best possible global bounds for jensen s inequality in this article we found the form of best possible global upper bound for jensen s inequality thereby previous results on this topic are essentially improved we also give some applications in analysis and information theory

novel summation inequalities and their applications to stability analysis for systems with time varying delay the inequality technique plays an important role in stability analysis for time delay systems this technical note presents a new sequence of novel summation inequalities by introducing some free matrices which includes the newly developed wirtinger based and free matrix based summation inequalities as special cases moreover the idea can be easily extended to the multiple summation inequality case based on the proposed inequalities relaxed stability conditions are obtained for systems with time varying delay numerical examples are given to demonstrate the effectiveness of the proposed approach

exponential stability of impulsive systems with application to uncertain sampled data systems we establish exponential stability of nonlinear time varying impulsive systems by employing lyapunov functions with discontinuity at the impulse times our stability conditions have the property that when specialized to linear impulsive systems the stability tests can be formulated as linear matrix inequalities lmis then we consider lti uncertain sampled data systems in which there are two sources of uncertainty the values of the process parameters can be unknown while satisfying a polytopic condition and the sampling intervals can be uncertain and variable we model such systems as linear impulsive systems and we apply our theorem to the analysis and state feedback stabilization we find a positive constant which determines an upper bound on the sampling intervals for which the stability of the closed loop is guaranteed the control design lmis also provide controller gains that can be used to stabilize the process we also consider sampled data systems with constant sampling intervals and provide results that are less conservative than the ones obtained for variable sampling intervals

an iqc approach to robust stability of aperiodic sampled data systems conditions for robust stability of sampled data systems with non uniform sampling patterns and structural uncertainties are derived the problem is tackled under the integral quadratic constraint iqc framework where the aperiodic sampling operation is modelled by an delay integration operator characterization based on integral quadratic constrains iqc is identified for this operator and the iqc theory is applied to derive convex stability criteria compared to the dominating lyapunov approach where the candidate lyapunov krasovskii functionals or looped functionals need to be tailored for the systems under consideration and therefore the stability conditions need to be re derived whenever additional uncertainties are considered the proposed approach has the advantage of avoiding such endeavor numerical examples are given to illustrate this main point and effectiveness of the proposed approach

a note on relationship between two classes of integral inequalities this technical note firstly introduces two classes of integral inequalities with and without free matrices respectively and points out that they although in different forms are actually equivalent in the sense of conservatism i e the two corresponding ones produce the same tight upper bounds secondly the relationship between the method of integral inequalities with free matrices and the free weighting matrix technique is intensively investigated it is shown that these two different methods are actually equivalent in assessing the stability of time delay systems

stability analysis of sampled data systems using sum of squares this technical brief proposes a new approach to stability analysis of linear systems with sampled data inputs or channels the method based on a variation of the discrete time lyapunov approach provides stability conditions using functional variables subject to convex constraints these stability conditions can be solved using the sum of squares methodology with little or no conservatism in both the case of synchronous and asynchronous sampling numerical examples are included to show convergence

affine bessel legendre inequality application to stability analysis for systems with time varying delays abstract recently some novel inequalities have been proposed such as the auxiliary function based integral inequality and the bessel legendre inequality which can be obtained from the former by choosing legendre polynomials as auxiliary functions these inequalities have been successfully applied to systems with constant delays but there have been some difficulties in application to systems with time varying delays since the resulting bounds contain the reciprocal convexity which may not be tractable as it is this paper proposes an equivalent form of the bessel legendre inequality which has the advantage of being easily applied to systems with time varying delays without the reciprocal convexity

recent developments on the stability of systems with aperiodic sampling an overview this article presents basic concepts and recent research directions about the stability of sampled data systems with aperiodic sampling we focus mainly on the stability problem for systems with arbitrary time varying sampling intervals which has been addressed in several areas of research in control theory systems with aperiodic sampling can be seen as time delay systems hybrid systems input output interconnections discrete time systems with time varying parameters etc the goal of the article is to provide a structural overview of the progress made on the stability analysis problem without being exhaustive which would be neither possible nor useful we try to bring together results from diverse communities and present them in a unified manner for each of the existing approaches the basic concepts fundamental results converse stability theorems when available and relations with the other approaches are discussed in detail results concerning extensions of lyapunov and frequency domain methods for systems with aperiodic sampling are recalled as they allow to derive constructive stability conditions furthermore numerical criteria are presented while indicating the sources of conservatism the problems that remain open and the possible directions of improvement at last some emerging research directions such as the design of stabilizing sampling sequences are briefly discussed

polynomial lpv approach to robust h control of nonlinear sampled data systems this paper addresses l2 gain analysis and robust h control of nonlinear sampled data systems described by a polynomial linear parameter varying plpv model first conditions for l2 gain analysis of sampled data plpv systems are derived using a modified krasovskii functional in which the distance between the real and measured parameters are considered as uncertainties the resulting conditions are formulated in terms of parameterised linear matrix inequalities plmi second a solution approach for satisfying plmi conditions over the set of whole parameters and uncertainties is proposed using the sum of squares decomposition method third conditions for robust h sampled data controller synthesis for nonlinear systems described by plpv models are derived in terms of parameterised bilinear matrix inequalities pbmi with the maximum allowable sampling period as the parameter to make the resulting bilinear matrix inequality bmi problems tractable an iterative algorithm is developed and utilised to s

noncommutative plurisubharmonic polynomials part ii local assumptions abstract we say that a symmetric noncommutative nc polynomial is nc plurisubharmonic nc plush on an nc open set if it has an nc complex hessian that is positive semidefinite when evaluated on open sets of matrix tuples of sufficiently large size in this paper we show that if an nc polynomial is nc plurisubharmonic on an nc open set then the polynomial is actually nc plurisubharmonic everywhere and has the form 0 1 p f j t f j k j k j t f f t where the sums are finite and f j k j f are all nc analytic greene et al 2011 1 has shown that if p is nc plurisubharmonic everywhere then p has the form in eq 0 1 in other words 1 makes a global assumption while the current paper makes a local assumption but both reach the same conclusion we show that if p is nc plurisubharmonic on an nc open set local then p is in fact nc plurisubharmonic everywhere global and has the form expressed in eq 0 1 this paper requires a technique that is not used in 1 we use a gram like vector and matrix representation called the border vector and middle matrix for homogeneous degree 2 nc polynomials we then analyze this representation for the nc complex hessian on an nc open set and positive semidefiniteness forces a very rigid structure on the border vector and middle matrix this rigid structure plus the theorems in 1 ultimately force the form in eq 0 1

stability analysis of linear systems with two delays of overlapping ranges in this paper a delay dependent stability criterion is derived for linear time delay systems with two constant delays having overlapping ranges this overlapping feature of the delays is exploited in delay dependent analysis of the system using lyapunov krasovskii approach the developed treatment of the delays reduces conservatism in analysis compared to the approaches treating the delays individually numerical examples are used to demonstrate the less conservativeness of the derived criterion

convergence and equivalence results for the jensen s inequality application to time delay and sampled data systems the jensen s inequality plays a crucial role in the analysis of time delay and sampled data systems its conservatism is studied through the use of the gruss inequality it has been reported in the literature that fragmentation or partitioning schemes allow to empirically improve the results we prove here that the jensen s gap can be made arbitrarily small provided that the order of uniform fragmentation is chosen sufficiently large nonuniform fragmentation schemes are also shown to speed up the convergence in certain cases finally a family of bounds is characterized and a comparison with other bounds of the literature is provided it is shown that the other bounds are equivalent to jensen s and that they exhibit interesting well posedness and linearity properties which can be exploited to obtain better numerical results

an extended reciprocally convex matrix inequality for stability analysis of systems with time varying delay abstract the reciprocally convex combination lemma rccl is an important technique to develop stability criteria for the systems with a time varying delay this note develops an extended reciprocally convex matrix inequality which reduces the estimation gap of the rccl based matrix inequality and reduces the number of decision variables of the recently proposed delay dependent rccl a stability criterion of a linear time delay system is established through the proposed matrix inequality finally a numerical example is given to demonstrate the advantage of the proposed method

stability analysis of linear impulsive delay dynamical systems via looped functionals this article investigates the asymptotic stability of impulsive delay dynamical systems idds by using the lyapunovkrasovskii method and looped functionals the proposed conditions reduce the conservatism of the results found in the literature by allowing the functionals to grow during both the continuous dynamics and the discrete dynamics sufficient conditions for asymptotic stability in the form of linear matrix inequalities lmi are provided for the case of impulsive delay dynamical systems with linear and time invariant lti base systems non impulsive actions several numerical examples illustrate the effectiveness of the method

brief paper robust h filtering for a class of uncertain linear systems with time varying delay this paper deals with the problem of delay dependent robust h filtering for uncertain linear systems with time varying delay two cases of time varying delays are fully considered one is the time varying delay being continuous uniformly bounded while the other is the time varying delay being differentiable uniformly bounded with delay derivative bounded by a constant a stable linear filter is designed to ensure that the filtering error system is asymptotically stable with a prescribed level of h noise attenuation based on a new integral inequality delay dependent sufficient conditions for the existence of such a filter are established in terms of linear matrix inequalities lmis through deriving these conditions neither model transformation nor bounding technique for cross terms is employed moreover the relationship between the sufficient conditions for the two cases of time varying delay is disclosed finally two examples are also given to illustrate the effectiveness of the proposed methodology

a generalization of gruss s inequality in inner product spaces and applications a generalization of the classical gruss s integral inequality in inner product spaces is given some applications for positive linear functionals and integrals are also pointed out

stability analysis of uncertain sampled data systems with incremental delay using looped functionals the robust stability analysis of asynchronous and uncertain sampled data systems with constant incremental input delay is addressed in the looped functional framework these functionals have been shown to be suitable for the analysis of impulsive systems as they allow one to express discrete time stability conditions in an affine way enabling then the consideration of uncertain and time varying systems the stability conditions are obtained by first reformulating the sampled data system as an impulsive system and by then considering a tailored looped functional along with wirtinger s inequality a recently introduced inequality that has been shown to be less conservative than jensen s inequality several examples are given for illustration

generalized jensen inequalities with application to stability analysis of systems with distributed delays over infinite time horizons the jensen inequality has been recognized as a powerful tool to deal with the stability of time delay systems recently a new inequality that encompasses the jensen inequality was proposed for the stability analysis of systems with finite delays in this paper we first present a generalized integral inequality and its double integral extension it is shown how these inequalities can be applied to improve the stability result for linear continuous time systems with gamma distributed delays then for the discrete time counterpart we provide an extended jensen summation inequality with infinite sequences which leads to less conservative stability conditions for linear discrete time systems with poisson distributed delays the improvements obtained by the introduced generalized inequalities are demonstrated through examples

technical communique absolute stability of time delay systems with sector bounded nonlinearity this paper deals with the problem of absolute stability of time delay systems with sector bounded nonlinearity some new delay dependent stability criteria are obtained and formulated in the form of linear matrix inequalities lmis neither model transformation nor bounding technique for cross terms is involved through derivation of the stability criteria numerical examples show that the results obtained in this paper improve the estimate of the stability limit over some existing result

quadratic separation for feedback connection of an uncertain matrix and an implicit linear transformation topological separation is investigated in the case of an uncertain time invariant matrix interconnected with an implicit linear transformation a quadratic separator independent of the uncertainty is shown to prove losslessly the closed loop well posedness several applications for lti descriptor system analysis are then given first some known results for stability and pole location of descriptor systems are demonstrated in a new way second contributions to robust stability analysis and time delay systems stability analysis are exposed these prove to be new even when compared to results for usual lti systems not in descriptor form all results are formulated as linear matrix inequalities lmis

cal h _ infty delay scheduled control of linear systems with time varying delays this paper deals with h infin delay scheduled control of linear systems with time varying delays first a new model transformation is given which allows to provide a unified approach to stability analysis and state feedback control synthesis for time delay systems represented in linear fractional transformation lft form a new type of controller is then synthesized where the state feedback is scheduled by the value of the delay delay scheduled controllers the results are provided in terms of linear matrix inequalities lmis which are known to be efficiently solvable

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

a generalized free matrix based integral inequality for stability analysis of time varying delay systems abstract this paper focuses on the delay dependent stability problem of time varying delay systems a generalized free matrix based integral inequality gfmbii is presented this inequality is able to deal with time varying delay systems without using the reciprocal convexity lemma it overcomes the drawback that the bessel legendre inequality is inconvenient to cope with a time varying delay system as the resultant bound contains a reciprocal convexity through the use of the derived inequality and by constructing a suitable lyapunov krasovskii function lkf improved stability criteria are presented in the form of linear matrix inequalities lmis two numerical examples are carried out to demonstrate that the results outperform the state of the art in the literature

sampled data fuzzy stabilization of nonlinear systems under nonuniform sampling this paper investigates the sampled data fuzzy stabilization problem for a class of nonlinear systems that is exactly modeled in takagi sugeno fuzzy form at least locally a new method for designing parallel distribution compensation fuzzy controller is proposed which just requires that the nonlinear function is locally lipschitz by considering the sample and hold behavior of the system and using jensen s integral inequality an inequality constrain condition is derived from the locally lipschitz property further by defining a time dependent lyapunov krasovskii functional term a new technique instead of the use of s procedure is developed and stabilization conditions for state feedback and observer based output feedback under nonuniform sampling are obtained compared with the existing ones the new design method not only avoids the difficulty of finding exact upper bounds of asynchronous errors of mismatch membership functions but contains less conservatism and less numerical complexity as well finally some illustrative examples are given to show the effectiveness of the proposed design method and the significant improvement over the existing results

noncommutative convexity arises from linear matrix inequalities this paper concerns polynomials in g noncommutative variables x x1 xg inverses of such polynomials and more generally noncommutative rational expressions with real coefficients which are formally symmetric and analytic near 0 the focus is on rational expressions r r x which are matrix convex near 0 i e those rational expressions r for which there is an 0 such that if x x1 xg is a g tuple of n n symmetric matrices satisfying in x12 xg2 is positive definite and y is also then the symmetric matrix tr x 1 t r y r tx 1 t y is positive semidefinite for all numbers t 0 t 1 this article gives a complete classification of matrix convex rational expressions see theorem 3 3 by representing such r in terms of a symmetric linear pencil l x id jajxj 0d 100 1 r 0 in the noncommuting variables xj where aj are symmetric d d matrices namely for a real number r is a schur complement of the linear pencil l moreover given a matrix convex r the set consisting of g tuples x of n n symmetric matrices 0 1 x r x iis negative definite has component containing 0 which is the same as the negativity set 0 2 x l x is negative definite for l conditions like l x is negative definite are known as linear matrix inequalities lmis in the engineering literature and arguably the main advance in linear systems theory in the 1990s was the introduction of lmi techniques in this language what we have shown in 0 1 vs 0 2 is that the set of solutions to a convex matrix inequality with noncommutative unknowns is the same as the set of solutions to some lmi in many engineering systems problems convexity would have all of the advantages of lmis indeed convexity guarantees that solutions are global and convexity bodes well for reliability of the numerics since lmis have a structure which is seemingly much more rigid than convexity there is the continual hope that a convexity based theory will be more far reaching than lmis but will it there are two natural situations one where the unknowns are scalars and one where the unknowns are matrices appearing in formulas which respect matrix multiplication these latter problems mathematically yield expressions with noncommutative unknowns and they arise in engineering systems problems which are dimensionless in the sense that they scale automatically with dimension as do most of the classics of control theory that is the case we study here and the result stated above suggests the surprising conclusion that for dimensionless systems problems convexity offers no greater generality than lmis indeed the result proves this for a class of model problems furthermore we show that existing algorithms together with algorithms described here construct the lmis above which are equivalent to the matrix inequalities based on the given matrix convex rational function r in a very different direction we prove that a symmetric polynomial p in g noncommutative symmetric variables has a symmetric determinantal representation namely there are symmetric matrices a0 ag in srd d with a0 invertible such that 0 3 detp x det a0 la x for each x a g tuple of symmetric n n matrices of course taking n 1 implies immediately that a commuting variables polynomial p on rg has a symmetric determinantal representation for g 2 much stronger commutative results can be obtained using tools of algebraic geometry but these do not seem to generalize to the higher dimensional case on the other hand a nonsymmetric commutative determinantal representation for any g is due to valiant universality of determinant in algebraic complexity theory our determinantal representation theorem is a bi product of the theory of systems realizations of noncommutative rational functions and can be read independently of much of the rest of the paper while the notion of noncommutative rational functions is standard the equivalence relation we use on rational expressions in our construction based on evaluating rational expressions on matrices is new and gives a new approach to noncommutative rational functions

state feedback synchronization control of impulsive neural networks with mixed delays and linear fractional uncertainties abstract this study examines the synchronization problem of impulsive neural networks with mixed time varying delays and linear fractional uncertainties the mixed time varying delays include distributed leakage discrete and distributed time varying delays moreover the restrictions on derivatives of time varying delays with upper bounds to smaller than one is relaxed by introducing free weight matrices based on suitable lyapunov krasovskii functionals and integral inequalities linear matrix inequality approach is used to derive the sufficient conditions that guarantee the synchronization criteria of impulsive neural networks via delay dependent state feedback control finally three numerical examples are given to show the effectiveness of the theoretical results

stability and stabilization of discrete time tźs fuzzy systems with time varying delay via cauchyźschwartz based summation inequality this paper proposes new stability and stabilization conditions for discrete time fuzzy systems with time varying delays by constructing a suitable lyapunov krasovskii functional and introducing a new summation inequality based on the inequality of cauchy schwartz form which enhances the feasible region of the stability criterion for discrete time systems with time varying delay a stability criterion for such systems is established in order to show the effectiveness of the proposed inequality which provides more tight lower bound of a summation term of quadratic form a delay dependent stability criterion for such systems is derived within the framework of linear matrix inequalities which can be easily solved by various effective optimization algorithms going one step forward the proposed inequality is applied to a stabilization problem in discrete time fuzzy systems with time varying delays the advantages of the proposed stability and stabilization criteria are illustrated via two numerical examples

noncommutative plurisubharmonic polynomials part i global assumptions abstract we consider symmetric polynomials p in the noncommutative nc free variables x 1 x 2 x g we define the nc complex hessian of p as the second directional derivative replacing x t by y q x x t h h t 2 p s t x t h y s k t s 0 y x t k h t we call an nc symmetric polynomial nc plurisubharmonic nc plush if it has an nc complex hessian that is positive semidefinite when evaluated on all tuples of n n matrices for every size n i e q x x t h h t 0 for all x h r n n g for every n 1 in this paper we classify all symmetric nc plush polynomials as convex polynomials with an nc analytic change of variables i e an nc symmetric polynomial p is nc plush if and only if it has the form 0 1 p f j t f j k j k j t f f t where the sums are finite and f j k j f are all nc analytic in this paper we also present a theory of noncommutative integration for nc polynomials and we prove a noncommutative version of the frobenius theorem a subsequent paper j m greene preprint 6 proves that if the nc complex hessian q of p takes positive semidefinite values on an nc open set then q takes positive semidefinite values on all tuples x h thus p has the form in eq 0 1 the proof in j m greene preprint 6 draws on most of the theorems in this paper together with a very different technique involving representations of noncommutative quadratic functions

universality class for domain growth in random magnets a 2d ising model with random ferromagnetic bonds is studied by monte carlo simulation following a quench from t infinity to t tc the domain size grows as l t approximately in t to x at late times the data are consistent with the theoretical prediction chi 4 the exponent defined by si 0 si t approximately l t and the scaling functions for the spatial correlations are very close to those of the pure system suggesting that pure and random systems belong to the same universality class

delay dependent robust stabilization of uncertain state delayed systems this paper concerns a problem of robust stabilization of uncertain state delayed systems a new delay dependent stabilization condition using a memoryless controller is formulated in terms of matrix inequalities an algorithm involving convex optimization is proposed to design a controller guaranteeing a suboptimal maximal delay such that the system can be stabilized for all admissible uncertainties

factorization of operator valued polynomials in several non commuting variables abstract a version of fejer riesz factorization and factorization of positive operator valued polynomials in several non commuting variables holds the proofs use arveson s extension theorem and matrix completions

ellipsoidal sets for resilient and robust static output feedback the implementation of a controller if not exact may lead to the so called fragility problem i e the loss of expected closed loop properties in the present note this difficult problem is dealt with considering robust static output feedback sof control for uncertain linear time invariant systems by analogy with robust analysis theory based on quadratic separation a new formulation for the sof design is shown to be a valuable way to tackle fragility issues indeed the use of a quadratic separator for design purpose allows to define a whole resilient nonfragile set of sof control laws results are formulated as matrix inequalities one of which is nonlinear a numerical algorithm based on nonconvex optimization is provided ant its running is illustrated on classical examples from literature

direct delay decomposition approach to synchronization of chaotic fuzzy cellular neural networks with discrete unbounded distributed delays and markovian jumping parameters in this paper the problem of synchronization of chaotic fuzzy cellular neural networks fcnns with discrete unbounded distributed delays and markovian jumping parameters mjps is investigated sufficient delay dependent stability criteria are obtained in terms of linear matrix inequalities lmis to ensure the chaotic delayed fcnns to be stochastic asymptotically synchronous with the help of free weighting matrix and some inequality techniques the information of the delayed plant states can be taken into full consideration here the delay interval is decomposed into two subintervals by using the tuning parameter such that 0 1 by developing a delay decomposition approach and constructing suitable lyapunov krasovskii functional lkf sufficient conditions for synchronization are established for each subinterval numerical example and its simulations are provided to demonstrate the effectiveness and less conservatism of the derived results

event based h filtering for sampled data systems this paper is concerned with event based h filtering for sampled data systems first an event based data packet processor is introduced to release sampled measurement outputs only if an event condition is violated as a result communication resources can be saved significantly while preserving the desired h performance second the resulting filtering error system is modeled as a system with an interval time varying delay by employing the lyapunov krasovskii functional approach a new bounded real lemma brl is established such that the filtering error system is asymptotically stable with the prescribed h performance third by performing an invertible linear transformation on the filtering error system a linear matrix inequality lmi based sufficient condition which is equivalent to the condition in the brl is obtained on the feasibility of the event based h filtering problem consequently suitable h filters and the event parameters in the event condition can be co designed provided that a set of lmis are satisfied finally a mechanical system with two masses and two springs is given to show the effectiveness of the proposed method

pre specified performance based model reduction for time varying delay systems in fuzzy framework this paper attempts to provide a new solution to the model approximation problem for dynamic systems with time varying delays under the fuzzy framework for a given high order system our focus is on the construction of a reduced order model which approximates the original one in a prescribed error performance level and guarantees the asymptotic stability of the corresponding error system based on the reciprocally convex technique a less conservative stability condition is established for the dynamic error system with a given error performance index furthermore the reduced order model is eventually obtained by applying the projection approach which converts the model approximation into a sequential minimization problem subject to linear matrix inequality constraints by employing the cone complementary linearization algorithm finally two numerical examples are provided to illustrate the effectiveness of the proposed method

a note on equivalence between two integral inequalities for time delay systems jensen s inequality and extended jensen s inequality are two important integral inequalities when problems of stability analysis and controller synthesis for time delay systems are considered the extended jensen s inequality introduces two additional free matrices and is generally regarded to be less conservative than jensen s inequality the equivalence between jensen s inequality and extended jensen s inequality in bounding the quadratic term h t h t x t s z x s d s in lyapunov functional of time delay systems is presented and theoretically proved it is shown that the extended jensen s inequality does not decrease the lower bound of this quadratic term obtained using jensen s inequality and then it does not reduce the conservativeness though two additional free matrices m 1 and m 2 are involved

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

a note on wirtinger type integral inequalities for time delay systems different integral inequalities play important role when problems of stability analysis and controller synthesis for time delay systems are considered the connection of jensen s inequality and extended jensen s inequality is well understood now to reduce the conservativeness introduced by the application of jensen s inequality several versions of the wirtinger integral inequality have been published recently this note presents a comparison between some of these inequalities