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Piecewise linear function

About: Piecewise linear function is a research topic. Over the lifetime, 8133 publications have been published within this topic receiving 161444 citations.


Papers
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Journal ArticleDOI
TL;DR: In this paper, the authors considered a distributed optimal control problem governed by an elliptic partial differential equation and showed that the Lagrange multiplier associated with the state constraints is indeed more regular under quite general assumptions.
Abstract: In this paper we are concerned with a distributed optimal control problem governed by an elliptic partial differential equation. State constraints of box type are considered. We show that the Lagrange multiplier associated with the state constraints, which is known to be a measure, is indeed more regular under quite general assumptions. We discretize the problem by continuous piecewise linear finite elements and we are able to prove that, for the case of a linear equation, the order of convergence for the error inL 2 (Ω) of the control variable is h| log h| in dimensions 2 and 3.

44 citations

Journal ArticleDOI
TL;DR: In this paper, an anisotropic error estimator involving only first order derivatives is proposed for the Laplace problem and continuous, piecewise linear finite elements, and upper and lower bounds are presented, the involved constants being independent of the mesh aspect ratio provided the error gradient is equidistributed in the directions of maximum and minimum stretching.

44 citations

Journal ArticleDOI
TL;DR: Conditions in the form of linear matrix inequalities (LMIs) that guarantee asymptotic stability in a reasonably large region around a limit cycle for a class of systems known as piecewise linear systems (PLS) are given.
Abstract: Oscillations appear in numerous applications from biology to technology. However, besides local results, rigorous stability and robustness analysis of oscillations are rarely done due to their intrinsic nonlinear behavior. Poincare/spl acute/ maps associated with the system cannot typically be found explicitly and stability is estimated using extensive simulations and experiments. This paper gives conditions in the form of linear matrix inequalities (LMIs) that guarantee asymptotic stability in a reasonably large region around a limit cycle for a class of systems known as piecewise linear systems (PLS). Such conditions, based on recent results on impact maps and surface Lyapunov functions (SuLF), allow a systematic and efficient analysis of oscillations of PLS or arbitrarily close approximations of nonlinear systems by PLS. The methodology applies to any locally stable limit cycle of a PLS, regardless of the dimension and the number of switching surfaces of the system, and is illustrated with a biological application: a fourth-order neural oscillator, also used in many robotics applications such as juggling and locomotion.

44 citations

15 Dec 1994
TL;DR: In this article, the authors provide the details for the conversion of the medial axis transform (MAT) of a set of two-dimensional and three-dimensional objects to a boundary representation.
Abstract: The medial axis transform (MAT) has potential as a powerful representation for a conceptual design tool for objects with inherent symmetry or near-symmetry. The medial axis of two-dimensional objects or medial surface of three-dimensional objects provides a conceptual design base, with transition to a detailed design occurring when the radius function is added to the medial axis or surface, since this additional information completely specifies a particular object. To make such a design tool practicable, however, it is essential to be able to convert from an MAT format to a boundary representation of an object. In this thesis, we provide the details for the conversion of the MAT of a set of two- and three-dimensional objects to a boundary representation. We demonstrate certain smoothness properties of the MAT and show the relationship between the tangent to the MAT at a point and the boundary points related to that MAT point. We classify the MAT points based on the tangency conditions at the point, and for each type of point, we detail the method for obtaining the boundary points related to it. We discuss requirements for an MAT to be locally valid in the sense that the given curves could actually be the MAT of an allowable object. We also provide a theoretical error bound on the computation for the two-dimensional case. Finally, we discuss an implementation of our algorithm both for piecewise linear two-dimensional MATs and for piecewise planar and linear three-dimensional MATs, and demonstrate some results we have obtained.

44 citations

Journal ArticleDOI
TL;DR: This work presents an extension of a second order cell-centered hydrodynamics scheme on unstructured polyhedral cells 13 toward higher order, to reduce dissipation, especially for smooth flows.

44 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023179
2022377
2021312
2020353
2019329
2018297