Reach Control on Simplices by Piecewise Affine Feedback

We study the reach control problem for affine systems on simplices, and the focus is on cases when it is known that the problem is not solvable by continuous state feedback. We examine from a geometric viewpoint the structural properties of the system which make continuous state feedbacks fail. This structure is encoded by so-called reach control indices, which are defined and developed in the paper. Based on these indices, we propose a subdivision algorithm and associated piecewise affine feedback. The method is shown to solve the reach control problem in all remaining cases, assuming it is solvable by open-loop controls

New Orbits for the n-Body Problem

In this paper, we consider minimizing the action functional as a method for numerically discovering periodic solutions to the $n$-body problem. With this method, we can find a large number of choreographies and other more general solutions. We show that most of the solutions found, including all but one of the choreographies, are unstable. It appears to be much easier to find unstable solutions to the $n$-body problem than stable ones. Simpler solutions are more likely to be stable than exotic ones.Comment: 16 pages, 2 tables, 6 figure

A convolutional neural-network model of human cochlear mechanics and filter tuning for real-time applications

Auditory models are commonly used as feature extractors for automatic speech-recognition systems or as front-ends for robotics, machine-hearing and hearing-aid applications. Although auditory models can capture the biophysical and nonlinear properties of human hearing in great detail, these biophysical models are computationally expensive and cannot be used in real-time applications. We present a hybrid approach where convolutional neural networks are combined with computational neuroscience to yield a real-time end-to-end model for human cochlear mechanics, including level-dependent filter tuning (CoNNear). The CoNNear model was trained on acoustic speech material and its performance and applicability were evaluated using (unseen) sound stimuli commonly employed in cochlear mechanics research. The CoNNear model accurately simulates human cochlear frequency selectivity and its dependence on sound intensity, an essential quality for robust speech intelligibility at negative speech-to-background-noise ratios. The CoNNear architecture is based on parallel and differentiable computations and has the power to achieve real-time human performance. These unique CoNNear features will enable the next generation of human-like machine-hearing applications

Curve Shortening and the Rendezvous Problem for Mobile Autonomous Robots

If a smooth, closed, and embedded curve is deformed along its normal vector field at a rate proportional to its curvature, it shrinks to a circular point. This curve evolution is called Euclidean curve shortening and the result is known as the Gage-Hamilton-Grayson Theorem. Motivated by the rendezvous problem for mobile autonomous robots, we address the problem of creating a polygon shortening flow. A linear scheme is proposed that exhibits several analogues to Euclidean curve shortening: The polygon shrinks to an elliptical point, convex polygons remain convex, and the perimeter of the polygon is monotonically decreasing.Comment: 15 pages, 18 figure