controlled invariance for discrete time nonlinear systems with an application to the disturbance decoupling problems invariant distributions are defined for discrete time nonlinear control systems and necessary and sufficient conditions are given for their controlled invariance this extends to discrete time systems the basic tool which has been so important in solving the various synthesis problems for continuous time systems to indicate their utility in the discrete time setting they are used to locally solve the disturbance decoupling problem

a hybrid pseudo force laplace transform method for non linear transient response of a suspended cable a hybrid numerical scheme involving the combination of the laplace transform technique and the pseudo force method is proposed to analyze the non linear transient response of a suspended cable subjected to arbitrary dynamic loading a theoretical model of the cable with multi degree of freedom is first obtained through discretization of the partial differential equations by finite difference approximation the non linear governing equations take into account the effects of quadratic and cubic geometric non linearities the proposed method deals with the non linear effects as pseudo forces and then establishes an iterative solution scheme in the alternating laplace time domain by means of fast numerical laplace transform this method eschews a time stepping process and therefore is computationally efficient it also readily deals with the viscoelastic damping with frequency dependent model parameters and the hysteresis damping in terms of complex stiffness models numerical examples are presented to evaluate the dynamic responses of suspended cables under a concentrated sinusoidal force and a distributed random excitation and to identify the non linear response properties by comparison with the linear vibration the validity and accuracy of the proposed method is also verified by comparing the results with those obtained by using the direct time integration

remarks on linearization of discrete time autonomous systems and nonlinear observer design abstract this paper presents necessary and sufficient conditions under which a discrete time autonomous system with outputs is locally state equivalent to an observable linear system or a system in the nonlinear observer form krener and isidori 1983 in particular an open problem raised in lee and nam 1991 namely the observer linearization problem is solved for a nonlinear system which may not be invertible i e the mapping f may not be a local diffeomorphism as a consequence the nonlinear observer design problem is solved by means of exact linearization techniques

coupling fluid flow with porous media flow the transport of substances back and forth between surface water and groundwater is a very serious problem we study herein the mathematical model of this setting consisting of the stokes equations in the fluid region coupled with the darcy equations in the porous medium coupled across the interface by the beavers joseph saffman conditions we prove existence of weak solutions and give a complete analysis of a finite element scheme which allows a simulation of the coupled problem to be uncoupled into steps involving porous media and fluid flow subproblems this is important because there are many legacy codes available which have been optimized for uncoupled porous media and fluid flow

linearization of discrete time nonlinear systems and a canonical structure linearizable nonlinear discrete time systems are described as a composition of diffeomorphisms linearizability of a general nonlinear system is obtained as an extension of linearizability of the system expressed in terms of the diffeomorphisms

normal forms and approximated feedback linearization in discrete time abstract the paper discusses approximated feedback linearization of nonlinear discrete time dynamics which are controllable in first approximation and introduces two types of normal forms the study is set in the context of differential difference representations of discrete time dynamics proposed in monaco normand cyrot in normand cyrot ed perspectives in control a tribute to ioan dore landau springer londres 1998 pp 191 205

iterative techniques for time dependent stokes problems abstract in this paper we consider solving the coupled systems of discrete equations which arise from implicit time stepping procedures for the time dependent stokes equations using a mixed finite element spatial discretization at each time step a two by two block system corresponding to a perturbed stokes problem must be solved although there are a number of techniques for iteratively solving this type of block system to be effective they require a good preconditioner for the resulting pressure operator schur complement in contrast to the time independent stokes equations where the pressure operator is well conditioned the pressure operator for the perturbed system becomes more ill conditioned as the time step is reduced and or the reynolds number is increased in this paper we shall describe and analyze preconditioners for the resulting pressure systems these preconditioners give rise to iterative rates of convergence which are independent of both the mesh size h as well as the time step and reynolds number parameter k

finite element analysis of a three dimensional underwater cable with time dependent length three dimensional underwater vibrations of a geometrically non linear cable with a weight at the lower end are investigated the length of the cable is time dependent a set of non linear time varying differential equations describing this system is derived by hamilton s principle and the variable domain finite element method the results of numerical simulation are presented for constant speed and sinusoidal axial motions of the cable the vibration responses of three initial conditions are shown the effects of initial displacements initial tensions due to gravity and the hydrodynamic forces on the non linear longitudinal and transverse amplitudes are presented finally some conclusions are drawn

modeling studies and efficient numerical methods for proton exchange membrane fuel cell abstract in this paper a three dimensional 3d nonisothermal multiphysics two phase steady state transport model and its efficient numerical methods are systematically studied for a full proton exchange membrane fuel cell pemfc in the sense of efficiency and accuracy the conservation equations of mass momentum species charge and energy are fully addressed in view of nonisothermality and multiphase characteristics in addition from an accurate numerical discretization s point of view we present some new formulations for species equations by investigating the interactions among the species in a framework of the combined finite element upwind finite volume method some efficient numerical methods are developed in terms of kirchhoff transformation for the sake of a fast and convergent numerical simulation the 3d simulations demonstrate that the convergent solutions can be attained within 80 nonlinear iterations in contrast to the oscillating and nonconvergent iterations conducted by commercial flow solvers or in house code with standard finite element volume methods numerical convergence tests are carried out to verify the efficiency and accuracy of our numerical algorithms and techniques

local input output decoupling of discrete time non linear systems a local treatment of the restricted block input output decoupling problem is given the major tools employed are the invariant and locally controlled invariant distributions which have recently been extended to the discrete time domain

a note on preconditioning for indefinite linear systems preconditioners are often conceived as approximate inverses for nonsingular indefinite matrices of saddle point or kkt form we show how preconditioners incorporating an exact schur complement lead to preconditioned matrices with exactly two or exactly three distinct eigenvalues thus approximations of the schur complement lead to preconditioners which can be very effective even though they are in no sense approximate inverses

robust preconditioning for coupled stokes darcy problems with the darcy problem in primal form abstract the coupled darcy stokes problem is widely used for modeling fluid transport in physical systems consisting of a porous part and a free part in this work we consider preconditioners for monolithic solution algorithms of the coupled darcy stokes problem where the darcy problem is in primal form we employ the operator preconditioning framework and utilize a fractional solver at the interface between the problems to obtain order optimal schemes that are robust with respect to the material parameters i e the permeability viscosity and beavers joseph saffman condition our approach is similar to that of holter et al 2020 but since the darcy problem is in primal form expressing mass conservation at the interface involves the normal derivative which introduces some mathematical challenges these challenges will be specifically addressed in this paper in particular we will employ fractional laplacians at the interface numerical experiments illustrating the performance are provided the preconditioner is posed in non standard sobolev spaces which may be perceived as an obstacle for its use in applications however we detail the implementational aspects and show that the preconditioner is quite feasible to realize in practice

a note on preconditioning nonsymmetric matrices the preconditioners for indefinite matrices of kkt form in m f murphy g h golub and a j wathen siam j sci comput 21 2000 pp 1969 1972 are extended to general nonsymmetric matrices

analysis of preconditioners for saddle point problems mixed finite element formulations give rise to large sparse block linear systems of equations the solution of which is often sought via a preconditioned iterative technique in this work we present a general analysis of block preconditioners based on the stability conditions inherited from the formulation of the finite element method the babuska brezzi or inf sup conditions the analysis is motivated by the notions of norm equivalence and field of values equivalence of matrices in particular we give sufficient conditions for diagonal and triangular block preconditioners to be norm and field of values equivalent to the system matrix

a modified two grid decoupling method for the mixed navier stokes darcy model in this paper a modified two grid decoupling method for the mixed navier stokes darcy model is proposed and analyzed the solution on interface condition is approximated by the solution obtained on the fine grid stability and convergence of the method are proved the convergence rate for the l 2 norm of u and p is improved from o h 3 2 to o h 2 the numerical results are consistent with the theoretical results and illustrate that the proposed method is quite efficient

a functional equations approach to nonlinear discrete time feedback stabilization through pole placement abstract the present work proposes a new approach to the nonlinear discrete time feedback stabilization problem with pole placement the problem s formulation is realized through a system of nonlinear functional equations and a rather general set of necessary and sufficient conditions for solvability is derived using tools from functional equations theory one can prove that the solution to the above system of nonlinear functional equations is locally analytic and an easily programmable series solution method can be developed under a simultaneous implementation of a nonlinear coordinate transformation and a nonlinear discrete time state feedback control law that are both computed through the solution of the system of nonlinear functional equations the feedback stabilization with pole placement design objective can be attained under rather general conditions the key idea of the proposed single step design approach is to bypass the intermediate step of transforming the original system into a linear controllable one with an external reference input associated with the classical exact feedback linearization approach however since the proposed method does not involve an external reference input it cannot meet other control objectives such as trajectory tracking and model matching

preconditioning techniques for a mixed stokes darcy model in porous media applications we study numerical methods for a mixed stokes darcy model in porous media applications the global model is composed of two different submodels in a fluid region and a porous media region coupled through a set of interface conditions the weak formulation of the coupled model is of a saddle point type the mixed finite element discretization applied to the saddle point problem leads to a coupled indefinite and nonsymmetric linear system of algebraic equations we apply the preconditioned gmres method to solve the discrete system and are particularly interested in efficient and effective decoupled preconditioning techniques several decoupled preconditioners are proposed theoretical analysis and numerical experiments show the effectiveness and efficiency of the preconditioners effects of physical parameters on the convergence performance are also investigated

field of values analysis of preconditioned linearized rayleigh benard convection problems abstract in this paper we use the notion of field of values fov equivalence of matrices to study a class of block triangular preconditioners for the fixed point linearization of the rayleigh benard convection problem discretized with inf sup stable finite element spaces first sufficient conditions on the nondimensional parameters of the problem are determined in order to establish the fov equivalence between the system matrix and the preconditioners four upper triangular block preconditioners belonging to the general proposed class are then considered numerical experiments show that the generalized minimal residual gmres convergence is robust with respect to the mesh size for these preconditioned systems we also compare the performance of the different preconditioners in terms of computational time

the immersion under feedback of a multidimensional discrete time non linear system into a linear system this paper deals with the problem of finding conditions which ensure the reproducibility of the input output behaviour of a linear analytic discrete time system by means of a linear discrete time system on this basis the problem of modifying the system by feedback in order for it to enjoy such a property is posed and solved

global external linearization of nonlinear systems via feedback this note presents necessary conditions and sufficient conditions for an affine nonlinear system to be globally feedback equivalent to a controllable linear system over an open subset v of rn when v equals rn necessary and sufficient conditions are obtained

a unified approach for uzawa algorithms we present a unified approach in analyzing uzawa iterative algorithms for saddle point problems we study the classical uzawa method the augmented lagrangian method and two versions of inexact uzawa algorithms the target application is the stokes system but other saddle point systems e g arising from mortar methods or lagrange multipliers methods can benefit from our study we prove convergence of uzawa algorithms and find optimal rates of convergence in an abstract setting on finite or infinite dimensional hilbert spaces the results can be used to design multilevel or adaptive algorithms for solving saddle point problems the discrete spaces do not have to satisfy the lbb stability condition

long time stability and convergence rate of maccormack rapid solver method for nonstationary stokes darcy problem abstract we propose and study a combination of two second order implicit explicit imex methods for the coupled stokes darcy system that governs flows in karst aquifers the first is a second order explicit two step maccormack scheme and the second is a second order implicit crank nicolson method both algorithms only require the solution of two decoupled problems at each time step one stokes and the other darcy this combination so called the maccormack rapid solver method is very efficient faster at least of second order accuracy in time and space and can be easily implemented using legacy codes under time step limitation of the form t c h where h t are mesh size and time step respectively and c is a physical parameter we prove both long time stability and the rate of convergence of the method some numerical experiments are presented and discussed

fast nonsymmetric iterations and preconditioning for navier stokes equations discretization and linearization of the steady state navier stokes equations gives rise to a nonsymmetric indefinite linear system of equations in this paper we introduce preconditioning techniques for such systems with the property that the eigenvalues of the preconditioned matrices are bounded independently of the mesh size used in the discretization we confirm and supplement these analytic results with a series of numerical experiments indicating that krylov subspace iterative methods for nonsymmetric systems display rates of convergence that are independent of the mesh parameter in addition we show that preconditioning costs can be kept small by using iterative methods for some intermediate steps performed by the preconditioner

block triangular preconditioners for nonsymmetric saddle point problems field of values analysis a preconditioned minimal residual method for nonsymmetric saddle point problems is analyzed the proposed preconditioner is of block triangular form the aim of this article is to show that a rigorous convergence analysis can be performed by using the field of values of the preconditioned linear system as an example a saddle point problem obtained from a mixed finite element discretization of the oseen equations is considered the convergence estimates obtained by using a field of values analysis are independent of the discretization parameter h several computational experiments supplement the theoretical results and illustrate the performance of the method

control systems in neural networks with local memory abstract based on a new paradigm of neural networks consisting of neurons with local memory nnlm we discuss the representation of a control system by neural networks using this representation the basic issues of complete controllability and observability for the system are addressed a separation principle of learning and control is presented for nnlm the result shows that the weights of the network will not affect its dynamics some results about local linearization via a regular static feedback and nonlinear transformation are also given

receding horizon control and discontinuous state feedback stabilization this paper addresses three aspects of receding horizon control in discrete time 1 feedback stabilization of general nonlinear systems with receding horizon control 2 the generation of stabili

constraint preconditioning for the coupled stokes darcy system the use of a constraint indefinite preconditioner for the iterative solution of the linear system arising from the finite element discretization of coupled stokes darcy flow is studied spectral and field of values bounds for the preconditioned system which are independent of the underlying mesh size are presented numerical experiments on several two and three dimensional problems illustrate the effectiveness of our approach

quadratic forms and approximate feed back linearization in discrete time quadratic feedback linearization is discussed for discrete time systems i e approximate feedback linearization up to the second order with respect to input and state variables after introducing in discrete time two equivalence classes invariant under quadratic static state feedback and diffeomorphism two canonical quadratic representations are proposed then it is shown that any nonlinear discrete time dynamics which are controllable in their first approximations can be linearized up to the second order under dynamic feedback the results stated here can be considered as the general discrete time analogues of the continuous time ones kang and krener 1992

feedback linearization of discrete time systems abstract we give necessary and sufficient conditions for a nonlinear discrete time system to be locally linearizable by a change of state coordinates and a feedback the conditions are expressed in terms of controllability distributions associated to the system

a two grid decoupling method for the mixed stokes darcy model in this paper we consider the mixed stokes darcy problem which describes a fluid flow coupled with a porous media we present a modified two grid method for decoupling this mixed model stability is proved and optimal error estimates are derived the numerical results show that the modified two grid method is effective and has the same accuracy as the coupling scheme when we choose h h 2

on the linear equivalents of nonlinear systems this paper studies feedback equivalence of nonlinear systems an equivalence relation is established for the systems x f x u a complete characterization of the class of nonlinear systems which are equivalent to controllable linear systems is obtained

analysis of long time stability and errors of two partitioned methods for uncoupling evolutionary groundwater surface water flows the most effective simulations of the multiphysics coupling of groundwater to surface water must involve employing the best groundwater codes and the best surface water codes partitioned methods which solve the coupled problem by successively solving the subphysics problems have recently been studied for the stokes darcy coupling with convergence established over bounded time intervals with constants growing exponentially in t this report analyzes and tests two such partitioned noniterative domain decomposition methods for the fully evolutionary stokes darcy problem under a modest time step restriction of the form delta t leq c where c c physical parameters we prove unconditional long time over 0 leq t infty stability of both partitioned methods from this we derive an optimal error estimate that is uniform in time over 0 leq t infty

dynamic and energetic analyses of a string slider non linear coupling system by variable grid finite difference in this paper a string slider non linear coupling system with time dependent boundary condition is considered one partial differential equation pde describing the transverse small amplitude vibration of the string non linearly coupled with one ordinary differential equation ode describing the horizontal displacement of the slider are derived by hamilton s principle this is a moving boundary problem since the unknown position of the slider has to be determined as a part of the solutions a transformation of the variable that converts the original non stationary boundary conditions to a set of fixed boundary conditions is proposed to avoid the increased complication and loss of accuracy associated with unequal space intervals near the moving boundary the finite difference method with variable grid is employed to show the numerical results of the coupling effect between the string and slider finally some periodic motions of the moving boundary are assigned to show the divergence of the string vibrations

robin robin domain decomposition methods for the steady state stokes darcy system with the beavers joseph interface condition domain decomposition methods for solving the coupled stokes darcy system with the beavers joseph interface condition are proposed and analyzed robin boundary conditions are used to decouple the stokes and darcy parts of the system then parallel and serial domain decomposition methods are constructed based on the two decoupled sub problems convergence of the two methods is demonstrated and the results of computational experiments are presented to illustrate the convergence

some numerical aspects of approximate linearization of single input non linear systems we characterize approximate linearization to order m of a single input non linear system in terms of the existence of an integrating factor to order m 2 which is equivalent to the well known condition that a system of vector fields be order m 1 involutive we present a constructive method to obtain the required transformation for approximate linearization which results in substantial computational savings

uzawa smoother in multigrid for the coupled porous medium and stokes flow system the multigrid solution of coupled porous media and stokes flow problems is considered the darcy equation as the saturated porous medium model is coupled to the stokes equations by means of appropriate interface conditions we focus on an efficient multigrid solution technique for the coupled problem which is discretized by finite volumes on staggered grids giving rise to a saddle point linear system special treatment is required regarding the discretization at the interface an uzawa smoother is employed in multigrid which is a decoupled procedure based on symmetric gauss seidel smoothing for velocity components and a simple richardson iteration for the pressure field since a relaxation parameter is part of a richardson iteration local fourier analysis is applied to determine the optimal parameters highly satisfactory multigrid convergence is reported and moreover the algorithm performs very well for small values of the hydraulic conductivity and fluid viscosity which are relevant for applications

approximate and local linearizability of nonlinear discrete time systems we consider a single input non linear discrete time system of the form where x n u and f x u n 1 n is a c n valued function necessary and sufficient conditions for approximate linearizability are given for we also give necessary and sufficient conditions for local linearizability finally we present analogous results for multi input non linear discrete time systems

immersion and immersion by nonsingular feedback of a discrete time nonlinear system into a linear system an analytic discrete time nonlinear system is considered necessary and sufficient conditions for immersion of the system into a linear system are given local immersion is defined by nonsingular feedback into a linear system and given necessary and sufficient conditions for this problem it is shown that a similar approach can also be applied to continuous time affine nonlinear systems

on block preconditioners for nonsymmetric saddle point problems we discuss a class of preconditioning methods for an iterative solution of algebraic nonsymmetric saddle point problems arising from a mixed finite element discretization of partial differential equations in particular the navier stokes equation we prove that block diagonal and block triangular preconditioners based on symmetric positive definite blocks guarantee that the convergence rate of the method is independent of the mesh parameter h

a two grid method of a mixed stokes darcy model for coupling fluid flow with porous media flow we study numerical methods for solving a coupled stokes darcy problem in porous media flow applications a two grid method is proposed for decoupling the mixed model by a coarse grid approximation to the interface coupling conditions error estimates are derived for the proposed method both theoretical analysis and numerical experiments show the efficiency and effectiveness of the two grid approach for solving multimodeling problems potential extensions and future directions are discussed

some observations on babu vs ka and brezzi theories summary some observations are made on abstract error estimates for galerkin approximations based on babuska brezzi conditions a basic error estimate due to babuska is sharpened by means of an identity that p i p for any nontrivial idempotent operator p some remarks are also made on the brezzi s theory for mixed variational problems and their galerkin approximations

long time stability of four methods for splitting the evolutionary stokes darcy problem into stokes and darcy subproblems this report analyzes the long time stability of four methods for non iterative sub physics uncoupling for the evolutionary stokes darcy problem the four methods uncouple each timestep into separate stokes and darcy solves using ideas from splitting methods three methods uncouple sequentially while one is a parallel uncoupling method we prove long time stability of four splitting based partitioned methods under timestep restrictions depending on the problem parameters the methods include those that are stable uniformly in s 0 the storativity coefficient for moderate k m i n the minimum hydraulic conductivity uniformly in k m i n for moderate s 0 and with no coupling between the timestep and the spacial meshwidth

a modified local and parallel finite element method for the mixed stokes darcy model abstract based on two grid discretizations a modified local and parallel finite element algorithm for the mixed stokes darcy problem is proposed and analyzed in this paper the idea is that on the coarse grid we solve a coupled stokes darcy problem on the fine grid we first consider the discrete darcy problem with a coarse grid approximation u h to the interface coupling conditions then solve a series of independent local stokes subproblems with the numerical solution of darcy problem on fine grid to the interface coupling conditions in parallel optimal error estimates are obtained and numerical experiments demonstrate the accuracy and efficiency of the modified local and parallel algorithm

a multilevel decoupled method for a mixed stokes darcy model this paper studies decoupled numerical methods for a mixed stokes darcy model for coupling fluid and porous media flows a two level algorithm is proposed and analyzed in mu and xu 2007 10 we generalize the two level algorithm to a multilevel algorithm in this paper and present numerical analysis on the error estimates for the multilevel algorithm the multilevel algorithm solves the mixed stokes darcy system by applying efficient legacy code for single model solvers to solve two decoupled stokes and darcy subproblems on all the subsequently refined meshes except for a much smaller global problem only on a very coarse initial mesh numerical experiments are conducted for both the two level and multilevel algorithms to illustrate their effectiveness and efficiency and validate the related theoretical analysis

a rectangular mixed element method with continuous flux approximation for coupling stokes and darcy flows we construct and analyze herein a mixed element method with continuous flux approximation based on staggered rectangular meshes for solving the model of stokes equation coupled with darcy equation with beavers joseph saffman interface condition we prove the inf sup condition and the approximate solution existence uniquely the l 2 error estimates are established for both the velocity and pressure numerical experiments are given to verify the convergence rate for the problem

uniform preconditioners for the time dependent stokes problem implicit time stepping procedures for the time dependent stokes problem lead to stationary singular perturbation problems at each time step these singular perturbation problems are systems of saddle point type which formally approach a mixed formulation of the poisson equation as the time step tends to zero preconditioners for discrete analogous of these systems are discussed the preconditioners uses standard positive definite elliptic preconditioners as building blocks and lead to condition numbers which are bounded uniformly with respect to the time step and the spatial discretization the construction of the discrete preconditioners is related to the mapping properties of the corresponding continuous system