the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

construction of lyapunov krasovskii functionals for switched nonlinear systems with input delay abstract this paper studies the stability issue for switched nonlinear systems with input delay and disturbance it is assumed that for the nominal system an exponential stabilizing controller is predesigned such that the switched system is stable under a certain switching signal and a piecewise lyapunov function for the corresponding closed loop system is known however in the presence of input delay and disturbance the system may be unstable under the same switching signal for this case a new lyapunov krasovskii functional is firstly constructed based on the known lyapunov function then by employing this new functional a new switching signal satisfying the new average dwell time conditions is constructed to guarantee the input to state stability of the system under a certain delay bound the bound on the average dwell time is closely related to the bound on the input delay finally numerical examples are given to illustrate the effectiveness of the proposed theory

on the deficiency indices and self adjointness of symmetric hamiltonian systems abstract the main purpose of this paper is to investigate the formal deficiency indices n of a symmetric first order system jf bf h f on an interval i where i r or i r here j b h are n n matrix valued functions and the hamiltonian h 0 may be singular even everywhere we obtain two results for such a system to have minimal numbers n 0 if i r resp n n if i r and a criterion for their maximality n 2n for i r as well as the quasi regularity this covers the kac krein and de branges trans amer math soc 99 1961 118 theorems on 2 2 canonical systems and some results from kogan and rofe beketov proc roy soc edinburgh sect a 74 1974 75 5 some conditions for a canonical system to have intermediate formal deficiency indices are presented too we also obtain a generalization of the well known titchmarsh sears theorem for second order sturm liouville type equations this contains results due to lidskii and krein as special cases we present two approaches to the above problems one dealing with formal deficiency indices and one dealing with ordinary deficiency indices our main non formal approach is based on the investigation of a symmetric linear relation smin which is naturally associated to a first order system this approach works in the framework of extension theory and therefore we investigate in detail the domain d s min of s min in particular we prove the so called regularity theorem for d s min as a byproduct of the regularity result we obtain very short proofs of generalizations of the main results of the paper by kogan and rofe beketov 1974 75

resolvent convergence and spectral approximations of sequences of self adjoint subspaces abstract this paper studies resolvent convergence and spectral approximations of sequences of self adjoint subspaces relations in complex hilbert spaces concepts of strong resolvent convergence norm resolvent convergence spectral inclusion and spectral exactness are introduced fundamental properties of resolvents of subspaces are studied by applying these properties several equivalent and sufficient conditions for convergence of sequences of self adjoint subspaces in the strong and norm resolvent senses are given it is shown that a sequence of self adjoint subspaces is spectrally inclusive under the strong resolvent convergence and spectrally exact under the norm resolvent convergence a sufficient condition is given for spectral exactness of a sequence of self adjoint subspaces in an open interval lacking essential spectral points in addition criteria are established for spectral inclusion and spectral exactness of a sequence of self adjoint subspaces that are defined on proper closed subspaces

robust adaptive output feedback tracking for a class of nonlinear time delayed plants within the model reference adaptive control mrac framework a continuous adaptive output feedback control scheme is developed for a class of nonlinear siso dynamic systems with time delays which is robust with respect to unknown time varying plant delays and to an external disturbance with unknown bounds a special form of the lyapunov krasovskii functional with a virtual adaptation gain vector is introduced to prove stability

exponential synchronization of a class of neural networks with sampled data control abstract this paper investigates the problem of the master slave synchronization for a class of neural networks with discrete and distributed delays under sampled data control by introducing some new terms a novel piecewise time dependent lyapunov krasovskii functional lkf is constructed to fully capture the available characteristics of real sampling information and nonlinear function vector of the system based on the lkf and wirtinger based inequality less conservative synchronization criteria are obtained to guarantee the exponential stability of the error system and then the slave system is synchronized with the master system the designed sampled data controller can be obtained by solving a set of linear matrix inequalities lmis which depend on the maximum sampling period and the decay rate the criteria are less conservative than the ones obtained in the existing works a numerical example is presented to illustrate the effectiveness and merits of the proposed method

backward differentiation type formulas for volterra integral equations of the second kind numerical integration formulas are discussed which are obtained by differentiation of the volterra integral equation and by applying backward differentiation formulas to the resulting integro differential equation in particular the stability of the method is investigated for a class of convolution kernels the accuracy and stability behaviour of the method proposed in this paper is compared with that of i a block implicit runge kutta scheme and ii the scheme obtained by applying directly a quadrature rule which is reducible to the backward differentiation formulas the present method is particularly advantageous in the case of stiff volterra integral equations

convergence of regular approximations to the spectra of singular fourth order sturm liouville problems abstractwe prove some new results which justify the use of interval truncation as a means of regu larising a singular fourth order sturm liouville problem near a singular endpoint of particularinterest are the results in the so called lim 3 case which has no analogue in second order singularproblems 1 introduction in 1978 bailey gordon and shampine 4 released a code sleign for computing eigenvalues ofsturm liouville problems this code was remarkable both for its reliability and for the fact that itwas able to handle singular as well as regular problems with a minimum of user input sleign sstrategy for dealing with problems having singular endpoints was to truncate the interval near asingular endpoint thereby regularising the problem at the time no rigorous proofs were given forthe universal validity of this strategy recently however a new code sleign2 has appeared capable of dealing with much more general singular second order problems in particular sleign2can deal with non friedrichs boundary conditions near a lim 2 singular endpoint and can dealwith cases where there is an in nite sequence of eigenvalues tending to the development ofsleign2 followed the work of bailey everitt weidmann and zettl 2 in proving rigorously thetypes of spectral convergence which could be expected from the interval truncation process nearsingular endpoints in 1994 greenberg and marletta produced a code sleuth for solving fourth order regularsturm liouville problems 12 this was the culmination of three years of work on higher orderself adjoint odes 10 11 and numerical solution of eigenproblems for hamiltonian systems 14 the new code gave su cient increases in speed over the hamiltonian systems code 14 for it to befeasible to solve singular fourth order problems such problems were treated on an heuristic basis in 12 the purpose of the present work is to prove results similar to those produced by bailey everitt weidmann and zettl 2 but in the context of fourth order problems for lim 2 and lim 4 singularendpoints we use methods which are direct adaptations to the fourth order case of the methodsof 2 however for the lim 4 case a new complication is the existence of complex as well as realboundary conditions for the lim 3 case new di culties are present as this case does not arise forsecond order problems and cannot be treated by any of the methods described in 2 we overcomethese di culties by using the oscillation theory described in 11

smooth dynamic output feedback control for multiple time delay systems with nonlinear uncertainties the dynamic output feedback control problem is considered for a class of systems with multiple time delays and nonlinear uncertainties based on the control input matrix and output matrix we decompose the system into two subsystems the dynamic compensator is designed for the first subsystem and then the output feedback controller is constructed based on the compensator and the second subsystem by using the introduced new lyapunov krasovskii functional we show that the solution of the resultant closed loop system converges exponentially to an adjustable bounded region compared with the previous works the developed controller in this paper is memoryless and smooth which only uses the system output the control design conditions are relaxed because of the developed dynamic compensator the result is further extended to the general nonlinear case the corresponding dynamic nonlinear output feedback control method is proposed finally simulations are performed to show the potential of the proposed methods

output feedback model reference adaptive control for multi input multi output plants with state delay two new output feedback adaptive control schemes based on model reference adaptive control mrac and adaptive laws for updating the controller parameters are developed for a class of linear multi input multi output mimo systems with state delay an effective controller structure established on a new error equation parametrization is proposed to achieve tracking with the error tending to zero asymptotically to achieve exact asymptotical tracking we introduce in the standard mrac structure for plants without delay a new additional adaptive feedforward control component as an output of a dynamical system driven by the reference signal adaptive laws are developed using the spr lyapunov design approach and two assumptions regarding the prior knowledge of the high frequency matrix kp this work is the first asymptotic exact zero tracking results for this class of systems in the framework of the certainty equivalence approach 2005 elsevier b v all rights reserved

self adjoint extensions for second order symmetric linear difference equations abstract in this paper self adjoint extensions for second order symmetric linear difference equations with real coefficients are studied by applying the glazman krein naimark theory for hermitian subspaces both self adjoint subspace extensions and self adjoint operator extensions of the corresponding minimal subspaces are completely characterized in terms of boundary conditions where the two endpoints may be regular or singular

convergence properties of the runge kutta chebyshev method the runge kutta chebyshev method is ans stage runge kutta method designed for the explicit integration of stiff systems of ordinary differential equations originating from spatial discretization of parabolic partial differential equations method of lines the method possesses an extended real stability interval with a length proportional tos 2 the method can be applied withs arbitrarily large which is an attractive feature due to the proportionality of withs 2 the involved stability property here is internal stability internal stability has to do with the propagation of errors over the stages within one single integration step this internal stability property plays an important role in our examination of full convergence properties of a class of 1st and 2nd order schemes full convergence means convergence of the fully discrete solution to the solution of the partial differential equation upon simultaneous space time grid refinement for a model class of linear problems we prove convergence under the sole condition that the necessary time step restriction for stability is satisfied these error bounds are valid for anys and independent of the stiffness of the problem numerical examples are given to illustrate the theoretical results

runge kutta theory for volterra integral equations of the second kind the present paper develops the theory of general runge kutta methods for volterra integral equations of the second kind the order conditions are derived by using the theory of p series which for our problem reduces to the theory of v series these results are then applied to two special classes of runge kutta methods introduced by pouzet and by bel tyukov

pouzet runge kutta chebyshev method for volterra integral equations of the second kind we construct a stabilized runge kutta method of pouzet type pouzet runge kutta chebyshev method prkc for nonlinear stiff volterra integral equations of the second kind the prkc method is an explicit second order scheme which possesses extended stability region due to the first kind chebyshev polynomial used to establish the stability function

memoryless static output feedback sliding mode control for nonlinear systems with delayed disturbances in this technical note a local stabilization problem for nonlinear systems with disturbances which are dependent on time varying delay phenomena is studied an appropriate transformation is introduced to express the system in regular form which facilitates both design and analysis then sufficient conditions are developed based on a lyapunov razumikhin approach such that the sliding motion is uniformly asymptotically or exponentially stable a static output feedback control law independent of the time delay is synthesized to guarantee reachability of the sliding mode the accessible bounds on the disturbances are fully used in the control design to reduce the conservatism a simulation example shows the effectiveness of the approach

regular approximations of spectra of singular discrete linear hamiltonian systems with one singular endpoint abstract this paper is concerned with regular approximations of spectra of singular discrete linear hamiltonian systems with one singular endpoint for any given self adjoint subspace extension sse of the corresponding minimal subspace its spectrum can be approximated by eigenvalues of a sequence of induced regular sses generated by the same difference expression on smaller finite intervals it is shown that every sse of the minimal subspace has a pure discrete spectrum and the k th eigenvalue of any given sse is exactly the limit of the k th eigenvalues of the induced regular sses that is spectral exactness holds in the limit circle case furthermore error estimates for the approximations of eigenvalues are given in this case in addition in the limit point and intermediate cases spectral inclusive holds

brief paper stabilization for state input delay systems via static and integral output feedback this paper addresses the static and integral output feedback stabilization problems of continuous time linear systems with an unknown state input delay by combining an augmentation approach and the delay partitioning technique criteria for static and integral output feedback stabilizability are proposed in terms of nonlinear matrix inequalities with a free parameter matrix introduced these new characterizations possess a special structure which leads to linearized iterative computation the effectiveness and merits of the proposed approach are shown through numerical examples

computation of the m matrix for fourth order problems this paper is concerned with the numerical computation of the titchmarsh weyl m matrix we show how an algorithm may be developed which relates solutions of an initial value problem to an approximation of the m matrix in some special cases we compare the results from our algorithm with results obtained using special functions

memoryless state feedback controller design for time delay systems with matched uncertain nonlinearities the robust control problem for a class of uncertain time delay systems is investigated the systems include multiple time delays and uncertain nonlinear functions to our best knowledge no controller design method for the systems is reported in the literature except for sliding mode control smc however the sliding mode controller must use the delayed state of the system so that the precise time delays have to be known and also a large memory is needed for controller implementation to overcome the shortcomings we employ the backstepping method to design a memoryless state feedback controller by constructing a novel lyapunov krasovskii functional we show that the resulting closed loop system is asymptotically stable simulations are done to verify the effectiveness of the proposed method

defect indices and definiteness conditions for a class of discrete linear hamiltonian systems abstract this paper is concerned with a class of discrete linear hamiltonian systems in finite or infinite intervals a definiteness condition and its equivalent statements are discussed and three sufficient conditions for the definiteness condition are given a precise relationship between the defect index of the minimal subspace generated by the system and the number of linearly independent square summable solutions of the system is established in particular they are equal if and only if the definiteness condition is satisfied finally two criteria for the limit point case and one criterion for the limit circle case are obtained

a return to the hardy littlewood integral inequality this paper is concerned with the extension and generalization of a classic integral inequality first considered by hardy and littlewood in 1932 the integral inequality is big int b a p f 2 q f 2 big 2 leqslant k int b a w f 2 int b a w w 1 pf qf 2 f in delta where k is a positive number the coefficients p q and w are real valued and w 0 on the interval a b infty a b leqslant infty and p 1 q w in l loc a b delta is the maximal linear manifold of functions f for which the two integrals on the right hand side are both finite the hardy littlewood inequality is the special case a 0 b infty p w 1 and q 0 on 0 infty here k 4 and all the cases of equality can be identified in this paper the results of everitt in 1971 connecting the above inequality with the titchmarsh weyl m coefficient of the associated differential equation py qy lambda wy on a b are restated in an improved form and are shown to hold under the above minimal conditions on the coefficients p q and w it is proved that the best possible value for the number k satisfies 1 k leqslant infty and that within these bounds all values of k can be realized in the course of establishing these results new information of independent interest is obtained about the behaviour of the m coefficient near the real axis examples are given to exhibit the wide range of possibilities for the integral inequality recent results on the asymptotic form of the m coefficient and on the transformation theory of the associated differential equation are shown to have applications for the general properties of the inequality all recent contributions to this general integral inequality are listed in the references and their content is surveyed during the course of the paper

monotone convergence theorems for semi bounded operators and forms with applications let hn be a monotone sequence of non negative self adjoint operators or relations in a hilbert space then there exists a self adjoint relation h such that hn converges to h in the strong resolvent sense this result and related limit results are explored in detail and new simple proofs are presented the corresponding statements for monotone sequences of semi bounded closed forms are established as immediate consequences applications and examples illustrating the general results include sequences of multiplication operators sturm liouville operators with increasing

regular approximation of linear hamiltonian operators with two singular endpoints abstract in this paper we investigate regular approximation of linear hamiltonian operators with two singular endpoints for any given singular self adjoint hamiltonian operator its spectrum can be approximated with the eigenvalues of a sequence of regular operators the inherited restriction operators are constructed by the limit circle solutions and the limit point solutions and proved to be self adjoint via the abstract operator theory we obtain the strongly resolvent convergence of the inherited regular operators and spectral inclusion of spectrum moreover for the case with both endpoints being limit circle spectral exactness of spectrum is obtained by hilbert schmidt norm convergence in the case of friedrichs extension of singular hamiltonian systems spectral exactness also holds due to variational principle of eigenvalues

rkc time stepping for advection diffusion reaction problems the original explicit runge kutta chebyshev rkc method is a stabilized second order integration method for pure diffusion problems recently it has been extended in an implicit explicit manner to also incorporate highly stiff reaction terms this implicit explicit rkc method thus treats diffusion terms explicitly and the highly stiff reaction terms implicitly the current paper deals with the incorporation of advection terms for the explicit method thus aiming at the implicit explicit rkc integration of advection diffusion reaction equations in a manner that advection and diffusion terms are treated simultaneously and explicitly and the highly stiff reaction terms implicitly

output feedback stabilization for a discrete time system with a time varying delay this study employs the free weighting matrix approach to investigate the output feedback control of a linear discrete time system with an interval time varying delay first the delay dependent stability is analyzed using a new method of estimating the upper bound on the difference of a lyapunov function without ignoring any terms and based on the results a design criterion for a static output feedback sof controller is derived since the conditions thus obtained for the existence of admissible controllers are not expressed strictly in terms of linear matrix inequalities a modified cone complementarity linearization algorithm is employed to solve the nonconvex feasibility sof control problem furthermore the problem of designing a dynamic output feedback controller is formulated as one of designing an sof controller numerical examples demonstrate the effectiveness of the method and its advantage over existing methods

robust output feedback model reference adaptive control of siso plants with multiple uncertain time varying state delays within the model reference adaptive control framework a simple adaptive output feedback control scheme is developed for a class of linear single input single output dynamic systems with multiple state delays which is robust with respect to multiple unknown time varying plant delays and to an external disturbance with unknown bounds a special form of the lyapunov krasovskii functional with ldquovirtualrdquo adaptation gain is introduced to prove stability

spectral properties of self adjoint subspaces abstract this paper focuses on spectral properties of self adjoint subspaces in the product space of a hilbert space several classifications of the spectrum of a self adjoint subspace are introduced including discrete spectrum essential spectrum continuous spectrum singular continuous spectrum absolutely continuous spectrum and singular spectrum their relationships between the subspace and its operator part are established respectively it is shown that all self adjoint subspace extensions of a closed hermitian subspace have the same essential spectrum as a simple application it is shown that every self adjoint subspace extension of the corresponding minimal subspace to a singular second order symmetric linear difference equation has a bounded spectrum under some conditions this shows that there is an essential difference between spectral properties of singular symmetric difference operators and classical regular or singular symmetric differential operators

adaptive robust tracking and model following of uncertain dynamical systems with multiple time delays the problem of robust tracking and model following is considered for a class of linear systems with multiple delayed state perturbations time varying uncertain parameters and disturbance in this note it is assumed that the upper bounds of the delayed state perturbations uncertainties and external disturbances are unknown an improved adaptation law with spl sigma modification is proposed to estimate such unknown bounds and on the basis of the updated values of these unknown bounds a class of continuous memoryless state feedback controllers is constructed for robust tracking of dynamical signals the proposed adaptive robust tracking controller can guarantee that the tracking error decreases asymptotically to zero in the presence of multiple delayed state perturbations time varying uncertain parameters and disturbance in addition it is also shown that this improved adaptation law with spl sigma modification can be applied to the general adaptive control problems to obtain some more exact control results finally a numerical example is given to demonstrate the validity of the results

regular approximations of spectra of singular second order symmetric linear difference equations abstract this paper is concerned with regular approximations of spectra of singular second order symmetric linear difference equations for any given self adjoint subspace extension of the corresponding minimal subspace its spectrum can be approximated by eigenvalues of a sequence of regular self adjoint subspace extensions generated by the same difference expression on smaller finite intervals

a general version of the hardy littlewood polya everitt help inequality the inequality 0 1 below is naturally associated with the equation pu qu u by assuming that one end point of the interval a b is regular and the other limit point for this equation everitt characterized the best constant k in tems of spectral properties of the equation this paper sketches a theory for more general inequalities 0 2 0 3 similarly related to the equation su tu here s and t are ordinary symmetric differential expressions a characterization of the best constants in 0 2 0 3 is given which generalises that of everitt for the case when s is of order 1 and t is multiplication by a positive function all possible inequalities are given together with the best constants and cases of equality furthermore an example is given of a valid inequality 0 1 on an interval with both end points regular for the corresponding differential equation this contradicts a conjecture by everitt and evans finally the general theory for the left definite inequality 0 3 is specialised to the case when s is a sturm liouville expression a family of examples is given for which the best constants can be explicitly calculated

a numerical method for the determination of the titchmarsh weyl m coefficient a numerical method for determining the titchmarsh weyl m function for the singular eigenvalue equation py9 9 qy wy on a where a is finite is presented the algorithm based on weyl s theory utilizes a result first used by atkinson to map a point on the real line onto the weyl circle in the complex plane in the limit point case these circles nest and tend to the limit point m using weyl s result for the diameter of the circles error estimates for m are obtained in 1971 w n everitt obtained an extension of an integral inequality of hardy littlewood namely the help inequality he showed that the existence of that inequality is determined by the properties of the null set of im 2 m in view of the major difficulties in analysing m even in the rare cases when it is given explicitly very few examples of the help inequality are known the computational techniques discussed in this paper have been applied to the problem of finding best constants in these inequalities

numerical determination of the titchmarsh weyl m coefficient and its applications to help inequalities this paper is concerned with numerical methods for finding m lambda the titchmarsh weyl m coefficient for the singular eigenvalue equation y qy lambda y on 0 infty and the results are applied to the problem of finding best constants for everitt s extension to the hardy little wood polya help integral inequality

on a conjecture of bennewitz and the behaviour of the titchmarsh weyl matrix near a pole for any real limitn 2nthorder selfadjoint linear differential expression on 0 titchmarshweyl matrices m can be defined two matrices of particular interest are the matrices md and mn ass

robust sliding mode control for uncertain time delay systems an lmi approach this note is devoted to robust sliding mode control for time delay systems with mismatched parametric uncertainties a delay independent sufficient condition for the existence of linear sliding surfaces is given in terms of linear matrix inequalities based on which the corresponding reaching motion controller is also developed the results are illustrated by an example

output feedback control of large scale nonlinear time delay systems in lower triangular form this paper is concerned with the stabilization problem for a class of large scale nonlinear time delay systems in lower triangular form the uncertain nonlinearities are assumed to be bounded by continuous functions of the outputs or delayed outputs multiplied by unmeasured states or delayed states an observer based output feedback control scheme is proposed using the dynamic gain control design approach based on lyapunov stability theory global asymptotic stability of the closed loop control system is proved contrary to many existing control designs for lower triangular nonlinear systems the celebrated backstepping method is not utilized here an example is finally given to demonstrate the effectiveness of the proposed design procedure