arXiv subjects

Phan Thanh An

Publications and source records attributed to Phan Thanh An.

5 recordsLinked to original sources

arXiv2026

Shortest Paths in Domains with Crack-Induced Constraints

We formulate the shortest-path problem in a domain containing a crack as a Hamilton--Jacobi equation carrying a state constraint on the crack, which on the binding part of the crack takes the form of a variational inequality: the crack is a two-sided barrier that may be touched and followed but not traversed, so that it acts as a lower-dimensional state constraint, carried separately by each of its two lips, which admits tangential propagation but forbids crossing. We establish a comparison principle and uniqueness of the value function through trajectory-based arguments that circumvent the failure of the classical doubling technique of Crandall, Ishii, and Lions at the crack. The variational inequality induces a free boundary on the crack separating a sliding regime, where optimal trajectories run along the crack over a finite segment, from an illuminated regime, where the crack is reached transversally by direct rays; we give a complete geometric characterization of both regimes, including grazing (tangential) contact, the absence of reflection at regular crack points, and a diffraction law at crack tips, all derived from optimality alone. A simulation of the forward problem on an exactly solvable configuration reproduces the three regimes and the tip-diffraction law to the accuracy of the closed-form solution.

math.AP↗

arXiv2026

Point Feature Descriptor via Directional Partition of Unity on Maps

We develop a functional-analytic framework for smooth directional point descriptors in GPS-free map-based localization. Given a query point $\mathbf{p}$ in a map $\mathcal{M} \subset \mathbb{R}^d$, the descriptor integrates an environment signal against a partition-of-unity weight family built from a softmax kernel, yielding a $\mathcal{C}^\infty$ alternative to hard angular binning. Our main contributions are: (i) a totality theorem showing that the associated linear functionals form a total family in $L^2(\mathbb{S}^{d-1})$, establishing asymptotic injectivity of the descriptor map; and (ii) a descriptor-induced seminorm $|f|_{\mathcal{D},n} = \|P_n f\|_{L^2}$, identified via the Gram matrix of the weights, which satisfies a Parseval-type identity $|f|_{\mathcal{D},n} \to \|f\|_{L^2(\mathbb{S}^{d-1})}$ as $n \to \infty$. Complementary results include Fréchet differentiability, lower semicontinuity under occlusion, and explicit Lipschitz stability bounds with constants depending on the kernel and temperature. These properties underpin a localization theory in which the descriptor grid enables nearest-neighbor position recovery with a certifiable static error bound, while robot motion generates an observability Gramian whose smallest eigenvalue controls a dynamic error bound and generically resolves symmetry-induced ambiguities that persist under single-observation matching.

math.FA↗

arXiv2023

Multiple Shooting Approach for Finding Approximately Shortest Paths for Autonomous Robots in Unknown Environments in 2D

An autonomous robot with a limited vision range finds a path to the goal in an unknown environment in 2D avoiding polygonal obstacles. In the process of discovering the environmental map, the robot has to return to some positions marked previously, the regions where the robot traverses to return are defined as sequences of bundles of line segments. This paper presents a novel algorithm for finding approximately shortest paths along the sequences of bundles of line segments based on the method of multiple shooting. Three factors of the approach including bundle partition, collinear condition, and update of shooting points are presented. We then show that if the collinear condition holds, the exactly shortest paths of the problems are determined, otherwise, the sequence of paths obtained by the update of the method converges to the shortest path. The algorithm is implemented in Python and some numerical examples show that the running time of path-planning for autonomous robots using our method is faster than that using the rubber band technique of Li and Klette in Euclidean Shortest Paths, Springer, 53-89 (2011).

cs.RO↗

arXiv2023

Finding Approximately Convex Ropes in the Plane

The convex rope problem is to find a counterclockwise or clockwise convex rope starting at the vertex a and ending at the vertex b of a simple polygon P, where a is a vertex of the convex hull of P and b is visible from infinity. The convex rope mentioned is the shortest path joining a and b that does not enter the interior of P. In this paper, the problem is reconstructed as the one of finding such shortest path in a simple polygon and solved by the method of multiple shooting. We then show that if the collinear condition of the method holds at all shooting points, then these shooting points form the shortest path. Otherwise, the sequence of paths obtained by the update of the method converges to the shortest path. The algorithm is implemented in C++ for numerical experiments.

math.OC↗

arXiv2022

Blaschke and Separation Theorems for Orthogonally Convex Sets

In this paper, we deal with analytic and geometric properties of orthogonally convex sets. We establish a Blaschke-type theorem for path-connected and orthogonally convex sets in the plane using orthogonally convex paths. The separation of these sets is established using suitable grids. Consequently, a closed and orthogonally convex set is represented by the intersection of staircase-halfplanes in the plane. Some topological properties of orthogonally convex sets in dimensional spaces are also given.

math.OC↗