discrete time orthogonal spline collocation methods for schrodinger equations in two space variables two discrete time orthogonal spline collocation schemes are formulated and analyzed for solving the linear time dependent schrodinger equation in two space variables these are crank nicolson and alternating direction implicit adi schemes employing c1 piecewise polynomial spaces of arbitrary degree ge 3 in each space variable the stability of the schemes is examined and optimal order a priori l2 and h1 error estimates at each time step are derived parallel implementation of the adi scheme is discussed

structural modification part 2 assignment of natural frequencies and antiresonances by an added beam in this paper the inverse structural modification problem is solved in order to determine the dimensions of the cross section of a beam that when added to an original structure will assign natural frequencies or antiresonances as specified in order for this to be accomplished rotational receptances must be measured as presented in the companion paper when the modification is cast as an additional forcing term on the original structure a system of multivariate polynomials in the parameters of the beam cross section are revealed the solution of this system yields the beam parameters that assign the specified vibration behaviour

an orthogonal spline collocation alternating direction implicit crank nicolson method for linear parabolic problems on rectangles we formulate an alternating direction implicit crank nicolson scheme for solving a general linear variable coefficient parabolic problem in nondivergence form on a rectangle with the solution subject to nonhomogeneous dirichlet boundary condition orthogonal spline collocation with piecewise hermite bicubics is used for spatial discretization we show that for sufficiently small time stepsize the scheme is stable and of optimal order accuracy in time and the h1 norm in space we also describe an efficient implementation of the scheme and present numerical results demonstrating the accuracy and convergence rates in various norms

incomplete iterations in multistep backward difference methods for parabolic problems with smooth and nonsmooth data backward difference methods for the discretization of parabolic boundary value problems are considered in this paper in particular we analyze the case when the backward difference equations are only solved approximately by a preconditioned iteration we provide an analysis which shows that these methods remain stable and accurate if a suitable number of iterations often independent of the spatial discretization and time step size are used results are provided for the smooth as well as nonsmooth initial data cases finally the results of numerical experiments illustrating the algorithms performance on model problems are given

vibration nodes and the cancellation of poles and zeros by unit rank modifications to structures abstract the main focus of this paper is on pole zero cancellation in structures by unit rank modifications the modifications may be achieved by passive stiffnesses or by active control techniques a review of the classical theory on unit rank modification is included and new results are obtained to explain the existence of coincident zeros in point and cross receptances the necessary and sufficient conditions are established for the production of a vibration node by pole zero cancellation the classical theory is used to bring about the cancellation of poles and zeros and numerical examples are used to illustrate the application of the technique

robust output tracking for nonlinear uncertain systems abstract the problem of robust output tracking for nonlinear systems in the presence of uncertainties is further studied by extending the results of behtash 1990 to systems with mismatched uncertainties a new design method which is called the stable combined variable perturbation method scvpm is proposed and a robust controller is presented using high gain feedback our results are compatible with those of behtash 1990 in a sense that the new controller will degenerate to the one of behtash 1990 if the so called generalized matching condition is satisfied simulation results verified the effectiveness of the newly proposed method and controller

robust right coprime factorization and robust stabilization of nonlinear feedback control systems robust right coprime factorization and robust stabilization of nonlinear feedback control systems are studied the concept of robust right coprime factorization of nonlinear operators for feedback control systems is introduced some conditions for the robustness of a right coprime factorization of a nonlinear plant under unknown but bounded perturbations are derived an example with a closed form solution is included to illustrate the general theory and a step by step construction of the robust factorization and the robust stabilization

on the youla kucera parametrization for nonlinear systems abstract we consider the problem of stabilizing a nonlinear plant through the use of a possibly unstable pre compensator and a stable feedback compensator and parameterizing a class of such stabilizers in terms of a bibo stable map q in the linear case by means of superposition the pre and feedback compensator can be combined in such a way that the bibo map q occurs in just one feedback loop within the controller feeding back output residuals prediction errors to the control inputs the main contribution of this paper is to develop a nonlinear generalization of this property building on earlier work in forming the youla kucera parametrization for nonlinear systems we show the equivalence of the class of all bounded input stabilizing nonlinear pre and feedback compensators to a class of possibly unstable feedback controllers in which a map q s is present in only the one feedback loop we then show that necessary and sufficient condition to achieve stability of the system is that q s be bibo stable one advantage of the new formulation is that differential boundedness assumptions do not involve the parametrization q s in any way just as the linear versions of our results have applications in the areas of optimal control and adaptive control of linear systems it is conjectured that the present results will underlie more general results for adaptive control and nonlinear systems

a function space approach to sampled data control systems and tracking problems this paper presents a new framework for hybrid sampled data control systems instead of considering the state only at sampling instants this paper introduces a function piece during the sampling period as the state and gives an infinite dimensional model with such a state space this gives the advantage that sampled data systems with built in intersample behavior can be regarded as linear time invariant discrete time systems as a result the approach makes it possible to introduce such time invariant concepts as transfer functions poles and zeros to the sampled data systems even with the presence of the intersample behavior in particular tracking problems can be studied in this setting in a simple and unified way and ripples are completely characterized as a mismatch between the intersample reference signal and transmission zero directions this leads to the internal model principle for sampled data systems

fast direct solvers for piecewise hermite bicubic orthogonal spline collocation equations fast direct methods are proposed for the solution of linear systems arising when orthogonal spline collocation with piecewise hermite bicubics is employed for the approximate solution of poisson s

orthogonal spline collocation methods for the subdiffusion equation we develop two kinds of numerical schemes to efficiently solve the subdiffusion equation which is used to describe anomalous subdiffusive transport processes the time fractional derivative is first discretized by l1 approximation and the grunwald letnikov approximation respectively then we use the orthogonal spline collocation method to approximate the two semi discretized subdiffusion equations the stability and convergence of time semi discretization and full discretization schemes are both established strictly for the two schemes both of them are unconditionally stable numerically the convergent orders in space including the solution and its first derivative are four for the hermite cubic spline approximation and theoretically we get that at least the solution itself has a fourth order convergent rate extensive numerical results are presented to show the convergent order and robustness of the numerical schemes

operator based robust nonlinear control system design for mimo nonlinear plants with unknown coupling effects in this article operator based robust nonlinear control system design for multi input multi output mimo plants with unknown coupling effects is considered that is by using operator based robust nonlinear control design coupling effects existing in the mimo nonlinear plants can be decoupled based on a feedback design and robust right coprime factorisation approach the coupling effects caused by controllers and plant outputs can be stabilised by using definition of lipschitz operator and contraction mapping theorem and output tracking performance can be realised by a tracking design scheme finally a simulation example about temperature control process of 3 input 3 output aluminum plate is given to support the theoretical analysis

an adi extrapolated crank nicolson orthogonal spline collocation method for nonlinear reaction diffusion systems on evolving domains we consider the approximate solution of nonlinear reaction diffusion systems on evolving domains that arise in a variety of areas including biology chemistry ecology and physics by mapping a fixed domain onto the evolving domain at each time level we generalize to evolving domains the adi extrapolated crank nicolson orthogonal spline collocation technique developed in 8 9 for fixed domains the new method is tested on the schnakenberg model and we demonstrate numerically that it preserves the second order accuracy in time and optimal accuracy in space for piecewise hermite cubics in various norms moreover the efficacy of the method is demonstrated on several test problems from the literature which involve various types of domain evolution but for which exact solutions are not known

robust control for nonlinear systems using passivity based robust right coprime factorization the robust control for nonlinear feedback system with unknown perturbations is discussed based on the robust right coprime factorization and passivity property firstly based on passivity property the right factorization of the perturbed plant is realized secondly a robust control scheme is proposed considering the tracking property for the nonlinear system with nominal plant and perturbed plant finally we show the connection between the passivity and robust right coprime factorization for the obtained nonlinear system a numerical example is given to show the effectiveness of the proposed design scheme

an adi extrapolated crank nicolson orthogonal spline collocation method for nonlinear reaction diffusion systems an alternating direction implicit adi orthogonal spline collocation osc method is described for the approximate solution of a class of nonlinear reaction diffusion systems its efficacy is demonstrated on the solution of well known examples of such systems specifically the brusselator gray scott gierer meinhardt and schnakenberg models and comparisons are made with other numerical techniques considered in the literature the new adi method is based on an extrapolated crank nicolson osc method and is algebraically linear it is efficient requiring at each time level only o n operations where n is the number of unknowns moreover it is shown to produce approximations which are of optimal global accuracy in various norms and to possess superconvergence properties

fraction representations of non linear systems a simplified approach abstract a non linear system has a right fraction representation if it can be represented as pq 1 where p and q are stable systems and it has a left fraction representation if it can be represented as g 1 t where g and t are stable systems we develop here a theory of right and of left fraction representations for discrete time non linear systems with bounded input sequences we indicate the connection between fraction representations and the stabilization problem for non linear systems

convergence analysis of orthogonal spline collocation for elliptic boundary value problems existence uniqueness and optimal order h2 h1 and l2 error bounds are established for the orthogonal spline collocation solution of a dirichlet boundary value problem on the unit square the linear elliptic nonself adjoint partial differential equation is given in nondivergence form the approximate solution which is a tensor product of continuously differentiable piecewise polynomials of degree r geq 3 is determined by satisfying the partial differential equation at the nodes of a composite gauss quadrature

an alternating direction implicit backward differentiation orthogonal spline collocation method for linear variable coefficient parabolic equations we present a new fully discrete method to solve a linear variable coefficient parabolic equation on a rectangular region the scheme is an alternating direction implicit method which uses the third order backward differentiation formula for the time discretization and piecewise hermite bicubic orthogonal spline collocation for the spatial discretization the l 2 norm stability and convergence analysis is carried out for the heat equation the stability analysis reveals that the scheme is stable with respect to the right hand sides but it is unstable with respect to the initial conditions we explain using a damping property of the scheme and round off error analysis why this instability is harmless we also show how a different choice of initial values leads to optimal convergence orders numerical results demonstrate the third order accuracy in time and optimal order accuracy in space using maximum and sobolev norms respectively

stabilization of nonlinear systems based on a generalized be zout identity the relationship between the bezout and input output approaches to right coprimeness is investigated we introduce a generalized bezout identity and analyze its relationship to right coprimeness in terms of the generalized bezout identity a right factorization of a nonlinear plant is coprime according to the input output definition if and only if it satisfies the generalized bezout identity therefore the stability of an observer controller configuration is a necessary and sufficient condition for a nonlinear plant to have a right coprime factorization finally we present generalized bezout maps for some nonlinear plants with a state space representation

operator based nonlinear feedback control design using robust right coprime factorization in this note robust stabilization and tracking performance of operator based nonlinear feedback control systems are studied by using robust right coprime factorization specifically a new condition of robust right coprime factorization of nonlinear systems with unknown bounded perturbations is derived using the new condition a broader class of nonlinear plants can be controlled robustly when the spaces of the nonlinear plant output and the reference input are different a space change filter is designed and in this case this note considers tracking controller design using the exponential iteration theorem

perturbed nonlinear systems control using extended robust right coprime factorisation abstractin this paper extended operator based robust right coprime factorisation is investigated for dealing with a class of nonlinear systems with unknown bounded perturbations first a new kind of operator is introduced by which operator based right coprime factorisation approach is extended to consider the perturbed nonlinear systems by regulating the exponent of the proposed operator a broader class of nonlinear systems can be handled using the extended right coprime factorisation approach second for guaranteeing robust stability of the perturbed nonlinear systems feasible design schemes based on some sufficient conditions are discussed which can reduce complicated calculation in control processing using a different unimodular operator finally a simulation example is involved to illustrate the proposed design scheme for confirming the effectiveness of the proposed method

alternating direction multistep methods for parabolic problems iterative stabilization efficient multistep procedures for time stepping galerkin methods for parabolic partial differential equations are presented and analyzed the procedures involve using an alternating direction preconditioned iterative method for approximately solving the linear equations arising at each timestep in a discrete galerkin method optimal order convergence rates are obtained for the iterative methods work estimates of almost optimal order are obtained

operator based robust control design for nonlinear plants with perturbation in this article operator based robust control design for nonlinear plants with perturbation is considered by using robust right coprime factorisation approach in detail the property of unimodular operator is studied and a control design structure is proposed based on the proposed design scheme the designed system is robustly stable also output tracking performance can be realised simultaneously finally the effectiveness of the proposed design scheme is demonstrated by a simulation example

an alternating direction implicit orthogonal spline collocation scheme for nonlinear parabolic problems on rectangular polygons we consider a nonlinear parabolic initial boundary value problem on a rectangular polygon with the solution satisfying robin boundary conditions with variable coefficients an approximation to the solution at the desired time value is obtained using an alternating direction implicit extrapolated crank nicolson scheme in which orthogonal spline collocation with piecewise polynomials of an arbitrary degree is used for spatial discretization at each time level the scheme determines the intermediate solution along horizontal lines and the approximate solution along vertical lines passing through the gauss points only at the last time level is the approximate solution along vertical lines converted into the approximate solution defined on the entire rectangular polygon this property of our approach leads to its efficient implementation and its applicability to rectangular polygons