stability analysis of continuous time periodic systems via the harmonic analysis asymptotic stability of finite dimensional linear continuous time periodic fdlcp systems is studied by harmonic analysis it is first shown that stability can be examined with what we call the harmonic lyapunov equation another necessary and sufficient stability criterion is developed via this generalized lyapunov equation which reduces the stability test into that of an approximate fdlcp model whose transition matrix can be determined explicitly by extending the gerschgorin theorem to linear operators on the linear space l sub 2 yet another disc group criterion is derived which is only sufficient stability of the lossy mathieu equation is analyzed as a numerical example to illustrate the results

some analysis methods for rotating systems with periodic coefficients two of the more common procedures for analyzing the stability and forced response of equations with periodic coefficients are reviewed the use of floquet methods and the use of multiblade coordinate and harmonic balance methods the analysis procedures of these periodic coefficient systems are compared with those of the more familiar constant coefficient systems previously announced in star as n82 23702

blow up problems for the heat equation with a local nonlinear neumann boundary condition abstract this paper estimates the blow up time for the heat equation u t u with a local nonlinear neumann boundary condition the normal derivative u n u q on 1 one piece of the boundary while on the rest part of the boundary u n 0 the motivation of the study is the partial damage to the insulation on the surface of space shuttles caused by high speed flying subjects we show the finite time blow up of the solution and estimate both upper and lower bounds of the blow up time in terms of the area of 1 in many other work they need the convexity of the domain and only consider the problem with 1 in this paper we remove the convexity condition and only require to be c 2 in addition we deal with the local nonlinearity namely 1 can be just part of

trace formula of linear continuous time periodic systems via the harmonic lyapunov equation in the former part of this paper a trace formula is established for the h 2 norms of a class of finite dimensional linear continuous time periodic fdlcp systems based on the solution of the harmonic lyapunov equations this trace formula is quite similar in form to what we have for the h 2 norm of an lti continuous time system apart from the fact that infinite dimensional matrices are involved in the fdlcp setting based on this formula in the latter part of this paper some trace formulas are developed via the approximate modelling and truncating approach which are numerically implementable in most practical fdlcp systems there are numerical examples to illustrate the efficacy of the trace formulas suggested

fr operator approach to the h sub 2 analysis and synthesis of sampled data systems recently a frequency domain operator called frequency response fr operator was defined and shown to represent the transfer characteristics of a stable sampled data system using this novel frequency domain notion and introducing its extended notion called hybrid fr operator we define an h sub 2 norm for sampled data systems in this paper then sampled data h sub 2 control problems are formulated and solved whereby the usefulness of these frequency domain notions is demonstrated both in the analysis and synthesis aspects of sampled data systems for the case of sampled data systems with hybrid i e both continuous time and discrete time input and output signals the h sub 2 norm defined by a hybrid fr operator turns out to be slightly different from that defined in previous studies the source of the discrepancy is also identified

a convex concave problem with a nonlinear boundary condition in this paper we study the existence of nontrivial solutions of the problem u u u p 2uin u u q 2uon with 1 q 2 n 1 n 2 and 1 p 2n n 2 in the concave convex case i e 1 q 2 p if is small there exist two positive solutions while for large there is no positive solution when p is critical and q subcritical we obtain existence results using the concentration compactness method finally we apply the implicit function theorem to obtain solutions for small near u0 1

brief h2 and h norm computations of linear continuous time periodic systems via the skew analysis of frequency response operators the h 2 and h norm computations of finite dimensional linear continuous time periodic fdlcp systems through the frequency response operators defined by steady state analysis are discussed by the skew truncation the h 2 norm can be reached to any degree of accuracy by that of an asymptotically equivalent linear time invariant lti continuous time system the h norm can be approximated by the maximum singular value of the frequency response of an asymptotically equivalent lti continuous time system over a certain frequency range via the modified skew truncation by the latter result a hamiltonian test is proved for fdlcp systems in an lti fashion based on which a modified bisection algorithm is developed

global blow up for a localized nonlinear parabolic equation with a nonlocal boundary condition abstract this paper deals with the blow up properties of positive solutions to a nonlinear parabolic equation with a localized reaction source and a nonlocal boundary condition under certain conditions the blowup criteria is established furthermore when f u u p 0 p 1 the global blowup behavior is shown and the blowup rate estimates are also obtained

h 2 performance of continuous time periodically time varying controllers abstract this paper uses a frequency domain approach to the analysis of h 2 performance of continuous time periodically time varying controllers for control of linear time invariant plants it is shown that the time varying dynamics of the periodic controller deteriorates the closed loop system h 2 performance and a linear time invariant controller can be found to provide strictly better h 2 control of the system

a maximum likelihood estimate for two variable fractal surface the fractal dimension estimate for two variable fractional brownian motion using the maximum likelihood estimate mle is developed we formulate a model to describe the two variable fractional brownian motion then derive the likelihood function for that model and estimate the fractal dimension by maximizing the likelihood function we then compare the mle with the box dimension estimation method

spectral representation of fractional brownian motion in n dimensions and its properties fractional brownian motion fbm provides a useful model for processes with strong long term dependence such as 1 f sup spl beta spectral behavior however fbm s are nonstationary processes so that the interpretation of such a spectrum is still a matter of speculation to facilitate the study of this problem another model is provided for the construction of fbm from a white noise like process by means of a stochastic or ito integral in frequency of a stationary uncorrelated random process also a generalized power spectrum of the nonstationary fbm process is defined this new approach to fbm can be used to compute all of the correlations power spectra and other properties of fbm in this paper a number of these fbm properties are developed from this model such as the t sup h law of scaling the power law of fractional order the correlation of two arbitrary fbm s and the evaluation of the fractal dimension under various transformations this new treatment of fbm using a spectral representation is extended also for the first time to two or more topological dimensions in order to analyze the features of isotropic n dimensional fbm

existence conditions and properties of the frequency response operators of continuous time periodic systems the definition of the frequency response operator via the steady state analysis in finite dimensional linear continuous time periodic fdlcp systems is revisited it is shown that the frequency response operator is guaranteed to be well defined only densely on the linear space l2 which is different from the usual understanding fortunately however it turns out that this frequency response operator can have an extension onto l2 so that the equivalence between the time domain h2 norm respectively the l2 induced norm and the frequency domain h2 norm respectively the h infty norm of the frequency response operator is recovered under some stronger assumptions it is also shown that the frequency response operator can be viewed as a bounded operator from l1 to l1 which can also be established via the steady state analysis

survey invariant representations of discrete time periodic systems the correspondence between periodic and time invariant systems can take a variety of forms the purpose of this survey paper is to present a complete and clear picture on time invariant representations for discrete time periodic systems the core of the treatment evolves around the concept of periodic transfer function

a novel extended precise integration method based on fourier series expansion for the h2 norm of linear time varying periodic systems abstracta new reliable algorithm for computing the h2 norm of linear time varying periodic ltp systems via the periodic lyapunov differential equation plde is proposed by taking full advantage of the periodicity the transition matrix of the underlying ltp system associated with the plde is effectively computed by developing a novel extended precise integration method based on fourier series expansion where the time consuming work for the computation of the matrix exponential and its related integrals in every sub interval is avoided then a highly accurate and efficient algorithm for the plde is derived using the block form of the transition matrix thus the h2 norm is evaluated by solving a simple first order ordinary differential equation finally two numerical examples are presented and compared with other algorithms to verify the numerical accuracy and efficiency of the proposed algorithm

frequency response of sampled data systems this paper introduces the concept of frequency response for sampled data systems and explores some basic properties as well as its computational procedures it is shown that 1 by making use of the lifting technique the notion of frequency response can be naturally introduced to sampled data systems in spite of their time varying characteristics 2 it represents a frequency domain steady state behaviour and 3 it is also closely related to the original transfer function representation via an integral formula it is shown that the computation of the frequency response can be reduced to a finite dimensional eigenvalue problem and some examples are presented to illustrate the results

brief paper h2 norm of linear time periodic systems a perturbation analysis we consider a class of linear time periodic systems in which the dynamical generator a t represents the sum of a stable time invariant operator a 0 and a small amplitude zero mean t periodic operator ea p t we employ a perturbation analysis to develop a computationally efficient method for determination of the h 2 norm up to second order in the perturbation parameter e we show that a the h 2 norm can be obtained from a conveniently coupled system of lyapunov and sylvester equations that are of the same dimension as a 0 b there is no coupling between different harmonics of a p t in the expression for the h 2 norm these two properties do not hold for arbitrary values of e and their derivation would not be possible if we tried to determine the h 2 norm directly without resorting to perturbation analysis our method is well suited for identification of the values of period t that lead to the largest increase reduction of the h 2 norm two examples are provided to motivate the developments and illustrate the procedure

improvements on lower bounds for the blow up time under local nonlinear neumann conditions abstract this paper studies the heat equation u t u in a bounded domain r n n 2 with positive initial data and a local nonlinear neumann boundary condition the normal derivative u n u q on partial boundary 1 for some q 1 while u n 0 on the other part we investigate the lower bound of the blow up time t of u in several aspects first t is proved to be at least of order q 1 1 as q 1 since the existing upper bound is of order q 1 1 this result is sharp secondly if is convex and 1 denotes the surface area of 1 then t is shown to be at least of order 1 1 n 1 for n 3 and 1 1 ln 1 1 for n 2 as 1 0 while the previous result is 1 for any 1 n 1 finally we generalize the results for convex domains to the domains with only local convexity near 1

h 2 norm computation of linear time varying periodic systems via the periodic lyapunov differential equation a new reliable algorithm for computing norm of linear time varying periodic ltp systems in continuous time domain is proposed in this article the solving of norm is firstly transformed into the solving of the periodic lyapunov differential equation plde then the key point in this article is that a new method based on the interval mixed energy matrix and the interval transition matrix computed by the structure preserving magnus series method is proposed for solving the plde with the solutions of plde the norm of ltp systems is evaluated by a simple first order ordinary differential equation finally the effectiveness and the high accuracy of the proposed algorithms are demonstrated by two numerical examples

an improved box counting method for image fractal dimension estimation fractal dimension fd is a useful feature for texture segmentation shape classification and graphic analysis in many fields the box counting approach is one of the frequently used techniques to estimate the fd of an image this paper presents an efficient box counting based method for the improvement of fd estimation accuracy a new model is proposed to assign the smallest number of boxes to cover the entire image surface at each selected scale as required thereby yielding more accurate estimates the experiments using synthesized fractional brownian motion images real texture images and remote sensing images demonstrate this new method can outperform the well known differential boxing counting dbc method

the balance between nonlinear inwards and outwards boundary flux for a nonlinear heat equation abstract we study nonnegative solutions of u t u m xx x t 0 l 0 t u m x 0 t u p 0 t t 0 t u m x l t u q l t t 0 t u x 0 u 0 x x 0 l where m p q and l are positive parameters for different values of the parameters three situations may occur 1 all solutions of this problem exist for all t 0 2 for certain initial data functions the solution exists for all t 0 while for others the solution blows up as t t for some finite t 3 excepting the trivial solution when u0 0 all solutions blow up as t t for some finite t we identify in terms of the parameters which of them actually happens for solutions which blow up we find the blow up rate and the blow up set

technical communique approximation of frequency response for sampled data control systems this paper proves that the frequency response gains of fast sample fast hold approximations of a sampled data system converge to that of the original system as the sampling rate gets faster while this may appear to hold trivially there is a serious technical difficulty and the proof is indeed nontrivial it is also guaranteed that this convergence is uniform on the total frequency range the latter property is necessary to guarantee that a single approximant can be used for frequency response computation for the overall frequency range

fractional brownian motion a maximum likelihood estimator and its application to image texture fractals have been shown to be useful in characterizing texture in a variety of contexts use of this methodology normally involves measurement of a parameter h which is directly related to fractal dimension in this work the basic theory of fractional brownian motion is extended to the discrete case it is shown that the power spectral density of such a discrete process is only approximately proportional to f a instead of in direct proportion as in the continuous case an asymptotic cramer rao bound is derived for the variance of an estimate of h subsequently a maximum likelihood estimator mle is developed to estimate h it is shown that the variance of this estimator nearly achieves the minimum bound a generation algorithm for discrete fractional motion is presented and used to demonstrate the capabilities of the mle when the discrete fractional brownian process is contaminated with additive gaussian noise the results show that even at signal to noise ratios of 30 db significant errors in estimation of h can result when noise is present the mle is then applied to x ray images of the human calcaneus to demonstrate how the line to line formulation can be applied to the two dimensional case these results indicate that it has strong potential for quantifying texture

on monodromy matrix computation abstract we present a study on the critical time step for the numerical integration based on the runge kutta method of the monodromy matrix the fundamental matrix solution associated with a set of n first order linear ordinary differential equations with periodic coefficients by applying the lyapunov schmidt method for any dimension n and systems which are perturbations of autonomous systems we give an approximation to the critical time step which involves the autonomous part as well as the periodic perturbations

frequency response of sampled data systems this paper develops a frequency domain theory that provides a method to analyze and design sampled data control systems including their intersample behaviors the key idea is to consider the signal space x x t x t n x xn exp j t jn st n x xn 2 where s is the sampling angular frequency it is shown that a stable sampled data system equipped with a strictly proper pre filter before the sampler maps x into x l2 in the steady state that mapping is denoted by q j and is referred to as an fr operator it is proved that the norm of the sampled data system as an operator from l2 to l2 is given by max q j 2 2 where 12 s 12 s a set of equations relating outputs of a closed loop system to its inputs in the frequency domain is derived and their solution is given in an explicit form based on that solution the sensitivity fr operator y j and the complementary sensitivity fr operator t j are defined for feedback control systems and it is shown that y j gives the improvement of the sensitivity of the transfer characteristics from the reference to the controlled output and also represents the ability of rejecting disturbances and that t j represents the degree of robust stability and at the same time gives the effect of detection noises it is also shown that y j t j i and thus a frequency domain paradigm for the design of sampled data control systems which is exactly parallel to the continuous time case is established

multiple resolution texture analysis and classification textures are classified based on the change in their properties with changing resolution the area of the gray level surface is measured at serveral resolutions this area decreases at coarser resolutions since fine details that contribute to the area disappear fractal properties of the picture are computed from the rate of this decrease in area and are used for texture comparison and classification the relation of a texture picture to its negative and directional properties are also discussed

characteristics of natural scenes related to the fractal dimension many objects in images of natural scenes are so complex and erratic that describing them by the familiar models of classical geometry is inadequate in this paper we exploit the power of fractal geometry to generate global characteristics of natural scenes in particular we are concerned with the following two questions 1 can we develop a measure which can distinguish between different global backgrounds e g mountains and trees and 2 can we develop a measure that is sensitive to change in distance or scale we present a model based on fractional brownian motion which will allow us to recover two characteristics related to the fractal dimension from silhouettes the first characteristic is an estimate of the fractal dimension based on a least squares linear fit we show that this feature is stable under a variety of real image conditions and use it to distinguish silhouettes of trees from silhouettes of mountains next we introduce a new theoretical concept called the average holder constant and relate it mathematically to the fractal dimension it is shown that this measurement is sensitive to scale in a predictable manner and hence provides the potential for use as a range indicator corroborating experimental results are presented

statistical analysis of stable fdlcp systems by parametric transfer matrices this paper considers the problem of determining the statistical characteristics of the response of internally stable finite dimensional continuous time periodic fdlcp systems to stationary stochastic input signals closed formulae for calculating the variance matrix of the state and the output vectors are derived as well as for the mean variance of the output and the 2 norm of the systems where only matrices of finite dimension are used the method bases on application of the parametric transfer matrix ptm and needs a precalculation of the transition matrix of the system on the interval 0 t t where t is the period of the system

a new approach to the discretization of continuous time controllers a new method for discretizing continuous time controllers is derived using principles of controller approximation it focuses on the closed loop use of the discrete time controller the resulting approximation criterion is a measure for stability of the control system provides an upper bound on the sampling time for which stability can be guaranteed and because it is based on continuous time controller approximation it indicates the cost of discretization in terms of performance degradation the discrete time controller is obtained through minimization of this criterion which can be performed with standard software used in h sub infinity controller design

blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition abstract in this paper we study the large time behavior of positive solutions of the heat equation under the nonlinear boundary condition u f u where is the outward normal and f is nondecreasing with f u 0 for u 0 we show that if b r and 1 f is integrable at infinity there is finite time blow up for any initial datum in the two dimensional case we show that this is true for any smooth simply connected domain in the radially symmetric case if f c2 is convex and satisfies the properties above we show that blow up occurs only at the boundary

a general framework for linear periodic systems with applications to h sup infinity sampled data control the authors present a framework for dealing with continuous time periodic systems the main tool is a lifting technique which provides a strong correspondence between continuous time periodic systems and certain types of discrete time time invariant systems with infinite dimensional input and output spaces despite the infinite dimensionality of the input and output spaces a lifted system has a finite dimensional state space if the original system does this fact permits rather constructive methods for analyzing these systems as a demonstration of the utility of this framework the authors use it to describe the continuous time i e intersample behavior of sampled data systems and to obtain a complete solution to the problem of parameterizing all controllers that constrain the l sup 2 induced norm of a sampled data system to within a certain bound

using the fredholm resolvent for computing the ℋ 2 norm of linear periodic systems this article presents a method for the 2 norm computation of single loop finite dimensional linear continuous periodic systems which is based on the parametric transfer function concept and the calculus of fredholm integral equations of the second kind general formulae for the computation of the mean output variance are provided as relations of two integral functions of an auxiliary parameter on the basis of the general solution an approximate computation method is derived and the deviation of the approximate solution is estimated in principle this approach allows one to perform the computation with any desired accuracy

the h 2 problem for sampled data systems m for sampled data systems in this paper we pose and give a complete solution to an analog of the h2 problem for sampled data system we motivate and develop a natural generalization of the h2 cost to periodic systems which is then applied to the continuous time closed loop mapping in a sampled data control system it is shown that the cost criterion developed is actually a norm in an space of hilbert schmidt operator valued functions we give state space solutions to the optimal and suboptimal controllers synthesis problems in this new norm by establishing an equivalent standard h2 problem

input output stability of sampled data systems the authors study the l sub p input output stability of a continuous time controller using the usual arrangement of periodic sampling and zero order hold it is noted that even if the hybrid system is exponentially stable this arrangement does not yield l sub p 1

boundary fluxes for nonlocal diffusion abstract we study a nonlocal diffusion operator in a bounded smooth domain prescribing the flux through the boundary this problem may be seen as a generalization of the usual neumann problem for the heat equation first we prove existence uniqueness and a comparison principle next we study the behavior of solutions for some prescribed boundary data including blowing up ones finally we look at a nonlinear flux boundary condition