energy conserving explicit local time stepping for second order wave equations locally refined meshes impose severe stability constraints on explicit time stepping methods for the numerical simulation of time dependent wave phenomena to overcome that stability restriction local time stepping methods are developed which allow arbitrarily small time steps precisely where small elements in the mesh are located when combined with a symmetric finite element discretization in space with an essentially diagonal mass matrix the resulting discrete numerical scheme is explicit is inherently parallel and exactly conserves a discrete energy starting from the standard second order leap frog scheme time stepping methods of arbitrary order of accuracy are derived numerical experiments illustrate the efficiency and usefulness of these methods and validate the theory

an immersed free surface boundary treatment for seismic wave simulation abstractfinite difference methods are popular for wave simulation within the seismic exploration community thanks to their efficiency however difficulties arise when encountering complex topography due to the regular grid pattern of the finite difference schemes despite alternatives that can handle the free surface with little effort such as the spectral element or discontinuous galerkin s methods incorporating a free surface boundary condition within the finite difference framework is still appealing even at the cost of extra algorithm complexity and stronger requirement of computational resources we present a free surface boundary treatment within the finite difference framework belonging to the family of the immersed boundary methods inherently the presented boundary treatment is separated from the rest of the wave simulation which makes it easy to be integrated in existing finite difference codes specifically we construct an extrapolation operator for each grid point above the free surfa

switched nonlinear systems with state dependent dwell time the asymptotic convergence of a switched nonlinear system in the presence of disturbances is studied the system switches among a family of integral input to state stable systems the time between two consecutive switchings is not less than a value d this dwell time d is allowed to take different values according to a function whose argument is the state of the system at the switching times we propose a dwell time function which depends on the comparison functions which characterize the integral input to state stability property and guarantees the state of the switched system to converge to zero under the action of disturbances with bounded energy the main feature of the analysis is that it does not rely on the property for the switching to stop in finite time the two important cases of locally exponentially stable and feedforward systems are analyzed in detail

stabilization by means of state space depending switching rules abstract in this paper we consider switched systems defined by a pair of linear systems x a 1 x x a 2 x such that there exists a neutrally stable linear combination of the matrices a 1 a 2 under an additional assumption we prove that there exists a state space depending switching rule which stabilizes the system in a very general sense the result is illustrated by some simulations and examples

on a common quadratic lyapunov function for widely distant systems a sufficient condition concerning the construction of a common quadratic lyapunov function for widely distant linear systems is presented the condition is set on a mutual relation matrix whose size is just equal to the number of the coefficient matrices

wave propagation in anisotropic elastic materials and curvilinear coordinates using a summation by parts finite difference method we develop a fourth order accurate finite difference method for solving the three dimensional elastic wave equation in general heterogeneous anisotropic materials on curvilinear grids the proposed method is an extension of the method for isotropic materials previously described in the paper by sjogreen and petersson 2012 11 the proposed method discretizes the anisotropic elastic wave equation in second order formulation using a node centered finite difference method that satisfies the principle of summation by parts the summation by parts technique results in a provably stable numerical method that is energy conserving we also generalize and evaluate the super grid far field technique for truncating unbounded domains unlike the commonly used perfectly matched layers pml the super grid technique is stable for general anisotropic material because it is based on a coordinate stretching combined with an artificial dissipation as a result the discretization satisfies an energy estimate proving that the numerical approximation is stable we demonstrate by numerical experiments that sufficiently wide super grid layers result in very small artificial reflections applications of the proposed method are demonstrated by three dimensional simulations of anisotropic wave propagation in crystals

estimation of sojourn time distributions for cyclic semi markov processes in equilibrium summary the em algorithm is used to develop procedures for estimating the interoccurrence distributions when n independent and identically distributed cyclic semi markov processes each being ergodic irreducible and in equilibrium are observed over finite windows this work is an extension of vardi 1982a b who considers a similar problem for renewal processes and develops the rt algorithm for estimation of the common interoccurrence distribution the results are illustrated using simulations for alternating renewal processes and compared with an extension of an approximate approach of denby vardi 1985

a stability result for linear parameter varying systems abstract a negative real part eigenvalue assumption on a linear parameter varying system implies exponential stability under a condition that includes either slowly varying parameters or slowly accelerating parameters

stability of discrete linear inclusion let m ai be a set of linear operators on rn the discrete linear inclusion dli m is the set of possible trajectories xi i 0 such that xn ainain 1 ai1x0 where x0 rn and aij m we study several notions of stability for dli m including absolute asymptotic stability aas which is that all products ain ai1 0 as n we mainly study the case that m is a finite set we give criteria for the various forms of stability two new approaches are taken one relates the question of aas of dli m to formal language theory and finite automata while the second connects the aas property to the structure of a lie algebra associated to the elements of m more generally the discrete linear inclusion dli m makes sense for m contained in a banach algebra b we prove some results for aas in this case and give counterexamples showing that some results valid for finite sets of operators on rn are not true for finite sets m in a general banach algebras b

dynamic earthquake rupture simulations on nonplanar faults embedded in 3d geometrically complex heterogeneous elastic solids dynamic propagation of shear ruptures on a frictional interface in an elastic solid is a useful idealization of natural earthquakes the conditions relating discontinuities in particle velocities across fault zones and tractions acting on the fault are often expressed as nonlinear friction laws the corresponding initial boundary value problems are both numerically and computationally challenging in addition seismic waves generated by earthquake ruptures must be propagated for many wavelengths away from the fault therefore reliable and efficient numerical simulations require both provably stable and high order accurate numerical methods we present a high order accurate finite difference method for a enforcing nonlinear friction laws in a consistent and provably stable manner suitable for efficient explicit time integration b dynamic propagation of earthquake ruptures along nonplanar faults and c accurate propagation of seismic waves in heterogeneous media with free surface topography we solve the first order form of the 3d elastic wave equation on a boundary conforming curvilinear mesh in terms of particle velocities and stresses that are collocated in space and time using summation by parts sbp finite difference operators in space boundary and interface conditions are imposed weakly using penalties by deriving semi discrete energy estimates analogous to the continuous energy estimates we prove numerical stability the finite difference stencils used in this paper are sixth order accurate in the interior and third order accurate close to the boundaries however the method is applicable to any spatial operator with a diagonal norm satisfying the sbp property time stepping is performed with a 4th order accurate explicit low storage runge kutta scheme thus yielding a globally fourth order accurate method in both space and time we show numerical simulations on band limited self similar fractal faults revealing the complexity of rupture dynamics on rough faults

p sv wave propagation in heterogeneous media velocity stress finite difference method i present a finite difference method for modeling p sv wave propagation in heterogeneous media this is an extension of the method i previously proposed for modeling sh wave propagation by using velocity and stress in a discrete grid the two components of the velocity cannot be defined at the same node for a complete staggered grid the stability condition and the p wave phase velocity dispersion curve do not depend on the poisson s ratio while the s wave phase velocity dispersion curve behavior is rather insensitive to the poisson s ratio therefore the same code used for elastic media can be used for liquid media where s wave velocity goes to zero and no special treatment is needed for a liquid solid interface typical physical phenomena arising with p sv modeling such as surface waves are in agreement with analytical results the weathered layer and corner edge models show in seismograms the same converted phases obtained by previous authors this method gives stable results for step discontinuities as shown for a liquid layer above an elastic half space the head wave preserves the correct amplitude finally the corner edge model illustrates a more complex geometry for the liquid solid interface as the poisson s ratio v increases from 0 25 to 0 5 the shear converted phases are removed from seismograms and from the time section of the wave field

difference approximations for the second order wave equation difference approximations are derived for the second order wave equation in one and two space dimensions without first writing it as a first order system both the dirichlet and the neumann problems are treated for the one dimensional case relations between the boundary error and the interior phase error are derived for a fully second order accurate discretization as well as a scheme that is fourth order accurate in the interior and second order accurate at the boundary general two dimensional domains are considered for the dirichlet problem where the domain is embedded in a cartesian grid and the boundary conditions are approximated by interpolation a stable conservative scheme is derived where the time step is determined only by the interior discretization formula discretization cells cut by the boundary are treated implicitly but the resulting scheme becomes explicit because the implicit dependence only is pointwise numerical examples are provided to verify the stability and accuracy of the proposed method

an extension of lasalle s invariance principle for switched systems in this paper we address invariance principles for a certain class of switched nonlinear systems we provide an extension of lasalle s invariance principle for these systems and state asymptotic stability criteria we also present some related results on the compactness of the trajectories of these switched systems

a converse lyapunov theorem for nonlinear switched systems abstract in this paper we present a converse lyapunov theorem for uniform asymptotic stability of switched nonlinear systems its proof is a simple consequence of some results on converse lyapunov theorems for systems with bounded disturbances obtained by lin et al siam j control optim 34 1996 124 160 once an association of the switched system with a nonlinear system with disturbances is established

stability robustness for linear state space models a lyapunov mapping approach some pythagorean like conditions for the existence of a common lyapunov solution for two given matrices are presented these conditions are set on the lyapunov mappings associated with given matrices the key is the semipositiveness of linear mappings defined on symmetric matrices from the viewpoint of the existence of a common lyapunov solution some known stability robustness results are reviewed and refined

stability criteria for switched and hybrid systems 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 them 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 in determining stability of switched systems in general which is exemplified by np hardness 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 are 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

on the kalman yakubovich popov lemma the purpose of this note is to present a new elementary proof for the multivariable k y p lemma a minimum of linear algebra and finite dimensional convexity theory is used

stability analysis of switched systems using variational principles an introduction many natural and artificial systems and processes encompass several modes of operation with a different dynamical behavior in each mode switched systems provide a suitable mathematical model for such processes and their stability analysis is important for both theoretical and practical reasons we review a specific approach for stability analysis based on using variational principles to characterize the most unstable solution of the switched system we also discuss a link between the variational approach and the stability analysis of switched systems using lie algebraic considerations both approaches require the use of sophisticated tools from many different fields of applied mathematics the purpose of this paper is to provide an accessible and self contained review of these topics emphasizing the intuitive and geometric underlying ideas

a lagrange duality approach to state feedback stabilisability in switched positive linear systems lagrange duality is used in order to obtain a novel and useful state feedback exponential stabilisability characterisation for discrete time switched positive linear systems a switched positive linear system is proved to be state feedback exponentially stabilisable if and only if there is a set of lagrange multipliers whose image via a linear mapping of the system s modes is a schur matrix this characterisation directly leads to a simple and novel method for the synthesis of exponentially stabilising state feedback mappings a particular set of lagrange multipliers is explicitly obtained as the result of constructively establishing an upper bound on the minimum number of lagrange multipliers required to attain through the aforementioned linear mapping a schur matrix this construction of a particular set of lagrange multipliers can always be used to complete the first step of the obtained synthesis method it is further shown that for a stabilisable switched positive linear system with rank one mo

the design of a panel study under an alternating poisson process assumption we discuss the design of a panel study for the estimation of the average durations in the two states of an alternating poisson process two types of designs are examined the first fixes the number of follow up waves but permits the total length of follow up to vary and the second varies the number of follow up waves with fixed length simple expressions for nearly optimal designs are presented and we compare these with designs allowing for continuous observation both equilibrium and nonequilibrium cases are examined an example is given using morbidity data to illustrate how these results can be applied

stable and high order accurate boundary treatments for the elastic wave equation on second order form stable and high order accurate boundary treatments for the elastic wave equation on second order form

stability and stabilization of discrete time switched systems this paper addresses two strategies for stabilization of discrete time linear switched systems the first one is of open loop nature trajectory independent and is based on the determination of an upper bound of the minimum dwell time by means of a family of quadratic lyapunov functions the relevant point on dwell time calculation is that the proposed stability condition does not require the lyapunov function be uniformly decreasing at every switching time the second one is of closed loop nature trajectory dependent and is designed from the solution of what we call lyapunov metzler inequalities from which the stability condition is expressed being non convex a more conservative but simpler to solve version of the lyapunov metzler inequalities is provided the theoretical results are illustrated by means of examples

an iterative method of lyapunov function construction for differential inclusions abstract the problem of choosing the lyapunov function from the class of quadratic forms for differential inclusions is reduced to that of searching saddle points of a suitable function an iterative algorithm is used for this purpose

the boundedness of all products of a pair of matrices is undecidable 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 rho sigma is not computable because testing whether rho sigma less than or equal to1 is an undecidable problem as a consequence the robust stability of linear systems under time varying 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 so called finiteness conjecture our results are based on a simple reduction from the emptiness problem for probabilistic finite automata which is known to be undecidable c 2000 elsevier science b v all rights reserved

a lagrange duality characterisation for stability under arbitrary switching in switched positive linear systems abstractthe present communication is concerned with uniform exponential stability under arbitrary switching in discrete time switched positive linear systems lagrange duality is used in order to obtain a new characterisation for uniform exponential stability which is in terms of sets of inequalities involving each of the matrices that represent the modes of the system these sets of inequalities are shown to generalise the classical linear lyapunov inequality that characterises in positive matrices the property of being schur each solution to these sets of inequalities is shown to provide a representation in terms of a number of linear functionals for a common lyapunov function for the switched positive linear system a result is further presented which conveys to a conservative upper bound on the minimum required number of linear functionals in the above mentioned representation and also to a method for computing them our proof for the aforementioned characterisation is based on another equ

a short cut method for estimation in renewal processes this article proposes a simple noniterative estimator of the lifetime distribution from stationary renewal processes data which is a variation of the kaplan meier estimator the method is compared with the nonparametric maximum likelihood estimator using real and simulated data and the results show that as long as the censoring level is not too high the two estimates are remarkably close a heuristic asymptotic argument is given to support the simulation results and to justify the rationale behind the estimator the estimator s connection with the kaplan meier estimator is exploited to suggest a variance estimate and associated confidence interval which through a simulation study are found to be conservative

general lemmas for stability analysis of linear continuous time systems with slowly time varying parameters this note investigates stability conditions for some dynamic systems with slowly time varying parameters or with unknown internal or external disturbances the models and conditions are motivated by adaptive control problems the results presented here provide some sufficient conditions for the exponential convergence of the state sequence of systems without system disturbance and for the uniform boundedness of the state sequence of such systems when they are subject to uniformly bounded system disturbances

fourth order finite difference p sv seismograms i describe the properties of a fourth order accurate space second order accurate time two dimensional p sk finite difference scheme based on the madariagavirieux staggered grid formulation the numerical scheme is developed from the first order system of hyperbolic elastic equations of motion and constitutive laws expressed in particle velocities and stresses the madariaga virieux staggered grid scheme has the desirable quality that it can correctly model any variation in material properties including both large and small poisson s ratio materials with minimal numerical dispersion and numerical anisotropy dispersion analysis indicates that the shortest wavelengths in the model need to be sampled at 5 gridpoints wavelength the scheme can be used to accurately simulate wave propagation in mixed acoustic elastic media making it ideal for modeling marine problems explicitly calculating both velocities and stresses makes it relatively simple to initiate a source at the free surface or within a layer and to satisfy free surface boundary conditions benchmark comparisons of finite difference and analytical solutions to lamb s problem are almost identical as are comparisons of finite difference and reflectivity solutions for elastic elastic and acoustic elastic layered models

lyapunov function construction by linear programming in this paper the lyapunov function given by the infinity norm is considered a new proof of the theorem of molchanov and pyatintskii 1986 is given the presented proof enables us to give an explicit calculation of the matrix which was only demonstrated to exist in the original paper combining the obtained result with the idea of scaling faces of a polytope gives the method of fitting the shape of level surfaces of the lyapunov function given by the infinity norm to the specified demands

interface waves in almost incompressible elastic materials we study the problem of two elastic half planes in contact and the stoneley interface wave that may exist at the interface between two different elastic materials emphasis being put on the case when the half planes are almost incompressible we show that numerical simulations involving interface waves require an unexpectedly high number of grid points per wavelength as the materials become more incompressible let and be the lame parameters and densities of the first and second half plane respectively a theoretical study shows that if k is a real constant k k k and 0 then for an accurate solution the required number of grid points per wavelength scales as 1 p where p is the order of accuracy of the numerical method this requirement becomes very restrictive close to the incompressible limit especially for lower order methods i e a small p the theoretical findings are supported by numerical experiments that illustrate the demanding resolution requirement as well as the superiority of higher order methods the scaling is also seen to hold for a more general choice of lame parameters numerical experiments when one of the half planes is a vacuum are also presented where the higher resolution requirement is illustrated in a numerical solution of lamb s problem

boundary waves and stability of the perfectly matched layer for the two space dimensional elastic wave equation in second order form boundary waves and stability of the perfectly matched layer for the two space dimensional elastic wave equation in second order form

a converse lyapunov theorem for linear parameter varying and linear switching systems we study families of linear time varying systems where time variations have to satisfy restrictions on the dwell time that is on the minimum distance between discontinuities as well as on the derivative in between discontinuities such classes of systems may be formulated as linear flows on vector bundles the main objective of this paper is to construct parameter dependent lyapunov functions which characterize the exponential growth rate this is possible in the generic irreducible case as an application the gelfand formula is generalized to the class of systems studied here in other words the maximal exponential growth rate may be approximated by only considering the periodic systems in the family of time varying systems a perspective on the question of continuous dependence of the exponential growth rate on the data is given

accuracy of finite difference and finite element modeling of the scalar and elastic wave equations numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media and they promise to be invaluable in solving the full inverse problem this paper quantitatively compares finite difference and finite element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time domain and frequency domain techniques in addition to versatility and ease of implementation it is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy to be of value a solution technique must be able to minimize 1 numerical attenuation or amplification 2 polarization errors 3 numerical anisotropy 4 errors in phase and group velocities 5 extraneous numerical parasitic modes 6 numerical diffraction and scattering and 7 errors and transmission coefficients this paper shows that in homogeneous media the

a family of numerical schemes for the computation of elastic waves a family of numerical schemes based on finite difference operators is introduced for the computation of elastic waves we use a displacement stress formulation of the model after deriving some stability results we give an analysis of the computational cost imposing an accuracy criterion on the phase velocity we derive the numerical parameters we also show that an optimum order of approximation exists for a given precision

sufficient conditions for stability of linear time varying systems abstract in this paper we consider sufficient conditions for the exponential stability of linear time varying systems of the form dot x t a t x t t 0 stability guaranteeing upper bounds for different measures of parameter variations are derived

a numerical technique for the stability analysis of linear switched systems in this paper the ray gridding approach a new numerical technique for the stability analysis of linear switched systems is presented it is based on uniform partitions of the state space in terms of ray directions which allow refinable families of polytopes of adjustable complexity to be examined for invariance in this framework the existence of a polyhedral lyapunov function that is common to a family of asymptotically stable subsystems can be checked efficiently via simple iterative algorithms the technique can be used to prove the stability of switched linear systems classes of linear time varying systems and linear differential lnclusions we also present preliminary results on another related problem namely the construction of multiple polyhedral lyapunov functions for the specification of stabilizing switching sequences for a switched system constructed from a family of stable linear subsystems

boundary estimates for the elastic wave equation in almost incompressible materials we study the half plane problem for the elastic wave equation subject to a free surface boundary condition with particular emphasis on almost incompressible materials a normal mode analysis is developed to estimate the solution in terms of the boundary data showing that the problem is boundary stable the dependence on the material properties which is difficult to analyze by the energy method is made transparent by our estimates the normal mode technique is used to analyze the influence of truncation errors in a finite difference approximation our analysis explains why the number of grid points per wave length must be increased when the shear modulus mu becomes small compared to the first lame parameter lambda that is for almost incompressible materials when the surface waves are scaled to have unit wave length our analysis predicts that the grid size must be proportional to mu lambda 1 2 for a second order method for a fourth order method the grid size can be proportional t

the convergence rate for difference approximations to general mixed initial boundary value problems the convergence rate is investigated for the solutions to difference schemes approximating the mixed initial boundary value problem for general systems of differential equations it is shown that if an energy estimate holds then the extra boundary conditions can be of one order lower accuracy without destroying the convergence rate expected from the approximation at inner points if the maximal order of the derivatives occurring in the boundary conditions is low enough then even lower accuracy can be permitted for the extra boundary conditions

technical communique analysis and design of aimd congestion control algorithms in communication networks we present a model of a network of synchronised sources operating additive increase multiplicative decrease aimd congestion control algorithms we show i that networks of such devices in the presence of a drop tail bottleneck buffer may be modelled as a positive linear system ii that such networks possess a unique stationary point and iii that this stationary point is globally exponentially stable we use these results to establish conditions for the fair co existence of traffic in networks employing heterogeneous aimd algorithms and to design a new protocol for operation over high speed and long distance links

stable difference approximations for the elastic wave equation in second order formulation we consider the three dimensional elastic wave equation for an isotropic heterogeneous material subject to a stress free boundary condition building on our recently developed theory for difference methods for second order hyperbolic systems h o kreiss n a petersson j ystrom siam j numer anal 40 2002 pp 1940 1967 we develop an explicit second order accurate technique which is stable for all ratios of longitudinal over transverse phase velocities the spatial discretization is self adjoint and the stability is obtained through an energy estimate seismic events are often modeled using singular source terms and we devise a technique to place sources independently of the grid while retaining second order accuracy away from the source several numerical examples are given

criteria of asymptotic stability of differential and difference inclusions encountered in control theory abstract for a class of differential inclusions to which many of the practically important control systems can be reduced necessary and sufficient conditions for asymptotic stability of the zero solution are established by the method of lyapunov functions similar results are obtained for the corresponding class of difference inclusions

the finiteness conjecture for the generalized spectral radius of a set of matrices the generalized spectral radius g9 of a set of n n matrices is g9 lim supk g9k 1k where g9k sup a1a2 ak each ai the joint spectral radius g9 is g9 lim supk g9k 1k where g9k sup a1 ak each ai it is known that g9 g9 holds for any finite set of n n matrices the finiteness conjecture asserts that for any finite set of real n n matrices there exists a finite k such that g9 g9 g9k 1k the normed finiteness conjecture for a given operator norm asserts that for any finite set a1 am having all ai op 1 either g9 1 or g9 g9 g9k 1k 1 for some finite k it is shown that the finiteness conjecture is true if and only if the normed finiteness conjecture is true for all operator norms the normed finiteness conjecture is proved for a large class of operator norms extending results of gurvits in particular for polytope norms and for the euclidean norm explicit upper bounds are given for the least k having g9 g9k 1k these results imply upper bounds for generalized critical exponents for these norms

an efficient lower bound for the generalized spectral radius of a set of matrices the generalized spectral radius gsr is a fundamental concept in studying the regularity of compactly supported wavelets here we describe an efficient method for estimating a lower bound for the gsr let mq be the set of all q q matrices with complex entries suppose a0 am 1 is a collection of m matrices in mq let ln be the set of all products of length n of the elements of define n maxa ln a in where a is the spectral radius of a the generalized spectral radius of is then lim supn n the standard method for estimating from below and at level n is to calculate the spectral radii of all mn products in ln and select the largest here we use three elementary theorems from linear algebra combinatorics and number theory to show that the same result can be obtained with no more than mnn matrix calculations

lie algebraic stability criteria for switched systems it was recently shown that a family of exponentially stable linear systems whose matrices generate a solvable lie algebra possesses a quadratic common lyapunov function which implies that the corresponding switched linear system is exponentially stable for arbitrary switching in this paper we prove that the same properties hold under the weaker condition that the lie algebra generated by given matrices can be decomposed into a sum of a solvable ideal and a subalgebra with a compact lie group the corresponding local stability result for nonlinear switched systems is also established moreover we demonstrate that if a lie algebra fails to satisfy the above condition then it can be generated by a family of stable matrices such that the corresponding switched linear system is not stable relevant facts from the theory of lie algebras are collected at the end of the paper for easy reference

stable and high order accurate difference methods for the elastic wave equation in discontinuous media in this paper we develop a stable and systematic procedure for numerical treatment of elastic waves in discontinuous and layered media we consider both planar and curved interfaces where media parameters are allowed to be discontinuous the key feature is the highly accurate and provably stable treatment of interfaces where media discontinuities arise we discretize in space using high order accurate finite difference schemes that satisfy the summation by parts rule conditions at layer interfaces are imposed weakly using penalties by deriving lower bounds of the penalty strength and constructing discrete energy estimates we prove time stability we present numerical experiments in two space dimensions to illustrate the usefulness of the proposed method for simulations involving typical interface phenomena in elastic materials the numerical experiments verify high order accuracy and time stability

computing the joint spectral radius this paper presents algorithms for finding an arbitrarily small interval that contains the joint spectral radius of a finite set of matrices it also presents a numerical criterion for verifying in certain cases that the joint spectral radius is the maximum of the spectral radii of the given matrices error bounds are derived for the case where calculations are done with finite precision and the matrices are not known exactly the algorithms are implemented and applied to estimate holder exponents of the orthonormal wavelets n constructed by daubechies for 3 n 8

the generalized spectral radius theorem an analytic geometric proof abstract let be a bounded set of complex matrices m a 1 a m a i the generalized spectral radius theorem states that where and are defined as follows lim sup m 1 m where m sup a a m with a the spectral radius lim sup m 1 m where m sup a a m with any matrix norm we give an elementary proof based on analytic and geometric tools which is in some ways simpler than the first proof by berger and wang

brief on absolute stability analysis by polyhedral lyapunov functions this paper addresses the problem of computing a polyhedral lyapunov function plf for a linear differential inclusion ldi a numerically efficient algorithm is presented to compute a plf for an ldi it is also pointed out that for the case of a polyhedron defined by vertices rather than by faces one can state algebraic stability conditions analogous to those given by molchanov and pyatnitskiy 1986

output error estimation for summation by parts finite difference schemes the paper develops a posteriori error estimates of integral output functionals for summation by parts finite difference methods the error estimates are based on the adjoint weighted residual method and take advantage of a variational interpretation of summation by parts discretizations the estimates are computed on a fixed grid and do not require an embedded grid or explicit interpolation operators for smooth boundary value problems containing first and second derivatives the error estimates converge to the exact error as the mesh is refined the theory is verified using linear boundary value problems and the euler equations

convex necessary and sufficient stabilisability conditions in switched linear systems with rank one modes abstracta novel convex necessary and sufficient condition for state feedback exponential stabilisability in discrete time switched linear systems whose modes are described by rank one matrices is

the application of high order differencing to the scalar wave equation the second order central difference is often used to approximate the derivatives of the wave equation it is demonstrated that gains in computational efficiency can be made by using high order approximations for these derivatives a one dimensional model is used to illustrate the relative accuracy of o t2 x2 o t2 x4 o t4 x4 ando t4 x10 central difference schemes for comparison o t2 ando t4 pseudospectral schemes are used as an additional measure of performance the results indicate that o t4 x10 differencing can achieve similar accuracy as the o t4 spectral scheme for practical illustration a two dimensional form of the o t4 x10 algorithm is used to compute the exploding reflector response of a salt dome model and compared with a fine grid o t2 x4 result transmissive sponge like boundary conditions are also examined and shown to be effective

explicit local time stepping methods for maxwell s equations explicit local time stepping methods are derived for time dependent maxwell equations in conducting and non conducting media by using smaller time steps precisely where smaller elements in the mesh are located these methods overcome the bottleneck caused by local mesh refinement in explicit time integrators when combined with a finite element discretisation in space with an essentially diagonal mass matrix the resulting discrete time marching schemes are fully explicit and thus inherently parallel in a non conducting source free medium they also conserve a discrete energy which provides a rigorous criterion for stability starting from the standard leap frog scheme local time stepping methods of arbitrarily high accuracy are derived for non conducting media numerical experiments with a discontinuous galerkin discretisation in space validate the theory and illustrate the usefulness of the proposed time integration schemes

high order finite difference methods for wave propagation in discontinuous media high order finite difference approximations are derived for the second order wave equation with discontinuous coefficients on rectangular geometries the discontinuity is treated by splitting the domain at the discontinuities in a multi block fashion each sub domain is discretized with compact second derivative summation by parts operators and the blocks are patched together to a global domain using the projection method this guarantees a conservative strictly stable and high order accurate scheme the analysis is verified by numerical simulations in one and two spatial dimensions

stable and accurate wave propagation in discontinuous media a time stable discretization is derived for the second order wave equation with discontinuous coefficients the discontinuity corresponds to inhomogeneity in the underlying medium and is treated by splitting the domain each homogeneous sub domain is discretized using narrow diagonal summation by parts operators and then patched to its neighbors by using a penalty method leading to fully explicit time integration this discretization yields a time stable and efficient scheme the analysis is verified by numerical simulations in one dimension using high order finite difference discretizations and in three dimensions using an unstructured finite volume discretization

an invariance principle for nonlinear switched systems abstract in this paper we address the problem of extending lasalle invariance principle to switched system we prove an extension of the invariance principle relative to dwell time switched solutions and a second one relative to constrained switched systems

stable and accurate schemes for the compressible navier stokes equations minimal stencil width discretizations of combined mixed and non mixed second order derivatives are analyzed with respect to accuracy and stability we show that these discretizations lead to stability for cauchy problems with a careful boundary treatment we also show that the stability holds for initial boundary value problems the analysis is verified by numerical simulations of burgers and navier stokes equations in two and three space dimensions

positive matrices associated with synchronised communication networks we study the spectrum of a positive matrix that arises in the study of certain communication networks bounds are given on the rate of convergence of the networks to the equilibrium condition

finding the observed information matrix when using the em algorithm a procedure is derived for extracting the observed information matrix when the em algorithm is used to find maximum likelihood estimates in incomplete data problems the technique requires computation of a complete data gradient vector or second derivative matrix but not those associated with the incomplete data likelihood in addition a method useful in speeding up the convergence of the em algorithm is developed two examples are presented

bounded semigroups of matrices in this note are proved two conjectures of daubechies and lagarias the first asserts that if z is a bounded set of matrices such that all left infinite products converge then 8 generates a bounded semigroup the second asserts the equality of two differently defined joint spectral radii for a bounded set of matrices one definition involves the conventional spectral radius and one involves the operator norm

robust stability of time varying polytopic systems abstract this paper provides global asymptotic stability conditions for time varying continuous time polytopic systems using a parameter dependent lyapunov function the time varying parameter uncertainty as well as its time derivative are modelled as belonging to polytopic convex sets and their dependence is made explicit to get less conservative results a particular case characterized by the parameter uncertainty satisfying a linear differential equation is analyzed and a simpler version of the aforementioned stability conditions is presented the results are expressed in terms of linear matrix inequalities being thus numerically solvable with no big difficulty the theory is illustrated by the determination of the asymptotic stability region of a mathieu s type equation with an uncertain time varying parameter

asymptotic stability and generalized gelfand spectral radius formula let be a set of n n complex matrices for m 1 2 let m be the set of all products of matrices in of length m denote by the multiplicative semigroup generated by is said to be asymptotically stable in the sense of dynamical systems if there is 0 1 such that there are bounded neighborhoods u v cn of the origin for which av mu for all a m m 1 2 for a bounded set of n n complex matrices it is shown that the following conditions are mutually equivalent i is asymptotically stable ii g9 lim supm supa m a 1 m 1 iii lim supm supa m a 1 m 1 where a stands for the spectral radius of a and iv there exists a positive number such that a 1 for all a this fact answers an open question raised by brayton and tong the generalized gelfand spectral radius formula that is g9 conjectured by daubechies and lagarias and proved by berger and wang using advanced tools from ring theory and then by elsner using analytic geometric tools follows immediately form the above asymptotic stability theorem

asymptotic height optimization for topical ifs tetris heaps and the finiteness conjecture given an iterated function system ifs of topical maps verifying some conditions we prove that the asymptotic height optimization problems are equivalent to finding the extrema of a continuous functional the average height on some compact space of measures we give general results to determine these extrema and then apply them to two concrete problems first we give a new proof of the theorem that the densest heaps of two tetris pieces are sturmian second we construct an explicit counterexample to the lagarias wang finiteness conjecture

vector norms as lyapunov functions for linear systems a unified theory of quadratic and piecewise linear lyapunov functions for continuous and discrete time linear systems is presented the key to this work is the description of these lyapunov functions by vector norms the main results are sufficient and necessary conditions for a vector norm to be a lyapunov function as well as a method based on these conditions of constructing such lyapunov functions