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76
Coordination of Groups of Mobile Autonomous Agents Using Nearest Neighbor Rules
, 2002
"... In a recent Physical Review Letters paper, Vicsek et. al. propose a simple but compelling discretetime model of n autonomous agents fi.e., points or particlesg all moving in the plane with the same speed but with dierent headings. Each agent's heading is updated using a local rule based on ..."
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Cited by 636 (42 self)
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In a recent Physical Review Letters paper, Vicsek et. al. propose a simple but compelling discretetime model of n autonomous agents fi.e., points or particlesg all moving in the plane with the same speed but with dierent headings. Each agent's heading is updated using a local rule based on the average of its own heading plus the headings of its \neighbors." In their paper, Vicsek et. al. provide simulation results which demonstrate that the nearest neighbor rule they are studying can cause all agents to eventually move in the same direction despite the absence of centralized coordination and despite the fact that each agent's set of nearest neighbors change with time as the system evolves. This paper provides a theoretical explanation for this observed behavior. In addition, convergence results are derived for several other similarly inspired models.
A Survey of Computational Complexity Results in Systems and Control
, 2000
"... The purpose of this paper is twofold: (a) to provide a tutorial introduction to some key concepts from the theory of computational complexity, highlighting their relevance to systems and control theory, and (b) to survey the relatively recent research activity lying at the interface between these fi ..."
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Cited by 118 (20 self)
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The purpose of this paper is twofold: (a) to provide a tutorial introduction to some key concepts from the theory of computational complexity, highlighting their relevance to systems and control theory, and (b) to survey the relatively recent research activity lying at the interface between these fields. We begin with a brief introduction to models of computation, the concepts of undecidability, polynomial time algorithms, NPcompleteness, and the implications of intractability results. We then survey a number of problems that arise in systems and control theory, some of them classical, some of them related to current research. We discuss them from the point of view of computational complexity and also point out many open problems. In particular, we consider problems related to stability or stabilizability of linear systems with parametric uncertainty, robust control, timevarying linear systems, nonlinear and hybrid systems, and stochastic optimal control.
Wavelet transforms versus Fourier transforms
 Department of Mathematics, MIT, Cambridge MA
, 213
"... Abstract. This note is a very basic introduction to wavelets. It starts with an orthogonal basis of piecewise constant functions, constructed by dilation and translation. The "wavelet transform " maps each f(x) to its coefficients with respect to this basis. The mathematics is simple and t ..."
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Cited by 69 (2 self)
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Abstract. This note is a very basic introduction to wavelets. It starts with an orthogonal basis of piecewise constant functions, constructed by dilation and translation. The "wavelet transform " maps each f(x) to its coefficients with respect to this basis. The mathematics is simple and the transform is fast (faster than the Fast Fourier Transform, which we briefly explain), but approximation by piecewise constants is poor. To improve this first wavelet, we are led to dilation equations and their unusual solutions. Higherorder wavelets are constructed, and it is surprisingly quick to compute with them — always indirectly and recursively. We comment informally on the contest between these transforms in signal processing, especially for video and image compression (including highdefinition television). So far the Fourier Transform — or its 8 by 8 windowed version, the Discrete Cosine Transform — is often chosen. But wavelets are already competitive, and they are ahead for fingerprints. We present a sample of this developing theory. 1. The Haar wavelet To explain wavelets we start with an example. It has every property we hope for, except one. If that one defect is accepted, the construction is simple and the computations are fast. By trying to remove the defect, we are led to dilation equations and recursively defined functions and a small world of fascinating new problems — many still unsolved. A sensible person would stop after the first wavelet, but fortunately mathematics goes on. The basic example is easier to draw than to describe: W(x)
The Lyapunov exponent and joint spectral radius of pairs of matrices are hard  when not impossible  to compute and to approximate
, 1997
"... We analyse the computability and the complexity of various definitions of spectral radii for sets of matrices. We show that the joint and generalized spectral radii of two integer matrices are not approximable in polynomial time, and that two related quantities  the lower spectral radius and th ..."
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Cited by 63 (17 self)
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We analyse the computability and the complexity of various definitions of spectral radii for sets of matrices. We show that the joint and generalized spectral radii of two integer matrices are not approximable in polynomial time, and that two related quantities  the lower spectral radius and the largest Lyapunov exponent  are not algorithmically approximable.
The Boundedness of All Products of a Pair of Matrices is Undecidable
, 2000
"... We show that the boundedness of the set of all products of a given pair Sigma of rational matrices is undecidable. Furthermore, we show that the joint (or generalized) spectral radius #(#) is not computable because testing whether #(#)61 is an undecidable problem. As a consequence, the robust stabil ..."
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Cited by 49 (13 self)
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We show that the boundedness of the set of all products of a given pair Sigma of rational matrices is undecidable. Furthermore, we show that the joint (or generalized) spectral radius #(#) is not computable because testing whether #(#)61 is an undecidable problem. As a consequence, the robust stability of linear systems under timevarying perturbations is undecidable, and the same is true for the stability of a simple class of hybrid systems. We also discuss some connections with the socalled "finiteness conjecture". Our results are based on a simple reduction from the emptiness problem for probabilistic finite automata, which is known to be undecidable.
Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture
 J. of the American Mathematical Society
, 2001
"... A topical map is a map from Rn into itself verifying some conditions (see §1.2) and which, roughly speaking, behaves like a translation along some line, the amount of which is measured by a real number, called the average height (or average displacement) of the map. Then we look at a topical Iterate ..."
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Cited by 38 (3 self)
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A topical map is a map from Rn into itself verifying some conditions (see §1.2) and which, roughly speaking, behaves like a translation along some line, the amount of which is measured by a real number, called the average height (or average displacement) of the map. Then we look at a topical Iterated Function System (IFS),
Stability criteria for switched and hybrid systems
 SIAM Review
, 2007
"... The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving these problems in a number of diverse communities, an ..."
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Cited by 36 (4 self)
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The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving these problems in a number of diverse communities, and to review some problems that remain open. An important contribution of our work is to bring together material from several areas of research and to present results in a unified manner. We begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions. Closely related to the concept of stability are the notions of exponential growth rates and converse Lyapunov theorems, both of which are discussed in detail. In particular, results on common quadratic Lyapunov functions and piecewise linear Lyapunov functions are presented, as they represent constructive methods for proving stability, and also represent problems in which significant progress has been made. We also comment on the inherent difficulty of determining stability of switched systems in general which is exemplified by NPhardness and undecidability results. We then proceed by considering the stability of switched systems in which there are constraints on the switching rules, through both dwell time requirements and state dependent switching laws. Also in this case the theory of Lyapunov functions and the existence of converse theorems is reviewed. We briefly comment on the classical Lur’e problem and on the theory of stability radii, both of which contain many of the features of switched systems and are rich sources of practical results on the topic. Finally we present a list of questions and open problems which provide motivation for continued research in this area.
An Elementary Counterexample to the Finiteness Conjecture
 SIAM JOURNAL ON MATRIX ANALYSIS
, 2001
"... ..."
The Generalized Spectral Radius and Extremal Norms
, 2000
"... The generalized spectral radius, also known under the name of joint spectral radius, or (after taking logarithms) maximal Lyapunov exponent of a discrete inclusion is examined. We present a new proof for a result of Barabanov, which states that for irreducible sets of matrices an extremal norm alway ..."
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Cited by 26 (4 self)
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The generalized spectral radius, also known under the name of joint spectral radius, or (after taking logarithms) maximal Lyapunov exponent of a discrete inclusion is examined. We present a new proof for a result of Barabanov, which states that for irreducible sets of matrices an extremal norm always exists. This approach lends itself easily to the analysis of further properties of the generalized spectral radius. We prove that the generalized spectral radius is locally Lipschitz continuous on the space of compact irreducible sets of matrices and show a strict monotonicity property of the generalized spectral radius. Sufficient conditions for the existence of extremal norms are obtained.
Characterizations of Scaling Functions: Continuous Solutions
 SIAM J. Matrix Anal. Appl
, 1994
"... A dilation equation is a functional equation of the form f(t) = � N k=0 ck f(2t − k), and any nonzero solution of such an equation is called a scaling function. Dilation equations play an important role in several fields, including interpolating subdivision schemes and wavelet theory. This paper ..."
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Cited by 18 (4 self)
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A dilation equation is a functional equation of the form f(t) = � N k=0 ck f(2t − k), and any nonzero solution of such an equation is called a scaling function. Dilation equations play an important role in several fields, including interpolating subdivision schemes and wavelet theory. This paper obtains sharp bounds for the Hölder exponent of continuity of any continuous, compactly supported scaling function in terms of the joint spectral radius of two matrices determined by the coefficients {c0,..., cN}. The arguments lead directly to a characterization of all dilation equations that have continuous, compactly supported solutions.