arXiv subjects

Niles Armstrong

Publications and source records attributed to Niles Armstrong.

4 recordsLinked to original sources

arXiv2022

Kinodynamic control systems and discontinuities in clearance

We investigate the structure of discontinuities in clearance (or minimum time) functions for nonlinear control systems with general, closed obstacles (or targets). We establish general results regarding interactions between admissible trajectories and clearance discontinuities: e.g. instantaneous increases in clearance when passing through a discontinuity, and propagation of discontinuity along optimal trajectories. Then, investigating sufficient conditions for discontinuities, we explore a common directionality condition for velocities at a point, characterized by strict positivity of the minimal Hamiltonian. Elementary consequences of this common directionality assumption are explored before demonstrating how, in concert with corresponding obstacle configurations, it gives rise to clearance discontinuities both on the surface of the obstacle and propagating out into free space. Minimal assumptions are made on the topological structure of obstacle sets.

math.OC↗

arXiv2019

A Uniqueness Theorem for Mean Value Sets for Elliptic Divergence Form Operators

We give background which shows the connection between the mean value theorem and the obstacle problem, and then we prove that a set is a mean value set for an elliptic operator of the form $Lu := \partial_i (a^{ij}(x) \partial_j u(x))$ if and only if it arises as the noncontact set of an obstacle problem involving the Green's function of the operator.

math.AP↗

arXiv2018

Nondegenerate Motion of Singular Points in Obstacle Problems with Varying Data

Recent work by Serfaty and Serra give a formula for the velocity of the free boundary of the obstacle problem at regular points [Serfaty-Serra 2018], and much older work by King, Lacey, and Vazquez gives an example of a singular free boundary point (in the Hele-Shaw flow) that remains stationary for a positive amount of time [King-Lacey-Vazquez 1995]. The authors show how singular free boundaries in the obstacle problem in some settings move immediately in response to varying data. Three applications of this result are given, and in particular, the authors show a uniqueness result: For sufficiently smooth elliptic divergence form operators on domains in $\mathrm{I \! R}^n$ and for the Laplace-Beltrami operator on a smooth manifold, the boundaries of distinct mean value sets (of the type found in [Blank-Hao 2015] and [Benson-Blank-LeCrone 2018]) which are centered at the same point do not intersect.

math.AP↗

arXiv2018

Properties of Mean Value Sets: Angle Conditions, Blowup Solutions, and Nonconvexity

We study the mean values sets of the second order divergence form elliptic operator with principal coefficients defined as $$a^{ij}_k(x):= \begin{cases} α_k δ^{ij}(x) &x_n>0 β_k δ^{ij}(x) &x_n<0. \end{cases}$$ In particular, we will show that the mean value sets associated to such an operator need not be convex as $α_k$ and $β_k$ converge to 1. This example then leads to an example of nonconvex mean value sets for smooth $a^{ij}(x)$.

math.AP↗