abel lemma based finite sum inequality and its application to stability analysis for linear discrete time delay systems this paper is concerned with stability of linear discrete time delay systems note that a tighter estimation on a finite sum term appearing in the forward difference of some lyapunov functional leads to a less conservative delay dependent stability criterion by using abel lemma a novel finite sum inequality is established which can provide a tighter estimation than the ones in the literature for the finite sum term applying this abel lemma based finite sum inequality a stability criterion for linear discrete time delay systems is derived it is shown through numerical examples that the stability criterion can provide a larger admissible maximum upper bound than stability criteria using a jensen type inequality approach and a free weighting matrix approach

new results on stability analysis for systems with discrete distributed delay the integral inequality technique is widely used to derive delay dependent conditions and various integral inequalities have been developed to reduce the conservatism of the conditions derived in this study a new integral inequality was devised that is tighter than existing ones it was used to investigate the stability of linear systems with a discrete distributed delay and a new stability condition was established the results can be applied to systems with a delay belonging to an interval which may be unstable when the delay is small or nonexistent three numerical examples demonstrate the effectiveness and the smaller conservatism of the method

technical communique improved stability criterion and its applications in delayed controller design for discrete time systems this paper is concerned with the problem of delay dependent stability analysis for discrete time systems with interval like time varying delays and the problem of stabilization for discrete time linear systems via time delayed controllers the first problem is solved by applying a novel lyapunov functional and an improved delay dependent stability criterion is obtained in terms of a linear matrix inequality based on this a sufficient condition for the solvability of the second problem is presented the reduced conservatism of the proposed stability result is shown through a numerical example while the applicability of the time delayed controller design method is demonstrated by an inverted pendulum system

stochastic finite time h state estimation for discrete time semi markovian jump neural networks with time varying delays in this article the finite time h infty state estimation problem is addressed for a class of discrete time neural networks with semi markovian jump parameters and time varying delays the focus is mainly on the design of a state estimator such that the constructed error system is stochastically finite time bounded with a prescribed h infty performance level via finite time lyapunov stability theory by constructing a delay product type lyapunov functional in which the information of time varying delays and characteristics of activation functions are fully taken into account and using the jensen summation inequality the free weighting matrix approach and the extended reciprocally convex matrix inequality some sufficient conditions are established in terms of linear matrix inequalities to ensure the existence of the state estimator finally numerical examples with simulation results are provided to illustrate the effectiveness of our proposed results

new results on stability of discrete time systems with time varying state delay this note is concerned with the stability analysis of discrete time systems with time varying state delay by defining new lyapunov functions and by making use of novel techniques to achieve delay dependence several new conditions are obtained for the asymptotic stability of these systems the merit of the proposed conditions lies in their less conservativeness which is achieved by circumventing the utilization of some bounding inequalities for cross products between two vectors and by paying careful attention to the subtle difference between the terms sigmam k dk k 1 middot and sigma m k dm k 1 middot which is largely ignored in the existing literature these conditions are shown via several examples to be much less conservative than some existing result

a novel lyapunov functional for stability of time varying delay systems via matrix refined function this paper is concerned with the development of novel stability criteria for time varying delayed systems to this end a new function which consisted of two quadratic functions with a special structural matrix is established to be a lyapunov functional candidate the proposed lyapunov functional plays key role to decrease the conservatism of derived conditions which is verified by three numerical examples

further results on stability analysis of discrete time systems with time varying delays via the use of novel convex combination coefficients this paper focuses on deriving an improved stability criterion for discrete time delay systems via a lyapunov krasovskii functional approach that takes advantage of triple summation terms to this end novel convex combination coefficients coupled with second order time varying delay terms are introduced and the corresponding reciprocally convex approach is established with the use of the minimal number of slack matrix variables finally three numerical examples are provided to illustrate the effectiveness of the proposed method with particular regard to robustness and h performance

coloring trees in reverse mathematics abstract the tree theorem for pairs tt 2 2 first introduced by chubb hirst and mcnicholl asserts that given a finite coloring of pairs of comparable nodes in the full binary tree 2 there is a set of nodes isomorphic to 2 which is homogeneous for the coloring this is a generalization of the more familiar ramsey s theorem for pairs rt 2 2 which has been studied extensively in computability theory and reverse mathematics we answer a longstanding open question about the strength of tt 2 2 by showing that this principle does not imply the arithmetic comprehension axiom aca 0 over the base system recursive comprehension axiom rca 0 of second order arithmetic combined with a recent result of patey s that tt 2 2 is strictly stronger than rt 2 2 this establishes tt 2 2 as the first known example of a natural combinatorial principle to occupy the interval strictly between aca 0 and rt 2 2 the proof of this fact uses an extension of the bushy tree forcing method and develops new techniques for dealing with combinatorial statements formulated on trees rather than on

brief paper delay dependent stability for discrete systems with large delay sequence based on switching techniques this paper considers the exponential stability problem for a class of discrete time varying delay systems with large delay sequences ldss a new method based on a switching technique is presented to solve this problem a switched delay system in which one of the discrete subsystems may be unstable is firstly employed to describe such a system and then some new concepts about ldss are introduced next using a novel piecewise lyapunov functional explicit delay dependent conditions are developed to show how long and how frequent the ldss can be and still maintain the system exponentially stable without ldss the criterion obtained in this paper includes the established one as a special case two numerical examples are given to show the effectiveness of the proposed method

wirtinger based integral inequality application to time delay systems in the last decade the jensen inequality has been intensively used in the context of time delay or sampled data systems since it is an appropriate tool to derive tractable stability conditions expressed in terms of linear matrix inequalities lmis however it is also well known that this inequality introduces an undesirable conservatism in the stability conditions and looking at the literature reducing this gap is a relevant issue and always an open problem in this paper we propose an alternative inequality based on the fourier theory more precisely on the wirtinger inequalities it is shown that this resulting inequality encompasses the jensen one and also leads to tractable lmi conditions in order to illustrate the potential gain of employing this new inequality with respect to the jensen one two applications on time delay and sampled data stability analysis are provided

free matrix based integral inequality for stability analysis of systems with time varying delay the free weighting matrix and integral inequality methods are widely used to derive delay dependent criteria for the stability analysis of time varying delay systems because they avoid both the use of a model transformation and the technique of bounding cross terms this technical note presents a new integral inequality called a free matrix based integral inequality that further reduces the conservativeness in those methods it includes well known integral inequalities as special cases using it to investigate the stability of systems with time varying delays yields less conservative delay dependent stability criteria which are given in terms of linear matrix inequalities two numerical examples demonstrate the effectiveness and superiority of the method

h control of discrete time linear systems with norm bounded uncertainties and time delay in state this paper deals with h control problem for discrete time linear systems with norm bounded uncertainties and time delay in state in terms of a modified riccati inequality for discrete time linear systems a sufficient condition is presented and a state feedback control law is also given the proposed control law is a memoryless state feedback control law

comparison of some bounding inequalities applied in stability analysis of time delay systems abstract this paper gives comparisons of several bounding inequalities proposed recently for the estimation of integrals and sums of quadratic functions which plays an important role in stability analysis of time delay systems via the lyapunov krasovskii approach the equivalences of several known variants of the free matrix based inequalities and their generalized and simplified forms are shown secondly the relationship between the simplified free matrix based inequality and the combination of the bessel based inequality with different bounding inequalities affine in the length of the intervals are investigated finally we shall demonstrate the theoretical results with the application of the various estimations for the stability analysis of a linear system with time varying delay using the same lyapunov krasovskii functional

stability analysis of systems with time varying delay via novel augmented lyapunov krasovskii functionals and an improved integral inequality abstract delay dependent stability analysis of linear systems with a time varying delay is investigated in this paper firstly instead of developing a lyapunov krasovskii functional lkf including augmented non integral quadratic terms as usual this paper proposes two augmented integral function based lkfs which can reflect closer relationships among system states secondly to bound the derivative of lkf more accurately a new integral inequality with several free matrices is developed compared with the free matrix based integral inequality this inequality can provide additional freedom due to the free matrices introduced then by employing those lkfs and the improved integral inequality several delay dependent stability criteria are established for two types of delays finally two numerical examples are given to demonstrate the superiority of the proposed method

extracting information is hard a turing degree of non integral effective hausdorff dimension abstract we construct a 2 0 infinite binary sequence with effective hausdorff dimension 1 2 that does not compute a sequence of higher dimension introduced by lutz effective hausdorff dimension can be viewed as a measure of the information density of a sequence in particular the dimension of a 2 is the lim inf of the ratio between the information content and length of initial segments of a thus the main result demonstrates that it is not always possible to extract information from a partially random source to produce a sequence that has higher information density

new stability criteria for linear discrete time systems with interval like time varying delays this technical note is concerned with stability of linear discrete time systems with interval like time varying delays two new stability criteria are derived by constructing a new lyapunov functional and utilizing two novel techniques for estimating the forward difference of the lyapunov functional the relationship between these stability criteria and some existing ones is established through the relationship it is clear that these stability criteria are less conservative than some existing ones which is confirmed by a numerical example

reciprocally convex approach to stability of systems with time varying delays whereas the upper bound lemma for matrix cross product introduced by park 1999 and modified by moon park kwon and lee 2001 plays a key role in guiding various delay dependent criteria for delayed systems the jensen inequality has become an alternative as a way of reducing the number of decision variables it directly relaxes the integral term of quadratic quantities into the quadratic term of the integral quantities resulting in a linear combination of positive functions weighted by the inverses of convex parameters this paper suggests the lower bound lemma for such a combination which achieves performance behavior identical to approaches based on the integral inequality lemma but with much less decision variables comparable to those based on the jensen inequality lemma

output feedback stabilization for a discrete time system with a time varying delay this study employs the free weighting matrix approach to investigate the output feedback control of a linear discrete time system with an interval time varying delay first the delay dependent stability is analyzed using a new method of estimating the upper bound on the difference of a lyapunov function without ignoring any terms and based on the results a design criterion for a static output feedback sof controller is derived since the conditions thus obtained for the existence of admissible controllers are not expressed strictly in terms of linear matrix inequalities a modified cone complementarity linearization algorithm is employed to solve the nonconvex feasibility sof control problem furthermore the problem of designing a dynamic output feedback controller is formulated as one of designing an sof controller numerical examples demonstrate the effectiveness of the method and its advantage over existing methods

on equivalence and efficiency of certain stability criteria for time delay systems in recent years there have been a number of new delay dependent stability criteria based on linear matrix inequalities published in the literature this note aims to theoretically establish the equivalence of seven of these stability criteria moreover the efficiency of these stability criteria is assessed based on the number of unknowns in the linear matrix inequalities

finite dimensional approximation of nonlinear problems in the first two papers of this series 4 5 we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a banach space we derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point the abstract theory is applied to the galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von karman equations

delay variation dependent stability of delayed discrete time systems this note is concerned with the stability analysis of linear discrete time system with a time varying delay a generalized free weighting matrix gfwm approach is proposed to estimate summation terms in the forward difference of lyapunov functional and theoretical study shows that the gfwm approach encompasses several frequently used estimation approaches as special cases moreover an augmented lyapunov functional with a delay product type term is constructed to take into account delay changing information as a result the proposed gfwm approach together with the augmented lyapunov functional leads to a less conservative delay variation dependent stability criterion finally numerical examples are given to illustrate the advantages of the proposed criterion

technical communique stability and robust stability for systems with a time varying delay to concern the stability and robust stability criteria for systems with time varying delays this note uses not only the time varying delayed state x t h t but also the delay upper bounded state x email protected to exploit all possible information for the relationship among a current state x t an exactly delayed state x t h t a marginally delayed state x email protected and the derivative of the state email protected t when constructing lyapunov krasovskii functionals and some appropriate integral inequalities originally suggested by park 1999 a delay dependent stability criterion for systems with uncertain time invariant delays ieee transactions on automatic control 44 4 876 877 two fundamental criteria are provided for the cases where no bound of delay derivative is assumed and where an upper bound of delay derivative is assumed examples show the resulting criteria outperform all existing ones in the literature

on a numerical liapunov schmidt spectral method and its application to biological pattern formation spectral expansions are used to provide a basis which preserves continuous symmetries we show that spectral methods satisfy the conditions for convergence of numerical liapunov schmidt methods an explicit algorithm for the calculation of stationary bifurcation scenarios near primary instabilities in general continuous symmetric equations is given the above convergence is extended to gamma equivalent discrete and original bifurcation scenarios the method is applied to a biologically motivated reaction diffusion system with spherical symmetry forming patterns a specific singularity of a generic steady state bifurcation is investigated in detail

stability of discrete time systems with time varying delays via a novel summation inequality this technical note is concerned with the stability analysis of discrete linear systems with time varying delays the novelty of the technical note comes from the consideration of a new inequality which is less conservative than the celebrated jensen inequality employed in the context of discrete time delay systems this inequality is a discrete time counterpart of the wirtinger based integral inequality that was recently employed for the improved analysis of continuous tine systems with delays however differently from the continuous time case the proof of the new inequality is not based on the wirtinger inequality the method is also combined with an efficient representation of the improved reciprocally convex combination inequality in order to reduce the conservatism induced by the lmis optimization setup the effectiveness of the proposed result is illustrated by some classical examples from the literature

stability analysis for delayed neural networks considering both conservativeness and complexity this paper investigates delay dependent stability for continuous neural networks with a time varying delay this paper aims at deriving a new stability criterion considering tradeoff between conservativeness and calculation complexity a new lyapunov krasovskii functional with simple augmented terms and delay dependent terms is constructed and its derivative is estimated by several techniques including free weighting matrix and inequality estimation methods then the influence of the techniques used on the conservativeness and the complexity is analyzed one by one moreover useful guidelines for improving criterion and future work are briefly discussed finally the advantages of the proposed criterion compared with the existing ones are verified based on three numerical examples

pigeons do not jump high abstract the infinite pigeonhole principle for 2 partitions asserts the existence for every set a of an infinite subset of a or of its complement in this paper we develop a new notion of forcing enabling a fine analysis of the computability theoretic features of the pigeonhole principle we deduce various consequences such as the existence for every set a of an infinite subset of it or its complement of non high degree we also prove that every 0 3 set has an infinite low3 solution and give a simpler proof of liu s theorem that every set has an infinite subset in it or its complement of non pa degree

extended dissipativity analysis for discrete time delayed neural networks based on an extended reciprocally convex matrix inequality abstract in this paper the extended dissipativity analysis for discrete time neural networks with a time varying delay is investigated first a novel lyapunov krasovskii functional lkf is constructed with a delay product type term introduced then in the forward difference of the lkf the sum terms are bounded via an extended reciprocally convex matrix inequality as a result an extended dissipativity criterion is established in terms of linear matrix inequalities meanwhile this criterion is extended to the stability analysis of the counterpart system without disturbance finally two numerical examples are given to demonstrate the effectiveness and improvements of the presented criterion

summation inequalities to bounded real lemmas of discrete time systems with time varying delay summation inequality is an important technique for analysis of discrete time systems with a time varying delay it seems that from the literature a tighter inequality usually leads to a less conservative criterion based on h infty performance analysis problem this note presents different findings on the relationship between the conservatism of bounded real lemma brl and the tightness of summation inequality firstly the brl obtained by the wirtinger based inequality wbi is not always less conservative than the one by the jensen based inequality although the wbi is tighter secondly the wbi is tighter than a general free matrix based inequality gfmbi developed in this note while the brl obtained via the gfmbi is less conservative than the wbi based brl finally a numerical example is given to demonstrate those findings

exploiting symmetry in boundary element methods linear operator equations mathcal l f g are considered in the context of boundary element methods where the operator mathcal l is equivariant i e commutes with the actions of a given finite symmetry group by introducing a generalization of reynolds projectors a decomposition of the identity operator is constructed which in turn allows the decomposition of the problem mathcal l f g into a finite number of symmetric subproblems the data function g does not need to possess any symmetry properties it is shown that analogous reductions are possible for discretizations an explicit construction of the corresponding reduced system matrices is given this effects a considerable reduction in the computational complexity for example in the case of the isometry group of the 3 cube the computational complexity of a direct linear equation solver for full matrices is reduced by 99 65 percent specific decompositions of the identity are given for most of the significant finite isometry group

a numerical liapunov schmidt method with applications to hopf bifurcation on a square we discuss an iterative method for calculating the reduced bifurcation equation of the liapunov schmidt method and its numerical approximation using appropriate genericity assumptions with symmetry we derive a taylor series for the reduced equation where the bifurcation behavior is determined by its numerical approximation at a finite order of truncation this method is used to calculate reduced equations at hopf bifurcation of the two dimensional brusselator equations on a square with neumann and dirichlet boundary conditions we examine several hopf bifurcations within the three parameter space there are regions where we observe direct bifurcation to branches of periodic solutions with submaximal symmetry

computation of cusp singularities for operator equations and their discretizations we discuss the direct calculation of cusp singularities as solutions of a minimally augmented defining system which is nonsingular under the canonical cusp conditions the underlying operator equation may have any finite fredholm index and bounds on the discretization error are derived for the case of projection methods

notes on stability of time delay systems bounding inequalities and augmented lyapunov krasovskii functionals the bounding inequalities and the lyapunov krasovskii functionals lkfs are important for the stability analysis of time delay systems much attention has been paid to develop tighter inequalities for improving stability criteria while the contribution of the lkfs has not been considered when discussing the relationship between the tightness of inequalities and the conservatism of criteria this note is concerned with this issue firstly it is proved that when a simple lkf is applied the stability criteria obtained by the wirtinger based inequality and the jensen inequality are equivalent although the wirtinger based inequality is tighter it means that the tighter inequality does not always lead to a less conservative criterion secondly it is found that a suitable augmented lkf with necessary integral vectors in its derivative is required to achieve the advantage of the wirtinger based inequality based on this observation two delay product type terms are introduced into the lkf to establish new stability criteria finally a numerical example is given to verify the equivalence statements and to show the benefit of the proposed criteria

technical communique a delay partitioning approach to the stability analysis of discrete time systems this paper revisits the problem of stability analysis for linear discrete time systems with time varying delay in the state by utilizing the delay partitioning idea new stability criteria are proposed in terms of linear matrix inequalities lmis these conditions are developed based on a novel lyapunov functional in addition to delay dependence the obtained conditions are also dependent on the partitioning size we have also established that the conservatism of the conditions is a non increasing function of the number of partitions numerical examples are given to illustrate the effectiveness and advantage of the proposed methods

pattern selection with o 3 symmetry abstract for each irreducible representation of so 3 and o 3 we determine up to conjugacy all isotropy subgroups and identify in particular the maximal isotropy subgroups each isotropy subgroup corresponds to a possible planform for the spherical benard problem using an equivariant branching lemma of cicogna 1 we prove for each of these representations the existence of solutions corresponding to a number of different planforms thus extending substantially the work of busse 2 3 and sattinger 4 we also give a useful criterion for showing when solutions obtained by the equivariant branching lemma must be unstable