anomalous diffusion in a field of randomly distributed scatterers we consider the motion of particles which are scattered by randomly distributed obstacles in between scattering events the particles move uniformly the governing master equation is obtained by mapping the problem onto a master equation which was previously devised for the description of anomalous diffusion of particles with inertia r friedrich phys rev lett 96 230601 2006 we show that for a scale free distance distribution of scatterers a time fractional master equation arises the corresponding diffusion equation which exhibits a power law diffusion coefficient is solved in d dimensions via the method of subordination

joint probability distributions for a class of non markovian processes we consider joint probability distributions for the class of coupled langevin equations introduced by fogedby h c fogedby phys rev e 50 1657 1994 we generalize well known results for the single time probability distributions to the case of n time joint probability distributions it is shown that these probability distribution functions can be obtained by an integral transform from distributions of a markovian process the integral kernel obeys a partial differential equation with fractional time derivatives reflecting the non markovian character of the process

on common diagonal lyapunov solutions abstract several recent results regarding common diagonal lyapunov solutions are further explored here the first one attributed to redheffer and revisited by shorten and narendra reduces the diagonal stability of a matrix to common diagonal lyapunov solutions on two matrices of order one less we present a shorter purely matrix theoretic proof of this result along with its extensions the second one concerns two different necessary and sufficient conditions due to oleng narendra and shorten for a pair of 2 2 matrices to share a common diagonal lyapunov solution we show that these two conditions are connected directly to each other

stabilization of positive switched linear systems and its application in consensus of multiagent systems in this note we study the stabilization problem of positive switched linear system psls with disturbance by introducing a class of weak common linear copositive lyapunov functions we design time dependent switching rules under which the system is asymptotically stable both continuous time and discrete time psls are taken into consideration and the obtained results are also extended to nonpositive switched systems finally the theoretical results are applied to consensus problem of multiagent systems with switching topologies and disturbance

improved results on stability of continuous time switched positive linear systems in this paper the problems of stability for switched positive linear systems splss under arbitrary switching are investigated in a continuous time context the so called copositive polynomial lyapunov function cplf giving a generalization of copositive types of lyapunov function is first proposed which is formulated in a higher order form of the positive states of the underlying systems it is illustrated in this paper that some classical types of lyapunov functions can be seen as special cases of the proposed cplf then new stability conditions are developed by the new lyapunov function approach it is also proved that the conservativeness of the obtained criteria can be further reduced as the degree of the lyapunov function increases a numerical example is given to demonstrate the effectiveness and less conservativeness of the developed techniques

fractional fokker planck equation for nonlinear stochastic differential equations driven by non gaussian levy stable noises the fokker planck equation has been very useful for studying dynamic behavior of stochastic differential equations driven by gaussian noises however there are both theoretical and empirical reasons to consider similar equations driven by strongly non gaussian noises in particular they yield strongly non gaussian anomalous diffusion which seems to be relevant in different domains of physics in this paper we therefore derive a fractional fokker planck equation for the probability distribution of particles whose motion is governed by a nonlinear langevin type equation which is driven by a levy stable noise rather than a gaussian we obtain in fact a general result for a markovian forcing we also discuss the existence and uniqueness of the solution of the fractional fokker planck equation

on exponential stability of switched homogeneous positive systems of degree one in this paper we investigate the stability problem of switched homogeneous positive systems of degree one we first show that the positiveness of the considered system is guaranteed when each switched subsystem is cooperative then an integral excitation condition is formulated to ensure the exponential stability of the switched homogeneous positive system of degree one in particular the proposed condition allows that some or even all switched subsystems are only lyapunov stable rather than asymptotically stable in the second part of this paper we further consider the cases of sub homogeneous systems time varying delay systems and discrete time systems in the sub homogeneous case we first show that the previous integral excitation condition is also sufficient such that the exponential stability is achieved we also derive an exponential stability result by relaxing the cooperative and integral excitation conditions in the time varying delay case a delay independent exponential stability result is shown under the extended integral excitation condition finally in the discrete time case based on the fact that each switched subsystem is order preserving instead of cooperative the exponential stability of the considered system is shown under a weak excitation condition

limit theorems for coupled continuous time random walks scaling limits of continuous time random walks are used in physics to model anomalous diffusion in which a cloud of particles spreads at a different rate than the classical brownian motion governing equations for these limit processes generalize the classical diffusion equation in this article we characterize scaling limits in the case where the particle jump sizes and the waiting time between jumps are dependent this leads to an efficient method of computing the limit and a surprising connection to fractional derivatives

common diagonal lyapunov function for third order linear switched system consider the stability problem for the following linear switched system differential inclusion x a x a a 1 a 2 a n here a i i 1 2 n are n i n dimensional hurwitz stable real matrices in this study for this system we investigate the problem of the existence and construction of a common diagonal lyapunov function of the form v x x t d x where d is a positive diagonal matrix in the case of n 3 i e third order system we suggest a simple elimination algorithm which gives a common d in the case of existence highlights we consider n dimensional linear switched systems construction of a common diagonal lyapunov function for the systems is considered a simple elimination algorithm for third order system is given for the general case we apply kelly s cutting plane method to the given problem

langevin equations for continuous time levy flights we consider the combined effects of a power law l evy step distribution characterized by the step index f and a power law waiting time distribution characterized by the time index g on the long time behavior of a random walker the main point of our analysis is a formulation in terms of coupled langevin equations which allows in a natural way for the inclusion of external force fields in the anomalous case for f2 and g1 the dynamic exponent z locks onto the ratio f g drawing on recent results on l evy flights in the presence of a random force field we also find that this result is independent of the presence of weak quenched disorder for d below the critical dimension mathit d mathit c 2f 2 the disorder is relevant corresponding to a nontrivial fixed point for the force correlation function

a general stability test for switched positive systems based on a multidimensional system analysis the main goal of this technical note is to present a new sufficient condition for the stability of switched positive linear systems this condition is general in two senses first of all because it applies to switching systems of order composed of subsystems for any and secondly because it applies to both discrete and continuous time systems furthermore it is easy to verify as it amounts to check for the stability of a certain matrix the new condition is derived from a multidimensional system analysis where the relation between the stability of multidimensional and switched systems is investigated a comparison with other existing tests is also presented showing that the one proposed here allows to infer about the stability of a system in cases where other tests fail or do not apply at all

on the simultaneous diagonal stability of a pair of positive linear systems in this paper we derive a necessary and sufficient condition for the existence of a diagonal common quadratic lyapunov function cqlf for a pair of positive linear time invariant lti systems

subordinated stochastic processes with aged operational time in this paper subordinated stochastic processes are considered where the renewal process acting as the operational time it is assumed that the observation of the process begins at a certain time after the start of the renewal process a recurrence formula was derived for calculating the multipoint probability density functions of the aged renewal process two point correlation functions for certain subordinated stochastic processes particularly for the generalized ornstein uhlenbeck process were calculated a model of relaxation in a disordered medium with traps and obstacles is proposed

continuous time random walk with correlated waiting times based on the langevin description of the continuous time random walk ctrw we consider a generalization of ctrw in which the waiting times between the subsequent jumps are correlated we discuss the cases of exponential and slowly decaying persistent power law correlations between the waiting times as two generic examples and obtain the corresponding mean squared displacements as functions of time in the case of exponential type correlations the sub diffusion at short times is slower than in the absence of correlations at long times the behavior of the mean squared displacement is the same as in uncorrelated ctrw for power law correlations we find subdiffusion characterized by the same exponent at all times which appears to be smaller than the one in uncorrelated ctrw interestingly in the limiting case of an extremely long power law correlations the sub diffusion exponent does not tend to zero but is bounded from below by the subdiffusion exponent corresponding to a short time behavior in the case of exponential correlations

extension of diagonal stability and stabilization for continuous time fractional positive linear systems abstract this paper considers an extension of diagonal stability for continuous fractional positive linear systems fpls with the fractional order 0 1 based on diagonal stability of metzler matrix an extension that involves the combination of a diagonal positive definite matrix and a skew symmetric anti diagonal matrix related a hurwitz and metzler matrix is established combining this extension and the linear matrix inequalities lmis criteria of stability for fractional order systems fos a necessary and sufficient condition for extension diagonal stability of fpls is presented a state feedback controller is given which ensures the stabilization and positivity of the closed loop systems numerical examples are provided to demonstrate the effectiveness and applicability of the proposed methods

strong stability of discrete time systems the paper introduces a new notion of stability for internal state space autonomous system descriptions in discrete time referred to as strong stability which extends a parallel notion introduced in the continuous time case this is a stronger notion of stability compared to alternative definitions asymptotic lyapunov which prohibits systems described by natural coordinates to have overshooting responses for arbitrary initial conditions in state space three finer notions of strong stability are introduced and necessary and sufficient conditions are established for each one of them the class of discrete time systems for which strong and asymptotic stability coincide is characterized and links between the skewness of the eigen frame and the violation of strong stability property are obtained connections between the notions of strong stability in the continuous and discrete domains are briefly discussed finally strong stabilization problems under state and output feedback are studied the results of the paper are illustrated with a numerical example

stability analysis of positive descriptor systems abstract in this paper we discuss stability properties of positive descriptor systems in the continuous time as well as in the discrete time case we present characterisations of positivity and establish generalised stability criteria for the case of positive descriptor systems we show that if the spectral projector onto the finite deflating subspace of the matrix pair e a is nonnegative then all stability criteria for standard positive systems take a comparably simple form in the positive descriptor case furthermore we provide sufficient conditions that guarantee entry wise nonnegativity along with positive semidefiniteness of solutions of generalised projected lyapunov equations as an application of the framework established throughout this paper we exemplarily generalise two criteria for the stability of two switched standard positive systems under arbitrary switching to the descriptor case

technical communique preservation of lyapunov functions under bilinear mapping bilinear mapping preserves quadratic lyapunov functions between continuous time and discrete time lti systems this note studies if this property holds for other types of lyapunov functions specified by vector norms of the system state some implications of the obtained results in the common lyapunov function problems are discussed

stabilization of switched positive system with impulse and marginally stable subsystems a mode dependent dwell time method abstract in this article the problem of stabilization for the switched positive linear system spls with impulse and marginally stable subsystems is studied in which both the continuous time case and the discrete time case are taken into account by the usage of a piecewise weak linear copositive lyapunov function lclf and the mode dependent dwell time mdt time dependent switching control laws are designed to guarantee the asymptotic stability of the system furthermore based on the comparison principal for positive systems the main results are generalized to the time variant spls two examples are worked out to indicate that our results have advantages over some known results in the literature

a krein like formula for singular perturbations of self adjoint operators and applications abstract given a self adjoint operator a d a h h and a continuous linear operator d a x with range h 0 x a banach space we explicitly construct a family a of self adjoint operators such that any a coincides with the original a on the kernel of such a family is obtained by giving a krein like formula where the role of the deficiency spaces is played by the dual pair x x the parameter belongs to the space of symmetric operators from x to x when x c one recovers the h 2 construction of kiselev and simon and so to some extent our results can be regarded as an extension of it to the infinite rank case considering the situation in which h l 2 r n and is the trace restriction operator along some null subset we give various applications to singular perturbations of non necessarily elliptic pseudo differential operators thus unifying and extending previously known results

on easily verifiable conditions for the existence of common linear copositive lyapunov functions this technical note addresses the existence of a common linear copositive lyapunov function clclf for continuous time switched positive linear systems two necessary and sufficient conditions for the existence of a clclf are provided one condition shows that the existence of a clclf for a family of n n dimensional metzler matrices can only be determined by n 1 lower dimensional hurwitz matrices the other condition further shows that the existence of a clclf is also determined by n 1 characteristic constants which gives an easily verifiable computation algorithm without checking a large amount of metzler hurwitz matrices

a characterization of lyapunov diagonal stability using hadamard products abstract several new characterizations of lyapunov diagonal stability are presented one of the characterizations is used to derive a simple necessary and sufficient condition for lyapunov diagonal stability of a 3 3 matrix

on the simultaneous diagonal stability of linear discrete time systems abstract this paper studies the set of time invariant linear discrete time systems in which each system has a diagonal quadratic lyapunov function first it is shown that there is generally no common diagonal quadratic lyapunov function for such a set of systems even if the set is assumed to be commutative it is also shown that the commutativity assures the existence of a common diagonal quadratic lyapunov function inside the set of 2 2 systems or the set of nonnegative systems then two simple topological results are presented concerning the simultaneous diagonal stability on the set of nonnegative systems the first is a measure of the difference of matrices that assures the simultaneous diagonal stability the second is a measure of the commutativity of matrices

langevin formulation of a subdiffusive continuous time random walk in physical time systems living in complex nonequilibrated environments often exhibit subdiffusion characterized by a sublinear power law scaling of the mean square displacement one of the most common models to describe such subdiffusive dynamics is the continuous time random walk ctrw stochastic trajectories of a ctrw can be described in terms of the subordination of a normal diffusive process by an inverse levy stable process here we propose an equivalent langevin formulation of a force free ctrw without subordination by introducing a different type of non gaussian noise we are able to express the ctrw dynamics in terms of a single langevin equation in physical time with additive noise we derive the full multipoint statistics of this noise and compare it with the scaled brownian motion sbm an alternative stochastic model describing subdiffusive dynamics interestingly these two noises are identical up to the second order correlation functions but different in the higher order statistics we extend our formalism to general waiting time distributions and force fields and compare our results with those of the sbm in the presence of external forces our proposed noise generates a different class of stochastic processes resembling a ctrw but with forces acting at all times

diagonal riccati stability and positive time delay systems abstract we consider a class of algebraic riccati equations arising in the study of positive linear time delay systems we show that this class admits diagonal positive definite solutions this implies that exponentially stable positive linear time delay systems possess lyapunpov krasovskii functionals of a simple quadratic form we also show that for this class of equations the existence of positive definite solutions is equivalent to a simple spectral condition on the coefficient matrices

operator levy motion and multiscaling anomalous diffusion the long term limit motions of individual heavy tailed power law particle jumps that characterize anomalous diffusion may have different scaling rates in different directions operator stable motions y t tg 0 are limits of d dimensional random jumps that are scale invariant according to c h y t y mathrm ct where h is a d ifmmode times else texttimes fi d matrix the eigenvalues of the matrix have real parts 1 ensuremath alpha j with each positive ensuremath alpha j l 2 in each of the j principle directions the random motion has a different fickian or super fickian diffusion dispersion rate proportional to t 1 ensuremath alpha j these motions have a governing equation with a spatial dispersion operator that is a mixture of fractional derivatives of different order in different directions subsets of the generalized fractional operator include i a fractional laplacian with a single order ensuremath alpha and a general directional mixing measure m ensuremath theta and ii a fractional laplacian with uniform mixing measure the riesz potential the motivation for the generalized dispersion is the observation that tracers in natural aquifers scale at different super fickian rates in the directions parallel and perpendicular to mean flow

fredholm indices and the phase diagram of quantum hall systems the quantized hall conductance in a plateau is related to the index of a fredholm operator in this paper we describe the generic phase diagram of fredholm indices associated with bounded and toeplitz operators we discuss the possible relevance of our results to the phase diagram of disordered integer quantum hall systems

essentially negative news about positive systems abstract in this paper the discretisation of switched and non switched linear positive systems using pade approximations is considered pade approximations to the matrix exponential are sometimes used by control engineers for discretising continuous time systems and for control system design we observe that this method of approximation is not suited for the discretisation of positive dynamic systems for two key reasons first certain types of lyapunov stability are not in general preserved secondly and more seriously positivity need not be preserved even when stability is finally we present an alternative approximation to the matrix exponential which preserves positivity and linear and quadratic stability

a result on common quadratic lyapunov functions in this note we define strong and weak common quadratic lyapunov functions cqlfs for sets of linear time invariant lti systems we show that the simultaneous existence of a weak cqlf of a special form and the nonexistence of a strong cqlf for a pair of lti systems is characterized by easily verifiable algebraic conditions these conditions are found to play an important role in proving the existence of strong cqlfs for general lti systems

singular perturbations of abstract wave equations given on the hilbert space h0 the self adjoint operator b and the skew adjoint operators c1 and c2 we consider on the hilbert space h d b h0 the skew adjoint operator w c21 b2c1 corresponding to the abstract wave equation c1 c2 b2 c1c2 given then an auxiliary hilbert space h and a linear map d b2 h with a kernel k dense in h0 we explicitly construct skew adjoint operators w on a hilbert space h d b h0 h which coincide with w on n k d b the extension parameter ranges over the set of positive bounded and injective self adjoint operators on h in the case c1 c2 0 our construction allows a natural definition of negative strongly singular perturbations a of a b2 such that the diagram w w a a is commutative

distributed order diffusion equations and multifractality models and solutions we study distributed order time fractional diffusion equations characterized by multifractal memory kernels in contrast to the simple power law kernel of common time fractional diffusion equations based on the physical approach to anomalous diffusion provided by the seminal scher montroll weiss continuous time random walk we analyze both natural and modified form distributed order time fractional diffusion equations and compare the two approaches the mean squared displacement is obtained and its limiting behavior analyzed we derive the connection between the wiener process described by the conventional langevin equation and the dynamics encoded by the distributed order time fractional diffusion equation in terms of a generalized subordination of time a detailed analysis of the multifractal properties of distributed order diffusion equations is provided

fractional fokker planck equation solution and application recently metzler et al phys rev lett 82 3563 1999 introduced a fractional fokker planck equation ffpe describing a subdiffusive behavior of a particle under the combined influence of external nonlinear force field and a boltzmann thermal heat bath in this paper we present the solution of the ffpe in terms of an integral transformation the transformation maps the solution of ordinary fokker planck equation onto the solution of the ffpe and is based on levy s generalized central limit theorem the meaning of the transformation is explained based on the known asymptotic solution of the continuous time random walk ctrw we investigate in detail i a force free particle ii a particle in a uniform field and iii a particle in a harmonic field we also find an exact solution of the ctrw and compare the ctrw result with the corresponding solution of the ffpe the relation between the fractional first passage time problem in an external nonlinear field and the corresponding integer first passage time is given an example of the one dimensional fractional first passage time in an external linear field is investigated in detail the ffpe is shown to be compatible with the scher montroll approach for dispersive transport and thus is applicable in a large variety of disordered systems the simple ffpe approach can be used as a practical tool for a phenomenological description of certain types of complicated transport phenomena

generalized joint spectral radius and stability of switching systems abstract this paper extends the notion of generalized joint spectral radius with exponents originally defined for a finite set of matrices to probability distributions we show that under a certain invariance condition the radius is calculated as the spectral radius of a matrix that can be easily computed extending the classical counterpart using this result we investigate the mean stability of switching systems in particular we establish the equivalence of mean square stability simultaneous contractibility in square mean and the existence of a quadratic lyapunov function also the stabilization of positive switching systems is studied numerical examples are given to illustrate the results

kirchhoff s rule for quantum wires we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the hamiltonian to be minus the laplace operator with general boundary conditions at the vertices this results in a scattering theory with n channels the corresponding on shell s matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy e 0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines it is shown to be unitary which may be viewed as the quantum version of kirchhoff s law we exhibit covariance and symmetry properties it is symmetric if the boundary conditions are real also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory finally we provide a composition rule by which the on shell s matrix of a graph is factorizable in terms of the s matrices of its subgraphs all proofs use only known facts from the theory of self adjoint extensions standard linear algebra complex function theory and elementary arguments from the theory of hermitian symplectic forms

non markovian stochastic liouville equation and its markovian representation extensions of the continuous time random walk approach some specific features and extensions of the continuous time random walk ctrw approach are analyzed in detail within the markovian representation mr and ctrw based non markovian stochastic liouville equation sle in the mr ctrw processes are represented by multidimensional markovian ones in this representation the probability density function pdf w t of fluctuation renewals is associated with that of reoccurrences in a certain jump state of some markovian controlling process within the mr the non markovian sle which describes the effect of ctrw like noise on the relaxation of dynamic and stochastic systems is generalized to take into account the influence of relaxing systems on the statistical properties of noise some applications of the generalized non markovian sle are discussed in particular it is applied to study two modifications of the ctrw approach one of them considers cascaded ctrws in which the controlling process is actually a ctrw like one controlled by another ctrw process controlled in turn by a third one etc within the mr a simple expression for the pdf w t of the total controlling process is obtained in terms of markovian variants of controlling pdfs in the cascade the expression is shown to be especially simple and instructive in the case of anomalous processes determined by the long time tailed w t the cascaded ctrws can model the effect of the complexity of a system on the relaxation kinetics in glasses fractals branching media ultrametric structures etc another ctrw modification describes the kinetics of processes governed by fluctuating w t within the mr the problem is analyzed in a general form without restrictive assumptions on the correlations of pdfs of consecutive renewals the analysis shows that fluctuations of w t can strongly affect the kinetics of the process possible manifestations of this effect are discussed

a new characterization of simultaneous lyapunov diagonal stability via hadamard products abstract a well known characterization by kraaijevanger 14 for lyapunov diagonal stability states that a real square matrix a is lyapunov diagonally stable if and only if a s is a p matrix for any positive semidefinite s with nonzero diagonal entries this result is extended here to a new characterization involving similar hadamard multiplications for simultaneous lyapunov diagonal stability on a set of matrices among the main ingredients for this extension are a new concept called p sets and a recent result regarding simultaneous lyapunov diagonal stability by berman goldberg and shorten 2

wave equations with point interactions in finite energy spaces given the abstract wave equation 0 where is the laplace operator with a point interaction of strength we define and study w the associated wave generator in the phase space of finite energy states we prove the existence of the phase flow generated by w and describe its most relevant properties with a particular emphasis on the associated symplectic structure and scattering theory

stability and stabilizability of switched linear systems a survey of recent results during the past several years there have been increasing research activities in the field of stability analysis and switching stabilization for switched systems this paper aims to briefly survey recent results in this field first the stability analysis for switched systems is reviewed we focus on the stability analysis for switched linear systems under arbitrary switching and we highlight necessary and sufficient conditions for asymptotic stability after a brief review of the stability analysis under restricted switching and the multiple lyapunov function theory the switching stabilization problem is studied and a variety of switching stabilization methods found in the literature are outlined then the switching stabilizability problem is investigated that is under what condition it is possible to stabilize a switched system by properly designing switching control laws note that the switching stabilizability problem has been one of the most elusive problems in the switched systems literature a necessary and sufficient condition for asymptotic stabilizability of switched linear systems is described here

continuous time random walks simulation of continuous trajectories continuous time random walks have been developed as a straightforward generalization of classical random walk processes some ten years ago fogedby introduced a continuous representation of these processes by means of a set of langevin equations h c fogedby phys rev e 50 1657 1994 the present work is devoted to a detailed discussion of fogedby s model and presents its application for the robust numerical generation of sample paths of continuous time random walk processes

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

on the stability of switched positive linear systems it was recently conjectured that the hurwitz stability of the convex hull of a set of metzler matrices is a necessary and sufficient condition for the asymptotic stability of the associated switched linear system under arbitrary switching in this note we show that 1 this conjecture is true for systems constructed from a pair of second order metzler matrices 2 the conjecture is true for systems constructed from an arbitrary finite number of second order metzler matrices and 3 the conjecture is in general false for higher order systems the implications of our results both for the design of switched positive linear systems and for research directions that arise as a result of our work are discussed toward the end of the note

strong stability of internal system descriptions this article introduces a new notion of stability for internal autonomous system descriptions that is referred to as strong stability and characterises the case of avoiding overshoots for all initial conditions within a given sphere this is a stronger notion of stability compared with alternative definitions asymptotic lyapunov which allows the analysis and design of control systems described by natural coordinates to have no overshooting response for arbitrary initial conditions for the case of linear time invariant systems necessary and sufficient conditions for strong stability are established in terms of the negative definiteness of the symmetric part of the state matrix the invariance of strong stability under orthogonal transformations is established and this enables the characterisation of the property in terms of the invariants of the schur form of the state matrix some interesting relations between strong stability and special forms of coordinate frames are examined and links between th