Piecewise

About: Piecewise is a research topic. Over the lifetime, 21064 publications have been published within this topic receiving 432096 citations. The topic is also known as: piecewise-defined function & hybrid function.

Convergence of the lax-friedrichs scheme and stability for conservation laws with a discontinuous space-time dependent flux

Abstract: The authors give the first convergence proof for the Lax-Friedrichs finite difference scheme for non-convex genuinely nonlinear scalar conservation laws of the form where the coefficient k(x,t) is allowed to be discontinuous along curves in the (x,t) plane. In contrast to most of the existing literature on problems with discontinuous coefficients, here the convergence proof is not based on the singular mapping approach, but rather on the div-curl lemma (but not the Young measure) and a Lax type entropy estimate that is robust with respect to the regularity of k(x,t). Following [14], the authors propose a definition of entropy solution that extends the classical Kružkov definition to the situation where k(x,t) is piecewise Lipschitz continuous in the (x,t) plane, and prove the stability (uniqueness) of such entropy solutions, provided that the flux function satisfies a so-called crossing condition, and that strong traces of the solution exist along the curves where k(x,t) is discontinuous. It is shown that a convergent subsequence of approximations produced by the Lax-Friedrichs scheme converges to such an entropy solution, implying that the entire computed sequence converges.

Approximating and learning unknown mappings using multilayer feedforward networks with bounded weights

Abstract: It is shown that feedforward networks having bounded weights are not undesirable restricted, but are in fact universal approximators, provided that the hidden-layer activation function belongs to one of several suitable broad classes of functions: polygonal functions, certain piecewise polynomial functions, or a class of functions analytic on some open interval. These results are obtained by trading bounds on network weights for possible increments to network complexity, as indexed by the number of hidden nodes. The hidden-layer activation functions used include functions not admitted by previous universal approximation results, so the present results also extend the already broad class of activation functions for which universal approximation results are available. A theorem which establishes the approximate ability of these arbitrary mappings to learn when examples are generated by a stationary ergodic process is given

Stabilization of a Class of Switched Linear Neutral Systems Under Asynchronous Switching

Abstract: This technical note concerns the stabilization problem for a class of switched linear neutral systems in which time delays appear in both the state and the state derivatives. In addition, the switching signal of the switched controller also involves time delays, which makes the switching between the controller and the system asynchronous. Based on a new integral inequality and the piecewise Lyapunov-Krasovskii functional technique, a condition for global uniform exponential stability of the switched neutral system under an average dwell time (ADT) scheme is proposed. Then, the corresponding solvability condition for the controller is established. Finally, a numerical example is given to illustrate the effectiveness of the proposed theory.

Stability of statistical properties in two-dimensional piecewise hyperbolic maps

Abstract: We investigate the statistical properties of a piecewise smooth dynamical system by studying directly the action of the transfer operator on appropriate spaces of distributions. We accomplish such a program in the case of two-dimensional maps with uniformly bounded second derivative, but we are confident that the present approach can be successful in much greater generality (we hope including higher dimensional billiards). For the class of systems at hand, we obtain a complete description of the SRB measures, their statistical properties and their stability with respect to many types of perturbations, including deterministic and random perturbations and holes.