integral versions of iss for sampled data nonlinear systems via their approximate discrete time models two integral versions of input to state stability are considered integral input to state stability iiss and integral input to integral state stability iiiss we present sufficient conditions that guarantee that if a controller achieves semiglobal practical iiss respectively iiiss of an ap proximate discrete time model of a nonlinear sampled data system then the same controller achieves semiglobal practical iiss respectively iiiss of the exact discrete time model by reducing the sampling period recent results on numerical methods for systems with measurable disturbances can be used to generate approximate models that we consider results are presented for arbitrary dynamic controllers that can be discontinuous in general

a quadrature formula for integrals with nearby singularities the purpose of this paper is to propose a new quadrature formula for integrals with nearby singularities in the boundary element method the integrands of nearby singular boundary integrals vary drastically with the distance between the field and the source point especially field variables and their derivatives at a field point near a boundary cannot be computed accurately in the present paper a quadrature formula for l isolated singularities near the integration interval based on lagrange interpolatory polynomials is obtained the error estimation of the proposed formula is also given quadrature formulas for regular and singular integrals with conjugate poles are derived numerical examples are given and the proposed quadrature rules present the expected polynomial accuracy

robustness of nonlinear systems with respect to delay and sampling of the controls we consider continuous time nonlinear time varying systems that are globally asymptotically stabilizable by state feedbacks we study the stability of these systems in closed loop with controls that are corrupted by both delay and sampling we establish robustness results through a lyapunov approach of a new type

a characterization of compact perturbations of toeplitz operators let x be a bounded operator on the hardy space h 2 of the unit circle it has been a longstanding problem to determine whether the condition that t u xt u x is compact for every inner function u implies that x is a compact perturbation of a toeplitz operator we show that the answer is affirmative

nystrom type methods for fredholm integral equations with weak singularities nystrom type methods are constructed and justified for a class of fredholm integral equations of the second kind with kernels which may have weak diagonal and boundary singularities the proposed approach is based on a suitable smoothing change of variables and product integration techniques global convergence estimates are derived and a collection of numerical results is given

ap weights for nondoubling measures in rn and applications we study an analogue of the classical theory of ap weights in r n without assuming that the underlying measure is doubling then we obtain weighted norm inequalities for the centered hardy littlewood maximal function and corresponding weighted estimates for nonclassical calderon zygmund operators in the sense of ntv1 we also consider commutators of those calderon zygmund operators with bounded mean oscillation functions bmo extending the main result from crw finally we study self improving properties of poincare b m o type inequalities within this context more precisely we show that if f is a locally integrable function satisfying 1 q r q f fq d a q for all cubes q then it is possible to deduce higher l p integrability result of f assuming certain simple geometric condition on the functional a

semi sigmoidal transformations for evaluating weakly singular boundary element integrals accurate numerical integration of line integrals is of fundamental importance to reliable implementation of the boundary element method usually the regular integrals arising from a boundary element method implementation are evaluated using standard gaussian quadrature however the singular integrals which arise are often evaluated in another way sometimes using a different integration method with different nodes and weights here a co ordinate transformation technique is introduced for evaluating weakly singular integrals which after some initial manipulation of the integral uses the same integration nodes and weights as those of the regular integrals the transformation technique is based on newly defined semi sigmoidal transformations which cluster integration nodes only near the singular point the semi sigmoidal transformations are defined in terms of existing sigmoidal transformations and have the benefit of evaluating integrals more accurately than full sigmoidal transformations as the clustering is restricted to one end point of the interval comparison of this new method with existing coordinate transformation techniques shows that more accurate evaluation of weakly singular integrals can be obtained based on observation of several integrals considered guidelines are suggested for the type of semi sigmoidal transformation to use and the degree to which nodes should be clustered at the singular points copyright 2000 john wiley sons ltd

input to state stability for discrete time time varying systems with applications to robust stabilization of systems in power form input to state stability iss of a parameterized family of discrete time time varying nonlinear systems is investigated a converse lyapunov theorem for such systems is developed we consider parameterized families of discrete time systems and concentrate on a semiglobal practical type of stability that naturally arises when an approximate discrete time model is used to design a controller for a sampled data system an application of our main result to time varying periodic systems is presented and this is used to solve a robust stabilization problem namely to design a control law for systems in power form yielding semiglobal practical iss sp iss

a new method for the numerical evaluation of nearly singular integrals on triangular elements in the 3d boundary element method a new method the sinh sigmoidal method is proposed for the numerical evaluation of both nearly weakly and nearly strongly singular integrals on triangular boundary elements these integrals arise in the 3d boundary element method when the source point is very close to the element of integration the new polar coordinate based method introduces a sinh transformation in the radial direction and a sigmoidal transformation in the angular direction before the application of gaussian quadrature it also uses approximately twice as many quadrature points in the angular direction as in the radial direction in response to a finding that the evaluation of these types of integrals is particularly sensitive to the placement of the quadrature points in the angular direction comparisons with various other methods demonstrate its accuracy and competitiveness a major advantage of the new method is its ease of implementation and applicability to a wide class of integrals

on non linear transformations for accurate numerical evaluation of weakly singular boundary integrals this paper presents a study of the performance of the non linear co ordinate transformations in the numerical integration of weakly singular boundary integrals a comparison of the smoothing property numerical convergence and accuracy of the available non linear polynomial transformations is presented for two dimensional problems effectiveness of generalized transformations valid for any type and location of singularity has been investigated it is found that weakly singular integrals are more efficiently handled with transformations valid for end point singularities by partitioning the element at the singular point further transformations which are excellent for cpv integrals are not as accurate for weakly singular integrals connection between the maximum permissible order of polynomial transformations and precision of computations has also been investigated cubic transformation is seen to be the optimum choice for single precision and quartic or quintic one for double precision computations a new approach which combines the method of singularity subtraction with non linear transformation has been proposed this composite approach is found to be more accurate efficient and robust than the singularity subtraction method and the non linear transformation methods copyright 2001 john wiley sons ltd

a double commutant relation in the calkin algebra on the bergman space abstract let t be the toeplitz algebra on the bergman space l a 2 b d v of the unit ball in c n we show that the image of t in the calkin algebra satisfies the double commutant relation t t this is a surprising result for it is the opposite of what happens in the hardy space case 16 17

optimization based stabilization of sampled data nonlinear systems via their approximate discrete time models we present results on numerical regulator design for sampled data nonlinear plants via their approximate discrete time plant models the regulator design is based on an approximate discrete time plant model and is carried out either via an infinite horizon optimization problem or via a finite horizon with terminal cost optimization problem in both cases we discuss situations when the sampling period t and the integration period h used in obtaining the approximate discrete time plant model are the same or they are independent of each other we show that using this approach practical and or semiglobal stability of the exact discrete time model is achieved under appropriate conditions

improvement of the asymptotic behaviour of the euler maclaurin formula for cauchy principal value and hadamard finite part integrals in the recent works commun numer meth engng 2001 17 881 to appear the superiority of the non linear transformations containing a real parameter b 0 has been demonstrated in numerical evaluation of weakly singular integrals based on these transformations we define a so called parametric sigmoidal transformation and employ it to evaluate the cauchy principal value and hadamard finite part integrals by using the euler maclaurin formula better approximation is expected due to the prominent properties of the parametric sigmoidal transformation of whose local behaviour near x 0 is governed by parameter b through the asymptotic error analysis of the euler maclaurin formula using the parametric sigmoidal transformation we can observe that it provides a distinct improvement on its predecessors using traditional sigmoidal transformations numerical results of some examples show the availability of the present method

asymptotic controllability implies feedback stabilization it is shown that every asymptotically controllable system can be globally stabilized by means of some discontinuous feedback law the stabilizing strategy is based on pointwise optimization of a smoothed version of a control lyapunov function iteratively sending trajectories into smaller and smaller neighborhoods of a desired equilibrium a major technical problem and one of the contributions of the present paper concerns the precise meaning of solution when using a discontinuous controller

on the essential commutant of the toeplitz algebra on the bergman space let tt be the c c algebra generated by the toeplitz operators tf f l b dv tf f l b dv on the bergman space of the unit ball we show that the essential commutant of tt equals tg g vobdd k tg g vobdd k where vobddvobdd is the collection of bounded functions of vanishing oscillation on b and kk denotes the collection of compact operators on la2 b dv

the simple chemostat with wall growth a model of the simple chemostat which allows for growth on the wall or other marked surface is presented as three nonlinear ordinary differential equations the organisms which are attached to the wall do not wash out of the chemostat this destroys the basic reduction of the chemostat equations to a monotone system a technique which has been useful in the analysis of many chemostat like equations the adherence to and shearing from the wall eliminates the boundary equilibria for a reasonably general model the basic properties of invariance dissipation and uniform persistence are established for two important special cases global asymptotic results are obtained finally a perturbation technique allows the special results to be extended to provide the rest point as a global attractor for nearby growth functions

recent developments on the stability of systems with aperiodic sampling an overview this article presents basic concepts and recent research directions about the stability of sampled data systems with aperiodic sampling we focus mainly on the stability problem for systems with arbitrary time varying sampling intervals which has been addressed in several areas of research in control theory systems with aperiodic sampling can be seen as time delay systems hybrid systems input output interconnections discrete time systems with time varying parameters etc the goal of the article is to provide a structural overview of the progress made on the stability analysis problem without being exhaustive which would be neither possible nor useful we try to bring together results from diverse communities and present them in a unified manner for each of the existing approaches the basic concepts fundamental results converse stability theorems when available and relations with the other approaches are discussed in detail results concerning extensions of lyapunov and frequency domain methods for systems with aperiodic sampling are recalled as they allow to derive constructive stability conditions furthermore numerical criteria are presented while indicating the sources of conservatism the problems that remain open and the possible directions of improvement at last some emerging research directions such as the design of stabilizing sampling sequences are briefly discussed

further results on robustness of possibly discontinuous sample and hold feedback we demonstrate that sample and hold state feedback control possibly discontinuous with respect to the state is robust when the closed loop system possesses an appropriate lyapunov function we first show that if a lyapunov decrease over sampling periods exists for the nominal system this decrease can be maintained with some degradation relative to a sufficiently small additive perturbation we then proceed to catalog several applications of this robustness e g robustness to measurement noise computational delays or fast actuator dynamics

a mathematical model of the chemostat with a general class of functions describing nutrient uptake a model of the chemostat involving n microorganisms competing for a single essential growth limiting substrate is considered instead of assuming the familiar michaelis menten kinetics for nutrient uptake a general class of functions is used which includes all monotone increasing uptake functions but which also allows uptake functions that describe inhibition by the substrate at high concentrations the qualitative behaviour of this generalized model is determined analytically it is shown that the behaviour depends intimately upon certain parameters provided that all the parameters are distinct which is a biologically reaonable assumption at most one competitor survives the substrate and the surviving competitor if one exists approach limiting values thus there is competitive exclusion however unlike the standard model in certain cases the outcome is initial condition dependent

the method of steepest descent for estimating quadrature errors this work presents an application of the method of steepest descent to estimate quadrature errors the method is used to provide a unified approach to estimating the truncation errors which occur when gauss legendre quadrature is used to evaluate the nearly singular integrals that arise as part of the two dimensional boundary element method the integrals considered here are of the form 1 1 h x d x x a 2 b 2 where h x is a well behaved function 0 and 1 a 1 since 0 b 1 the integral is nearly singular with a sharply peaked integrand the method of steepest descent is used to estimate the generally large truncation errors that occur when gauss legendre quadrature is used to evaluate these integrals as well as to estimate the much lower errors that occur when gauss legendre quadrature is performed on such integrals after a sinh transformation has been applied the new error estimates are highly accurate in the case of the transformed integral and are shown to be comparable to those found in previous work by elliott and johnston 2007 one advantage of the new estimates is that they are given by just one formula each for the un transformed and the transformed integrals rather than the much larger set of formulae in the previous work another advantage is that the new method applies over a much larger range of values than the previous method

nearly hypersingular integrals by double arctan transformation in higher order geometry modeling in order to directly utilize the dyadic green s functions in surface integral equations sies a novel double arctan transformation for nearly hypersingular integrals is proposed in this communication this new transformation is flexible and applicable to nearly hypersingular integrals in the forms of hat r hat r r 3 hat r r 3 and 1 r 3 over the curved surfaces by a fully numerical method with the help of the sigmoidal transformation to improve the stability of this new singularly handling method there results an efficient solution for the third order near singularity problems in sies moreover the proposed method is also effective for the lower orders of the nearly singular integral kernels with typical testing cases the performance of this method is fairly evaluated and its validity and stability is well demonstrated

error estimation of quadrature rules for evaluating singular integrals in boundary element problems the efficient numerical evaluation of integrals arising in the boundary element method is of considerable practical importance the superiority of the use of sigmoidal and semi sigmoidal transformations together with gauss legendre quadrature in this context has already been well demonstrated numerically by one of the authors in this paper the authors obtain asymptotic estimates of the truncation errors for these algorithms these estimates allow an informed choice of both the transformation and the quadrature error in the evaluation of boundary element integrals with algebraic or algebraic logarithmic singularities at an end point of the interval of integration copyright 2000 john wiley sons ltd

robust global stabilisability by means of sampled data control with positive sampling rate this work proposes a notion of robust reachability of one set from another set under constant control this notion is used to construct a control strategy involving sequential set to set reachability which guarantees robust global stabilisation of non linear sampled data systems with positive sampling rate sufficient conditions for robust reachability of one set from another under constant control are also provided the proposed method is illustrated through a number of examples including the study of the sampled data stabilisation problem of the chemostat

transformations for evaluating singular boundary element integrals accurate numerical integration of line integrals is of fundamental importance for the reliable implementation of the boundary element method usually the regular integrals arising from a boundary element method implementation are evaluated using standard gaussian quadrature however the singular integrals which arise are often evaluated in another way sometimes using a different integration method with different nodes and weights this paper presents a straightforward transformation to improve the accuracy of evaluating singular integrals the transformation is in a sense a generalisation of the popular method of telles with the underlying idea being to utilise the same gaussian quadrature points as used for evaluating nonsingular integrals in a typical boundary element method implementation the new transformation is also shown to be equivalent to other existing transformations in certain situations comparison of the new method with existing coordinate transformation techniques shows that a more accurate evaluation of weakly singular integrals can be obtained the technique can also be extended to evaluate certain hadamard finite part integrals based on the observation of several integrals considered guidelines are suggested for the best transformation order to use i e the degree to which nodes should be clustered near the singular point

application of sigmoidal transformations to weakly singular and near singular boundary element integrals accurate numerical determination of line integrals is fundamental to reliable implementation of the boundary element method for a source point distant from a particular element standard gaussian quadrature is adequate as well as being the technique of choice however when the integrals are weakly singular or nearly singular source point near the element this technique is no longer adequate here a co ordinate transformation technique based on sigmoidal transformations is introduced to evaluate weakly singular and near singular integrals a sigmoidal transformation has the effect of clustering the integration points towards the endpoints of the interval of integration the degree of clustering is governed by the order of the transformation comparison of this new method with existing co ordinate transformation techniques shows that more accurate evaluation of these integrals can be obtained based on observations of several integrals considered guidelines are suggested for the order of the sigmoidal transformations copyright 1999 john wiley sons ltd

further results on stabilization of periodic trajectories for a chemostat with two species we discuss an important class of problems involving the tracking of prescribed trajectories in the chemostat model we provide new tracking results for chemostats with two species and one limiting substrate based on lyapunov function methods in particular we use a linear feedback control of the dilution rate and an appropriate time varying substrate input concentration to produce a locally exponentially stable oscillatory behavior this means that all trajectories for the nutrient and corresponding species concentrations in the closed loop chemostat that stay near the oscillatory reference trajectory are attracted to the reference trajectory exponentially fast we also obtain a globally stable oscillatory reference trajectory for the species concentrations using a nonlinear feedback control depending on the dilution rate and the substrate input concentration this guarantees that all trajectories for the closed loop chemostat dynamics are attracted to the reference trajectory finally we construct an explicit lyapunov function for the corresponding global error dynamics we demonstrate the efficacy of our method in a simulation

a modified non linear transformation method for evaluating weakly singular boundary integrals accurate numerical evaluation of boundary integrals is fundamental to producing useful results with the boundary element method this paper introduces a generalisation of a recently introduced combined method subtraction of singularity followed by a non linear transformation which takes into account the effect of the basis functions the new method is applied to solve weakly singular integrals which arise in the solution of the two dimensional laplace equation the new method was found in the cases considered to be numerically superior to both the combined method and any of the non linear transformation methods

an algorithm for the approximate solution of integral equations of mellin type the cruciform crack problem of elasticity gives rise to an integral equation of the second kind on 0 1 whose kernel has a fixed singularity at 0 0 we introduce a transformation of 0 1 onto itself such that an arbitrary number of derivatives vanish at the end points 0 and 1 if the transformed kernel is dominated near the origin by a mellin kernel then we have given conditions under which the use of a modified euler maclaurin quadrature rule and the nystrom method gives an approximate solution which converges to the exact solution of the original equation the method is illustrated with a numerical example

global dynamics of a mathematical model of competition in the chemostat general response functions and differential death rates a model of exploitative competition of n species in a chemostat for a single essential nonreproducing growth limiting resource is considered s b hsu siam j appl math 34 1978 pp 760 763 applies lasalle s extension theorem of lyapunov stability theory to study the asymptotic behavior of solutions in the special case that the response functions are modeled by michaelis menten dynamics g j butler and g s k wolkowicz siam j appl math 45 1985 pp 138 151 on the other hand allow more general response functions including monotone and nonmonotone functions but their analysis requires the assumption that the death rates of all the species are negligible in comparison with the washout rate and hence can be ignored by means of lyapunov stability theory the global dynamics of the model for a large class of response functions are studied including both monotone and nonmonotone functions though it is not as general as the class studied by butler and wolkowicz and the results in

a comparison of transformation methods for evaluating two dimensional weakly singular integrals accurate numerical evaluation of integrals arising in the boundary element method is fundamental to achieving useful results via this solution technique in this paper a number of techniques are considered to evaluate the weakly singular integrals which arise in the solution of laplace s equation in three dimensions and poisson s equation in two dimensions both are two dimensional weakly singular integrals and are evaluated using in a product fashion methods which have recently been used for evaluating one dimensional weakly singular integrals arising in the boundary element method the methods used are based on various polynomial transformations of conventional gaussian quadrature points where the transformation polynomial has zero jacobian at the singular point methods which split the region of integration into sub regions are considered as well as non splitting methods in particular the newly introduced and highly accurate generalized composite subtraction of singularity and non linear transformation approach gssnt is applied to various two dimensional weakly singular integrals a study of the different methods reveals complex relationships between transformation orders position of the singular point integration kernel and basis function it is concluded that the gssnt method gives the best overall results for the two dimensional weakly singular integrals studied

stabilization in a chemostat with sampled and delayed measurements and uncertain growth functions we provide a new control design for chemostats under constant substrate input concentrations using piecewise constant delayed measurements of the substrate concentration our growth functions can be uncertain and are not necessarily monotone the dilution rate is the control we use a new lyapunov approach to derive conditions on the largest sampling interval and on the delay length to ensure asymptotic stabilization properties of a componentwise positive equilibrium point

a high order 3d boundary integral equation solver for elliptic pdes in smooth domains we present a high order boundary integral equation solver for 3d elliptic boundary value problems on domains with smooth boundaries we use nystrom s method for discretization and combine it with special quadrature rules for the singular kernels that appear in the boundary integrals the overall asymptotic complexity of our method is o n3 2 where n is the number of discretization points on the boundary of the domain and corresponds to linear complexity in the number of uniformly sampled evaluation points a kernel independent fast summation algorithm is used to accelerate the evaluation of the discretized integral operators we describe a high order accurate method for evaluating the solution at arbitrary points inside the domain including points close to the domain boundary we demonstrate how our solver combined with a regular grid spectral solver can be applied to problems with distributed sources we present numerical results for the stokes navier and poisson problems

a self adaptive co ordinate transformation for efficient numerical evaluation of general boundary element integrals almost all general purpose boundary element computer packages include a curved geometry modelling capability thus numerical quadrature schemes play an important role in the efficiency of programming the technique the present work discusses this problem in detail and introduces efficient means of computing singular or nearly singular integrals currently found in two dimensional axisymmetric and three dimensional applications emphasis is given to a new third degree polynomial transformation which was found greatly to improve the accuracy of gaussian quadrature scheme s within the near singularity range the procedure can easily be implemented into existing be codes and presents the important feature of being self adaptive i e it produces a variable lumping of the gauss stations toward the singularity depending on the minimum distance from the source point to the element the self adaptiveness of the scheme also makes it inactive when not useful large source distances which makes it very safe for general usage

sufficient conditions for stabilization of sampled data nonlinear systems via discrete time approximations given a parameterized by sampling period t family of approximate discrete time models of a sampled nonlinear plant and given a family of controllers stabilizing the family of plant models for all t sufficiently small we present conditions which guarantee that the same family of controllers semi globally practically stabilizes the exact discrete time model of the plant for sufficiently small sampling periods when the family of controllers is locally bounded uniformly in the sampling period the inter sample behavior can also be uniformly bounded so that the time varying sampled data model of the plant is uniformly semi globally practically stabilized the result justifies controller design for sampled data nonlinear systems based on the approximate discrete time model of the system when sampling is sufficiently fast and the conditions we propose are satisfied our analysis is applicable to a wide range of time discretization schemes and general plant models

a composite transformation for numerical integration of singular integrals in the bem in this paper we present a composite non linear transformation of a polynomial transformation pt and a sigmoidal transformation st by the name of ct ps composite transformation with polynomial and sigmoidal transformations this new transformation can be used for both cauchy principal value cpv integrals and weakly singular integrals with interior singularities moreover it is implied that the numerical evaluation using ct ps would be more efficient than either pt or st used alone by numerical examples we show that the presented ct ps method with the gauss legendre quadrature rule improves the approximation errors of the existing methods effectively in particular it is demonstrated that the results of the present method are excellent for cpv integrals with near end point singularities copyright 2003 john wiley sons ltd

a generalized non linear transformation for evaluating singular integrals in this paper we propose a function which according to a parameter included therein generates a new sigmoidal transformation and converges to the well known sato polynomial transformation following well established procedures in the literature we employ the present transformation in the numerical evaluation of weakly singular integrals cauchy principal value integrals and hadamard finite part integrals by several numerical examples it is shown that the present transformation is available for all kinds of singular integrals mentioned above and in some cases gives better results compared with those of traditional non linear transformations copyright 2005 john wiley sons ltd

relaxed lyapunov criteria for robust global stabilisation of non linear systems the notion of the restricted robust control lyapunov function rclf is introduced and is exploited for the design of robust feedback stabilisers for non linear systems particularly it is shown for systems with input constraints that restricted rclfs can be easily obtained while rclfs are not available moreover it is shown that the use of restricted rclfs usually results in different feedback designs from the ones obtained by the use of the standard rclf methodology using the restricted rclfs feedback design methodology a simple controller that guarantees robust global stabilisation of a perturbed chemostat model is provided

a framework for stabilization of nonlinear sampled data systems based on their approximate discrete time models a unified framework for design of stabilizing controllers for sampled data differential inclusions via their approximate discrete time models is presented both fixed and fast sampling are considered in each case sufficient conditions are presented which guarantee that the controller that stabilizes a family of approximate discrete time plant models also stabilizes the exact discrete time plant model for sufficiently small integration and or sampling periods previous results in the literature are extended to cover 1 continuous time plants modeled as differential inclusions 2 general approximate discrete time plant models 3 dynamical discontinuous controllers modeled as difference inclusions and 4 stability with respect to closed arbitrary not necessarily compact sets

asymptotic stabilization for feedforward systems with delayed feedbacks a problem of state feedback stabilization of time varying feedforward systems with a pointwise delay in the input is solved the approach relies on a time varying change of coordinates and lyapunov krasovskii functionals the result applies for any given constant delay and provides uniformly globally asymptotically stabilizing controllers of arbitrarily small amplitude although exact model knowledge and periodicity of the dynamics are required the closed loop systems enjoy input to state stability properties with respect to additive uncertainty on the controllers the work is illustrated through a tracking problem for a model for high level formation flight of unmanned air vehicles

coincidence of essential commutant and the double commutant relation in the calkin algebra let b be a von neumann algebra on a separable hilbert space h we show that if the dimension of b as a linear space is infinite then it has a proper c subalgebra a whose essential commutant in b h coincides with the essential commutant of b moreover if is the quotient map from b h to the calkin algebra b h k h then a b and a b

stability analysis of bilinear systems under aperiodic sampled data control this note considers the problem of local stability of bilinear systems with aperiodic sampled data linear state feedback control the sampling intervals are time varying and upper bounded it is shown that the feasibility of some linear matrix inequalities lmis implies the local asymptotic stability of the sampled data system in an ellipsoidal region containing the equilibrium the method is based on the analysis of contractive invariant sets and it is inspired by the dissipativity theory the results are illustrated by means of numerical examples

stabilization of a chemostat model with haldane growth functions and a delay in the measurements the stabilization of equilibria in chemostats with measurement delays is a complex and challenging problem and is of significant ongoing interest in bioengineering and population dynamics in this paper we solve an output feedback stabilization problem for chemostat models having two species one limiting substrate and either haldane or monod growth functions our feedback stabilizers depend on a given linear combination of the species concentrations which are both measured with a constant time delay the values of the delays are unknown instead one only knows an upper bound on the delays and we allow the upper bound to be arbitrarily large the stabilizing feedback depends on the known upper bound for the delays as well our work is based on a lyapunov krasovskii argument

robustness of nonlinear predictor feedback laws to time and state dependent delay perturbations much recent progress has been achieved for stabilization of linear and nonlinear systems with input delays that are long and dependent on either time or the plant state provided the dependence is known in this paper we consider the delay variations as unknown and study robustness of nominal constant delay predictor feedbacks under delay variations that depend on time and the state we show that when the delay perturbation and its rate have sufficiently small magnitude the local asymptotic stability of the closed loop system under the nominal predictor based design is preserved for the special case of linear systems and under only time varying delay perturbations we prove robustness of global exponential stability of the predictor feedback when the delay perturbation and its rate are small in any one of four different metrics we present two examples one that is concerned with the control of a dc motor through a network and one of a teleoperation like system

a new method for numerical evaluation of nearly singular integrals over high order geometry elements in 3d bem this work presents a new method for numerical computation of the two dimensional nearly singular integrals by using the eight node second order quadrilateral surface elements in 3d bem a new indirect regularized boundary element formulation excluding the cpv cauchy principal value and hfp hadamard finite part integrals is proposed based on this a new approximation formula of the distance from the fixed calculation point to a generic point of the aforementioned surface geometry elements is developed firstly and then the exponential transformation which has been widely employed in 2d bem is extended to 3d bem to remove the near singularities of integrands for considered integrals several numerical examples are given to verify the high efficiency and the stability of the proposed scheme

analysis of three dimensional anisotropic heat conduction problems on thin domains using an advanced boundary element method abstract in this paper an advanced boundary element method bem is developed for solving three dimensional 3d anisotropic heat conduction problems in thin walled structures the troublesome nearly singular integrals which are crucial in the applications of the bem to thin structures are calculated efficiently by using a nonlinear coordinate transformation method for the test problems studied promising bem results with only a small number of boundary elements have been obtained when the thickness of the structure is in the orders of micro scales 10 6 which is sufficient for modeling most thin walled structures as used in for example smart materials and thin layered coating systems the advantages disadvantages as well as potential applications of the proposed method as compared with the finite element method fem are also discussed

three dimensional stress analysis of thin structures using a boundary element method with sinh transformation for nearly singular integrals abstract in this work a three dimensional 3d boundary element method was established with an efficient nonlinear coordinate transformation scheme namely sinh transformation to evaluate nearly singular integrals in boundary integral formulations second order quadrilateral surface elements were developed based on this method to accurately describe the geometry of thin structures the elastic behaviors of selected thin structures were then computed by using the 3d boundary element model to demonstrate the accuracy and efficiency of this approach a number of testing examples i e the 3d kirsch problem the thin spherical shell problem the ellipsoidal vessel problem with non uniform thickness and the hollow circular cylinder problem were numerically studied to test the established method remarkable accuracy and efficiency were found in the developed approach through the comparison to the numerical results and analytical solutions reported in the literature

a generalisation of telles method for evaluating weakly singular boundary element integrals abstract the accurate numerical integration of line integrals is of fundamental importance for the reliable implementation of the boundary element method usually the regular integrals arising from a boundary element method implementation are evaluated using standard gaussian quadrature however the singular integrals which arise are often evaluated in another way sometimes using a different integration method with different nodes and weights this paper presents a simple transformation to improve the accuracy of evaluating weakly singular integrals the transformation is in a sense a generalisation of the popular method of telles internat j numer methods eng 24 1987 959 973 with the underlying idea being to utilise the same gaussian quadrature points as used for evaluating nonsingular integrals in a typical boundary element method implementation comparison of the generalised method with existing coordinate transformation techniques shows that a more accurate evaluation of weakly singular integrals can be obtained based on observation of several integrals considered guidelines are suggested for the best transformation order to use i e the degree to which nodes should be clustered at the singular points an analysis of the generalised approach is also presented where asymptotic estimates for the truncation error after evaluating the transformed integrals using gauss legendre quadrature are determined further numerical experimentation shows that these are very good estimates

reduction model approach for linear systems with sampled delayed inputs we propose a new construction of exponentially stabilizing sampled feedbacks for continuous time linear time invariant systems with an arbitrarily large constant pointwise delay in the inputs stability is guaranteed under an assumption on the size of the largest sampling interval the proposed design is based on an adaptation of the reduction model approach the stability of the closed loop systems is proved through a lyapunov functional of a new type from which is derived a robustness result

variable transformations and gauss legendre quadrature for integrals with endpoint singularities gauss legendre quadrature formulas have excellent convergence properties when applied to integrals 1 0 f x dx with f c 0 1 however their performance deteriorates when the integrands f x are in c 0 1 but are singular at x 0 and or x 1 one way of improving the performance of gauss legendre quadrature in such cases is by combining it with a suitable variable transformation such that the transformed integrand has weaker singularities than those of f x thus if x t is a variable transformation that maps 0 1 onto itself we apply gauss legendre quadrature to the transformed integral 1 0 f t t dt whose singularities at t 0 and or t 1 are weaker than those of f x at x 0 and or x 1 in this work we first define a new class of variable transformations we denote p q where p and q are two positive parameters that characterize it we also give a simple and easily computable representative of this class next by invoking some recent results by the author concerning asymptotic expansions of gauss legendre quadrature approximations as the number of abscissas tends to infinity we present a thorough study of convergence of the combined approximation procedure with variable transformations from p q we show how optimal results can be obtained by adjusting the parameters p and q of the variable transformation in an appropriate fashion we also give numerical examples that confirm the theoretical results

a sinh transformation for evaluating nearly singular boundary element integrals an implementation of the boundary element method requires the accurate evaluation of many integrals when the source point is far from the boundary element under consideration a straightforward application of gaussian quadrature suffices to evaluate such integrals when the source point is on the element the integrand becomes singular and accurate evaluation can be obtained using the same gaussian points transformed under a polynomial transformation which has zero jacobian at the singular point a class of integrals which lies between these two extremes is that of nearly singular integrals here the source point is close to but not on the element and the integrand remains finite at all points however instead of remaining flat the integrand develops a sharp peak as the source point moves closer to the element thus rendering accurate evaluation of the integral difficult this paper presents a transformation based on the sinh function which automatically takes into account the position of the projection of the source point onto the element which we call the nearly singular point and the distance from the source point to the element the transformation again clusters the points towards the nearly singular point but does not have a zero jacobian implementation of the transformation is straightforward and could easily be included in existing boundary element method software it is shown that for the two dimensional boundary element method several orders of magnitude improvement in relative error can be obtained using this transformation compared to a conventional implementation of gaussian quadrature asymptotic estimates for the truncation errors are also quoted copyright 2004 john wiley sons ltd

homogeneous state feedback stabilization of homogenous systems we show that for any asymptotically controllable homogeneous system in euclidean space not necessarily lipschitz at the origin there exists a homogeneous control lyapunov function and a homogeneous possibly discontinuous state feedback law stabilizing the corresponding sampled closed loop system if the system satisfies the usual local lipschitz condition on the whole space we obtain semiglobal stability of the sampled closed loop system for each sufficiently small fixed sampling rate if the system satisfies a global lipschitz condition we obtain global exponential stability for each sufficiently small fixed sampling rate the control lyapunov function and the feedback are based on the lyapunov exponents of a suitable auxiliary system and admit a numerical approximation

nonlinear predictors for systems with bounded trajectories and delayed measurements novel nonlinear predictors are studied for nonlinear systems with delayed measurements without assuming globally lipschitz conditions or a known predictor map but requiring instead bounded state trajectories the delay is constant and known these nonlinear predictors consists of a series of dynamic filters that generate estimates of the state vector and its maximum magnitude at different delayed time instants which differ from one another by a small fraction of the overall delay

zero dynamics of sampled nonlinear systems abstract the behaviour of the zero dynamics of a discrete time system obtained when sampling a continuous time nonlinear system is studied we give a result which generalizes the known behaviour of the zeros of a sampled linear system for small time intervals

formulas relating kl stability estimates of discrete time and sampled data nonlinear systems abstract we provide an explicit kl stability or input to state stability iss estimate for a sampled data nonlinear system in terms of the kl estimate for the corresponding discrete time system and a k function describing inter sample growth it is quite obvious that a uniform inter sample growth condition plus an iss property for the exact discrete time model of a closed loop system implies uniform iss of the sampled data nonlinear system our results serve to quantify these facts by means of comparison functions our results can be used as an alternative to prove and extend results in 1 or extend some results in 4 to a class of nonlinear systems finally the formulas we establish can be used as a tool for some other problems which we indicate

mpc for sampled data linear systems guaranteeing constraint satisfaction in continuous time model predictive controllers mpc designed for sampled data systems can be shown to violate the constraints in continuous time a reformulation of the initial problem will guarantee constraint satisfaction throughout the intersample period polytopic inclusions of the continuous trajectory are used in this technical note to establish additional constraints leading to a linearly constrained quadratic optimization problem continuous time asymptotic stability and continuous time positive invariance are proven for the reformulated problem

stabilization in a two species chemostat with monod growth functions we design feedback controllers for two species chemostats so that an equilibrium with arbitrary prescribed positive species concentrations becomes globally asymptotically stable we use a new global explicit strict lyapunov function construction which allows us to quantify the effects of disturbances using the input to state stability paradigm we assume that only a linear combination of the species concentrations is known we illustrate our approach using a numerical example

on automorphisms of the toeplitz algebra in this paper we completely determine those c automorphisms of the full toeplitz algebra which have the form inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i compact operator where is a bi lipschitz homeomorphism of the unit circle

on the essential commutant of let t qc resp t be the c algebra generated by the toeplitz operators t qc resp t l on the hardy space h 2 of the unit circle a well known theorem of davidson asserts that t qc is the essential commutant of t we show that the essential commutant of t qc is strictly larger than t thus the image of t in the calkin algebra does not satisfy the double commutant relation we also give a criterion for membership in the essential commutant of t qc

error analysis for a sinh transformation used in evaluating nearly singular boundary element integrals in the two dimensional boundary element method one often needs to evaluate numerically integrals of the form 1 1g x j x f x a 2 b 2 dx where j 2 is a quadratic g is a polynomial and f is a rational logarithmic or algebraic function with a singularity at zero the constants a and b are such that 1

estimating quadrature errors for an efficient method for quasi singular boundary integral abstract the numerical resolution of the boundary integral equations applied to the differential equations of laplace helmholtz and maxwell requires the handling of quasi singular integrals with different order of singularity the numerical approximation of the integral equations of different kinds is made by boundary finite elements in this paper we present a complete survey for estimating quadrature errors for the numerical techniques proposed by huang and cruse q huang t a cruse some notes on singular integral techniques in boundary element analysis int j numer methods eng 36 15 1993 2643 2659 to calculate the quasi singular integrals to validate the accuracy and efficiency of these techniques and approve our study some numerical examples are presented and discussed

system theoretic properties of sampled data representations of nonlinear systems obtained via taylor lie series it is possible to obtain a sampled data representation of a nonlinear dynamic system under zero order hold using taylor lie series within the context of taylor lie series theory explicit formulae for the resulting discrete time system are derived in terms of the system lie derivatives the effect of sampling on systemtheoretic properties such as equilibrium properties relative order stability zero dynamics and minimum phase characteristics is examined revealing the natural and transparent way in which the use of taylor lie series permeates the relevant theoretical aspects the results derived are also illustrated in two chemical engineering examples

a lyapunov proof of an improved maximum allowable transfer interval for networked control systems simple lyapunov proofs are given for an improved relative to previous results that have appeared in the literature bound on the maximum allowable transfer interval to guarantee global asymptotic or exponential stability in networked control systems and also for semiglobal practical asymptotic stability with respect to the length of the maximum allowable transfer interval

robustness of adaptive control under time delays for three dimensional curve tracking we analyze the robustness of a class of controllers that enable three dimensional curve tracking by a free moving particle the free particle tracks the closest point on the curve by building a strict lyapunov function and robustly forward invariant sets we show input to state stability under predictable tolerance and safety bounds that guarantee robustness under control uncertainty input delays and a class of polygonal state constraints including adaptive tracking and parameter identification under unknown control gains such an understanding may provide certified performance when the control laws are applied to real life systems

the iterated sinh transformation a sinh transformation has recently been proposed to improve the numerical accuracy of evaluating nearly singular integrals using gauss legendre quadrature it was shown that the transformation could improve the accuracy of evaluating such integrals which arise in the boundary element method by several orders of magnitude here this transformation is extended in an iterative fashion to allow the accurate evaluation of similar types of integrals that have more spiked integrands results show that one iteration of this sinh transformation is preferred for nearly weakly singular integrals whereas two iterations lead to several orders of magnitude improvement in the evaluation of nearly strongly singular integrals the same observation applies when considering integrals of derivatives of the two dimensional boundary element kernel however for these integrals more iterations are required as the distance from the source point to the boundary element decreases

a general algorithm for the numerical evaluation of nearly singular integrals on 3d boundary element a general numerical method is proposed to compute nearly singular integrals arising in the boundary integral equations bies the method provides a new implementation of the conventional distance transformation technique to make the result stable and accurate no matter where the projection point is located the distance functions are redefined in two local coordinate systems a new system denoted as a b is introduced here firstly its implementation is simpler than that of the polar system and it also performs efficiently then a new distance transformation is developed to remove or weaken the near singularities to perform integration on irregular elements an adaptive integration scheme is applied numerical examples are presented for both planar and curved surface elements the results demonstrate that our method can provide accurate results even when the source point is very close to the integration element and can keep reasonable accuracy on very irregular elements furthermore the accuracy of our method is much less sensitive to the position of the projection point than the conventional method

an extended sigmoidal transformation technique for evaluating weakly singular integrals without splitting the integration interval in this paper we present an efficient transformation technique for accurate numerical evaluation of weakly singular integrals with interior singularities the present transformation technique does not require any division of the integration interval it is composed of two parts of a sigmoidal transformation whose tails coincide with a singular point smoothly up to the order of the sigmoidal transformation employed therefore in using the standard gauss quadrature rule the present transformation has a feature in which the integration points are clustered into the interior singular point very closely as in the case of endpoint singular integrals the results of some numerical examples show the superiority of the present method over the existing methods

stability of a certain class of hybrid dynamical systems in this work we introduce a type of hybrid equation closely related to those that appear when a digital controller or a hybrid controller of a certain class is interconnected with a full non linear continuous time system for these equations we obtain results concerning existence uniqueness and noncontinuability of solutions this results are used to study the behaviour of the hybrid dynamical systems associated with those equations we establish relationships between the stability of these hybrid dynamical systems and the ones corresponding to their associated discretized systems on one hand and to their associated linearized systems on the other we also characterize their stability uniform asymptotic stability and exponential stability these results are applied to the exponential stabilization of a non linear control system by digital or hybrid controllers

a new sigmoidal transformation for weakly singular integrals in the boundary element method the power of the sigmoidal transformation in weakly singular integrals has been demonstrated by the recent works a sidi in numerical integration iv h brass and g hammerlin eds birkhauser verlag berlin 1993 pp 359 373 p r johnston internat j numer methods engrg 45 1999 pp 1333 1348 p r johnston internat j numer methods engrg 47 2000 pp 1709 1730 d elliott math methods appl sci 20 1997 pp 121 132 d elliott j austral math soc ser b 40 1998 pp e77 e137 especially application of this transformation is useful for efficient numerical evaluation of the singular integrals appearing in the usual boundary element method in this paper a new sigmoidal transformation containing a parameter b is presented it is shown that the present transformation with the gauss legendre quadrature rule can improve the asymptotic truncation error of the traditional sigmoidal transformations by controlling the parameter for some examples we compare the numerical results of the present method with those of the well known sidi and elliott transformations to show the superiority of the former

on finite gain lp stability of nonlinear sampled data systems abstract it is shown that uniform global exponential stability of the input free discrete time model of a globally lipschitz sampled data time varying nonlinear system with inputs implies finite gain l p stability of the sampled data system for all p 1 this result generalizes results on l p stability of sampled data linear systems and it is an important tool for analysis of robustness of sampled data nonlinear systems with inputs

a bi cubic transformation for the numerical evaluation of the cauchy principal value integrals in boundary methods the numerical strategies employed in the evaluation of singular integrals existing in the cauchy principal value cpv sense are undoubtedly one of the key aspects which remarkably affect the performance and accuracy of the boundary element method bem thus a new procedure based upon a bi cubic co ordinate transformation and oriented towards the numerical evaluation of both the cpv integrals and some others which contain different types of singularity is developed both the ideas and some details involved in the proposed formulae are presented obtaining rather simple and attractive expressions for the numerical quadrature which are also easily embodied into existing bem codes some illustrative examples which assess the stability and accuracy of the new formulae are included