are then applied to stochastic control problems. Under filter stability, we bound the difference in the expected cost incurred for implementing an incorrectly designed control policy compared to an optimal policy and relate filter stability, robustness, and unique ergodicity of non-linear filters.
http://hdl.handle.net/1974/26466Control of nonholonomic mechanical systems using virtual surfaces
http://hdl.handle.net/1974/26319
Control of nonholonomic mechanical systems using virtual surfaces
Kyle, Scott
In this report we study the modelling of simple mechanical systems evolving on trivial principal bundles, specifically \emph{locomotion} systems with nonholonomic constraints.
We show how we can model motion via group actions on configuration manifolds and assess the relationship between the constraints (and constrained variables) and the variables that physically induce motion on the vehicle by studying principal bundles.
With knowledge of the controllability (using the Lie algebra rank condition) of this formulation of a constrained simple mechanical system, we proceed to outlining a methodology to design a universal control algorithm for constrained mechanical systems using the method of virtual surfaces (or potential functions).
Lastly, we design a set of virtual surfaces to make a rolling disk (arguably the simplest practical nonholonomic system) stabilise to a point, track a path, and avoid a sequence of obstacles in the plane.
http://hdl.handle.net/1974/26319Special values of L-series, periodic coefficients and related themes
http://hdl.handle.net/1974/26145
Special values of L-series, periodic coefficients and related themes
Pathak, Siddhi
This thesis is centered around the theme of special values of L-functions and other infinite series, which are often expected to be transcendental numbers. More specifically, we focus on the following two questions in various scenarios: a) expressing the values in terms of certain special functions, b) determining their arithmetic nature (i.e., whether they are rational or irrational, algebraic or transcendental).
Motivated by the conjectures of S. Chowla and P. Erdos, we first study the L-series L(s,f) attached to a periodic arithmetical function f. Utilizing tools from transcendental number theory, we investigate the non-vanishing and the arithmetic nature of the values L(1,f) and L'(1,f). We introduce a probabilistic viewpoint towards the study of the values L(k,f) for any integer k greater than or equal to 1, especially in the case when f is an Erdos function.
On a related note, we explore the irrationality of the values of Dedekind zeta-functions at positive integers using elementary means. We also initiate the study of the sum over lattice points, of a rational function. This opens new doors for future research.
http://hdl.handle.net/1974/26145