arXiv subjects

Ethan Ebbighausen

Publications and source records attributed to Ethan Ebbighausen.

3 recordsLinked to original sources

arXiv2025

A Framework for Approximating Perturbed Optimal Control Problems

We consider trajectory optimal control problems in which parameter uncertainty limits the applicability of control trajectories computed prior to travel. Hence, efficient trajectory adjustment is needed to ensure successful travel. However, it is often prohibitive or impossible to recalculate the optimal control in-transit due to strict time constraints or limited onboard computing resources. Thus, we propose a framework for quick and accurate trajectory approximations by using post-optimality sensitivity information. This allows the reduction of uncertain parameter space and an instantaneous approximation of the new optimal controller while using sensitivity data computed and stored pretransit.

math.OC↗

arXiv2023

On the Convergence of the Density of Terras' Set

The Collatz Conjecture's connection to dynamical systems opens it to a variety of techniques aimed at recurrence and density results. First, we turn to density results and strengthen the result of Terras through finding a strict rate of convergence. This rate gives a preliminary result on the Triangle Conjecture, which describes a set nodes that would dominate $L^{C} = \{y \in \N \, | \, T^{k}(y) > y , \,\forall k \in \N \}$. Second, we extend prior arguments to show that the construction of several classes of measures imply the bounded trajectories piece of the Collatz Conjecture.

math.DS↗

arXiv2023

Syracuse Maps as Non-singular Power-Bounded Transformations and Their Inverse Maps

We prove that the dynamical system $(\mathbb{N}, 2^{\mathbb{N}}, T, μ)$, where $μ$ is a finite measure equivalent to the counting measure, is power-bounded in $L^1(μ)$ if and only if there exists one cycle of the map $T$ and for any $x \in \mathbb{N}$, there exists $k \in \mathbb{N}$ such that $T^k(x)$ is in some cycle of the map $T$. This result has immediate implications for the Collatz Conjecture, and we use it to motivate the study of number theoretic properties of the inverse image $T^{-1}(x)$ for $x \in \mathbb{N}$, where $T$ denotes the Collatz map here. We study similar properties for the related Syracuse maps, comparing them to the Collatz map. We also analyze some structural properties of the inverse image in relation to asymptotic density of the set $\{x \in \mathbb{N} \mid \exists k \in \mathbb{N}: T^k(x) < x\}$.

math.DS↗