symbolic computation of high order compact difference schemes for three dimensional linear elliptic partial differential equations with variable coefficients we present a symbolic computation procedure for deriving various high order compact difference approximation schemes for certain three dimensional linear elliptic partial differential equations with variable coefficients based on the maple software package we approximate the leading terms in the truncation error of the taylor series expansion of the governing equation and obtain a 19 point fourth order compact difference scheme for a general linear elliptic partial differential equation a test problem is solved numerically to validate the derived fourth order compact difference scheme this symbolic derivation method is simple and can be easily used to derive high order difference approximation schemes for other similar linear elliptic partial differential equations

adaptive predefined performance control for mimo systems with unknown direction via generalized fuzzy hyperbolic model an adaptive predefined performance control problem is investigated for a class of multiple input multiple output systems with unknown control direction and unknown backlash like hysteresis nonlinearities by using generalized fuzzy hyperbolic model gfhm compared with the existing methods the main features are as follows the prediction error is introduced to construct the adaptive laws which means that the approximate accuracy of the gfhm is solved the nussbaum type gain is utilized to deal with the unknown control direction which avoids the requirement of direction a priori and by transforming the tracking errors into new error variables the prescribed steady state and transient performance can be ensured it is shown that the proposed control approach can guarantee that all the signals of the resulting closed loop systems are bounded and the output tracks a desired trajectory while the tracking errors are confined all times within the prescribed bounds finally two simulation results and some comparisons are provided to verify the effectiveness of the proposed approach since the proposed control strategy is only implemented in a healthy case how to extend the strategy to a faulty case will be a further topic

analysis of a fourth order scheme for a three dimensional convection diffusion model problem we derive closed form expressions for the eigenvalues and discrete solution arising from a 19 point compact discretization of a three dimensional convection diffusion problem it is shown that the coefficient matrix is positive definite when the cell reynolds number is greater than some critical value by analyzing the terms composing the discrete solution we prove that an oscillation free discrete solution is guaranteed whenever the cell reynolds number exceeds a value which is grid size dependent an interesting result is that as the mesh size is refined this value approaches the golden mean

global stabilization of nonlinear systems with a class of unmatched uncertainties abstract global and asymptotic stabilization of uncertain dynamical systems which violate the matching conditions is investigated a class of unmatched uncertainties called equivalently matched uncertainties is defined a robust controller is proposed to stabilize the uncertain system asymptotically in the large provided that the nominal system is uniformly asymptotically stable and that the unmatched uncertainties are equivalently matched the required information about uncertain dynamics in the system is merely that the uncertainties are bounded in euclidean norm by known functions of the system state

error and extrapolation of a compact lod method for parabolic differential equations this paper is concerned with a compact locally one dimensional lod finite difference method for solving two dimensional nonhomogeneous parabolic differential equations an explicit error estimate for the finite difference solution is given in the discrete infinity norm it is shown that the method has the accuracy of the second order in time and the fourth order in space with respect to the discrete infinity norm a richardson extrapolation algorithm is developed to make the final computed solution fourth order accurate in both time and space when the time step equals the spatial mesh size numerical results demonstrate the accuracy and the high efficiency of the extrapolation algorithm

monotone iterative methods for numerical solutions of nonlinear integro elliptic boundary problems abstract the aim of this paper is to obtain various monotone iterative schemes for numerical solutions of a class of nonlinear nonlocal reaction diffusion convection equations under linear boundary conditions the boundary value problem under consideration is discretized into a system of nonlinear algebraic equations by the finite difference method and the iterative schemes are given for the finite difference system using upper and lower solutions as the initial iterations the construction of the monotone sequences and the definition of upper and lower solutions depend on the quasimonotone property of the reaction function and the iterative schemes are presented for each of the three types of quasimonotone functions an application of the monotone iterations including some numerical results is given to a modified logistic diffusive equation where the kernel in the integral term may be positive negative or changing sign in its domain

adaptive fuzzy identification and control for a class of nonlinear pure feedback mimo systems with unknown dead zones the adaptive fuzzy identification and control problems are considered for a class of multi input multi output nonlinear systems with unknown functions and unknown dead zone inputs the main characteristics of the considered systems are that 1 they are composed of n subsystems and each subsystem is in nested lower triangular form 2 dead zone inputs are in nonsymmetric nonlinear form and 3 dead zone inputs appear nonlinearly in the systems and their parameters are not required to be known the controller design for this class of systems is a difficult and complicated task because of the existences of unknown functions the couplings among the nested subsystems and the dead zone inputs in the controller design the fuzzy logic systems are employed to approximate the unknown functions and the differential mean value theorem is used to separate dead zone inputs to compensate for dead zone inputs the compensative terms are designed in the controllers the stability of the closed loop system is proved via the lyapunov stability theorem a simulation example is provided to validate the feasibility of the approach

h_ infty fuzzy control for a class of nonlinear coupled ode pde systems with input constraint this paper deals with the problem of h infty fuzzy control design with an input constraint for a class of coupled systems which consist of an n dimensional nonlinear subsystem of ordinary differential equations odes and a scalar linear parabolic subsystem of partial differential equation pde connected in feedback initially the nonlinear coupled system is represented by a takagi sugeno t s fuzzy coupled ode pde model then based on the fuzzy model and parallel distributed compensation scheme a fuzzy state feedback control design is developed via lyapunov s direct method such that the resulting closed loop fuzzy coupled system is exponentially stable and a prescribed h infty performance of disturbance attenuation is satisfied the existing condition of the proposed h infty fuzzy controllers is given in terms of linear matrix inequalities lmis moreover in order to make the attenuation level as small as possible while the input constraint is respected to avoid the high magnitude a suboptimal h infty constrained fuzzy control problem is also addressed which is formulated as an lmi optimization problem finally the proposed method is applied to the control of a hypersonic rocket car to illustrate its effectiveness

hybrid robust boundary and fuzzy control for disturbance attenuation of nonlinear coupled ode beam systems with application to a flexible spacecraft this paper introduces a hybrid robust boundary and fuzzy control design for disturbance attenuation of a class of coupled systems described by nonlinear ordinary differential equations odes and two nonlinear beam equations initially a takagi sugeno t s model is employed to exactly represent the nonlinear ode subsystem then a fuzzy controller is designed for the ode subsystem based on the t s fuzzy model and a robust boundary controller via beam boundary measurements is proposed for the nonlinear beam subsystem such a hybrid robust boundary and fuzzy controller is developed in terms of a set of space dependent bilinear matrix inequalities bmis by lyapunov s direct method which can exponentially stabilize the coupled system in the absence of disturbances and achieve an prescribed h infty performance of disturbance attenuation in the presence of disturbances furthermore in order to make the level of disturbance attenuation as small as possible a suboptimal h infty control problem is formulated as a bmi optimization problem a two step procedure is subsequently presented to solve this bmi optimization problem by the existing linear matrix inequality optimization techniques finally the proposed control method is applied to the control of a flexible spacecraft to illustrate its effectiveness

fourth order compact finite difference methods and monotone iterative algorithms for semilinear elliptic boundary value problems in this paper we study numerical methods for a class of two dimensional semilinear elliptic boundary value problems with variable coefficients in a union of rectangular domains a compact finite difference method with an anisotropic mesh is proposed for the problems the existence of a maximal and a minimal compact difference solution is proved by the method of upper and lower solutions and two sufficient conditions for the uniqueness of the solution are also given the optimal error estimate in the discrete l norm is obtained under certain conditions the error estimate shows the fourth order accuracy of the proposed method when two spatial mesh sizes are proportional by using an upper solution or a lower solution as the initial iteration an almost optimal picard type of monotone iterative algorithm is developed for solving the resulting nonlinear discrete system efficiently numerical results are presented to confirm our theoretical analysis

h _ infty fuzzy observer based control for a class of nonlinear distributed parameter systems with control constraints an hinfin fuzzy observer based control design is proposed for a class of nonlinear parabolic partial differential equation pde systems with control constraints for which the eigenspectrum of the spatial differential operator can be partitioned into a finite dimensional slow one and an infinite dimensional stable fast complement in the proposed control scheme galerkin s method is initially applied to the pde system to derive a nonlinear ordinary differential equation ode system that accurately describes the dynamics of the dominant slow modes of the pde system the resulting nonlinear ode system is subsequently represented by the takagi sugeno t s fuzzy model then based on the t s fuzzy model a fuzzy observer based controller is developed to stabilize the nonlinear pde system and achieve an optimized hinfin disturbance attenuation performance for the finite dimensional slow system while control constraints are respected the outcome of the hinfin fuzzy observer based control problem is formulated as a bilinear matrix inequality bmi optimization problem a local optimization algorithm that treats the bmi as a double linear matrix inequality is presented to solve this bmi optimization problem finally the proposed design method is applied to the control of the temperature profile of a catalytic rod to illustrate its effectiveness

fuzzy adaptive control with state observer for a class of nonlinear discrete time systems with input constraint in this paper an adaptive fuzzy controller is constructed for a class of nonlinear discrete time systems with unknown functions and bounded disturbances the main characteristics of the systems are that they take into account the effect of discrete time dead zone and the system states are not required to be measurable the stability problem of this class of systems is for the first time to be addressed in this paper due to the unavailability of the states and the presence of the discrete time dead zone the controller design becomes more difficult to stabilize the uncertain nonlinear discrete time systems the fuzzy logic systems are used to approximate the unknown functions a fuzzy state observer is designed to estimate the immeasurable states and the effect caused by discrete time dead zone can be solved via establishing an adaptation auxiliary signal based on the lyapunov approach it is proved that all the signals of the closed loop system are the semiglobal uniformly ultimately bounded and the tracking error is made within a small neighborhood around zero the feasibility of the developed control scheme is verified via two simulation examples

vibration control of an euler bernoulli beam under unknown spatiotemporally varying disturbance in this article the vibration suppression of an euler bernoulli beam system is considered by using the adaptive boundary control technique the dynamics of the beam are represented by a partial differential equation and four ordinary differential equations involving functions of both space and time by using lyapunov synthesis the robust boundary control with a disturbance observer is first proposed to suppress the vibration and attenuate the effect of the external disturbances to compensate for both the system parametric uncertainties and the disturbances uncertainties the adaptive boundary control is developed with the proposed boundary control the state of the beam system is proven to be uniformly ultimately bounded and converge to a small neighbourhood of zero by appropriately choosing the design parameters the effectiveness of the proposed control is successfully verified by numerical simulations

robust stabilization of the distributed parameter system with time delay via fuzzy control in this paper stabilization of the distributed parameter system dps with time delay is studied using galerkin s method and fuzzy control with the help of galerkin s method the dynamics of dps with time delay can be first converted into a group of low order functional ordinary differential equations which will be used for design of the robust fuzzy controller the fuzzy controller designed can guarantee exponential stability of the closed loop dps some sufficient conditions are derived for the stabilization together with the linear matrix inequality design approach the effectiveness of the proposed control design methodology is demonstrated in numerical simulations

convergence of fourth order compact difference schemes for three dimensional convection diffusion equations we consider a dirichlet boundary value problem for the three dimensional convection diffusion equations with constant coefficients in the unit cube a high order compact finite difference scheme is constructed on a 19 point stencil using the steklov averaging operators we prove that the finite difference scheme converges in discrete w 2 m omega norm with the convergence rate o h s m where the real parameter s satisfies the condition max 1 5 m s le m 4 m 0 1 2 and the exact solution belongs to the sobolev space w 2 s omega

on the stabilization of the disk beam system via torque and direct strain feedback controls this note deals with the stabilization problem of a rotating disk beam system the proposed feedback law consists of a nonlinear torque control exerted on the disk whereas a direct strain control is applied at the beam with a rigid offset as a consequence we end up with a nondissipative nonlinear hybrid pde ode system and hence the existence and stability of solutions of the system cannot be addressed by using standard methods such as energy multiplier or frequency domain to overcome this situation we adopt the riesz basis methodology and show that the closed loop system is well posed and stable as long as the angular velocity of the disk is less than a well defined value related to the beam the results are illustrated through a wide set of numerical examples

error estimates for approximate optimization by the extended ritz method an alternative to the classical ritz method for approximate optimization is investigated in the extended ritz method sets of admissible solutions are approximated by their intersections with sets of linear combinations of all n tuples of functions from a given basis this alternative scheme called variable basis approximation includes functions computable by trigonometric polynomials with free frequencies free node splines neural networks and other nonlinear approximating families estimates of rates of approximate optimization by the extended ritz method are derived upper bounds on rates of convergence of suboptimal solutions to the optimal one are expressed in terms of the degree n of variable basis functions the modulus of continuity of the functional to be minimized the modulus of tikhonov well posedness of the problem and certain norms tailored to the type of basis the results are applied to convex best approximation and to kernel methods in machine learning

continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems we consider a dynamic system containing uncertain elements only the set of possible values of these uncertainties is known based on this information a class of state feedback controls is proposed in order to guarantee uniform ultimate boundedness of every system response within an arbitrarily small neighborhood of the zero state these feedback controls are continuous functions of the state

uniform convergent monotone iterates for semilinear singularly perturbed parabolic problems this paper deals with discrete monotone iterative methods for solving semilinear singularly perturbed parabolic problems monotone sequences based on the accelerated monotone iterative method are constructed for a nonlinear difference scheme which approximates the semilinear parabolic problem this monotone convergence leads to the existence uniqueness theorem an analysis of uniform convergence of the monotone iterative method to the solutions of the nonlinear difference scheme and continuous problem is given numerical experiments are presented

an equivalence between sparse approximation and support vector machines in the first part of this paper we show a similarity between the principle of structural risk minimization principle srm vapnik 1982 and the idea of sparse approximation as defined in chen donoho and saunders 1995 and olshausen and field 1996 then we focus on two specific approximate implementations of srm and sparse approximation which have been used to solve the problem of function approximation for srm we consider the support vector machine technique proposed by v vapnik and his team at at t bell labs and for sparse approximation we consider a modification of the basis pursuit de noising algorithm proposed by chen donoho and saunders 1995 we show that under certain conditions these two techniques are equivalent they give the same solution and they require the solution of the same quadratic programming problem

barrier lyapunov functions for nussbaum gain adaptive control of full state constrained nonlinear systems abstract in this paper an adaptive controller design is studied for single input single output siso nonlinear systems with parameter uncertainties and the systems are enforced to subject to the full state constraints a remarkable feature of the constrained systems is that the so called control direction is unknown or in other words the signs of control gains are unknown in the existing results we discover that all the state constraint control results are required to determine a priori knowledge of control direction it will be inevitable to bring about a different design procedure and a difficult task when no a priori knowledge of control direction is known to stabilize this class of systems two parameter adaptive controllers with nussbaum gain technique are constructively framed to overcome the unknown control direction problem and the novel symmetric and asymmetric barrier lyapunov functions blfs are adopted to guarantee that the states are not to violate their constraints then the proposed blf strategy can be to conquer the conservatism of the traditional blf based controls for the full state constraints finally two theorems are provided to show that all the signals in the closed loop system are bounded the outputs are driven to follow the reference signals and all the states are ensured to remain in the predefined compact sets the effectiveness of the proposed scheme is performed via a simulation example

learning with boundary conditions kernel machines traditionally arise from an elegant formulation based on measuring the smoothness of the admissible solutions by the norm in the reproducing kernel hilbert space rkhs generated by the chosen kernel it was pointed out that they can be formulated in a related functional framework in which the green s function of suitable differential operators is thought of as a kernel in this letter our own picture of this intriguing connection is given by emphasizing some relevant distinctions between these different ways of measuring the smoothness of admissible solutions in particular we show that for some kernels there is no associated differential operator the crucial relevance of boundary conditions is especially emphasized which is in fact the truly distinguishing feature of the approach based on differential operators we provide a general solution to the problem of learning from data and boundary conditions and illustrate the significant role played by boundary conditions with examples it turns out that the degree of freedom that arises in the traditional formulation of kernel machines is indeed a limitation which is partly overcome when incorporating the boundary conditions this likely holds true in many real world applications in which there is prior knowledge about the expected behavior of classifiers and regressors on the boundary

the generalized lasso in the last few years the support vector machine svm method has motivated new interest in kernel regression techniques although the svm has been shown to exhibit excellent generalization properties in many experiments it suffers from several drawbacks both of a theoretical and a technical nature the absence of probabilistic outputs the restriction to mercer kernels and the steep growth of the number of support vectors with increasing size of the training set in this paper we present a different class of kernel regressors that effectively overcome the above problems we call this approach generalized lasso regression it has a clear probabilistic interpretation can handle learning sets that are corrupted by outliers produces extremely sparse solutions and is capable of dealing with large scale problems for regression functionals which can be modeled as iteratively reweighted least squares irls problems we present a highly efficient algorithm with guaranteed global convergence this defies a unique framework for sparse regression models in the very rich class of irls models including various types of robust regression models and logistic regression performance studies for many standard benchmark datasets effectively demonstrate the advantages of this model over related approaches

some global uniqueness and solvability results for linear complementarity problems over symmetric cones the purpose of this paper is to give a numerical treatment for a class of quasi linear elliptic equations under nonlinear boundary conditions including the three basic types of linear boundary conditions the quasi linear equation is discretized by the finite difference method and the method of upper lower solutions and its associated monotone iteration are used to compute the solutions of the finite difference system this method leads to monotone iterative schemes for the computation of numerical solutions as well as some comparison results among the monotone iterative schemes it also leads to the existence of a maximal and a minimal finite difference solution including the uniqueness of the solution and the convergence of the finite difference solution to the corresponding continuous solution applications are given to two physical problems in heat conduction and combustion theory and numerical results for the heat conduction problem are given and are compared with the known true continuous solution

multigrid solution of automatically generated high order discretizations for the biharmonic equation in this work we use a symbolic algebra package to derive a family of finite difference approximations for the biharmonic equation on a 9 point compact stencil the solution and its first derivatives are carried as unknowns at the grid points dirichlet boundary conditions are thus incorporated naturally since the approximations use the 9 point compact stencil no special formulas are needed near the boundaries both second order and fourth order discretizations are derived the fourth order approximation produce more accurate results than the 13 point classical stencil or the commonly used system of two second order equations coupled with the boundary condition the method suffers from slow convergence when classical iteration methods such as gauss seidel or sor are employed in order to alleviate this problem we propose several multigrid techniques that exhibit grid independent convergence and solve the biharmonic equation in a small amount of computer time test results from three different problems including stokes flow in a driven cavity are reported

the sensitivity and accuracy of fourth order finite difference schemes on nonuniform grids in one dimension abstract we construct local fourth order finite difference approximations of first and second derivatives on nonuniform grids in one dimension the approximations are required to satisfy symmetry relationships that come from the analogous higher dimensional fundamental operators the divergence the gradient and the laplacian for example we require that the discrete divergence and gradient be negative adjoint of each other div grad and the discrete laplacian is defined as lap divgrad the adjointness requirement on the divergence and gradient guarantees that the laplacian is a symmetric negative operator the discrete approximations we derive are fourth order on smooth grids but the approach can be extended to create approximations of arbitrarily high order we analyze the loss of accuracy in the approximations when the grid is not smooth and include a numerical example demonstrating the effectiveness of the higher order methods on nonuniform grids

high accuracy iterative solution of convection diffusion equation with boundary layers on nonuniform grids abstract a fourth order compact finite difference scheme and a multigrid method are employed to solve the two dimensional convection diffusion equations with boundary layers the computational domain is first discretized on a nonuniform stretched grid to resolve the boundary layers a grid transformation technique is used to map the nonuniform grid to a uniform one the fourth order compact scheme is applied to the transformed uniform grid a multigrid method is used to solve the resulting linear system numerical experiments are used to show that a graded mesh and a grid transformation are necessary to compute high accuracy solutions for the convection diffusion problems with boundary layers and dicretized by the fourth order compact scheme

a fourth order compact adi method for solving two dimensional unsteady convection diffusion problems n this article an exponential high order compact ehoc alternating direction implicit adi method in which the crank nicolson scheme is used for the time discretization and an exponential fourth order compact difference formula for the steady state 1d convection diffusion problem is used for the spatial discretization is presented for the solution of the unsteady 2d convection diffusion problems the method is temporally second order accurate and spatially fourth order accurate which requires only a regular five point 2d stencil similar to that in the standard second order methods the resulting ehoc adi scheme in each adi solution step corresponds to a strictly diagonally dominant tridiagonal matrix equation which can be inverted by simple tridiagonal gaussian decomposition and may also be solved by application of the one dimensional tridiagonal thomas algorithm with a considerable saving in computing time the unconditionally stable character of the method was verified by means of the discrete fourier or von neumann analysis numerical examples are given to demonstrate the performance of the method proposed and to compare mostly it with the high order adi method of karaa and zhang and the spatial third order compact scheme of note and tan

error estimate of fourth order compact scheme for linear schrodinger equations a fourth order compact difference scheme is proposed for two dimensional linear schrodinger equations with periodic boundary conditions by using the discrete energy method it is proven that the difference scheme is uniquely solvable unconditionally stable and convergent a maximum norm error estimate and thus an asymptotic expansion of the discrete solution are also obtained using the expansion of the difference solution high order approximations could be achieved by richardson extrapolations extension to three dimensional problems is also discussed numerical experiments are included to support the theoretical results and comparisons with the crank nicolson method are presented to show the effectiveness of our method

regularization techniques and suboptimal solutions to optimization problems in learning from data various regularization techniques are investigated in supervised learning from data theoretical features of the associated optimization problems are studied and sparse suboptimal solutions are searched for rates of approximate optimization are estimated for sequences of suboptimal solutions formed by linear combinations of n tuples of computational units and statistical learning bounds are derived as hypothesis sets reproducing kernel hilbert spaces and their subsets are considered

fixed point continuation for ell_1 minimization methodology and convergence we present a framework for solving the large scale ell 1 regularized convex minimization problem min x 1 mu f x our approach is based on two powerful algorithmic ideas operator splitting and continuation operator splitting results in a fixed point algorithm for any given scalar mu continuation refers to approximately following the path traced by the optimal value of x as mu increases in this paper we study the structure of optimal solution sets prove finite convergence for important quantities and establish q linear convergence rates for the fixed point algorithm applied to problems with f x convex but not necessarily strictly convex the continuation framework motivated by our convergence results is demonstrated to facilitate the construction of practical algorithms

stability and generalization we define notions of stability for learning algorithms and show how to use these notions to derive generalization error bounds based on the empirical error and the leave one out error the methods we use can be applied in the regression framework as well as in the classification one when the classifier is obtained by thresholding a real valued function we study the stability properties of large classes of learning algorithms such as regularization based algorithms in particular we focus on hilbert space regularization and kullback leibler regularization we demonstrate how to apply the results to svm for regression and classification

high order compact solvers for the three dimensional poisson equation new compact approximation schemes for the laplace operator of fourth and sixth order are proposed the schemes are based on a pade approximation of the taylor expansion for the discretized laplace operator the new schemes are compared with other finite difference approximations in several benchmark problems it is found that the new schemes exhibit a very good performance and are highly accurate especially on large grids they outperform noncompact schemes

robust adaptive fuzzy control for a class of nonlinear coupled ode beam systems with boundary uncertainty abstract this paper investigates a robust adaptive fuzzy control problem for a class of nonlinear ordinary differential equations odes coupled with a beam equation subject to boundary uncertainty a fuzzy control scheme based on the takagi sugeno t s fuzzy model is employed for the nonlinear ode subsystem with ode state feedback for the beam with boundary uncertainty a robust adaptive fuzzy boundary control scheme is adopted where a linear controller via boundary measurements is used to stabilize the beam and a robust adaptive fuzzy compensator is utilized to counteract the boundary uncertainty the asymptotic stabilization condition is derived for the nonlinear coupled ode beam system by means of the lyapunov s direct method which is provided in terms of a set of bilinear matrix inequalities bmis furthermore a two step procedure is presented to solve the bmi feasibility problem by the existing linear matrix inequality lmi optimization techniques finally a simulation study is conducted on a flexible spacecraft to demonstrate the effectiveness of the proposed method

multigrid method and fourth order compact scheme for 2d poisson equation with unequal mesh size discretization abstract a fourth order compact difference scheme with unequal mesh sizes in different coordinate directions is employed to discretize a two dimensional poisson equation in a rectangular domain multigrid methods using a partial semicoarsening strategy and line gauss seidel relaxation are designed to solve the resulting sparse linear systems numerical experiments are conducted to test the accuracy of the fourth order compact difference scheme and to compare it with the standard second order difference scheme convergence behavior of the partial semicoarsening and line gauss seidel relaxation multigrid methods is examined experimentally

parameterized linear matrix inequality techniques in fuzzy control system design this paper proposes different parameterized linear matrix inequality plmi characterizations for fuzzy control systems these plmi characterizations are in turn relaxed into pure lmi programs which provides tractable and effective techniques for the design of suboptimal fuzzy control systems the advantages of the proposed methods over earlier ones are then discussed and illustrated through numerical examples and simulations

distributed fuzzy control design of nonlinear hyperbolic pde systems with application to nonisothermal plug flow reactor this paper considers the problem of fuzzy control design for a class of nonlinear distributed parameter systems that is described by first order hyperbolic partial differential equations pdes where the control actuators are continuously distributed in space the goal of this paper is to develop a fuzzy state feedback control design methodology for these systems by employing a combination of pde theory and concepts from takagi sugeno t s fuzzy control first the t s fuzzy hyperbolic pde model is proposed to accurately represent the nonlinear first order hyperbolic pde system subsequently based on the t s fuzzy pde model a lyapunov technique is used to analyze the closed loop exponential stability with a given decay rate then a fuzzy state feedback control design procedure is developed in terms of a set of spatial differential linear matrix inequalities sdlmis from the resulting stability conditions furthermore utilizing the finite difference approximation method with a backward difference for the spatial derivative a recursive algorithm is presented to solve the sdlmis via the existing lmi optimization techniques finally the developed design methodology is successfully applied to the control of a nonisothermal plug flow reactor

the active disturbance rejection and sliding mode control approach to the stabilization of the euler bernoulli beam equation with boundary input disturbance in this paper we are concerned with the boundary feedback stabilization of a one dimensional euler bernoulli beam equation with the external disturbance flowing to the control end the active disturbance rejection control adrc and sliding mode control smc are adopted in investigation by the adrc approach the disturbance is estimated through an extended state observer and canceled online by the approximated one in the closed loop it is shown that the external disturbance can be attenuated in the sense that the resulting closed loop system under the extended state feedback tends to any arbitrary given vicinity of zero as the time goes to infinity in the second part we use the smc to reject the disturbance by removing the condition in adrc that the derivative of the disturbance is supposed to be bounded the existence and uniqueness of the solution for the closed loop via smc are proved and the monotonicity of the reaching condition is presented without the differentiation of the sliding mode function for which it may not always exist for the weak solution of the closed loop system the numerical simulations validate the effectiveness of both methods

fuzzy model based reliable static output feedback mathscr h _ infty control of nonlinear hyperbolic pde systems this paper investigates the problem of output feedback robust mathscr h infty control for a class of nonlinear spatially distributed systems described by first order hyperbolic partial differential equations pdes with markovian jumping actuator faults the nonlinear hyperbolic pde systems are first expressed by takagi sugeno fuzzy models with parameter uncertainties and then the objective is to design a reliable distributed fuzzy static output feedback controller guaranteeing the stochastic exponential stability of the resulting closed loop system with certain mathscr h infty disturbance attenuation performance based on a markovian lyapunov functional combined with some matrix inequality convexification techniques two approaches are developed for reliable fuzzy static output feedback controller design of the underlying fuzzy pde systems it is shown that the controller gains can be obtained by solving a set of finite linear matrix inequalities based on the finite difference method in space finally two examples are presented to demonstrate the effectiveness of the proposed methods

parameter estimation for the fractional fractal diffusion model based on its numerical solution in this paper we mainly consider a problem of parameter estimation for the fractional fractal diffusion model used to describe the anomalous diffusion in porous media the bayesian method is proposed to estimate parameters for the fractional fractal diffusion model based on its numerical solution firstly the center box difference method is employed to solve the fractional fractal diffusion equation subject to the initial boundary conditions then we apply the bayesian method to estimate three parameters for the model simultaneously that is the fractional order the fractal dimension d f and the structure parameter to testify the validity of the results for the model s parameter estimation experimental data from the fast desorption process of methane in coal are used it is shown that the numerical results modeled with parameters given by the bayesian method coincide well with the desorption experimental data indicating that the bayesian method is efficient and valid in identifying multi parameter for the fractional fractal diffusion model besides the fractional fractal diffusion model is demonstrated to be more capable than the classical fick s model to describe the anomalous diffusion behavior of gas in coal to clarify the parameters effect on the relative diffusion amount of methane for the fractional fractal diffusion model parameter sensitivity analysis is also carried out this paper provides a specific and efficient parameter estimation method for the fractional fractal diffusion model

time delayed finite difference reaction diffusion systems with nonquasimonotone functions this paper is concerned with finite difference solutions of a system of reaction diffusion equations with coupled nonlinear boundary conditions and time delays the reaction functions and the boundary functions are not necessarily quasimonotone and the time delays may appear in the reaction functions as well as in the boundary functions the investigation is devoted to the finite difference system for both the time dependent problem and its corresponding steady state problem some monotone iteration processes for the finite difference systems are given and the asymptotic behavior of the time dependent solution in relation to the steady state solution is discussed the asymptotic behavior result leads to some local and global attractors of the time dependent problem including the convergence of the time dependent solution to a unique steady state solution an application and some numerical results to an enzyme substrate reaction diffusion problem are given all the results are directly applicable to parabolic ordinary systems and to reaction diffusion systems without time delays

comparison of worst case errors in linear and neural network approximation sets of multivariable functions are described for which worst case errors in linear approximation are larger than those in approximation by neural networks a theoretical framework for such a description is developed in the context of nonlinear approximation by fixed versus variable basis functions comparisons of approximation rates are formulated in terms of certain norms tailored to sets of basis functions the results are applied to perceptron networks

on a variational norm tailored to variable basis approximation schemes a variational norm associated with sets of computational units and used in function approximation learning from data and infinite dimensional optimization is investigated for sets gk obtained by varying a vector y of parameters in a fixed structure computational unit k y e g the set of gaussians with free centers and widths upper and lower bounds on the gk variation norms of functions having certain integral representations are given in terms of the 1 norms of the weighting functions in such representations families of functions for which the two norms are equal are described

learning from examples as an inverse problem many works related learning from examples to regularization techniques for inverse problems emphasizing the strong algorithmic and conceptual analogy of certain learning algorithms with regularization algorithms in particular it is well known that regularization schemes such as tikhonov regularization can be effectively used in the context of learning and are closely related to algorithms such as support vector machines nevertheless the connection with inverse problem was considered only for the discrete finite sample problem and the probabilistic aspects of learning from examples were not taken into account in this paper we provide a natural extension of such analysis to the continuous population case and study the interplay between the discrete and continuous problems from a theoretical point of view this allows to draw a clear connection between the consistency approach in learning theory and the stability convergence property in ill posed inverse problems the main mathematical result of the paper is a new probabilistic bound for the regularized least squares algorithm by means of standard results on the approximation term the consistency of the algorithm easily follows

on the nystrom method for approximating a gram matrix for improved kernel based learning a problem for many kernel based methods is that the amount of computation required to find the solution scales as o n3 where n is the number of training examples we develop and analyze an algorithm to compute an easily interpretable low rank approximation to an n n gram matrix g such that computations of interest may be performed more rapidly the approximation is of the form gk cwk ct where c is a matrix consisting of a small number c of columns of g and wk is the best rank k approximation to w the matrix formed by the intersection between those c columns of g and the corresponding c rows of g an important aspect of the algorithm is the probability distribution used to randomly sample the columns we will use a judiciously chosen and data dependent nonuniform probability distribution let 2 and f denote the spectral norm and the frobenius norm respectively of a matrix and let gk be the best rank k approximation to g we prove that by choosing o k e4 columns g cwk ct g gk e i 1n gii2 both in expectation and with high probability for both 2 f and for all k 0 k rank w this approximation can be computed using o n additional space and time after making two passes over the data from external storage the relationships between this algorithm other related matrix decompositions and the nystrom method from integral equation theory are discussed

bounds on rates of variable basis and neural network approximation the tightness of bounds on rates of approximation by feedforward neural networks is investigated in a more general context of nonlinear approximation by variable basis functions tight bounds on the worst case error in approximation by linear combinations of n elements of an orthonormal variable basis are derived

fourth order compact finite difference method for fourth order nonlinear elliptic boundary value problems a compact finite difference method with non isotropic mesh is proposed for a two dimensional fourth order nonlinear elliptic boundary value problem the existence and uniqueness of its solutions are investigated by the method of upper and lower solutions without any requirement of the monotonicity of the nonlinear term three monotone and convergent iterations are provided for resolving the resulting discrete systems efficiently the convergence and the fourth order accuracy of the proposed method are proved numerical results demonstrate the high efficiency and advantages of this new approach

boundary control of transverse motion of marine risers with actuator dynamics this paper presents a boundary controller to reduce transverse motion of flexible marine risers driven by a hydraulic system at the top end of the risers under environmental disturbances induced by waves wind and ocean currents the boundary controller is designed based on lyapunov s direct method and the backstepping technique proof of existence and uniqueness of the solutions of the closed loop control system is carried out by using the galerkin approximation method simulation results illustrate the effectiveness of the proposed boundary controller

a compact multigrid solver for convection diffusion equations we introduce a multigrid algorithm to solve the convection diffusion equations using a nine point compact difference scheme we test the efficiency of the algorithm with various smoothers and intergrid transfer operators the algorithm displays a grid independent convergence rate and produces solutions with high accuracy numerical results are presented to validate the conclusions