on the best constants in inequalities of the markov and wirtinger types for polynomials on the half line abstract the main topic of the paper is best constants in markov type inequalities between the norms of higher derivatives of polynomials and the norms of the polynomials themselves the norm is the l 2 norm with laguerre weight the leading term of the asymptotics of the constants is determined and tight bounds for the principal coefficient in this term which is the operator norm of a volterra operator are given for best constants in inequalities of the wirtinger type the limit is computed and an asymptotic formula for the error term is presented

synchronization of chaotic lur e systems with time delays using sampled data control the asymptotical synchronization problem is investigated for two identical chaotic lur e systems with time delays the sampled data control method is employed for the system design a new synchronization condition is proposed in the form of linear matrix inequalities the error system is shown to be asymptotically stable with the constructed new piecewise differentiable lyapunov krasovskii functional lkf different from the existing work the new lkf makes full use of the information in the nonlinear part of the system the obtained stability condition is less conservative than some of the existing ones a longer sampling period is achieved with the new method the numerical examples are given and the simulations are performed on chua s circuit the results show the superiorities and effectiveness of the proposed control method

on discrete inequalities of wirtinger s type various discrete versions of wirtinger s type inequalities are considered a short account on the first results in this field given by fan taussky and todd 10 as well as some generalisations of these discrete inequalities are done also a general method for finding the best possible constants a n and b n in inequalities of the form a n sum k 1 n p kx k 2 leq sum k 0 n r k x k x k 1 2 leq b n sum k 1 n p kx k 2 where p p k and r r k are given weight sequences and x x k is an arbitrary sequence of the real numbers is presented two types of problems are investigated and several corollaries of the basic results are obtained further generalisations of discrete inequalities of wirtinger s type for higher differences are also treated

new proofs and a generalization of inequalities of fan taussky and todd discrete fourier analysis is used to obtain simple proofs of certain inequalities about finite number sequences determined by fan taussky and todd monatsh math 59 1955 73 90 and their converses determined by milovanovic and milovanovic j math anal appl 88 1992 378 387 using the same techniques the inequality 2 sin pi 2 n 1 4 sigma k 1 n b k 2 less than or equal to sigma k 0 n 1 b k 2b k 1 b k 2 2 less than or equal to 2 2 cos pi n 1 2 sigma k 1 n b k 2 is proved for all real numbers 0 b0 b1 b n b n 1 0 which answers a question raised by alzer j math anal appl 161 1991 142 147 second the method is used to obtain the eigenvalues and eigenvectors of matrices a ij that are rotation invariant i e that obey a ij a i 1 j 1 c 1994 academic press inc

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

boundary element collocation methods using splines with multiple knots we extend the theory of boundary element collocation methods by allowing reduced inter element smoothness or in other words by allowing trial functions that are splines with multiple knots our convergence analysis is based on a recurrence relation for the fourier coefficients of the numerical solution and so is restricted to uniform grids on smooth closed curves superconvergence is possible with special choices of the collocation points numerical experiments with a model problem confirm the convergence rates predicted by our theory

some discrete methods for boundary integral equations on smooth closed curves in this paper discrete trigonometric methods as well as quadrature methods for solution of boundary integral equations on smooth closed curves are presented and analyzed the analysis applies the general approach based on the consistency stability and convergence properties the trigonometric methods including e g variants based on product integration turn out to have high convergence even exponential convergence the family of quadrature methods covers operators with the even symbol and of order beta 0 they can be considered as fully discretized petrov galerkin methods that employ approximations both for the integral operator and the inner product the operator is approximated by using the trapezoidal rule either directly or after subtraction of the kernel singularity for instance for symm s equation beta 1 a method with the rate o h 5 of convergence is presented furthermore the quadrature solution of the cauchy singular integral equation beta 0 and the hypersingular

a note on opial and wirtinger type discrete inequalities abstract the aim of the present note is to establish some new discrete inequalities of the opial and wirtinger type involving sequences of real numbers and their forward differences the analysis used in the proofs is quite elementary and based on some simple observations and the applications of some fundamental inequalities

sampled data synchronization analysis of markovian neural networks with generally incomplete transition rates this paper investigates the problem of sampled data synchronization for markovian neural networks with generally incomplete transition rates different from traditional markovian neural networks each transition rate can be completely unknown or only its estimate value is known in this paper compared with most of existing markovian neural networks our model is more practical because the transition rates in markovian processes are difficult to precisely acquire due to the limitations of equipment and the influence of uncertain factors in addition the time dependent lyapunov krasovskii functional is proposed to synchronize drive system and response system by applying an extended jensen s integral inequality and wirtinger s inequality new delay dependent synchronization criteria are obtained which fully utilize the upper bound of variable sampling interval and the sawtooth structure information of varying input delay moreover the desired sampled data controllers are obtained finally two examples are provided to illustrate the effectiveness of the proposed method

e_7 wirtinger inequalities cayley 4 form and homotopy we study optimal curvature free inequalities of the type discovered by c loewner and m gromov using a generalisation of the wirtinger inequality for the comass using a model for the classifying space bs built inductively out of bs we prove that the symmetric metrics of certain two point homogeneous manifolds turn out not to be the systolically optimal metrics on those manifolds we point out the unexpected role played by the exceptional lie algebra e7 in systolic geometry via the calculation of wirtinger constants using a technique of pullback with controlled systolic ratio we calculate the optimal systolic ratio of the quaternionic projective plane modulo the existence of a joyce manifold with spin 7 holonomy and unit middle dimensional betti number

good lattice rules in weighted korobov spaces with general weights we study the problem of multivariate integration and the construction of good lattice rules in weighted korobov spaces with general weights these spaces are not necessarily tensor products of spaces of univariate functions sufficient conditions for tractability and strong tractability of multivariate integration in such weighted function spaces are found these conditions are also necessary if the weights are such that the reproducing kernel of the weighted korobov space is pointwise non negative the existence of a lattice rule which achieves the nearly optimal convergence order is proven a component by component cbc algorithm that constructs good lattice rules is presented the resulting lattice rules achieve tractability or strong tractability error bounds and achieve nearly optimal convergence order for suitably decaying weights we also study special weights such as finite order and order dependent weights for these special weights the cost of the cbc algorithm is polynomial numerical computations show that the lattice rules constructed by the cbc algorithm give much smaller worst case errors than the mean worst case errors over all quasi monte carlo rules or over all lattice rules and generally smaller worst case errors than the best korobov lattice rules in dimensions up to hundreds numerical results are provided to illustrate the efficiency of cbc lattice rules and korobov lattice rules with suitably chosen weights in particular for high dimensional finance problems

on l p norms of weighted mean matrices we study l p operator norms of weighted mean matrices using the approaches of kaluza szego and redheffer as an application we prove a conjecture of bennett

new insights on stability of sampled data systems with time delay abstract this paper analyses the stability of sampled data systems with time delay by employing a two sided looped functional approach some improved conditions are derived to guarantee the stability of the system under consideration then based the conditions some intrinsic relationships between sampled data period and time delay are obtained from the illustrative example being solved by the proposed approach it is observed that the results obtained are significantly better than those obtained by existing methods more importantly from the simulation being carried out it is discovered that contrary to the findings in previous studies time delays in a system may enlarge the interval of sampled data periods and accelerate the rate of convergence of the system states rather than deteriorate the system performance

properties of certain trigonometric series arising in numerical analysis h o f kdlsinn6 n l where 0 is real tx 0 and i 1 series of this kind have obvious connec tions with the riemann zeta function and with bernoulli polynomials and are important for fractional integration 6 xii sect 83 they have also arisen recently 2 4 in the study of certain numerical methods for bound ary integral equations on smooth curves in the plane for the applications in 2 we are interested firstly in knowing that g has a unique zero in the interval 0 rc so that a quadrature rule employing this zero is well defined and for later applications we need to know that this zero increases monotonically with c in the case where o 3 1 these questions were addressed in l as were the corresponding trickier questions about the partial sums for o b 1 we write g e p gol e b g e 4 me d fvbut fi f h e mi and for 0 p d f define k 4 b f p g 0 pi of crucial importance to the applications in 2 are the facts proved in theorem 2 below that h is nonnegative and h non positive and that k are both positive except in special cases in essence these properties ensure that the numerical method treated in 2 is stable the rate of convergence of the numerical methods of 2 is determined by the behaviour of g f i for small values of p the salient properties follow from 2 below in section 5 of the paper we indicate some extensions of our results to other classical orthogonal polynomials 2

a refinement of the discrete wirtinger inequality in this paper we obtain an improved discrete wirtinger inequality associated with a nonlinear second order differential equation we apply this result to prove a bonnesen style isoperimetric inequality for plane polygons and reinterpret the main theorem as a weighted exponential inequality

on the periodic orbits of the fourth order differential equation u qu u ef u u u u we provide sufficient conditions for the existence of periodic solutions of the fourth order differential equation u qu u ef u u u u where q and e are real parameters e is small and f is a nonlinear function

on the stability of the polygonal isoperimetric inequality abstract we obtain a sharp lower bound on the isoperimetric deficit of a general polygon in terms of the variance of its side lengths the variance of its radii and its deviation from being convex our technique involves a functional minimization problem on a suitably constructed compact manifold and is based on the spectral theory for circulant matrices

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

computation of periodic solutions in perturbed second order odes abstract the poincare lindstedt technique is used to compute periodic solutions in perturbed ordinary differential equations in this paper we give a general algorithm for its application on a special purpose symbolic computation package

on stability of systems with aperiodic sampling devices this technical note is concerned with stability analysis of aperiodic sampled data systems the stability problem is tackled from a pure discrete time point of view where the at sampling behavior of the system is modelled as the response of a nominal discrete time lti system in feedback interconnection with a structured uncertainty conditions under which the uncertainty is positive real pr are identified based on the pr property a number of integral quadratic constraints iqc are derived and the iqc theory is applied to derive stability conditions numerical examples are given to illustrate the effectiveness of the proposed approach

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

the periodic problem for curvature like equations with asymmetric perturbations we discuss existence and multiplicity of solutions of the periodic problem for the curvature like equation u 0 p a2 u02 0 f t u by means of variational techniques in the space of bounded variation functions as a 0 is allowed both the prescribed curvature equation and the 1 laplace equation are considered we are concerned with the case where the right hand side f of the equation interacts with the beginning of the spectrum of the 1 laplace operator with periodic boundary conditions on 0 t being mainly interested in the situation where ess sup 0 t r f t s may dier from ess inf 0 t r f t s

some remarks on the use of time varying delay to model sample and hold circuits this note revises the so called input delay approach to the control of sampled data systems with nonuniform sampling in which the sample and hold circuit is embedded into an analog system with a time varying input delay it is shown that the conservatism in calculating the maximal admissible sampling period is reduced by about 57 if the time varying delay embedding step is omitted

spline qualocation methods for variable coefficient elliptic equations on curves here the stability and convergence results of oqualocation methods providing additional orders of convergence are extended from the special class of pseudodifferential equations with constant coefficient symbols to general classical pseudodifferential equations of strongly and of oddly elliptic type the analysis exploits localization in the form of frozen coefficients pseudohomogeneous asymptotic symbol representation of classical pseudodifferential operators a decisive commutator property of the qualocation projection and requires qualocation rules which provide sufficiently many additional degrees of precision for the numerical integration of smooth remainders numerical examples show the predicted high orders of convergence

static output feedback stabilization necessary conditions for multiple delay controllers this note focuses on the static output feedback stabilization problem for a class of single input single output systems when the control law includes multiple distinct delays we are interested in giving necessary conditions for the existence of such stabilizing controllers illustrative examples second order system chain of integrators or chain of oscillators are presented and discussed

global branches of multi bump periodic solutions of the swift hohenberg equation in this paper we study the existence of single and multi bump periodic solutions of a class of fourth order ordinary differential equations arising in problems of pattern formation measuring the tendency to form patterns by a parameter q we view the problem as a nonlinear eigenvalue problem with the use of analytical as well as numerical methods branches of periodic solutions are investigated both locally and globally

periodic solutions for extended fisher kolmogorov and swift hohenberg equations by truncature techniques abstract combining truncature techniques with a variational approach we establish an existence result for nontrivial periodic solutions for a class of fourth order ordinary differential equations involving extended fisher kolmogorov and swift hohenberg equations

spline qualocation methods for boundary integral equations this paper further develops the qualocation method for the solution of integral equations on smooth closed curves qualocation is a petrov galerkin method in which the outer integrals are performed numerically by special quadrature rules here we allow the use of splines as trial and test functions and prove stability and convergence results for certain qualocation rules when the underlying petrov galerkin method is stable in the common case of a first kind equation with the logarithmic kernel a typical qualocation method employs piecewise linear splines and performs the outer integrals using just two points per interval this method with appropriate choice of the quadrature points and weights has ordero h 5 convergence in a suitable negative norm numerical experiments suggest that these rates of convergence are maintained for integral equations on open intervals when graded meshes are used to cope with unbounded solutions

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

a novel approach to stability and stabilization of fuzzy sampled data markovian chaotic systems abstract this paper studies the problems of stability and stabilization of a class of markovian chaotic systems via fuzzy sampled data control different from some existing results the markovian jumping with general uncertain transition rates is considered for such systems a novel approach i e input delay dependent vector approach is proposed for the first time based on this approach an input delay dependent lyapunov krasovskii functional lkf with cubic function of input delay is successfully constructed and the convex combination technique is used with ease to derive stability and stabilization criteria compared with some existing stability and stabilization criteria for chaotic systems our results are not only less conservative but also with reduced calculation complexity finally the superiorities of proposed results are illustrated by two numerical examples

brief paper wirtinger s inequality and lyapunov based sampled data stabilization discontinuous lyapunov functionals appeared to be very efficient for sampled data systems fridman 2010 naghshtabrizi hespanha teel 2008 in the present paper new discontinuous lyapunov functionals are introduced for sampled data control in the presence of a constant input delay the construction of these functionals is based on the vector extension of wirtinger s inequality these functionals lead to simplified and efficient stability conditions in terms of linear matrix inequalities lmis the new stability analysis is applied to sampled data state feedback stabilization and to a novel sampled data static output feedback problem where the delayed measurements are used for stabilization

high order collocation and quadrature methods for some logarithmic kernel integral equations on open arcs this paper is devoted to the solution of the dirichlet problem for the laplace and helmholtz equation in the complement of a smooth open curve in the plane the solution is looked for as a single layer potential the corresponding density being therefore the solution of an integral equation on the open arc this equation is transformed into an equivalent 1 periodic integral equation having existence and uniqueness of solution for any periodic data here we study the use of collocation and quadrature methods for solving this equation we show the convergence of both methods and prove the existence of an asymptotic expansion of the error in powers of the discretization parameter as a consequence we show that for some of them a superconvergence phenomenon occurs when computing the solution of the differential problem two numerical experiments are shown in order to illustrate the theoretical results introduced in this work

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

reconstruction algorithms in irregular sampling a constructive solution of the irregular sampling problem for band limited functions is given we show how a band limited function can be com pletely reconstructed from any random sampling set whose density is higher than the nyquist rate and give precise estimates for the speed of convergence of this iteration method variations of this algorithm allow for irregular sampling with derivatives reconstruction of band limited functions from local averages and irregular sampling of multivariate band limited functions in the irregular sampling problem one is asked whether and how a band limited function can be completely reconstructed from its irregularly sam pled values f xi this has many applications in signal and image processing seismology meteorology medical imaging etc finding constructive solutions of this problem has received considerable attention among mathematicians and engineers the mathematical literature provides several uniqueness results 1 2 17 18 19 it is now part of the folklore that for stable sampling the sampling rate must be at least the nyquist rate 18 these results as deep as they are have had little impact for the applied sciences because they were not constructive if the sampling set is just a perturbation of the regular oversampling then a reconstruction method has been obtained in a seminal paper by duffin and schaeffer 6 see also 29 if for some l 0 a 0 and o 0 the sampling points xk k e z satisfy a xk ok a k i then the norm equivalence a ir f x 2dx with w n o this norm equivalence implies that it is possible to reconstruct through an iterative procedure the so called frame method most of the later work on constructive methods consists of variations of this method 3 21 22 26 the above conditions on the sampling set exclude random irregular sampling sets e g sets with regions of higher sampling density a partial but undesirable remedy to handle highly irregular sampling sets would be to force the above conditions by throwing away information on part of the points and accept a very slow convergence of the iteration

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

an improved time delay implementation of derivative dependent feedback abstract we consider an lti system of relative degree r 2 that can be stabilized using r 1 output derivatives the derivatives are approximated by finite differences leading to a time delayed feedback we present a new method of designing and analyzing such feedback under continuous time and sampled measurements this method admits essentially larger time delay sampling period compared to the existing results and for the first time allows to use consecutively sampled measurements in the sampled data case the main idea is to present the difference between the derivative and its approximation in a convenient integral form the kernel of this integral is hard to express explicitly but we show that it satisfies certain properties these properties are employed to construct the lyapunov krasovskii functional that leads to lmi based stability conditions if the derivative dependent control exponentially stabilizes the system then its time delayed approximation stabilizes the system with the same decay rate provided the time delay for continuous time measurements or the sampling period for sampled measurements are small enough

discrete wirtinger type inequality under non periodic end conditions abstract in this paper we establish the following two alzer s type inequalities under non periodic end conditions which are discrete analogues of wirtinger s inequality let z z 1 z 2 z n n 2 be one complex n tuple with k 1 n z k 0 z k 1 n z k 2 z k 1 n 1 z k 1 z k 2 min z min z 2 z 1 z 3 z 2 z n z n 1 then z n max 1 k n z k z n min z where n 6 n n 1 2 n 1 and n 1 n n 2 4 the constants n and n are best possible

an invariance principle for diffusion in turbulence we prove an almost sure invariance principle for diffusion driven by velocities with unbounded stationary vector potentials the result generalizes to multiple particles motion driven by a common velocity field and independent molecular brownian motions

approximate methods based on order reduction for the periodic solutions of nonlinear third order ordinary differential equations four approximate methods based on order reduction the introduction of a book keeping parameter and power series expansions for the solution and the frequency of oscillation are used to analyze three autonomous nonlinear third order ordinary differential equations which have analytical periodic solutions the first method introduces the velocity in both sides of the equation if this linear term is not present the second one is based on the first one but employs a new independent variable whereas in the third and fourth techniques a term equal to the velocity times the square of the unknown frequency of oscillation is introduced in both sides of the equation the third method uses the original independent variable whereas the fourth one employs a new independent variable which depends linearly on the unknown frequency of oscillation it is shown that the second method provides accurate solutions only for initial velocities close to unity whereas the third one is found to yield very accurate results for the first and second equations considered here and only for large initial velocities for the third one the fourth technique provides as accurate results as or more accurate results than parameter perturbation techniques which deal with the third order equations directly and are based on the expansion of certain constants that appear in the differential equations in terms of a book keeping parameter

technical communique delay range dependent stability for systems with time varying delay this paper is concerned with the stability analysis for systems with time varying delay in a range an appropriate type of lyapunov functionals is proposed to investigate the delay range dependent stability problem the present results may improve the existing ones due to a method to estimate the upper bound of the derivative of lyapunov functional without ignoring some useful terms and the introduction of additional terms into the proposed lyapunov functional which take into account the range of delay numerical examples are given to demonstrate the effectiveness and the benefits of the proposed method

on discrete wirtinger northcott problems abstract some discrete wirtinger northcott inequalities are established in this paper let dx and x be the periodic difference and non periodic difference of a vector x r n respectively for 1 p we give the maximum and the minimum of d x x d x and x in the l p ball b p 0 n in particular we prove that for p the result min b 0 n d x 4 n n is even 4 n n 2 1 n is odd is equivalent to the well known discrete first northcott inequality established by fan taussky and todd

sampled data control for a class of linear time varying systems this paper is concerned with the sampled data control for a class of linear time varying system a classic halanay inequality is first extended to the time varying sampled data system then based on the comparison principle and the extended halanay inequality new criteria for globally uniformly exponential stability and globally uniformly asymptotic stability of the corresponding closed loop system are derived furthermore an algorithm is presented to solve the gain synthesis problem finally one example is given to show the effectiveness of the obtained results

periodic orbits of the fourth order non autonomous differential equation u qu pu ef t u u u u abstract we provide sufficient conditions for the existence of periodic solutions of the fourth order differential equation u qu pu e f t u u u u where q p and e are real parameters e is small and f is a nonlinear non autonomous periodic function with respect to t moreover we provide some applications

finite time synchronization of sampled data markovian jump complex dynamical networks with additive time varying delays based on dissipative theory abstract this paper is concerned with the problem of finite time synchronization issue of sampled data markovian jump complex dynamical networks mjcdns with additive time varying delays based on dissipative theory sufficient conditions to guarantee the finite time stability of mjcdns with additive time varying delays are presented sampled data control with stochastically varying sampling periods is considered the closed loop system is not just finite time bounded but also satisfies the dissipativity conditions further we handled a nonuniform sampled data controller by utilizing the properties of kronecker product combined with the appropriate lyapunov functionals technique a novel delay dependent finite time stability of dissipativity rule is derived in terms of linear matrix inequalities lmis to ensure that the delayed complex dynamical systems to be dissipative finally numerical examples are given to represent the applicability of the proposed approach

on backwards and forwards reachable sets bounding for perturbed time delay systems linear systems with interval time varying delay and unknown but bounded disturbances are considered in this paper we study the problem of finding outer bound of forwards reachable sets and inter bound of backwards reachable sets of the system firstly two definitions on forwards and backwards reachable sets where initial state vectors are not necessary to be equal to zero are introduced then by using the lyapunov krasovskii method two sufficient conditions for the existence of i the smallest possible outer bound of forwards reachable sets and ii the largest possible inter bound of backwards reachable sets are derived these conditions are presented in terms of linear matrix inequalities with two parameters need to tuned which therefore can be efficiently solved by combining existing convex optimization algorithms with a two dimensional search method to obtain optimal bounds lastly the obtained results are illustrated by four numerical examples

a class of nonlinear fourth order variant of a generalized camassa holm equation with compact and noncompact solutions abstract in this paper we study a class of nonlinear fourth order analogue of a generalized camassa holm equation it is shown that this class gives compactons conventional solitons solitary patterns and periodic solutions it is also found that the qualitative change in the physical structure of solutions depends mainly on the exponent of the wave function u x t positive or negative and on the coefficient of u n as well

numerical evaluation of some trigonometric series we present a method for the accurate numerical evaluation of a family of trigonometric series arising in the design of special purpose quadrature rules for boundary element methods the series converge rather slowly but can be expressed in terms of fourier chebyshev series that converge rapidly let r be a positive integer and define the functions gr and hr by gr t 2 e r cos 2rmt m 1

a volterra integral formulation for determining the periodic solutions of some autonomous nonlinear third order ordinary differential equations a volterra integral formulation based on the introduction of a term proportional to the velocity times the square of the unknown frequency of oscillation a new independent variable equal to the original one times the unknown frequency of oscillation the method of variation of parameters and series expansions of both the solution and the frequency of oscillation is used to determine the periodic solutions to three nonlinear autonomous third order ordinary differential equations it is shown that the first term of the series expansion of the frequency of oscillation coincides with that determined from a first order harmonic balance procedure whereas the two term approximation to the frequency of oscillation is shown to be more accurate than that of a parameter perturbation procedure for the second equation considered in this paper for the third equation it is shown that the two term approximation presented in this paper is more accurate than the corresponding one for one of the parameter perturbation methods and for initial velocities less than one for the other parameter perturbation approach