generating positive and stable solutions through delayed state feedback for time varying forward complete nonlinear systems with delay in the input a new reduction model approach is proposed it presents three advantages first the corresponding control laws do not include distributed terms second it yields closed loop systems with positive solutions that can be easily derived finally the stabilized systems possess some robustness properties that can be estimated

variational iteration method for autonomous ordinary differential systems in this paper a new iteration technique is proposed to solve autonomous ordinary differential systems in this method general lagrange multipliers are introduced to construct correction functionals for the systems the multipliers in the functionals can be identified by the variational theory the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary initial conditions some examples are given the results reveal that the method is very effective and convenient

on the convergence of he s variational iteration method in this work we will consider he s variational iteration method for solving second order initial value problems we will discuss the use of this approach for solving several important partial differential equations this method is based on the use of lagrange multipliers for identification of optimal value of a parameter in a functional this procedure is a powerful tool for solving the large amount of problems using the variational iteration method it is possible to find the exact solution or an approximate solution of the problem this technique provides a sequence of functions which converges to the exact solution of the problem our emphasis will be on the convergence of the variational iteration method in the current paper this scheme will be investigated in details and efficiency of the approach will be shown by applying the procedure on several interesting and important models

quasi norm a priori and a posteriori error estimates for the nonconforming approximation of p laplacian in this paper we derive quasi norm a priori and a posteriori error estimates for the crouzeix raviart type finite element approximation of the p laplacian sharper a priori upper error bounds are obtained for instance for sufficiently regular solutions we prove optimal a priori error bounds on the discretization error in an energy norm when 1 p leq 2 we also show that the new a posteriori error estimates provide improved upper and lower bounds on the discretization error for sufficiently regular solutions the a posteriori error estimates are further shown to be equivalent on the discretization error in a quasi norm

the adomian decomposition method for computing eigenelements of sturm liouville two point boundary value problems we will consider the adomian decomposition method for computing the eigenelements of sturm liouville two point boundary value problem the method proved to be very successful and powerful in computing such elements numerical examples showed the competitive nature of the method comparison with the results of others will also be presented

convergence of finite element approximations to state constrained convex parabolic boundary control problems this paper is concerned with the numerical solution of state constrained parabolic boundary control problems by finite element approximations error estimates for the optimal values and in the coercive case for the optimal solutions are derived these estimates are used to prove convergence results under rather weak assumptions in the noncoercive case a bang bang principle is shown to obtain convergence results for the discrete controls also a discussion of several numerical examples concludes the paper

variational iteration method new development and applications variational iteration method has been favourably applied to various kinds of nonlinear problems the main property of the method is in its flexibility and ability to solve nonlinear equations accurately and conveniently in this paper recent trends and developments in the use of the method are reviewed major applications to nonlinear wave equation nonlinear fractional differential equations nonlinear oscillations and nonlinear problems arising in various engineering applications are surveyed the confluence of modern mathematics and symbol computation has posed a challenge to developing technologies capable of handling strongly nonlinear equations which cannot be successfully dealt with by classical methods variational iteration method is uniquely qualified to address this challenge the flexibility and adaptation provided by the method have made the method a strong candidate for approximate analytical solutions this paper outlines the basic conceptual framework of variational iteration technique with application to nonlinear problems both achievements and limitations are discussed with direct reference to approximate solutions for nonlinear equations a new iteration formulation is suggested to overcome the shortcoming a very useful formulation for determining approximately the period of a nonlinear oscillator is suggested examples are given to illustrate the solution procedure

reduction model approach for linear time varying systems with delays we study stabilization problems for time varying linear systems with constant input delays our reduction method ensures input to state stability with respect to additive uncertainties under arbitrarily long delays it applies to rapidly time varying systems and gives a lower bound on the admissible rapidness parameters we also cover slowly time varying systems including upper bounds on the allowable slowness parameters we illustrate our work using a pendulum model

analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls an optimal control problem for impressed cathodic systems in electrochemistry is studied the control in this problem is the current density on the anode a matching objective functional is considered we first demonstrate the existence and uniqueness of solutions for the governing partial differential equation with a nonlinear boundary condition we then prove the existence of an optimal solution next we derive a necessary condition of optimality and establish an optimality system of equations finally we define a finite element algorithm and derive optimal error estimates

ritz galerkin approximation of the time optimal boundary control problem for parabolic systems with dirichlet boundary conditions this paper deals with ritz galerkin approximations of the following two problems i boundary value problems with l 2 boundary data given in the form of dirichlet boundary conditions ii time optimal control problems for parabolic systems with control acting on the boundary in both cases optimal rates of convergence of approximation are obtained also the specialization is considered to approximating with nonconformal elements which are not required to satisfy boundary conditions corresponding rates of convergence are also proved

predictor based sampled data exponential stabilization through continuous discrete observers the problem of stabilizing a linear continuous time system with discrete time measurements and a sampled input with a pointwise constant delay is considered in a first part we design a continuous discrete observer which converges when the maximum time interval between two consecutive measurements is sufficiently small in a second part we construct a dynamic output feedback by using a technique which is strongly reminiscent of the reduction model approach it stabilizes the system when the maximal time between two consecutive sampling instants is sufficiently small no limitation on the size of the delay is imposed and an iss property with respect to additive disturbances is established

variational iteration method for solving the epidemic model and the prey and predator problem in this article two problems are considered the first one is the problem of the spread of a non fatal disease in a population which is assumed to have constant size over the period of the epidemic and the other one is the problem of prey and predator the variational iteration method is employed to compute an approximation to the solutions of the systems of nonlinear ordinary differential equations governing on these problems the results are compared with the results obtained by adomian decomposition and power series methods some plots are presented to show the reliability and simplicity of the method

reduction model approach for linear time varying systems with input delays based on extensions of floquet theory we solve stabilization problems for linear time varying systems under input delays we show how changes of coordinates lead to systems with time invariant drifts which are covered by the reduction model method and which lead to the problem of stabilizing a time varying system without delay for continuous time periodic systems we can use floquet theory to find the changes of coordinates we also prove an analogue for discrete time systems through a discrete time extension of floquet theory

the variational iteration method for cauchy problems in this paper the variational iteration method is used to solve the cauchy problem some examples are given to elucidate the solution procedure and reliability of the obtained results the variational iteration algorithm leads to exact solutions in the present study

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

stabilization and robustness analysis for time varying systems with time varying delays using a sequential subpredictors approach we provide a new sequential predictors approach for the exponential stabilization of linear time varying systems our method circumvents the problem of constructing and estimating distributed terms in the control laws and allows arbitrarily large input delay bounds pointwise time varying input delays and uncertainties instead of using distributed terms our approach to handling longer delays is to increase the number of predictors we obtain explicit formulas to find lower bounds for the number of required predictors the formulas involve bounds on the delays and on the derivatives of the delays we illustrate our method in three examples

a posteriori error estimates for optimal control problems governed by parabolic equations in this paper we derive a posteriori error estimates for the finite element approximation of quadratic optimal control problem governed by linear parabolic equation we obtain a posteriori error estimates for both the state and the control approximation such estimates which are apparently not available in the literature are an important step towards developing reliable adaptive finite element approximation schemes for the control problem

variational iteration method for solving nonlinear boundary value problems abstract in this paper he s variational iteration method is implemented to give approximate and analytical solutions for a class of boundary value problems the variational iteration method which produces the solutions in terms of convergent series requiring no linearization or small perturbation numerical examples are given and comparisons are made with the adomian decomposition method the fact that this method solves nonlinear equations without using adomian polynomials can be considered as an advantage of this method over adomian decomposition method

accurate numerical bounds for the spectral points of singular sturm liouville problems over 0 x the eigenvalues of singular sturm liouville problems defined over the semi infinite positive real axis are examined on a truncated interval 0 x l as functions of the boundary point l as a basic theoretical result it is shown that the eigenvalues of the truncated interval problems satisfying dirichlet and neumann boundary conditions provide respectively upper and lower bounds to the eigenvalues of the original problem moreover the unperturbed system in a perturbation problem where l remains sufficiently small admits analytical solutions in terms of the bessel functions of the first kind applications to the schrodinger equations of diatomic molecules and a harmonic oscillator confirm the practical implementation of this approach in calculating highly accurate numerical eigenvalue enclosures it is worth mentioning that this study is therefore a completion of the paper j comput appl math 115 2000 535 where similar problems on the whole real axis x were treated along the same lines

variational iteration method for sturm liouville differential equations in this article he s variational iteration method is applied to linear sturm liouville eigenvalue and boundary value problems including the harmonic oscillator in this method solutions of the problems are approximated by a set of functions that may include possible constants to be determined from the boundary conditions by computing variations the lagrange multipliers are derived and the generalised expressions of variational iterations are constructed numerical results show that the method is simple however powerful and effective

global asymptotic stabilization of feedforward systems with delay in the input the problem of globally uniformly asymptotically and locally exponentially stabilizing a family of nonlinear feedforward systems when there is a delay in the input is solved no limitation on the size of the delay is imposed explicit expressions of bounded control laws are determined

an efficient method to approximate eigenfunctions and high index eigenvalues of regular sturm liouville problems the computation of the eigenvalues of a sturm liouville problem is a difficult task when high index eigenvalues are computed in most previous methods it can be seen that the uncertainty of the results increases as the estimated eigenvalues grow larger this paper is to present some new methods in which not only the error of calculating the higher eigenvalues does not grow but it also vanishes as eigenvalues tend to infinity moreover the proposed method gives good estimates of eigenfunctions corresponding to high eigenvalues

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

some a posteriori error estimators for elliptic partial differential equations we present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations the estimators are based on solving local neumann problems in each element the estimators differ in how they enforce consistency of the neumann problems we prove that as the mesh size decreases under suitable assumptions two of the error estimators approach upper bounds on the norm of the true error and all three error estimators are within multiplicative constants of the norm of the true error we present numerical results in which one of the error estimators appears to converge to the norm of the true error

local stabilization of nonlinear systems through the reduction model approach we study a general class of nonlinear systems with input delays of arbitrary size we adapt the reduction model approach to prove local asymptotic stability of the closed loop input delayed systems using feedbacks that may be nonlinear our lyapunov krasovskii functionals make it possible to determine estimates of the basins of attraction for the closed loop systems

variational iteration method some recent results and new interpretations this paper is an elementary introduction to the concepts of variational iteration method first the main concepts in variational iteration method such as general lagrange multiplier restricted variation correction functional are explained heuristically subsequently the solution procedure is systematically addressed in particular for nonlinear oscillators particular attention is paid throughout the paper to give an intuitive grasp for the method the main motivation is to put things together in a convenient form for later reference and systematic use

linear systems with delayed controls a reduction linear systems with delayed control action are transformed into systems without delays under an absolute continuity condition the new system is an ordinary differential control equation in the general case the new system is a measure differential control system it is shown how the controllability stabilization and various optimization problems can be analyzed via the reduced systems

optimal control for variational inequalities consider the problem of maximizing a functional which depends on a control function k and on the solution of an elliptic variational inequality with k appearing in the data the variational problem for k is nondiflerentiable and nonconvex we obtain necessary conditions on a maximizes k 0 and then use them to determine the structure of k 0 in some cases

local a posteriori error estimates for convex boundary control problems in this paper we derive some improved a posteriori error estimates for finite element approximation of neumann boundary control problems we first establish local upper a posteriori error estimates for both the state and the control approximation of the general convex boundary control problems we then derive local upper and lower a posteriori error estimates for a class of control problems that frequently appear in applications

approximate analytical solution for seepage flow with fractional derivatives in porous media abstract in this paper a new and more exact model for seepage flow in porous media with fractional derivatives has been proposed which has modified the well known darcy law and overcome the continuity assumption of seepage flow a new kind of analytical method of nonlinear problem called the variational iteration method is described and used to give approximate solutions of the problem the results show that the proposed iteration method requiring no linearization or small perturbation is very effective and convenient

finite dimensional approximation of a class of constrained nonlinear optimal control problems an abstract framework for the analysis and approximation of a class of nonlinear optimal control and optimization problems is constructed nonlinearities occur in both the objective functional and the constraints the framework includes an abstract nonlinear optimization problem posed on infinite dimensional spaces an approximate problem posed on finite dimensional spaces together with a number of hypotheses concerning the two problems the framework is used to show that optimal solutions exist to show that lagrange multipliers may be used to enforce the constraints to derive an optimality system from which optimal states and controls may be deduced and to derive existence results and error estimates for solutions of the approximate problem the abstract framework and the results derived from that framework are then applied to three concrete control or optimization problems and their approximation by finite element methods the first involves the von k arm an plate equations of nonlinear elasticity the second the ginzburg landau equations of superconductivity and the third the navier stokes equations for incompressible viscous flows

a posteriori error estimates for nonlinear problems l r o t p ω error estimates for finite element discretizations of parabolic equations using the abstract framework of 9 we analyze a residual a posteriori error estimator for space time finite element discretizations of quasilinear parabolic pdes the estimator gives global upper and local lower bounds on the error of the numerical solution the finite element discretizations in particular cover the so called 0 scheme which includes the implicit and explicit euler methods and the crank nicholson scheme

approximate solution of nonlinear differential equations with convolution product nonlinearities in this paper a new iteration method is proposed to solve nonlinear problems special attention is paid to nonlinear differential equations with convolution product nonlinearities the results reveal the approximations obtained by the proposed method are uniformly valid for both small and large parameters in nonlinear problems furthermore the first order of approximations are more accurate than perturbation solutions at high order of approximation