postprocessing the galerkin method the finite element case a postprocessing technique developed earlier for spectral methods is extended here to galerkin finite element methods for dissipative evolution partial differential equations the postprocessing amounts to solving a linear elliptic problem on a finer grid or higher order space once the time integration on the coarser mesh is completed this technique increases the convergence rate of the finite element method to which it is applied and this is done at almost no additional computational cost the numerical experiments presented here show that the resulting postprocessed method is computationally more efficient than the method to which it is applied say quadratic finite elements as well as standard methods of similar order of convergence as the postprocessed one say cubic finite elements the error analysis of the new method is performed in l2 and in l infty norms

stability estimation of high dimensional vibrating systems under state delay feedback control the paper presents a method of assessing the stability of high dimensional vibrating systems under state feedback control with a short time delay it is first proved that if the time delay is sufficiently short ann degree of freedom system with feedback delay maintains 2neigenvalues near those of the corresponding system without feedback delay a perturbation approach is then proposed to determine the first order variation of an arbitrary eigenvalue and corresponding eigenvector of the system with feedback delay by solving a set of linear algebraic equations only the computation in this approach can be simplified to a matrix multiplication provided that the product of the time delay and the modulus of the eigenvalue is much less than 1 furthermore the approach is directly related to the newton raphson iteration in the continuation of eigenvalues for long time delay the efficacy of the approach is demonstrated via a number of case studies on two feedback delay systems of two degrees of freedom and ten degrees of freedom respectively

brief paper a way to stabilize linear systems with delayed state a simple method is proposed for stabilizing linear systems with delayed state using linear feedback the method requires checking the negativity of a matrix containing two free parameters and solving a matrix lyapunov equation with these parameters a known stabilizability criterion and a simple stability theorem for the open loop system are obtained as special cases of the main result

design of robust controllers for time delay systems the problem of stabilizing linear dynamical time delay systems subject to bounded uncertainties is investigated two memoryless feedback controllers are considered it is established that when the matching conditions are met and certain bounding relations are satisfied then the linear controller renders the zero response of the system asymptotically stable saturation type controllers are shown to guarantee that all system responses are uniformly ultimately bounded

optimal error bounds for two grid schemes applied to the navier stokes equations abstract we consider two grid mixed finite element schemes for the spatial discretization of the incompressible navier stokes equations a standard mixed finite element method is applied over the coarse grid to approximate the nonlinear navier stokes equations while a linear evolutionary problem is solved over the fine grid the previously computed galerkin approximation to the velocity is used to linearize the convective term for the analysis we take into account the lack of regularity of the solutions of the navier stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data optimal error bounds are obtained

properties of some weighted sobolev spaces and application to spectral approximations the weighted sobolev spaces on a square whose weight is a given power of the product of the distances to the edges are introduced trace theorems related to these spaces are proved then a regularity result for the dirichlet problem for the laplace operator is proved next several projection operators with polynomial values are considered for which approximation results in weighted norms are stated finally a collocation spectral method for the dirichlet problem for the laplace operator with inhomogeneous boundary conditions is analyzed this work includes new results on chebyshev approximation

some practical experience with the time integration of dissipative equations different methods for the numerical integration of evolution dissipative partial differential equations are tested with the kuramoto sivashinsky equation discretizations in space include galerkin and nonlinear galerkin methods for integration in time three different codes are used including standard stiff ode methods numerical tests show that standard codes for stiff ode render a gain of computing time of several orders of magnitude with respect problem tailored methods

a spectral element method for the navier stokes equations with improved accuracy we present a procedure to enhance the accuracy of spectral element methods for the evolutionary navier stokes equations the approximations to the velocity and the pressure obtained using a spectral element method are postprocessed by solving a discrete stokes problem only once when the time evolution is completed the analysis shows that this procedure improves the order of convergence for the approximations to the velocity and the pressure the choice of a discrete pressure space endowed with an inf sup condition independent of the discretization parameter leads to optimal error bounds for the pressure

van der pol s oscillator under delayed feedback the effect of delayed feedback on oscillatory behaviour is investigated for the classical van der pol oscillator it is shown how the presence of delay can change the amplitude of limit cycle oscillations or suppress them altogether the result is compared to the conventional proportional and derivative feedback the derivative like effect of delay is also demonstrated in a modified equation where a delayed term provides the damping

postprocessing finite element methods for the navier stokes equations the fully discrete case an accuracy enhancing postprocessing technique for finite element discretizations of the navier stokes equations is analyzed the technique had been previously analyzed only for semidiscretizations and fully discrete methods are addressed in the present paper we show that the increased spatial accuracy of the postprocessing procedure is not affected by the errors arising from any convergent time stepping procedure further refined bounds are obtained when the time stepping procedure is either the backward euler method or the two step backward differentiation formula

delayed feedback control of oscillations in non linear planar systems the use of delayed feedback of position for the control of oscillatory behaviour is analysed it is shown that linear feedback can annihilate limit cycles and stabilize the origin furthermore it is possible to create an asymptotically stable limit cycle with any prescribed amplitude using a simple non linear feedback for almost all values of the delay these feats can be achieved without using derivative information in the feedback the results indicate a possible mechanism by which biological systems accomplish complex control tasks using delayed and partial state information

on the construction of families of approximate inertial manifolds abstract the concept of approximate inertial manifold is related to the study of the longtime behaviour of partial differential equations in this paper we present a general method in order to construct sequences of such manifolds providing better and better approximations to the attractor le concept de variete inertielle approximative intervient dans l etude du comportement pour les grands temps des equations aux derivees partielles nous presentons dans cet article une methode generale pour construire des suites de telles varietes qui approximent de mieux en mieux l attracteur

nonresonant hopf bifurcations of a controlled van der pol duffing oscillator abstract the trivial equilibrium of a van der pol duffing oscillator with a nonlinear feedback control may lose its stability via hopf bifurcations when the time delay involved in the feedback control reaches certain values nonresonant hopf hopf interactions may occur in the controlled van der pol duffing oscillator when the corresponding characteristic equation has two pairs of purely imaginary roots with the aid of normal form theory and centre manifold theorem as well as a perturbation method the dynamic behaviour of the nonresonant co dimension two bifurcation is investigated by studying the possible solutions and their stability of the four dimensional ordinary differential equations on the centre manifold in the vicinity of the nonresonant hopf bifurcation the oscillator may exhibit the initial equilibrium solution two periodic solutions as well as a quasi periodic solution on a two dimensional torus depending on the dummy unfolding parameters and nonlinear terms the analytical predictions are found to be in good agreement with the results of numerical integration of the original delay differential equation

non periodic averaging principles for measure functional differential equations and functional dynamic equations on time scales involving impulses abstract we present a non periodic averaging principle for measure functional differential equations and using the correspondence between solutions of measure functional differential equations and solutions of functional dynamic equations on time scales see federson et al 2012 8 we obtain a non periodic averaging result for functional dynamic equations on time scales moreover using the relation between measure functional differential equations and impulsive measure functional differential equations we get a non periodic averaging theorem for these equations also it is a known fact that we can relate impulsive measure functional differential equations and impulsive functional dynamic equations on time scales see federson et al 2013 9 therefore applying this correspondence to our averaging principle we obtain a non periodic averaging theorem for impulsive functional dynamic equations on time scales

a noniterative method for the design of linear robust controllers a procedure for determining a linear control law which guarantees asymptotic stability for an uncertain system is derived using an elementary matrix identity an alternative proof of the existence of stabilizing control law for the matched case is provided bounds for the rates of decay are established and by using these bounds it is shown that every trajectory of the closed loop system can be made to decay at a prespecified exponential rate the technique is simpler to use than existing techniques since it does not require a numerical procedure but allows the control law to be obtained from some simple formulas

quadratic stabilization of continuous time systems with state delay and norm bounded time varying uncertainties this paper develops a method for designing a feedback control law to stabilize a class of uncertain linear systems the class of systems under consideration is described by a continuous time model with variable state delay and depends on time varying unknown but bounded uncertain parameters it is shown that the construction of the stabilizing controller involves solving a certain algebraic riccati equation several previous results are derived as interesting special cases finally a simulation example is presented to illustrate the analytical developments

the postprocessed mixed finite element method for the navier stokes equations a postprocessing technique for mixed finite element methods for the incompressible navier stokes equations is studied the technique was earlier developed for spectral and standard finite element methods for dissipative partial differential equations the postprocessing amounts to solving a stokes problem on a finer grid or higher order space once the time integration on the coarser mesh is completed the analysis presented here shows that this technique increases the convergence rate of both the velocity and the pressure approximations numerical experiments are presented that confirm both this increase in the convergence rate and the corresponding improvement in computational efficiency

the postprocessed mixed finite element method for the navier stokes equations refined error bounds a postprocessing technique for mixed finite element methods for the incompressible navier stokes equations is analyzed the postprocess which amounts to solving a linear stokes problem is shown to increase the order of convergence of the method to which it is applied by one unit times the logarithm of the mesh diameter in proving the error bounds some superconvergence results are also obtained contrary to previous analysis of the postprocessing technique in the present paper we take into account the loss of regularity suffered by the solutions of the navier stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data as in h g heywood and r rannacher siam j numer anal 25 1988 pp 489 512 where the same hypothesis is assumed no better than fifth order convergence is achieved

vibrational feedback control of time delay systems this paper applies vibrational feedback control to time lag systems averaging theory for differential delay systems is presented and then applied to aid in the design of periodic controllers both stabilization and transient issues are discussed an illustrative example is given which demonstrates that the proposed controller 1 provides superior robustness properties in comparison to time invariant controllers when applied correctly 2 is not robust with respect to unmodeled delays and 3 does not have zero placement capabilities in the sense defined in the paper

controlling friction induced instability by recursive time delayed acceleration feedback a novel time delayed acceleration feedback method of controlling friction induced instability is proposed a single degree of freedom mechanical oscillator on a moving belt represents the basic friction driven system the control force is synthesized based on an infinite weighted sum of the acceleration of the vibrating mass measured at regular intervals in the past such a control force can be effectively produced by the recursive summation of the time delayed acceleration and the time delayed control signal and hence the technique is termed as the recursive time delayed acceleration feedback control the local stability analysis of the equilibrium reveals nontrivial and beneficial influences of the recursive gain on the system performance robustness of the control is shown to improve with the increasing value of the recursive gain a multiple time scale based analysis of the system elucidates the role of the recursive gain in enhancing the amount of dissipation produced by the control action the influences of the time delay and the control gain on the optimized performance of the system are also discussed numerical simulations of the system equations corroborate the analytical results the present method is believed to be applicable to any self excited system having a large degree of instability that is not removable by an ordinary time delayed feedback

memoryless stabilization of uncertain dynamic delay systems riccati equation approach the authors present a procedure for obtaining the memoryless linear state feedback control of uncertain dynamic delay systems the uncertainties are time varying and within a given compact set this method is an extension of the riccati equation approach proposed by i r petersen and c v hollot 1986 the extension is straightforward also the uncertainties do not need to satisfy the matching conditions

numerical solution of a nonlinear dissipative system using a pseudospectral method and inertial manifolds we consider the numerical solution of nonlinear dissipative partial differential equations using a pseudospectral method and the methodology of approximate inertial manifolds coarse and fine grids are employed with a nonlinear mapping to relate the solutions computed on these grids the approach is illustrated by consideration of the kuramoto sivashinsky equation subject to periodic solutions that are restricted to the invariant subspace of odd functions numerical results are presented on the ability of the method to mimic the time evolution of the infinite dimensional dynamical system the results demonstrate that the coarse fine grid method is more accurate than the standard pseudospectral method on the same coarse grid however time integration studies suggest that the increased accuracy is bought at an increase in computational cost to achieve a specified accuracy the computational cost of the two grid method exceeds that of a standard pseudospectral method for this spatially periodic problem

delay dependent robust stabilization of uncertain systems with multiple state delays this paper deals with the problem of robust stabilization for uncertain systems with multiple state delays the parameter uncertainties are time varying and unknown but are norm bounded and the delays are time varying a new method for achieving robust stabilization is presented for a class of uncertain time delay systems via linear memoryless state feedback control the results depend on the size of the delays and are given in terms of several linear matrix inequalities

the postprocessing galerkin and nonlinear galerkin methods a truncation analysis point of view we revisit the postprocessing algorithm and give a justification from a classical truncation analysis point of view we assume a perturbation expansion for the high frequency mode component of solutions to the underlying equation keeping terms to certain orders we then generate approximate systems which correspond to numerical schemes we show that the first two leading order methods are in fact the postprocessed galerkin and postprocessed nonlinear galerkin methods respectively hence postprocessed galerkin is a natural leading order method more natural than the standard galerkin method for approximating solutions of parabolic dissipative pdes the analysis is presented in the framework of the two dimensional navier stokes equation nse however similar analysis may be done for any parabolic dissipative nonlinear pde the truncation analysis is based on asymptotic estimates in time for the low and high mode components we also introduce and investigate an alternative postprocessing scheme whic

postprocessing the linear finite element method we extend the idea of the postprocessing galerkin methods for dissipative evolution equations to the case of the linear finite element method the postprocessing technique has been developed earlier for spectral methods and for higher order finite element methods the analysis shows that this procedure improves the order of convergence of the piecewise linear galerkin finite element approximation in the h1 norm we show by means of numerical experiments that there is no improvement in the order of convergence in the l2 norm

a riccati equation approach to the design of linear robust controllers abstract in this paper by the lyapunov stability criterion and the riccati equation we derive a new procedure for determining a linear control law to stabilize an uncertain system the main features of this approach are that no precompensator is needed the required feedback gains are small and the uncertain system can be exponentially stabilized with any prescribed exponential rate in addition the robust control law can be made linear quadratic optimal with respect to an index function for the nominal system two examples are given to illustrate the results

time delayed feedback stabilisation of nonlinear potential systems mechanical systems with nonlinear potential forces and delayed feedback are studied it is assumed that in the absence of control the trivial equilibrium positions of considered systems are stable but they are not attracting ones an approach for the constructing of nonlinear controllers providing the asymptotic stability of the equilibrium positions is proposed by the use of the lyapunov direct method and the razumikhin approach it is proved that for the corresponding closed loop systems the asymptotic stability can be guaranteed even in the cases when delay is unknown and time varying moreover estimates for solutions of closed loop systems are found an example and the results of a computer simulation are presented to demonstrate the effectiveness of the proposed approach

stabilization of vibration systems via delayed state difference feedback this paper presents a study of the stabilization problem via delayed state feedback of pyragas type for unstable vibration systems that have even number of characteristic roots with positive real parts on the basis of stability switches with respect to the time delay a new stabilization criterion is established and an effective procedure for determining the admissible values of the feedback gains and the delay is given two examples are given in detail to demonstrate the efficiency of the theory

an enhanced pseudospectral chebyshev method for dissipative partial differential equations we present an enhanced pseudospectral chebyshev method based on a postprocess previously introduced for the galerkin method de frutos et al applied mathematics and computation reports july 1996 when the regularity of the solution is limited the new method is shown to possess a superior rate of convergence and improved efficiency than the classical pseudospectral scheme in the particular case of smooth solutions the reduction of the error obtained by postprocessing is not so clear however the use of a filter in the numerical solution before applying the postprocess improves again the efficiency of the method the numerical analysis of the new method and some numerical experiments showing its superior efficiency are presented

controlling coherence of noisy and chaotic oscillators by a linear feedback abstract we analyze the dynamics of a noisy limit cycle oscillator coupled to a general passive linear system we analytically demonstrate that the phase diffusion constant which characterizes the coherence of the oscillations can be efficiently controlled theoretical analysis is performed in the framework of linear and gaussian approximations and is supported by numerical simulations we also demonstrate numerically the coherence control of a chaotic system

postprocessing the finite element method for semilinear parabolic problems in this paper we consider postprocessing of the finite element method for semilinear parabolic problems the postprocessing amounts to solving a linear elliptic problem on a finer grid or higher order space once the time integration on the coarser mesh is completed the convergence rate is increased at almost no additional computational cost this procedure was introduced and analyzed in garcia archilla and titi siam j numer anal 37 2000 pp 470 499 we extend the analysis to the fully discrete case and prove error estimates for both space and time discretization the analysis is based on error estimates for the approximation of time derivatives by difference quotients

robust stabilization of uncertain systems with time varying multistate delay the authors deal with the problem of stabilizing a class of uncertain linear systems with time varying multi state delay and subject to norm bounded parameter uncertainty via memoryless linear state feedback some sufficient conditions for the robust stabilizability are derived fur this class of uncertain systems if there exists a positive definite symmetric solution satisfying the algebraic riccati equation or inequality a suitable memoryless state feedback law can be derived also moreover all such parametric algebraic riccati inequalities have been transformed into some linear matrix inequality problems so there is no tuning of the parameters to gain a stabilizing solution

efficient methods using high accuracy approximate inertial manifolds summary we extend the idea of the post processing galerkin method in the context of dissipative evolution equations to the nonlinear galerkin the filtered galerkin and the filtered nonlinear galerkin methods in general the post processing algorithm takes advantage of the fact that the error committed in the lower modes of the nonlinear galerkin method and galerkin method for approximating smooth bounded solutions is much smaller than the total error of the method in each case an improvement in accuracy is obtained by post processing these more accurate lower modes with an appropriately chosen highly accurate approximate inertial manifold aim we present numerical experiments that support the theoretical improvements in accuracy both the theory and computations are presented in the framework of a two dimensional reaction diffusion system with polynomial nonlinearity however the algorithm is very general and can be implemented for other dissipative evolution systems the computations clearly show the post processed filtered galerkin method to be the most efficient method

vibrational control of chemical reactions in a cstr with delayed recycle stream vibrational control is a method of modification of dynamic properties obtained by introducing zero mean oscillations into system parameters this paper shows that by introducing oscillations into the flow rates of a continuous stirred tank reactor with delayed recycle stream it is possible to operate reactions in the vicinity of previously unstable steady states

gromov witten theory of deligne mumford stacks given a smooth complex deligne mumford stack cal x with a projective coarse moduli space we introduce gromov witten invariants of cal x and prove some of their basic properties including the wdvv equation

superconvergence of elliptic reconstructions of finite element solutions of parabolic problems in domains with piecewise smooth boundaries abstract this paper addresses the superconvergence properties of the elliptic reconstruction of semi discrete finite element solutions of linear parabolic problems on domains with piecewise smooth boundary numerical computations are presented and explained employing the optimal a priori error estimates proven in the appendix

log mirror symmetry and local mirror symmetry we study mirror symmetry of log calabi yau surfaces on one hand we consider the number of affine lines of each degree in 2 b where b is a smooth cubic on the other hand we consider coefficients of a certain expansion of a function obtained from the integrals of dx x dy y over 2 chains whose boundaries lie on b where b is a family of smooth cubics then for small degrees they coincide we discuss the relation between this phenomenon and local mirror symmetry for 2 in a calabi yau 3 fold ckyz

fundamental theorems of averaging for functional differential equations abstract in this paper we improve on classical averaging theorems for functional differential equations by proposing new averaged models specifically we present averaged models which retain functional dependence on state variables delays and additional parametric dependence as shown in three examples the new models often yield improvements in approximation accuracy

using stacks to impose tangency conditions on curves we define a deligne mumford stack xp r which depends on a scheme x an effective carrier divisor d c x and a positive integer r then we show that the abramovich vistoli mod uli stack of stable maps into xq r provides compactifications of the locally closed substacks of mgtn x 3 corresponding to relative stable maps 1 introduction this paper has two main purposes the first is to introduce the rth root construction which is a way of creating stack structure along a divisor given a scheme x an effective cartier divisor d c x and a positive integer r which is invertible on x it produces a deligne mumford stack xo r when r is not invertible the construction still makes sense and produces an artin stack which has finite diagonal over x since the output is in general an artin stack we give the definition and basic properties when x is an artin stack this simplifies the exposition since we need to apply the construction multiple times this construction was known to several mathematicians before we became aware of it but very little appeared in the literature aside from a brief mention in agv 3 5 3 secondly we compare twisted stable maps into xp r with ordinary stable maps into x the theory of twisted stable maps was developed by chen and ruan cr in the symplectic category and by abramovich and vistoli av in the algebraic category the latter showed that there is a deligne mumford stack xg n x 3 parametrizing morphisms x from twisted rc marked genus g nodal curves into a deligne mumford stack x such that 3 in the case where x is a scheme this is just the space of kontsevich stable maps mg n x 3 the theory of relative stable maps was developed by several symplectic ge ometers and the algebraic definition was worked out partially by andreas gath mann ga and in general by jun li li let x be a smooth complex projective veriety d c x a smooth divisor and 3 g b x such that d 3 0 here b x denotes the group of 1 cycles modulo algebraic equivalence choose a partition d 3 q qn where each qi is a nonnegative integer then a stable map c x from a smooth curve c having marked points x xn such that f c 3 and d y qixi ls a relative stable map the space of such mor phisms was compactified by ga and li and they used this to define relative gromov witten invariants

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

nonlinear galerkin method using chebyshev and legendre polynomials i the one dimensional case a new strategy stemming from the nonlinear galerkin method m marion and r temam siam j numer anal 26 1989 pp 1139 1157 for solving linear elliptic and nonlinear dissipative evolution equations by using chebyshev and legendre polynomials is presented the essential idea is to decompose the solution into a low mode part and a high mode part and to treat them separately the robustness of the method over the classical galerkin method is substantiated by rigorous error estimates and preliminary numerical experiments

a postprocessed galerkin method with chebyshev or legendre polynomials we present an approximate inertial manifold based postprocess to enhance chebyshev or legendre spectral galerkin methods we prove that the postprocess improves the order of convergence of the galerkin solution yielding the same accuracy as the nonlinear galerkin method numerical experiments show that the new method is computationally more efficient than galerkin and nonlinear galerkin methods new approximation results for chebyshev polynomials are presented

on the rate of convergence of the nonlinear galerkin methods in this paper we provide estimates to the rate of convergence of the nonlinear galerkin approximation method in particular and by means of an illustrative example we show that the nonlinear galerkin method converges faster than the usual galerkin method

total and partial amplitude death in networks of diffusively coupled oscillators networks of weakly nonlinear oscillators are considered with diffusive and time delayed coupling averaging theory is used to determine parameter ranges for which the network experiences amplitude death whereby oscillations are quenched and the equilibrium solution has a large domain of attraction the amplitude death is shown to be a common phenomenon which can be observed regardless of the precise nature of the nonlinearities and under very general coupling conditions in addition when the network consists of dissimilar oscillators there exist parameter values for which only parts of the network are suppressed sufficient conditions are derived for total and partial amplitude death in arbitrary network topologies with general nonlinearities coupling coefficients and connection delays

enumeration of rational plane curves tangent to a smooth cubic abstract we compute the number of rational degree d plane curves having prescribed fixed and moving contacts to a smooth plane cubic e we use twisted stable maps to the stack p e r 2 for r large where p e r 2 is the rth root of p 2 along e we prove that certain gromov witten invariants of this stack are enumerative and establish recursive formulas for these numbers

the intrinsic normal cone let x be an algebraic stack in the sense of deligne mumford we construct a purely 0 dimensional algebraic stack over x in the sense of artin the intrinsic normal cone frak c x the notion of perfect obstruction theory for x is introduced and it is shown how to construct given a perfect obstruction theory for x a pure dimensional virtual fundamental class in the chow group of x we then prove some properties of such classes both in the absolute and in the relative context via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory and sometimes a perfect one

on the analysis of the double hopf bifurcation in machining processes via centre manifold reduction the single degree of freedom model of orthogonal cutting is investigated to study machine tool vibrations in the vicinity of a double hopf bifurcation point centre manifold reduction and normal form calculations are performed to investigate the long term dynamics of the cutting process the normal form of the four dimensional centre subsystem is derived analytically and the possible topologies in the infinite dimensional phase space of the system are revealed it is shown that bistable parameter regions exist where unstable periodic and in certain cases unstable quasi periodic motions coexist with the equilibrium taking into account the non smoothness caused by loss of contact between the tool and the workpiece the boundary of the bistable region is also derived analytically the results are verified by numerical continuation the possibility of transient chaotic motions in the global non smooth dynamics is shown

memoryless stabilization of uncertain linear systems including time varying state delays the problem of stabilizing a class of uncertain time delay systems via memoryless linear feedback is examined the systems under consideration are linear systems with time varying state delays they also contain uncertain parameters possibly time varying whose values are known only to within a prescribed compact bounding set the main contribution given is to enlarge the class of time delay systems for which one can construct a stabilizing memoryless linear feedback controller within this framework a novel notion of robust memoryless stabilizability is first introduced via the method of lyapunov functionals then a sufficient condition for the stabilizability is proposed it is shown that solvability of a parameterized riccati equation can be used to determine whether the time delay system satisfies the sufficient condition if there exists a positive definite symmetric solution satisfying the riccati equation a suitable memoryless linear feedback law can be derived

robust stabilization to non linear delayed systems via delayed state feedback the averaging method the problem of robust stabilization to dynamical systems via delayed feedback is important in applications due to the fact that the characteristic quasi polynomial of a delayed system is difficult to analyze when uncertain parameters are involved this problem has been most frequently solved on the basis of the method of lyapunov functional by solving riccati equations or by solving linear matrix inequalities by applying the averaging method that reduces the delay differential equation of infinite dimensional to an ordinary differential equation this paper presents a simple method to stabilize the trivial solution or periodic solutions of a type of non linear delayed vibration systems via delayed state feedback in particular this method is applied to the robust stabilization of those systems when the system parameters are uncertain but fall into given intervals respectively in addition an extension is made to this problem for a general class of delayed systems that result from a small perturbation of a linear delay system with characteristic roots of non positive real parts only this can serve as a straightforward application to the hopf bifurcation control of delayed systems with weak non linearity

postprocessing the galerkin method a novel approach to approximate inertial manifolds a postprocess of the standard galerkin method for the discretization of dissipative equations is presented the postprocessed galerkin method uses the same approximate inertial manifold phi app to approximate the high wave number modes of the solution as in the nonlinear galerkin nlg method however in this postprocessed galerkin method the value of phi app is calculated only once after the time integration of the standard galerkin method is completed contrary to the nlg in which phi app evolves with time and affects the time evolution of the lower wave number modes the postprocessed galerkin method which is much cheaper to implement computationally than the nlg is shown in the case of fourier modes to possess the same rate of convergence accuracy as the simplest version of the nlg which is based on either the foias manley temam approximate inertial manifold or the euler galerkin approximate inertial manifold this is proved for some problems in one and two spatial dimensions including the navier stokes equations under periodic boundary conditions the advantages of postprocessing that we present here apply not only to the standard galerkin method but also to the computationally more efficient pseudospectral method

efficient spectral galerkin method ii direct solvers of second and fourth order equations using chebyshev polynomials efficient direct solvers based on the chebyshev galerkin methods for second and fourth order equations are presented they are based on appropriate base functions for the galerkin formulation that lead to discrete systems with special structured matrices that can be efficiently inverted numerical results indicate that the direct solvers presented in this paper are significantly more accurate and efficient than those based on the chebyshev tau method

an approximate inertial manifolds approach to postprocessing the galerkin method for the navier stokes equation in a recent paper we have introduced a postprocessing procedure for the galerkin method for dissipative evolution partial differential equations with periodic boundary conditions the postprocessing technique uses approximate inertial manifolds to approximate the high modes the small scale components in the exact solutions in terms of the galerkin approximations which in this case play the role of the lower modes large scale components this procedure can be seen as a defect correction technique but contrary to standard procedures the correction is computed only when the time evolution is completed here we extend these results to more realistic boundary conditions specifically we study in detail the two dimensional navier stokes equations subject to homogeneous nonslip dirichlet boundary conditions we also discuss other equations such as reaction diffusion systems and the cahn hilliard equations

averaging of linear systems with almost periodic coefficients a time delay approach abstract we study stability of linear systems with fast almost periodic coefficients that are piecewise continuous in time the classical averaging method guarantees the stability of such systems for small enough values of parameter provided the corresponding averaged system is stable however it is difficult to find an upper bound on the small parameter by using classical tools for asymptotic analysis in this paper we introduce an efficient constructive method for finding an upper bound on the value of the small parameter that guarantees a desired exponential decay rate we transform the system into a model with time delays of the length of the small parameter the resulting time delay system is a perturbation of the averaged system the averaged system is supposed to be exponentially stable the stability of the time delay system guarantees the stability of the original one we construct an appropriate lyapunov functional for finding sufficient stability conditions in the form of linear matrix inequalities lmis the upper bound on the small parameter that preserves the exponential stability is found from the lmis two numerical examples stabilization by vibrational control and by time dependent switching illustrate the efficiency of the method moreover we apply the time delay approach to persistently excited systems that leads to a novel quadratic time independent lyapunov functional for such systems we further extend our method to input to state stability iss analysis finally the results are extended to linear fast varying systems with time varying delays

vibrational control of nonlinear time lag systems with bounded delay averaging theory stabilizability and transient behavior this paper develops the theory of vibrational control of nonlinear time lag systems with arbitrarily large but bounded delay averaging theory for fast oscillating differential delay equations is presented and then applied to vibrational control conditions are given which ensure the existence of parametric vibrations that stabilize nonlinear time lag systems transient behavior is also discussed illustrative examples are given which show 1 the feasibility of the theory to important applications and 2 the differences in the theory presented and the existing known theory for vibrational control of ordinary differential equations

a riccati equation approach to the stabilization of uncertain linear systems this paper presents a method for designing a feedback control law to stabilize a class of uncertain linear systems the systems under consideration contain uncertain parameters whose values are known only to within a given compact bounding set furthermore these uncertain parameters may be time varying the method used to establish asymptotic stability of the closed loop system obtained when the feedback control is applied involves the use of a quadratic lyapunov function the main contribution of this paper involves the development of a computationally feasible algorithm for the construction of a suitable quadratic lyapunov function once the lyapunov function has been obtained it is used to construct the stabilizing feedback control law the fundamental idea behind the algorithm presented involves constructing an upper bound for the lyapunov derivative corresponding to the closed loop system this upper bound is a quadratic form by using this upper bounding procedure a suitable lyapunov function can be found by solving a certain matrix riccati equation another major contribution of this paper is the identification of classes of systems for which the success of the algorithm is both necessary and sufficient for the existence of a suitable quadratic lyapunov function

simple stability criteria for single and composite linear systems with time delays easy ways to test the stability of systems involving time delays have been sought in this paper several stability conditions with an extremely simple form are provided first some criteria for single linear systems with time delays are presented then these results are extended to the composite linear systems with time delays which exist in each subsystem and in the interconnections between subsystems among these criteria those which are expressed by scalar inequalities also permit us to assess the transient behaviour of systems

stability analysis of linear systems with time delay this paper presents asymptotic stability criteria for linear systems with time delay the results not only improve previous results but also provide a bound for the delay time such that if the system is asymptotically stable when the delay does not exist it retains the asymptotic stability when the delay time is within the bound

aeroelastic dynamic response and control of an airfoil section with control surface nonlinearities abstract nonlinearities in aircraft mechanisms are inevitable especially in the control system it is necessary to investigate the effects of them on the dynamic response and control performance of aeroelastic system in this paper based on the state dependent riccati equation method a state feedback suboptimal control law is derived for aeroelastic response and flutter suppression of a three degree of freedom typical airfoil section with the control law designed nonlinear effects of freeplay in the control surface and time delay between the control input and actuator are investigated by numerical approach a cubic nonlinearity in pitch degree is adopted to prevent the aeroelastic responses from divergence when the flow velocity exceeds the critical flutter speed for the system with a freeplay the responses of both open and closed loop systems are determined with runge kutta algorithm in conjunction with henon s method this method is used to locate the switching points accurately and efficiently as the system moves from one subdomain into another the simulation results show that the freeplay leads to a forward phase response and a slight increase of flutter speed of the closed loop system the effect of freeplay on the aeroelastic response decreases as the flow velocity increases the time delay between the control input and actuator may impair control performance and cause high frequency motion and quasi periodic vibration

legendre and chebyshev spectral approximations of burgers equation we give error estimates for both legendre and chebyshev spectral approximations of the steady state burgers problem u xx lambda uu x f 0 in i 1 1 u 1 u 1 0 to do that we prove some abstract approximation properties of orthogonal projection operators in some weighted sobolev spacesh s i for both legendre and chebyshev weights

a posteriori error estimation with the p version of the finite element method for nonlinear parabolic differential equations we analyze an a posteriori error estimator for nonlinear parabolic differential equations in several space dimensions the spatial discretization is carried out using the p version of the finite element method the error estimates are obtained by solving an elliptic problem at the desired times when the estimation is wanted some numerical experiments prove the efficiency of the error estimation

relative gromov witten invariants and the mirror formula let x be a smooth complex projective variety and let y subset x be a smooth very ample hypersurface such that k y is nef using the technique of relative gromov witten invariants we give a new short and geometric proof of a version of the mirror formula i e we show that the generating function of the genus zero 1 point gromov witten invariants of y can be obtained from that of x by a certain change of variables the so called mirror transformation moreover we use the same techniques to give a similar expression for the virtual numbers of degree d plane rational curves meeting a smooth cubic at one point with multiplicity 3d which play a role in local mirror symmetry

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