a tensor singular values and its symmetric embedding eigenvalues national natural science foundation of china 11201092 11261012 scientific research starting foundation for doctors of guizhou normal university china program of young scientists of jiangxi province 20122bcb23003

robust and adaptive design of numerical optimization based extremum seeking control we consider the employment of numerical optimization and state regulation to solve the extremum seeking control esc problem which does not assume the time scale separation between the plant dynamics and the extremum seeking loop extremum seeking is realized via a state regulator that drives the state traveling along a convergent set point sequence generated by a numerical optimization algorithm in this paper we propose a novel design of an asymptotic state regulator via output tracking for state feedback linearizable systems where we trade off finite time state regulation to obtain flexibility in designing a robust extremum seeking controller existing techniques such as nonlinear damping and nonlinear adaptive control are then used to deal with input disturbance and unmodeled plant dynamics simulation examples illustrate the effectiveness of the basic and robust extremum seeking schemes and some design guidelines are provided for engineering applications

nonlinearities enhance parameter convergence in output feedback systems while the parameter convergence properties of standard adaptive algorithms for linear systems are well established there are no similar results on the parameter convergence of adaptive controllers for nonlinear systems which have gained popularity in recent years in this paper we focus on a recently developed class of adaptive schemes for output feedback nonlinear systems and show that parameter convergence is guaranteed if and only if an appropriately defined signal vector which does not depend on closed loop signals is persistently exciting then we develop an analytic procedure which allows us given a specific nonlinear system and a specific reference signal to determine a priori whether or not this vector is persistently exciting pe and hence whether or not the parameter estimates will converge in the process we show that the presence of nonlinearities usually reduces the sufficient richness sr requirements on the reference signals and hence enhances parameter convergence this is the first result on the relationship between pe and sr for adaptive nonlinear control systems

singular values of a real rectangular tensor real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics in this paper we systematically study properties of singular values of a real rectangular tensor and give an algorithm to find the largest singular value of a nonnegative rectangular tensor numerical results show that the algorithm is efficient

two level method based on finite element and crank nicolson extrapolation for the time dependent navier stokes equations a fully discrete two level finite element method the two level method is presented for solving the two dimensional time dependent navier stokes problem the method requires a crank nicolson extrapolation solution u h tau 0 p h tau 0 on a spatial time coarse grid j h tau 0 and a backward euler solution u h tau p h tau on a space time fine grid j h tau the error estimates of optimal order of the discrete solution for the two level method are derived compared with the standard crank nicolson extrapolation method the one level method based on a space time fine grid j h tau the two level method is of the error estimates of the same order as the one level method in the h1 norm for velocity and the l2 norm for pressure however the two level method involves much less work than the one level method

brief stability of extremum seeking feedback for general nonlinear dynamic systems while the mainstream methods of adaptive control both linear and nonlinear deal only with regulation to known set points or reference trajectories in many applications the set point should be selected to achieve a maximum of an uncertain reference to output equilibrium map the techniques of the so called extremum control or self optimizing control developed for this problem in the 1950 1960s have long gone out of fashion in the theoretical control literature because of the difficulties that arise in a rigorous analytical treatment in this paper we provide the first proof of stability of an extremum seeking feedback scheme by employing the tools of averaging and singular perturbation analysis our scheme is much more general that the existing extremum control results which represent the plant as a static nonlinear map possibly cascaded with a linear dynamic block we allow the plant to be a general nonlinear dynamic system possibly non affine in control and open loop unstable whose reference to output equilibrium map has a maximum and whose equilibria are locally exponentially stabilizable

bounded extremum seeking with discontinuous dithers the analysis of discontinuous extremum seeking es controllers e g those applicable to digital systems has historically been more complicated than that of continuous controllers we establish a simple and general extension of a recently developed bounded form of es to a general class of oscillatory functions including functions discontinuous with respect to time such as triangle or square waves with dead time we establish our main results by combining a novel idea for oscillatory control with an extension of functional analytic techniques originally utilized by kurzweil jarnik sussmann and liu in the late 80s and early 90s and recently studied by durr et al we demonstrate the value of the result with an application to inverter switching control

z eigenvalue methods for a global polynomial optimization problem as a global polynomial optimization problem the best rank one approximation to higher order tensors has extensive engineering and statistical applications different from traditional optimization solution methods in this paper we propose some z eigenvalue methods for solving this problem we first propose a direct z eigenvalue method for this problem when the dimension is two in multidimensional case by a conventional descent optimization method we may find a local minimizer of this problem then by using orthogonal transformations we convert the underlying supersymmetric tensor to a pseudo canonical form which has the same e eigenvalues and some zero entries based upon these we propose a direct orthogonal transformation z eigenvalue method for this problem in the case of order three and dimension three in the case of order three and higher dimension we propose a heuristic orthogonal transformation z eigenvalue method by improving the local minimum with the lower dimensional z eigenvalue methods and a heuristic cross hill z eigenvalue method by using the two dimensional z eigenvalue method to find more local minimizers numerical experiments show that our methods are efficient and promising

fast extremum seeking for wiener hammerstein plants this paper considers an observer based extremum seeking es scheme acting on a wiener hammerstein wh plant the es scheme utilises a high frequency sinusoidal dither in order to achieve fast minimisation of the plant s static nonlinearity unlike the prevailing analyses of fast es schemes acting on wh plants the presented result is semi global and applicable to plants with fairly general unknown nonlinearities design of the es scheme requires knowledge of the relative orders of both the input and output dynamics of the plant two different tunings of the es parameters are considered one which forces the es dynamics to be fast and one which is less restrictive it is shown that approximate knowledge of the phase of the plant s input dynamics is required if the es scheme is to use the less restrictive tuning for both tunings it is shown how the es scheme may be designed in order to simultaneously achieve arbitrarily fast and accurate minimisation of the static nonlinearity from an arbitrarily large set of initial conditions moreover the result shows that a wh plant with unstable input dynamics can in fact be stabilised by the es scheme

multivariable extremum seeking for pde dynamic systems in this article we address the design and analysis of multivariable newton based extremum seeking es for static locally quadratic maps subject to actuation dynamics governed by diffusion partial differential equations pdes multi input systems with distinct diffusion coefficients in each individual input channel are dealt with the phase compensation of the dither signals is handled as a trajectory generation problem and the inclusion of a multivariable diffusion feedback controller with a perturbation based averaging based estimate of the hessian s inverse allow to obtain local exponential convergence results to a small neighborhood of the optimal point the stability analysis is carried out using backstepping transformation and averaging in infinite dimensions capturing the infinite dimensional state due the diffusion pdes in addition the generalization of the results for different classes of parabolic reaction advection diffusion pdes wave equations and or first order hyperbolic transport dominated pdes is also discussed as for ordinary differential equations case the proposed newton approach removes the dependence of the algorithm s convergence rate on the unknown hessian of the nonlinear map to be optimized being user assignable unlike the gradient algorithm a numerical example illustrates the performance of our multivariable es for compensating pde based systems

estimation based approach for real time optimisation of uncertain nonlinear systems abstractin this paper the minimisation of an unknown but measurable cost function with uncertain dynamics is considered the drift term of the uncertain dynamical system and the gradient of the objective function are treated as unknown time varying parameters a novel estimation scheme based on almost invariant manifolds is proposed to estimate the time varying parameters a direct gradient based adaptive extremum seeking controller is designed to solve the uncertain optimisation problem this approach is shown to improve the transient performance of real time optimisation control systems

on a class of generating vector fields for the extremum seeking problem lie bracket approximation and stability properties abstract in this paper we describe a broad class of control functions for extremum seeking problems we show that it unifies and generalizes existing extremum seeking strategies which are based on lie bracket approximations and allows to design new controls with favorable properties in extremum seeking and vibrational stabilization tasks the second result of this paper is a novel approach for studying the asymptotic behavior of extremum seeking systems it provides a constructive procedure for defining frequencies of control functions to ensure the practical asymptotic and exponential stability in contrast to many known results we also prove asymptotic and exponential stability in the sense of lyapunov for the proposed class of extremum seeking systems under appropriate assumptions on the vector fields

asymptotic behavior and time discretization analysis for the non stationary navier stokes problem the asymptotic behavior and the euler time discretization analysis are presented for the two dimensional non stationary navier stokes problem if the data and f t satisfy a uniqueness condition corresponding to the stationary navier stokes problem we then obtain the convergence of the non stationary navier stokes problem to the stationary navier stokes problem and the uniform boundedness stability and error estimates of the euler time discretization for the non stationary navier stokes problem

inertial manifolds for nonlinear evolutionary equations abstract in this paper we introduce the concept of an inertial manifold for nonlinear evolutionary equations in particular for ordinary and partial differential equations these manifolds which are finite dimensional invariant lipschitz manifolds seem to be an appropriate tool for the study of questions related to the long time behavior of solutions of the evolutionary equations the inertial manifolds contain the global attractor they attract exponentially all solutions and they are stable with respect to perturbations furthermore in the infinite dimensional case they allow for the reduction of the dynamics to a finite dimensional ordinary differential equation

iterative learning control based on extremum seeking this paper proposes a non model based approach to iterative learning control ilc via extremum seeking single input single output discrete time nonlinear systems are considered where the objective is to recursively construct an input such that the corresponding system output tracks a prescribed reference trajectory as closely as possible on finite horizon the problem is formulated in terms of extremum seeking control which is amenable to a range of local and global optimisation methods contrary to the existing ilc literature the formulation allows the initial condition of each iteration to be incorporated as an optimisation variable to improve tracking sufficient conditions for convergence to the reference trajectory are provided the main feature of this approach is that it does not rely on knowledge about the system s model to perform iterative learning control in contrast to most results in the literature

a conforming finite element method for the time dependent navier stokes equations we study a spatial discretization of the stokes problem in a domain of mathbb r 2 or mathbb r 3 by a finite element method the unknowns being the velocity and the pressure optimal error

stabilized finite element method based on the crank nicolson extrapolation scheme for the time dependent navier stokes equations this paper provides an error analysis for the crank nicolson extrapolation scheme of time discretization applied to the spatially discrete stabilized finite element approximation of the two dimensional time dependent navier stokes problem where the finite element space pair x h m h for the approximation u n h p n h of the velocity u and the pressure p is constructed by the low order finite element the q 1 p 0 quadrilateral element or the p 1 p 0 triangle element with mesh size h error estimates of the numerical solution u n h p n h to the exact solution u t n p t n with t n 0 t are derived

block tensors and symmetric embeddings abstract well known connections exist between the singular value decomposition of a matrix a and the schur decomposition of its symmetric embedding sym a 0 a a t 0 in particular if is a singular value of a then and are eigenvalues of the symmetric embedding the top and bottom halves of sym a s eigenvectors are singular vectors for a power methods applied to a can be related to power methods applied to sym a the rank of sym a is twice the rank of a in this paper we develop similar connections for tensors by building on l h lim s variational approach to tensor singular values and vectors we show how to embed a general order d tensor a into an order d symmetric tensor sym a through the embedding we relate power methods for a s singular values to power methods for sym a s eigenvalues finally we connect the multilinear and outer product rank of a to the multilinear and outer product rank of sym a

finite element approximation for the viscoelastic fluid motion problem in this article the variational formulation of the two dimensional viscoelastic fluid motion problem and its finite element approximation are considered an local error estimate for the velocity with h1 norm and the pressure with l2 norm is obtained and a uniform error estimate for the velocity and pressure with the above norms is provided if the given data satisfies the uniqueness condition

extremum seeking for static maps with delays in this paper we address the design and analysis of multi variable extremum seeking for static maps subject to arbitrarily long time delays both gradient and newton based methods are considered multi input systems with different time delays in each individual input channel as well as output delays are dealt with the phase compensation of the dither signals and the inclusion of predictor feedback with a perturbation based averaging based estimate of the hessian allow to obtain local exponential convergence results to a small neighborhood of the optimal point even in the presence of delays the stability analysis is carried out using backstepping transformation and averaging in infinite dimensions capturing the infinite dimensional state due the time delay in particular a new backstepping like transformation is introduced to design the predictor for the gradient based extremum seeking scheme with multiple and distinct input delays the proposed newton based extremum seeking approach removes the dependence of the convergence rate on the unknown hessian of the nonlinear map to be optimized being user assignable as in the literature free of delays a source seeking example illustrates the performance of the proposed delay compensated extremum seeking schemes

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

stability of galerkin and inertial algorithms with variable time step size this paper provides galerkin and inertial algorithms for solving a class of nonlinear evolution equations spatial discretization can be performed by either spectral or finite element methods time discretization is done by euler explicit or euler semi implicit difference schemes with variable time step size moreover the boundedness and stability of these algorithms are studied by comparison we find that the boundedness and stability of inertial algorithm are superior to the ones of galerkin algorithm in the case of explicit scheme and the boundedness and stability of two algorithms are same in the case of semi implicit scheme

continuous time gradient like descent algorithm for constrained convex unknown functions penalty method application abstract this paper suggests a novel continuous time gradient descent algorithm of a constrained convex unknown function with a stochastic noise in the observed data the penalty function approach is employed to introduce the restrictions of the system and to provide a successful optimization process the solution is restricted to a static scheme dealing with the class of strongly convex functions subject to a set of constraints to estimate the stochastic gradient we employ a modified version of the synchronous detection method all the parameters of the proposed approach are decreasing in time to both compensate noise effect in the observations and to provide the mean square convergence of the suggested approach we present two different numerical examples to validate the contributions of the paper

brief adaptive extremum seeking control of nonlinear dynamic systems with parametric uncertainties we pose and solve an extremum seeking control problem for a class of nonlinear systems with unknown parameters extremum seeking controllers are developed to drive system states to the desired set points that extremize the value of an objective function the proposed adaptive extremum seeking controller is inverse optimal in the sense that it minimizes a meaningful cost function that incorporates penalty on both the performance error and control action simulation studies are provided to verify the effectiveness of the proposed approach

extremum seeking control with an adaptive dither signal we present an output feedback law that steers a fully actuated control affine system with general drift vector field to a minimum of the output function the proposed control law is based on a suitable approximation of descent directions by lie brackets in the high frequency limit a novelty of our approach is an adaptive choice of the frequency parameter in this way the problem of determining a sufficiently large frequency parameter becomes obsolete the adaptive choice of the frequency parameter also prevents finite escape times in the presence of a drift the proposed control law does not only lead to convergence into a neighborhood of the set of minima but leads to exact convergence for the case of an output function with a global minimum and no other stationary points we prove global convergence

an eigenvalue method for testing positive definiteness of a multivariate form in this paper we present an eigenvalue method for testing positive definiteness of a multivariate form this problem plays an important role in the stability study of nonlinear autonomous systems via lyapunov s direct method in automatic control at first we apply the d andrea dickenstein version of the classical macaulay formulas of the resultant to compute the symmetric hyperdeterminant of an even order supersymmetric tensor by using the supersymmetry property we give detailed computation procedures for the bezoutians and specified ordering of monomials in this approach we then use these formulas to calculate the characteristic polynomial of a fourth order three dimensional supersymmetric tensor and give an eigenvalue method for testing positive definiteness of a quartic form of three variables some numerical results of this method are reported

on non local stability properties of extremum seeking control in this paper we consider several extremum seeking schemes and show under appropriate conditions that these schemes achieve extremum seeking from an arbitrarily large domain of initial conditions if the parameters in the controller are appropriately adjusted this non local stability result is proved by showing semi global practical stability of the closed loop system with respect to the design parameters we show that reducing the size of the parameters typically slows down the convergence rate of the extremum seeking controllers and enlarges the domain of the attraction our results provide guidelines on how to tune the controller parameters in order to achieve extremum seeking simulation examples illustrate our results

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

nonlinear galerkin methods and mixed finite elements two grid algorithms for the navier stokes equations a nonlinear galerkin method using mixed finite elements is presented for the two dimensional incompressible navier stokes equations the scheme is based on two finite element spaces x h and x h for the approximation of the velocity defined respectively on one coarse grid with grid size h and one fine grid with grid size h ll h and one finite element space m h for the approximation of the pressure nonlinearity and time dependence are both treated on the coarse space we prove that the difference between the new nonlinear galerkin method and the standard galerkin solution is of the order of h 2 both in velocity h 1 omega and pressure l 2 omega norm we also discuss a penalized version of our algorithm which enjoys similar properties

extremum seeking control of state constrained nonlinear systems an extremum seeking control problem is posed for a class of nonlinear systems with unknown dynamical parameters whose states are subject to convex pointwise inequality constraints using a barrier function approach an adaptive method is proposed for generating setpoints online which converge to the feasible minimizer of a convex objective function containing the unknown dynamic parameters a tracking controller regulates system states to the generated setpoint via state feedback while maintaining feasibility of the state constraints a simulation example demonstrates application of the method

convergence of three iterative methods based on the finite element discretization for the stationary navier stokes equations abstract this paper considers three iterative methods for solving the stationary navier stokes equations iterative method i consists in solving the stationary stokes equations iterative method ii consists in solving the stationary linearized navier stokes equations and iterative method iii consists in solving the stationary oseen equations under the finite element discretization respectively at each iterative step also we discuss the stability and convergence of three iterative methods the iterative methods i and ii are stability and convergence under the strong uniqueness conditions where the iterative method ii is the second order convergence furthermore the iterative method iii is uncondition stability and convergence under the uniqueness condition finally some numerical tests show that the efficiency of the theoretical analysis

semi global stability analysis of observer based extremum seeking for hammerstein plants this paper considers semi global stability of an observer based extremum seeking es scheme acting on a hammerstein plant unlike previous semi global analyses no restriction is placed on the speed of the es dynamics relative to the plant dynamics for any set of frequencies of a probing sinusoidal dither it is shown how the plant output can be made to converge to an arbitrarily small neighborhood of its minimum from an arbitrarily large set of initial conditions furthermore in the absence of noise it is shown how increasing the dither frequency can allow arbitrarily fast convergence of the plant input to a small neighborhood of its optimum practical application of the results requires no a priori knowledge of the plant nonlinearity however in order to design the es scheme to operate over a wide range of dither frequencies some knowledge of the plant dynamics is required simulation examples are used to demonstrate these results and to highlight practical considerations when using a high frequency dither

euler implicit explicit iterative scheme for the stationary navier stokes equations in this paper a new uniqueness assumption a2 of the solution for the stationary navier stokes equations is presented under assumption a2 the exponential stability of the solution bar u bar p for the stationary navier stokes equations is proven moreover the euler implicit explicit scheme based on the mixed finite element is applied to solve the stationary navier stokes equations finally the almost unconditionally stability is proven and the optimal error estimates uniform in time are provided for the scheme

brief paper parameter convergence in adaptive extremum seeking control this paper addresses the problem of parameter convergence in adaptive extremum seeking control design an alternate version of the popular persistence of excitation condition is proposed for a class of nonlinear systems with parametric uncertainties the condition is translated to an asymptotic sufficient richness condition on the reference set point since the desired optimal set point is not known a priori in this type of problem the proposed method includes a technique for generating perturbation signal that satisfies this condition in closed loop this demonstrates its superiority in terms of parameter convergence the method guarantees parameter convergence with minimal but sufficient level of perturbation the effectiveness of the proposed method is illustrated with a simulation example

dither re use in nash equilibrium seeking nash equilibrium seeking nes scheme consists of a number of decentralized extremum seeking es agents each controlling an input such that an associated selfish cost is regulated to its steady state nash equilibrium a non local stability result for the nes scheme is provided which allows two agents to use the same dither signal if the effect of each agent on the other s steady state cost function is sufficiently weak its application to plants with quadratic cost functions is presented as an example it is then demonstrated in simulation that by reducing the number of distinct dither signals the proposed scheme still has acceptable convergence properties while design effort is reduced

d eigenvalues of diffusion kurtosis tensors diffusion kurtosis imaging dki is a new model in medical engineering where a diffusion kurtosis dk tensor is involved a dk tensor is a fourth order three dimensional fully symmetric tensor in this paper we introduce d eigenvalues for a dk tensor the largest the smallest and the average d eigenvalues of a dk tensor correspond with the largest the smallest and the average apparent kurtosis coefficients akc of a water molecule in the space respectively we present their computational methods and discuss related anisotropy value of akc

unified frameworks for sampled data extremum seeking control global optimisation and multi unit systems two frameworks are proposed for extremum seeking of general nonlinear plants based on a sampled data control law within which a broad class of nonlinear programming methods is accommodated it is established that under some generic assumptions semi global practical convergence to a global extremum can be achieved in the case where the extremum seeking algorithm satisfies a stronger asymptotic stability property the converging sequence is also shown to be stable using a trajectory based proof as opposed to a lyapunov function type approach the former is more straightforward and insightful this allows for more general optimisation algorithms than considered in existing literature such as those which do not admit a state update realisation and or lyapunov functions lying at the heart of the analysis throughout is robustness of the optimisation algorithms to additive perturbations of the objective function multi unit extremum seeking is also investigated with the objective of accelerating the speed of convergence

convergence of an algorithm for the largest singular value of a nonnegative rectangular tensor abstract in this paper we present an iterative algorithm for computing the largest singular value of a nonnegative rectangular tensor we establish the convergence of this algorithm for any irreducible nonnegative rectangular tensor

fast extremum seeking on hammerstein plants partial plant knowledge may be used to develop model based extremum seekers however existing results rely on a type of time scale separation which leads to slow optimization relative to the plant dynamics in this work a fast model based extremum seeking scheme is proposed for a hammerstein plant and semi global stability results are provided the structure of a hammerstein plant is used to advantage in designing filters that enable the extremum seeker to act on a faster time scale than the plant dynamics this leads to fast convergence while maintaining semi global stability

on the question of turbulence modeling by approximate inertial manifolds and the nonlinear galerkin method this paper contributes to the discussion about the ability of the aim method to model turbulent flow and about the theoretical potential of the aim ngm to provide a computational basis for the calculation of turbulent flow

dual adaptive extremum control of hammerstein systems a novel dual adaptive controller for extremum control of stochastic and uncertain nonlinear hammerstein systems is proposed the design is based on the innovations dual control cost function originally developed for conventional adaptive control of linear systems however the design process is extensively modified and developed so as to cater for the extremum control scenario this is a more challenging problem because the reference input is itself a nonlinear function of the unknown system parameters rather than an independent and predefined external reference signal as in all dual adaptive schemes it leads to a control law that balances out the need for caution due to parameter uncertainty with the conflicting requirement of probing that acts to quickly reduce parameter uncertainty the proposed controller s performance is analysed through extensive monte carlo simulation trials and compared with several other non dual adaptive extremum controllers it is shown that the novel extremum innovations

a framework for extremum seeking control of systems with parameter uncertainties traditionally the design of extremum seeking algorithm treats the system as essentially a black box which for many applications means disregarding known information about the model structure in contrast to this approach there have been recent examples where a known plant structure with uncertain parameters has been used in the online optimization of plant operation however the results for these approaches have been restricted to specific classes of plants and optimization algorithms this paper seeks to provide general results and a framework for the design of extremum seekers applied to systems with parameter uncertainties general conditions for an optimization method and a parameter estimator are presented so that their combination guarantees convergence of the extremum seeker for both static and dynamic plants tuning guidelines for the closed loop scheme are also presented the generality and flexibility of the proposed framework is demonstrated through a number of parameter estimators and optimization algorithms that can be combined to obtain extremum seeking examples of anti lock braking and model reference adaptive control are used to illustrate the effectiveness of the proposed framework

eigenvalues and invariants of tensors abstract a tensor is represented by a supermatrix under a co ordinate system in this paper we define e eigenvalues and e eigenvectors for tensors and supermatrices by the resultant theory we define the e characteristic polynomial of a tensor an e eigenvalue of a tensor is a root of the e characteristic polynomial in the regular case a complex number is an e eigenvalue if and only if it is a root of the e characteristic polynomial we convert the e characteristic polynomial of a tensor to a monic polynomial and show that the coefficients of that monic polynomial are invariants of that tensor i e they are invariant under co ordinate system changes we call them principal invariants of that tensor the maximum number of principal invariants of mth order n dimensional tensors is a function of m and n we denote it by d m n and show that d 1 n 1 d 2 n n d m 2 m for m 3 and d m n m n 1 m for m n 3 we also define the rank of a tensor all real eigenvectors associated with nonzero e eigenvalues are in a subspace with dimension equal to its rank

stability analysis of the nonlinear galerkin method our object in this article is to describe some numerical schemes for the approximation of nonlinear evolution equations and to study the stability of the schemes spatial discretization can be performed by either spectral or pseudospectral methods finite elements or finite differences time discretization is done by two level schemes partly or fully explicit the algorithms that we present stem from the study of the evolution equations from the dynamical systems point of view they are based on a differentiated treatment of the small and large wave lengths and they are particularly adapted to the integration of such equations on large intervals of time introduction the long time integration of dissipative evolution equations has emerged as a new type of problem in numerical analysis and its appearance is due to the considerable increase in computing power that we have seen during the last years much effort has been devoted in the past to the approximation of evolution equations on finite intervals of time or in the dynamically simple case where the solution converges to a steady stationary state as time goes to infinity however in many physically relevant situations the solutions to a dissipative evolution equation do not converge to a stationary solution they rather remain time dependent and they converge to a compact attractor that encompasses the nonlinear dynamics hence and whether this is addressed implicitly or explicitly the numerical integration of the evolution equation is then closely related to the approximation of the attractor that may be a complicated set even a fractal inertial algorithms are new integration algorithms that are based on the approximation of the attractor by smooth simple manifolds namely the approximate inertial manifolds these algorithms will be recalled below but at this point let us point out that one of their main features is a differentiated treatment of small and large wave lengths an inertial manifold is an exact law of dependence of small wave lengths in terms of the large ones the small wave received april 3 1990 1980 mathematics subject classification 1985 revision primary 65f10 65f35 65n20 35a40 1991 american mathematical society 0025 5718 91 1 00 25 perpage

a perturbation based proportional integral extremum seeking control approach this paper proposes a proportional integral extremum seeking control technique the technique is a generalization of the standard perturbation based techniques which provides fast transient performance of the closed loop system to the optimum equilibrium of a measured cost function under the assumption that the equations describing the dynamics of the nonlinear system and the cost function to be minimized are unknown it is shown that for a class of minimum phase nonlinear system with a cost function of relative degree one the closed loop esc can practically stabilize the unknown equilibrium optimum the stabilization result provides an alternative approach to output feedback control of nonlinear systems the effectiveness of the proposed approach is demonstrated using simulation examples

a small eddy correction method for nonlinear dissipative evolutionary equations considering the interaction between the large and small eddy components of solutions and using the idea of the newton iteration a small eddy correction method is proposed for approximating and numerically solving nonlinear dissipative pdes of parabolic type in particular the navier stokes equations nse we assume that the large eddy approximation to the solution is known formally applying the newton iterative procedure to the small eddy equation we then generate approximate systems it is shown that the first two steps in fact lead to the standard galerkin method sgm and the so called optimum nonlinear galerkin method ong and therefore the small eddy correction method is actually a certain generalization of sgm and ong the boundedness and convergence analysis are presented in the framework of the two dimensional nse the results show that the small eddy correction method can greatly improve the accuracy of sgm approximate solutions

a penalty finite element method based on the euler implicit explicit scheme for the time dependent navier stokes equations a fully discrete penalty finite element method is presented for the two dimensional time dependent navier stokes equations where the time discretization is based on the euler implicit explicit scheme with some implicit linear terms and an explicit nonlinear term and the finite element spatial discretization is based on the p 1b p 1 element pair which satisfies the discrete inf sup condition this method allows us to separate the computation of the velocity from the computation of the pressure with a larger time step size dt so that the numerical velocity u e h n and the pressure p e h n are easily computed an optimal error estimate of the numerical velocity and the pressure is provided for the fully discrete penalty finite element method when the penalty parameter e the time step size dt and the mesh size h satisfy the following stability conditions ec 1 1 dt k 1 1 and h 2 b 1 dt respectively for some positive constants c 1 k 1 and b 1 finally some numerical tests to confirm the theoretical results of the penalty finite element method are provided

stability and convergence of the crank nicolson adams bashforth scheme for the time dependent navier stokes equations in this paper we study the stability and convergence of the crank nicolson adams bashforth scheme for the two dimensional nonstationary navier stokes equations a finite element method is applied for the spatial approximation of the velocity and pressure the time discretization is based on the crank nicolson scheme for the linear term and the explicit adams bashforth scheme for the nonlinear term moreover we present optimal error estimates and prove that the scheme is almost unconditionally stable and convergent i e stable and convergent when the time step is less than or equal to a constant

convergence and stability of finite element nonlinear galerkin method for the navier stokes equations in this article we present a new fully discrete finite element nonlinear galerkin method which are well suited to the long time integration of the navier stokes equations spatial discretization is based on two grid finite element technique time discretization is based on euler explicit scheme with variable time step size moreover we analyse the boundedness convergence and stability condition of the finite element nonlinear galerkin method our discussion shows that the time step constraints of the method depend only on the coarse grid parameter h and the time step constraints of the finite element galerkin method depend on the fine grid parameter h h under the same convergence accuracy

an always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor in this paper we propose an iterative method to calculate the largest eigenvalue of a nonnegative tensor we prove this method converges for any irreducible nonnegative tensor we also apply this method to study the positive definiteness of a multivariate form

brief paper on global extremum seeking in the presence of local extrema we propose a global extremum seeking scheme which can seek the global optimal value in the presence of local extrema it is shown that the proposed global extremum seeking scheme can converge to an arbitrarily small neighborhood of the global extremum from an arbitrarily large set of initial conditions if sufficient conditions are satisfied a simple example illustrates the effectiveness of the proposed scheme

technical communique on the choice of dither in extremum seeking systems a case study we discuss how the choice of dither excitation signal affects the performance of extremum seeking using a benchmark situation a static scalar map and a simple scalar extremum seeking scheme our comparisons are based on the performance of the system with different dithers in terms of three performance indicators the speed of convergence domain of attraction and accuracy i e the ultimate bound on trajectories our analysis explicitly shows how the dither shape affects each of these performance indicators our study suggests that the practitioners using extremum seeking control should consider the dither shape as an important design parameter computer simulations support our theoretical findings

nonlinear galerkin methods this article presents a new method of integrating evolution differential equations the non linear galerkin method that is well adapted to the long term integration of such equations while the usual galerkin method can be interpreted as a projection of the considered equation on a linear space the methods considered here are related to the projection of the equation on a nonlinear manifold from the practical point of view some terms have been identified as small and sometimes but not always disregarded