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.

H∞-Control of Markovian Jump Systems and Solutions to Associated Piecewise-Deterministic Differential Games

Abstract: A class of linear-quadratic piecewise deterministic soft-constrained zero-sum differential games is formulated and solved, where the minimizing player has access to perfect or imperfect (continuous) state measurements. Such systems are also known as jump linear-quadratic systems, and the underlying game problem can also be viewed as an H∞ optimal control problem, where the system and cost matrices depend on the outcome of a Markov chain. Both finite- and infinite-horizon cases are considered, and a set of sufficient, as well as a set of necessary, conditions are obtained for the upper value of the game to be bounded. Policies for the minimizing player that achieve this upper value (which is zero) are piecewise linear on each sample path of the stochastic process, and are obtained from solutions of linearly coupled generalized Riccati equations. For the associated H∞-optimal control problem, these policies guarantee an L 2 gain type inequality on the closed-loop system.

On the dynamic analysis of piecewise-linear networks

Abstract: Piecewise-linear (PL) modeling is often used to approximate the behavior of nonlinear circuits. One of the possible PL modeling methodologies is based on the linear complementarity problem, and this approach has already been used extensively in the circuits and systems community for static networks. In this paper, the object of study is dynamic electrical circuits that can be recast as linear complementarity systems, i.e. as interconnections of linear time-invariant differential equations and complementarity conditions (ideal diode characteristics). A mathematically precise framework is developed that formalizes the mixed discrete and continuous behavior of these switched networks. Within this framework, the fundamental question of well-posedness (existence and uniqueness of solution trajectories given an initial condition) is studied and additional properties of the behavior are derived. For instance, a full characterization is presented of the inconsistent states.

Stability conditions for multiclass fluid queueing networks

Abstract: We introduce a new method to investigate stability of work-conserving policies in multiclass queueing networks. The method decomposes feasible trajectories and uses linear programming to test stability. We show that this linear program is a necessary and sufficient condition for the stability of all work-conserving policies for multiclass fluid queueing networks with two stations. Furthermore, we find new sufficient conditions for the stability of multiclass queueing networks involving any number of stations and conjecture that these conditions are also necessary. Previous research had identified sufficient conditions through the use of a particular class of (piecewise linear convex) Lyapunov functions. Using linear programming duality, we show that for two-station systems the Lyapunov function approach is equivalent to ours and therefore characterizes stability exactly.

Inverse inequalities on non-quasi-uniform meshes and application to the mortar element method

Abstract: We present a range of mesh-dependent inequalities for piecewise constant and continuous piecewise linear finite element functions u defined on locally refined shape-regular (but possibly non-quasi-uniform) meshes. These inequalities involve norms of the form ∥h α u∥ W s,p (Ω) for positive and negative s and α, where h is a function which reflects the local mesh diameter in an appropriate way. The only global parameter involved is N, the total number of degrees of freedom in the finite element space, and we avoid estimates involving either the global maximum or minimum mesh diameter. Our inequalities include new variants of inverse inequalities as well as trace and extension theorems. They can be used in several areas of finite element analysis to extend results – previously known only for quasi-uniform meshes – to the locally refined case. Here we describe applications to (i) the theory of nonlinear approximation and (ii) the stability of the mortar element method for locally refined meshes.

Polynomial Methods for Separable Convex Optimization in Unimodular Linear Spaces with Applications

Abstract: We consider the problem of minimizing a separable convex objective function over the linear space given by a system Mx=0 with M a totally unimodular matrix. In particular, this generalizes the usual minimum linear cost circulation and cocirculation problems in a network and the problems of determining the Euclidean distance from a point to the perfect bipartite matching polytope and the feasible flows polyhedron. We first show that the idea of minimum mean cycle canceling originally worked out for linear cost circulations by Goldberg and Tarjan [J. Assoc. Comput. Mach., 36 (1989), pp. 873--886.] and extended to some other problems [T. R. Ervolina and S. T. McCormick, Discrete Appl. Math., 46 (1993), pp. 133--165], [A. Frank and A. V. Karzanov, Technical Report RR 895-M, Laboratoire ARTEMIS IMAG, Universite Joseph Fourier, Grenoble, France, 1992], [T. Ibaraki, A. V. Karzanov, and H. Nagamochi, private communication, 1993], [M. Hadjiat, Technical Report, Groupe Intelligence Artificielle, Faculte des Sciences de Luminy, Marseille, France, 1994] can be generalized to give a combinatorial method with geometric convergence for our problem. We also generalize the computationally more efficient cancel-and-tighten method. We then consider objective functions that are piecewise linear, pure and piecewise quadratic, or piecewise mixed linear and quadratic, and we show how both methods can be implemented to find exact solutions in polynomial time (strongly polynomial in the piecewise linear case). These implementations are then further specialized for finding circulations and cocirculations in a network. We finish by showing how to extend our methods to find optimal integer solutions, to linear spaces of larger fractionality, and to the case when the objective functions are given by approximate oracles.