the linear profile decomposition for the fourth order schrodinger equation abstract in this paper we establish the linear profile decomposition for the one dimensional fourth order schrodinger equation i u t u 2 u 0 t r x r u 0 x f x l 2 where 0 as an application we establish a dichotomy result on the existence of extremals to the symmetric schrodinger strichartz inequality

principal minor sums of a tb m abstract the question is raised whether the sum of the k k principal minors of the titled matrix is a polynomial in t with positive coefficients when a and b are positive definite this would generalize a conjecture made by bessis moussa villani as stated by e h lieb and r seiringer equivalent forms of the bessis moussa villani conjecture j stat phys 115 2004 185 190 we give a variety of evidence for this further question some of which is new

robust h stabilization for t s fuzzy systems with time varying delays and memory sampled data control abstract in this paper we address the robust h stabilization for t s fuzzy systems with time varying delays based on memory sampled data control by developing some new terms an improved piecewise lyapunov krasovskii functional lkf is constructed to take full advantage of characteristic about real sampling pattern furthermore some relaxed matrices proposed in the lkf are not necessarily positive definite by using the lkf and free matrix based fmb integral inequality some sufficient criteria are established to ensure the stability of fuzzy systems and reduce the influence of external disturbance with an h norm bound then the memory sampled data controller can be derived by solving a group of linear matrix inequalities lmis with the maximal sampling period finally a numerical example is given to demonstrate the benefits and the superiority of the approach proposed

strong instability of standing waves for a fourth order nonlinear schrodinger equation with the mixed dispersions abstract in this paper we consider the strong instability of standing waves for a fourth order nonlinear schrodinger equation with the mixed dispersions i t 2 p 0 t x 0 t r n where 0 and 0 this equation arises in describing the propagation of intense laser beams in a bulk medium with kerr nonlinearity we firstly obtain the variational characterization of ground state solutions by using the profile decomposition theory in h 2 then we deduce that if 2 s u 1 0 the ground state standing wave e i t u is strongly unstable by blow up where u x n 2 u x and s is the action this result is a complement to the result of bonheure et al 2019 where the strong instability of standing waves has been studied in the case 0

approaches to t s fuzzy affine model based reliable output feedback control for nonlinear ito stochastic systems this paper deals with the problem of reliable and robust mathscr h infty static output feedback sof controller synthesis for continuous time nonlinear stochastic systems with actuator faults the nonlinear stochastic plant is expressed by an ito type takagi sugeno fuzzy affine model with parametric uncertainties and a markov process is employed to model the occurrence of actuator fault the purpose is to design an admissible piecewise sof controller such that the resulting closed loop system is stochastically stable with a prescribed disturbance attenuation level in an mathscr h infty sense specifically based on a markovian lyapunov function combined with ito differential formula s procedure and some matrix inequality convexification procedures two new approaches to the reliable sof controller analysis and synthesis are proposed for the underlying stochastic fuzzy affine systems it is shown that the existence of desired reliable controllers is fully characterized in terms of strict linear matrix inequalities finally simulation examples are presented to illustrate the effectiveness and advantages of the developed methods

uniform stabilization of the fourth order schrodinger equation abstract we study both boundary and internal stabilization problems for the fourth order schrodinger equation in a smooth bounded domain of r n we first consider the boundary stabilization problem by introducing suitable dissipative boundary conditions we prove that the solution decays exponentially in an appropriate energy space in the internal stabilization problem by assuming that the damping term is effective on a neighborhood of a part of the boundary we prove the exponential decay of the l 2 energy of the solution both results are established by using multiplier techniques and compactness uniqueness arguments

sampled data fuzzy control for a class of nonlinear parabolic distributed parameter systems under spatially point measurements abstract in this paper the exponential stabilization problem is addressed for a class of nonlinear parabolic partial differential equation pde systems via sampled data fuzzy control approach initially the nonlinear pde system is accurately represented by the takagi sugeno t s fuzzy pde model then based on the t s fuzzy pde model a novel time dependent lyapunov functional is used to design a sampled data fuzzy controller under spatially point measurements such that the closed loop fuzzy pde system is exponentially stable with a given decay rate the stabilization conditions are presented in terms of a set of linear matrix inequalities lmis finally simulation results on the control of the diffusion equation and the fitzhugh nagumo fhn equation to illustrate the effectiveness of the proposed design method

global well posedness and scattering for the nonstandard defocusing beam equation abstract we consider the nonstandard defocusing beam equation u t t i 1 n i 4 u u u p 1 u 0 in the subcritical case we show the global well posedness and the scattering results in low dimensions by the compactness rigidity argument of kenig and merle 2006 2008 7 3 the new ingredient is that we establish the suitable strichartz estimate to overcome the difficulty caused by the complicated dispersive estimate

advances on the bessis moussa villani trace conjecture abstract a long standing conjecture asserts that the polynomial p t tr a tb m has nonnegative coefficients whenever m is a positive integer and a and b are any two n n positive semidefinite hermitian matrices the conjecture arises from a question raised by bessis et al d bessis p moussa m villani monotonic converging variational approximations to the functional integrals in quantum statistical mechanics j math phys 16 1975 2318 2325 in connection with a problem in theoretical physics their conjecture as shown recently by lieb and seiringer is equivalent to the trace positivity statement above in this paper we derive a fundamental set of equations satisfied by a and b that minimize or maximize a coefficient of p t applied to the bessis moussa villani bmv conjecture these equations provide several reductions in particular we prove that it is enough to show that 1 it is true for infinitely many m 2 a nonzero matrix coefficient of a tb m always has at least one positive eigenvalue or 3 the result holds for singular positive semidefinite matrices moreover we prove that if the conjecture is false for some m then it is false for all larger m finally we outline a general program to settle the bmv conjecture that has had some recent success

trace functions as laplace transforms we study trace functions on the form t trf a tb where f is a real function defined on the positive half line and a and b are matrices such that a is positive definite and b is positive semidefinite if f is non negative and operator monotone decreasing then such a trace function can be written as the laplace transform of a positive measure the question is related to the bessis moussa villani conjecture

robust h control of uncertain fuzzy systems under time varying sampling robust h sampled data control for uncertain fuzzy systems is discussed in many practical situations a system is modeled as a continuous time fuzzy system while the control input is the zero order hold which can be represented as a piecewise continuous delay here we take a delay system approach to the h sampled data control problem the closed loop system with a sampled data state feedback controller becomes a system with time varying delay first h performance conditions for the closed loop system are given in terms of linear matrix inequalities lmis such conditions are derived by using the leibniz newton formula and free weighting matrix method for fuzzy time delay systems under the assumption that sampling time is not greater than some prescribed number then a design method for a robust h sampled data state feedback controller for uncertain fuzzy systems is proposed numerical examples are given to illustrate our robust h sampled data state feedback control design

the cubic fourth order schrodinger equation fourth order schrodinger equations have been introduced by karpman and shagalov to take into account the role of small fourth order dispersion terms in the propagation of intense laser beams in a bulk medium with kerr nonlinearity in this paper we investigate the cubic defocusing fourth order schrodinger equation i tu 2u u 2u 0 in arbitrary space dimension rn for arbitrary initial data we prove that the equation is globally well posed when n 8 and ill posed when n 9 with the additional important information that scattering holds true when 5 n 8

a note on the nonexistence of sum of squares certificates for the bessis moussa villani conjecture the algebraic reformulation of the bessis moussa villani conjecture is equivalent to a family of dimension free tracial inequalities involving positive semidefinite matrices sufficient conditions for these to hold in the form of algebraic identities involving polynomials in noncommuting variables have been given by klep and schweighofer sums of hermitian squares and the bmv conjecture j stat phys 133 739 2008 later the existence of these certificates has been settled for all but one case which is resolved in this note

on the defect of compactness for the strichartz estimates of the schrodinger equations abstract in this paper we prove that every sequence of solutions to the linear schrodinger equation with bounded data in h1 r d d 3 can be written up to a subsequence as an almost orthogonal sum of sequences of the type h d 2 2nv t tn h2n x xn hn where v is a solution of the linear schrodinger equation with a small remainder term in strichartz norms using this decomposition we prove a similar one for the defocusing h1 critical nonlinear schrodinger equation assuming that the initial data belong to a ball in the energy space where the equation is solvable this implies in particular the existence of an a priori estimate of the strichartz norms in terms of the energy

geometric optics and instability for semi classical schrodinger equations we prove some instability phenomena for semi classical linear or nonlinear schrodinger equations for some perturbations of the data we show that for very small times we can neglect the laplacian and the mechanism is the same as for the corresponding ordinary differential equation our approach allows smaller perturbations of the data where the instability occurs for times such that the problem cannot be reduced to the study of an ordinary differential equation

loss of regularity for supercritical nonlinear schrodinger equations we consider the nonlinear schrodinger equation with defocusing smooth nonlinearity below the critical sobolev regularity it is known that the cauchy problem is ill posed we show that this is even worse namely that there is a loss of regularity in the spirit of the result due to g lebeau in the case of the wave equation as a consequence the cauchy problem for energy supercritical equations is not well posed in the sense of hadamard we reduce the problem to a supercritical wkb analysis for super cubic smooth nonlinearity this analysis is new and relies on the introduction of a modulated energy functional a la brenier

proof of the cases p 7 of the lieb seiringer formulation of the bessis moussa villani conjecture it is shown that the polynomial t tr a tb p has nonnegative coefficients when p 7 and a and b are any two complex positive semidefinite n n matrices with arbitrary n this proves a general nontrivial case of the lieb seiringer formulation of the bessis moussa villani conjecture which is a long standing problem in theoretical physics

an improved input delay approach to stabilization of fuzzy systems under variable sampling in this paper we investigate the problem of stabilization for sampled data fuzzy systems under variable sampling a novel lyapunov krasovskii functional lkf is defined to capture the characteristic of sampled data systems and an improved input delay approach is proposed by the use of an appropriate enlargement scheme new stability and stabilization criteria are obtained in terms of linear matrix inequalities lmis compared with the existing results the newly obtained ones contain less conservatism some illustrative examples are given to show the effectiveness of the proposed method and the significant improvement on the existing results

biharmonic nonlinear schrodinger equation and the profile decomposition abstract this paper is concerned with the cauchy problem for the biharmonic nonlinear schrodinger equation with l 2 super critical nonlinearity by establishing the profile decomposition of bounded sequences in h 2 r n the best constant of a gagliardo nirenberg inequality is obtained moreover a sufficient condition for the global existence of the solution to the biharmonic nonlinear schrodinger equation is given

asymptotics for the fourth order nonlinear schrodinger equation in the critical case abstract we consider the cauchy problem for the fourth order nonlinear schrodinger equation i t u 1 4 x 4 u u 4 u t x r r u 0 x u 0 x x r we prove the large time asymptotic behavior of solutions with a logarithmic correction under the conditions u 0 x d x 0 0

equivalent forms of the bessis moussa villani conjecture the bmv conjecture for traces which states that text tr text exp left a lambda b right is the laplace transform of a positive measure is shown to be equivalent to two other statements i the polynomial lambda mapsto text tr left a lambda b right p has only non negative coefficients for all a b geqslant 0 p in mathbb n and ii lambda mapsto text tr left a lambda b right p is the laplace transform of a positive measure for a b geqslant 0 p 0

large deviations and chernoff bound for certain correlated states on a spin chain in this paper we extend the results of lenci and rey bellet j stat phys 119 715 2005 on the large deviation upper bound of the distribution measures of local hamiltonians with respect to a gibbs state in the setting of translation invariant finite range interactions we show that a certain factorization property of the reference state is sufficient for a large deviation upper bound to hold and that this factorization property is satisfied by gibbs states of the above kind as well as finitely correlated states as an application of the methods the chernoff bound for correlated states with factorization property is studied in the specific case of the distributions of the ergodic averages of a one site observable with respect to an ergodic finitely correlated state the spectral theory of positive maps is applied to prove the full large deviation principle

self focusing with fourth order dispersion we analyze self focusing and singularity formation in the nonlinear schrodinger equation nls with high order dispersion i psi t pm delta q psi psi 2 sigma psi 0 in the isotropic mixed dispersion nls i psi t delta psi epsilon delta 2 psi psi 2 sigma psi 0 and in nonisotropic mixed dispersion nls equations which model propagation in fiber arrays

new approach to delay dependent stability analysis and stabilization for continuous time fuzzy systems with time varying delay this paper is concerned with delay dependent stability analysis and stabilization problems for continuous time takagi and sugeno t s fuzzy systems with a time varying delay a new method for the delay dependent stability analysis and stabilization is suggested which is less conservative than other existing ones first based on a fuzzy lyapunov krasovskii functional lkf a delay dependent stability criterion is derived for the open loop fuzzy systems in the derivation process some free fuzzy weighting matrices are introduced to express the relationships among the terms of the system equation and among the terms in the leibniz newton formula then a delay dependent stabilization condition based on the so called parallel distributed compensation pdc scheme is worked out for the closed loop fuzzy systems the proposed stability criterion and stabilization condition are represented in terms of linear matrix inequalities lmis and compared with the existing ones via two examples finally application to control of a truck trailer is also given to illustrate the effectiveness of the proposed design method

global well posedness and scattering for the energy critical nonlinear schrodinger equation in r 3 we obtain global well posedness scattering and global l 10 spacetime bounds for energy class solutions to the quintic defocusing schrodinger equa tion in r 1 3 which is energy critical in particular this establishes global existence of classical solutions our work extends the results of bourgain 4 and grillakis 20 which handled the radial case the method is similar in spirit to the induction on energy strategy of bourgain 4 but we perform the induction analysis in both frequency space and physical space simultaneously and replace the morawetz inequality by an interaction variant first used in 12 13 the principal advantage of the interaction morawetz estimate is that it is not localized to the spatial origin and so is better able to handle nonradial solutions in particular this interaction estimate together with an almost conservation argument controlling the movement of l 2 mass in fre quency space rules out the possibility of energy concentration

instabilities for supercritical schrodinger equations in analytic manifolds in this paper we consider supercritical nonlinear schrodinger equations in an analytic riemannian manifold md g where the metric g is analytic using an analytic wkb method we are able to construct an ansatz for the semiclassical equation for times independent of the small parameter these approximate solutions will help to show two different types of instabilities the first is in the energy space and the second is an immediate loss of regularity in higher sobolev norms

h control for sampled data nonlinear systems described by takagi sugeno fuzzy systems in this paper we consider the design problem of output feedback h controllers for sampled data fuzzy systems we first transfer them into equivalent jump fuzzy systems we establish the so called bounded real lemma for jump fuzzy systems and give a design method of suboptimal output feedback h controllers in terms of two riccati inequalities with jumps we then apply the main results to the sampled data fuzzy systems and obtain a design method of suboptimal output feedback h controllers we give a numerical example and construct a suboptimal output feedback h controller

fuzzy model based sampled data control of chaotic systems a fuzzy time dependent lyapunov krasovskii functional approach this paper addresses the sampled data stabilization problem for chaotic systems represented by takagi sugeno t s fuzzy models if the upper bounds for the time derivative of membership functions are available combining the fuzzy blending for some quadratic functions together with the introduction of some new useful terms a novel fuzzy time dependent lyapunov krasovskii functional lkf is proposed to fully capture the available characteristics of the actual sampling pattern and membership functions simultaneously based on the proposed lkf a new criterion dependent on the upper bounds for the time derivative of membership functions is presented to guarantee the asymptotic stability of the whole closed loop system moreover a stability criterion independent of the upper bounds is also provided based on the corresponding common time dependent lkf then the designed fuzzy sampled data controller can be synthesized by analyzing the corresponding stabilization conditions moreover a search algorithm is provided to find the optimal tuning parameters finally one practical example of the lorenz system is given to illustrate that much less conservativeness can be achieved compared with the earlier results by using the corresponding common lkf and the results can be further improved when adopting the fuzzy time dependent lkf within large upper bounds

global well posedness scattering and blow up for the energy critical focusing non linear wave equation we study the energy critical focusing non linear wave equation with data in the energy space in dimensions 3 4 and 5 we prove that for cauchy data of energy smaller than the one of the static solution w which gives the best constant in the sobolev embedding the following alternative holds if the initial data has smaller norm in the homogeneous sobolev space h1 than the one of w then we have global well posedness and scattering if the norm is larger than the one of w then we have break down in finite time

global well posedness for a l 2 critical nonlinear higher order schrodinger equation we prove the global well posedness for a l 2 critical defocusing cubic higher order schrodinger equation namely i partial tu lambda k u u 2 u where lambda sqrt delta and k geq 3 k in mathbb z in r k with initial data u 0 in h gamma gamma gamma k frac k 4k 1 14k 3

robust h sub spl infin control for fuzzy systems with frobenius norm bounded uncertainties in this paper we investigate the problem of robust h sub spl infin performance and stabilization for a class of uncertain fuzzy systems with frobenius norm bounded parameter uncertainties in all system matrices both continuous and discrete time uncertain fuzzy systems are considered under a unified treatment called bounded real lemma for fuzzy systems unlike the bounded real lemma in the linear theory of robust h sub spl infin control where necessary and sufficient conditions were obtained only sufficient condition based on lyapunov method is shown furthermore connection between robust h sub spl infin problems involving uncertainty and standard uncertainty free h sub spl infin problems is established via matrix algebra as for controller synthesis a state feedback fuzzy control law is designed via relaxed linear matrix inequality lmi formulations

delay dependent guaranteed cost control for t s fuzzy systems with time delays this study introduces a guaranteed cost control method for nonlinear systems with time delays which can be represented by takagi sugeno t s fuzzy models with time delays the state feedback and generalized dynamic output feedback approaches are considered the generalized dynamic output feedback controller is presented by a new fuzzy controller architecture which is of dual indexed rule base it considers both the dynamic part and the output part of t s fuzzy model which guarantees that the controller without any delay information can stabilize time delay t s fuzzy systems based on delay dependent lyapunov functional approach some sufficient conditions for the existence of state feedback controller are provided via parallel distributed compensation pdc first second the corresponding conditions are extended into the generalized dynamic output feedback closed loop system via so called generalized pdc technique the upper bound of time delay can be obtained using convex optimization such that the system can be stabilized for all time delays whose sizes are not larger than the bound the minimizing method is also proposed to search the suboptimal upper bound of guaranteed cost function the effectiveness of the proposed method can be shown by the simulation examples

on the positivity of the coefficients of a certain polynomial defined by two positive definite matrices it is shown that the polynomial p t tr a tb m has positive coefficients when m 6 and a and b are any two 3 by 3 complex hermitian positive definite matrices this case is the first that is not covered by prior general results this problem arises from a conjecture raised by bessis moussa and villani in connection with a long standing problem in theoretical physics the full conjecture as shown recently by lieb and seiringer is equivalent to p t having positive coefficients for any m and any two n by n positive definite matrices we show that generally the question in the real case reduces to that of singular a and b and this is a key part of our proof

blow up of rough solutions to the fourth order nonlinear schrodinger equation abstract this paper deals with the formation of singularities of rough blow up solutions to the fourth order nonlinear schrodinger equation the limiting profile and l 2 concentration of the rough blow up solutions are obtained in h s r 4 with s s 0 where s 0 9 721 20 1 793 the new ingredient relies on the refined compactness result developed by zhu et al s h zhu j zhang h yang limiting profile of the blow up solutions for the fourth order nonlinear schrodinger equation dyn partial differ equ 7 2010 187 205

strichartz estimates for the fractional schrodinger and wave equations on compact manifolds without boundary we firstly prove strichartz estimates for the fractional schrodinger equations on mathbb r d d geq 1 endowed with a smooth bounded metric g we then prove strichartz estimates for the fractional schrodinger and wave equations on compact riemannian manifolds without boundary m g this result extends the well known strichartz estimate for the schrodinger equation given in 7 we finally give applications of strichartz estimates for the local well posedness of the pure power type nonlinear fractional schrodinger and wave equations posed on m g

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

comments on robust stabilization of a class of uncertain nonlinear systems via fuzzy control quadratic stabilizability h sup spl infin control theory and linear matrix inequalities this paper presents stability analysis for a class of uncertain nonlinear systems and a method for designing robust fuzzy controllers to stabilize the uncertain nonlinear systems first a stability condition for takagi and sugeno s fuzzy model is given in terms of lyapunov stability theory next new stability conditions for a generalized class of uncertain systems are derived from robust control techniques such as quadratic stabilization h sup spl infin control theory and linear matrix inequalities the derived stability conditions are used to analyze the stability of takagi and sugeno s fuzzy control systems with uncertainty which can be regarded as a generalized class of uncertain nonlinear systems the design method employs the so called parallel distributed compensation important issues for the stability analysis and design are remarked finally three design examples of fuzzy controllers for stabilizing nonlinear systems and uncertain nonlinear systems are presented

wellposedness for the fourth order nonlinear schrodinger equations abstract we study the local smoothing effects and wellposedness of cauchy problem for the fourth order nonlinear schrodinger equations in 1d i t u x 4 u p x u 2 x u 2 t x r where p is a polynomial excluding constant or linear terms

robust h fuzzy static output feedback control of t s fuzzy systems with parametric uncertainties this paper addresses robust h fuzzy static output feedback control problem for t s fuzzy systems with time varying norm bounded uncertainties sufficient conditions for synthesis of a fuzzy static output feedback controller for t s fuzzy systems are derived in terms of a set of linear matrix inequalities lmis in comparison with the existing literatures the proposed approach not only simplifies the design procedure but also achieves a better h performance three drawbacks existing in the previous papers such as coordinate transformation same output matrices and bmi problem have been eliminated the effectiveness of the proposed design method is demonstrated by an example for the control of a truck trailer system

the cauchy problem for the fourth order nonlinear schrodinger equation related to the vortex filament abstract the cauchy problem of one dimensional fourth order nonlinear schrodinger equation related to the vortex filament is studied local well posedness for initial data in h s r s 1 2 is obtained by the fourier restriction norm method under certain coefficient condition

sampled data fuzzy control with space varying gains for nonlinear time delay parabolic pde systems abstract this paper introduces a sampled data fuzzy control sdfc with space varying gains for nonlinear time delay parabolic partial differential equation pde systems first a takagi sugeno t s fuzzy time delay parabolic pde model is employed to represent the nonlinear time delay pde system second with the aid of the t s fuzzy time delay pde model a sdfc design with space varying gains is developed in the formulation of space dependent linear matrix inequalities lmis by constructing an appropriate lyapunov functional which can stabilize exponentially the time delay pde system these stabilization conditions can be applied to either slowing varying time delay or fast varying one finally simulation results on two examples are presented to demonstrate the effectiveness of the proposed method

global existence and asymptotic behavior of the cauchy problem for fourth order schrodinger equations with combined power type nonlinearities abstract we study the global existence of solutions in sobolev spaces and the asymptotic behavior for the cauchy problem of the following fourth order schrodinger equation with combined power type nonlinearities i u t 2 u 1 u p u 2 u 8 n 4 u 0 where t r x r n 8 n p 8 n 4 1 2 are nonzero real numbers

stability analysis and design of fuzzy control systems abstract the stability analysis and the design technique of fuzzy control systems using fuzzy block diagrams are discussed first we show the concept of fuzzy blocks and consider the connection problems of fuzzy blocks diagrams we derive some theorems and corollaries with respect to two basic types of connections of fuzzy blocks in order to preserve some properties in a connection of fuzzy blocks continuous piecewise polynomial membership functions are defined secondly a sufficient condition which guarantees the stability of fuzzy systems is obtained in terms of lyapunov s direct method we give an important fact based on this condition thirdly we propose a new design technique of a fuzzy controller the fuzzy block diagrams and the stability analysis are applied to the design problems of a model based fuzzy controller

proof of the bmv conjecture we prove the bmv bessis moussa villani 1 conjecture which states that the function t mapsto mathop rm tr exp a tb t geqslant 0 is the laplace transform of a positive measure on 0 if a and b are n times n hermitian matrices and b is positive semidefinite a semi explicit representation for this measure is given

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

finite dimensional global attractor for a dissipative anisotropic fourth order schrodinger equation abstract we study the long time behavior of the solutions to a nonlinear damped anisotropic fourth order schrodinger type equation in r 2 we prove that this behavior is described by the existence of regular finite dimensional global attractor in the energy space

sums of hermitian squares and the bmv conjecture we show that all the coefficients of the polynomial mathop mathrm tr a tb m in mathbb r t are nonnegative whenever m 13 is a nonnegative integer and a and b are positive semidefinite matrices of the same size this has previously been known only for m 7 the validity of the statement for arbitrary m has recently been shown to be equivalent to the bessis moussa villani conjecture from theoretical physics in our proof we establish a connection to sums of hermitian squares of polynomials in noncommuting variables and to semidefinite programming as a by product we obtain an example of a real polynomial in two noncommuting variables having nonnegative trace on all symmetric matrices of the same size yet not being a sum of hermitian squares and commutators

stabilization of nonlinear systems under variable sampling a fuzzy control approach this paper investigates the problem of stabilization for a takagi sugeno t s fuzzy system with nonuniform uncertain sampling the sampling is not required to be periodic and the only assumption is that the distance between any two consecutive sampling instants is less than a given bound by using the input delay approach the t s fuzzy system with variable uncertain sampling is transformed into a continuous time t s fuzzy system with a delay in the state though the resulting closed loop state delayed t s fuzzy system takes a standard form the existing results on delay t s fuzzy systems cannot be used for our purpose due to their restrictive assumptions on the derivative of state delay a new condition guaranteeing asymptotic stability of the closed loop sampled data system is derived by a lyapunov approach plus the free weighting matrix technique based on this stability condition two procedures for designing state feedback control laws are given one casts the controller design into a convex optimization by introducing some over design and the other utilizes the cone complementarity linearization idea to cast the controller design into a sequential minimization problem subject to linear matrix inequality constraints which can be readily solved using standard numerical software an illustrative example is provided to show the applicability and effectiveness of the proposed controller design methodology

parameterized linear matrix inequality techniques in fuzzy control system design this paper proposes different parameterized linear matrix inequality plmi characterizations for fuzzy control systems these plmi characterizations are in turn relaxed into pure lmi programs which provides tractable and effective techniques for the design of suboptimal fuzzy control systems the advantages of the proposed methods over earlier ones are then discussed and illustrated through numerical examples and simulations

fuzzy guaranteed cost control for nonlinear systems with time varying delay this paper focuses on the problem of guaranteed cost control for takagi sugeno t s fuzzy systems with time varying delayed state a linear quadratic cost function is considered as a performance index of the closed loop fuzzy system then the guaranteed cost control of the closed loop fuzzy system is discussed and the sufficient conditions are provided for the construction of a guaranteed cost controller via state feedback and observer based output feedback when these conditions which are given in terms of the feasibility of linear matrix inequalities lmis are satisfied the designed state feedback controller and observer based controller gain matrices can be obtained via a convex optimization problem two illustrative examples are provided to demonstrate the effectiveness of the approaches proposed in this paper

local energy preserving algorithms for nonlinear fourth order schrdinger equation with trapped term based on the rule that numerical algorithms should preserve the intrinsic properties of the original problem as many as possible we propose two local energy preserving algorithms for the nonlinear fourth order schrdinger equation with a trapped term the local energy conservation law is preserved on any local time space region with appropriate boundary conditions the first algorithm will be both globally charge and energy preserving and the second one will be energy preserving numerical experiments show that the proposed algorithms provide more accurate solution than many existing methods and also exhibit excellent performance in preserving conservation laws

global well posedness and scattering for the defocusing energy critical nonlinear schrodinger equations of fourth order in dimensions d 9 abstract we consider the defocusing energy critical nonlinear schrodinger equation of fourth order i u t 2 u u 8 d 4 u we prove that any finite energy solution is global and scatters both forward and backward in time in dimensions d 9

high frequency approximation of solutions to critical nonlinear wave equations this work is devoted to the description of bounded energy sequences of solutions to the equation 1 u u 4 0 in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i up to remainder terms small in energy norm and in every strichartz norm the proof relies on scattering theory for 1 and on a structure theorem for bounded energy sequences of solutions to the linear wave equation in particular we infer the existence of an a priori estimate of strichartz norms of solutions to 1 in terms of their energy

well posedness and exact controllability of fourth order schrodinger equation with boundary control and collocated observation in this paper we study the well posedness and exact controllability of a system described by a fourth order schrodinger equation on a bounded domain of mathbb r n n geqslant 2 with boundary control and collocated observation the neumann boundary control problem is first discussed it is shown that the system is well posed in the sense of d salamon this implies the exponential stability of the closed loop system under proportional output feedback control the well posedness result is then generalized to the dirichlet boundary control problem in particular in order to conclude feedback stabilization from well posedness we discuss the exact controllability with the dirichlet boundary control which is similar to the neumann boundary control case in addition we show that both systems are regular in the sense of g weiss and their feedthrough operators are zero

stability and stabilization of takagi sugeno fuzzy systems via sampled data and state quantized controller in this paper we investigate the problem of stability and stabilization for sampled data fuzzy systems with state quantization by using an input delay approach the sampled data fuzzy systems with state quantization are transformed into a continuous time system with a delay in the state the transformed system contains nondifferentiable time varying state delay based on some integral techniques some new stability and stabilization criteria are first proposed by a modified lyapunov functional furthermore in the case of no quantization some new stability and stabilization criteria are also obtained it is shown that the new stability and stabilization criteria can provide a larger upper bound of the sampling interval than some existing ones in the literature two simulation examples are given to show the effectiveness of the proposed design method

delay dependent robust h sub spl infin control for t s fuzzy systems with time delay this paper focuses on the problem of delay dependent robust fuzzy control for a class of nonlinear delay systems via state feedback the takagi sugeno t s fuzzy model is adopted for representing a nonlinear system with time delayed state a delay dependent stabilization criterion is first presented then the methods of robust stabilization and robust h sub spl infin control are developed which are dependent on the size of the delay and are based on the solutions of linear matrix inequalities lmis finally a design example of robust h sub spl infin controller for uncertain nonlinear systems is given to illustrate the effectiveness of the approaches proposed in this paper

an improved result on exponential stabilization of sampled data fuzzy systems this brief paper studies the exponential stabilization problem for the takagi sugeno fuzzy systems with a variable sampling different from previous results the gains of fuzzy state feedback controller adopted in this paper are time varying during two consecutive sampling instants which can contribute to the enlargement of the allowable sampling interval by choosing a suitable design parameter to reduce the design conservativeness a novel fuzzy time dependent lyapunov functional ftdlf is put forward to fully exploit the accessible information about the sampling pattern and the fuzzy basis functions moreover a more relaxed constraint condition is presented to ensure the positive definiteness of the ftdlf on sampling intervals by resorting to the novel ftdlf and the relaxed constraint condition new exponential stabilization criteria dependent on and independent of upper bounds on time derivatives of fuzzy basis functions are established by which a larger sampling interval can be achieved one example is offered to demonstrate the validity and superiorities of the obtained new results

robust h_ infty control for uncertain takagi sugeno fuzzy systems with interval time varying delay this paper is concerned with the problem of robust hinfin control for a class of uncertain time delay fuzzy systems the time delay is assumed to be a time varying continuous function belonging to a given interval which means that the lower and upper bounds for the time varying delay are available no restriction on the derivative of the time varying delay is needed which allows the time delay to be a fast time varying function the takagi sugeno t s uncertain fuzzy model with interval time varying delay is adopted based on the lyapunov krasovskii functional approach some delay dependent conditions for the existence of robust hinfin controller are formulated in the form of linear matrix inequalities lmis when these lmis are feasible an hinfin controller is presented a numerical example is given to demonstrate the effectiveness of the proposed method

an approach to h control of fuzzy dynamic systems this paper presents an approach to h control of a class of continuous time nonlinear systems which can be represented by a fuzzy dynamic model a suitable piecewise differentiable quadratic lyapunov function is used to establish asymptotic stability and h performance of the closed loop fuzzy control system a constructive algorithm is developed to obtain the h feedback control law which involves solving a set of suitable algebraic riccati equations an example is given to illustrate the application of the method

global well posedness and scattering for the defocusing energy critical nonlinear schrodinger equation in r1 4 we obtain global well posedness scattering uniform regularity and global l6t x spacetime bounds for energy space solutions to the defocusing energy critical nonlinear schrodinger equation in a 4 our arguments closely follow those of colliander keel et al though our derivation of the frequency localized interaction morawetz estimate is somewhat simpler as a consequence our method yields a better bound on the l6t x norm

sum of squares results for polynomials related to the bessis moussa villani conjecture we show that the polynomial sm k a b that is the sum of all words in noncommuting variables a and b having length m and exactly k letters equal to b is not equal to a sum of commutators and hermitian squares in the algebra r x y where x2 a and y2 b for all even values of m and k with 6 k m 10 and also for m k 12 6 this leaves only the case m k 16 8 open this topic is of interest in connection with the lieb seiringer formulation of the bessis moussa villani conjecture which asks whether tr sm k a b 0 holds for all positive semidefinite matrices a and b these results eliminate the possibility of using descent sum of squares to prove the bmv conjecture

singular solutions of the biharmonic nonlinear schrodinger equation we consider singular solutions of the l 2 critical biharmonic nonlinear schrodinger equation we prove that the blowup rate is bounded by a quartic root the solution approaches a quasi self similar profile and a finite amount of l 2 norm which is no less than the critical power concentrates into the singularity we also prove the existence of a ground state solution we use asymptotic analysis to show that the blowup rate of peak type singular solutions is slightly faster than that of a quartic root and the self similar profile is given by the ground state standing wave these findings are verified numerically up to focusing levels of 10 8 using an adaptive grid method we also use the spectral renormalization method to compute the ground state of the standing wave equation and the critical power for collapse in one two and three dimensions

quasi invariant gaussian measures for the cubic fourth order nonlinear schrodinger equation we consider the cubic fourth order nonlinear schrodinger equation on the circle in particular we prove that the mean zero gaussian measures on sobolev spaces h s mathbb t s frac 3 4 are quasi invariant under the flow

stabilization of chaotic systems under variable sampling and state quantized controller abstract this paper investigates the problem of stabilization for chaotic systems based on a t s fuzzy model under sampled data control and state quantization a novel lyapunov krasovskii functional lkf is introduced to the sampled data systems the benefit of the new approach is that the lkf develops more information about the actual sampling pattern in addition some symmetric matrices involved in the lkf are not required to be positive definite based on a recently introduced wirtinger based integral inequality that has been shown to be less conservative than jensen s inequality much less conservative stabilization conditions are obtained to ensure the maximal sampling period then the corresponding sampled data controllers can be synthesized by solving a set of linear matrix inequalities lmis finally an illustrative example is given to show the feasibility and effectiveness of the proposed method

on d hagele s approach to the bessis moussa villani conjecture abstract the reformulation of the bessis moussa villani bmv conjecture given by lieb and seiringer asserts that the coefficient m k a b of t k in the polynomial tr a tb m with a b positive semidefinite matrices is nonnegative for all m k we propose a natural extension of a method of attack on this problem due to hagele and investigate for what values of m k the method is successful obtaining a complete determination when either m or k is odd

periodic fourth order cubic nls local well posedness and non squeezing property abstract in this paper we consider the cubic fourth order nonlinear schrodinger equation 4nls under the periodic boundary condition we prove two results one is the local well posedness in h s t with 1 3 s 0 for the cauchy problem of the wick ordered 4nls the other one is the non squeezing property for the flow map of 4nls in the symplectic phase space l 2 t to prove the former we used the ideas introduced in 36 and 27 and to prove the latter we used the ideas in 8

asymptotic positivity of hurwitz product traces two proofs abstract consider the polynomial tr a tb m in t for positive hermitian matrices a and b with m n the bessis moussa villani conjecture in the equivalent form of lieb and seiringer states that this polynomial has nonnegative coefficients only we prove that they are at least asymptotically positive for the nontrivial case of ab 0 more precisely we show once complex analytically once combinatorially that the k th coefficient is positive for all integer m m 0 where m 0 depends on a b and k

novel delay dependent robust h control of uncertain systems with distributed time varying delays this paper investigates the problem of delay dependent robust h control for a class of uncertain systems with distributed time varying delays the aim is to design a delay dependent robust h control which ensures robust asymptotic stability of the system the delay derivative dependent robust h control criteria are obtained in terms of linear matrix inequalities lmis numerical examples are given to illustrate the effectiveness of the proposed method the results are also compared with the existing results to show the conservativeness

h stabilization criterion with less complexity for nonuniform sampling fuzzy systems abstract this paper investigates the problem of h stabilization for nonuniform sampling fuzzy systems a method to design a fuzzy controller is proposed by taking the variation ranges of membership functions within sampling intervals into consideration to reduce the computational complexity jensen s integral inequality method is employed based on a well known inequality a convex combination technique is developed to deal with nonlinear time varying coefficients derived from jensen s integral inequality combining with capturing the characteristic of sampled data systems with a novel piecewise lyapunov krasovskii functional lkf a less complex and less conservative h stabilization criterion is formulated as linear matrix inequalities lmis which can be easily checked by using standard numerical software some illustrative examples are given to show the effectiveness of the proposed method and the significant improvement on the existing results

well posedness of cauchy problem for the fourth order nonlinear schrodinger equations in multi dimensional spaces abstract we study the well posedness of cauchy problem for the fourth order nonlinear schrodinger equations i t u e u 2 u p x u 2 x u 2 t r x r n where e 1 0 1 n 2 denotes the spatial dimension and p is a polynomial excluding constant and linear terms

stability of solitons described by nonlinear schrodinger type equations with higher order dispersion abstract steady solitons described by the fourth order nonlinear schrodinger type equations in one two and three dimensions with power law nonlinearities are studied conditions of existence and stability of such solitons are found numerical results demonstrating soliton existence stability and instability are described they agree with the developed theory

more on operator monotone and operator convex functions of several variables abstract let c 1 c 2 c k be positive matrices in m n and f be a continuous real valued function on 0 in addition consider as a positive linear functional on m n and define t 1 t 2 t 3 t k f t 1 c 1 t 2 c 2 t 3 c 3 t k c k as a k variables continuous function on 0 0 in this paper we show that if f is an operator convex function of order mn then is a k variables operator convex function of order n 1 n k such that m n 1 n 2 n k also if f is an operator monotone function of order n k 1 then is a k variables operator monotone function of order n in particular if f is a non negative operator decreasing function on 0 then the function t f a t b is an operator decreasing and can be written as a laplace transform of a positive measure

on relaxed lmi based designs for fuzzy regulators and fuzzy observers relaxed conditions for stability of nonlinear continuous and discrete time systems given by fuzzy models are presented a theoretical analysis shows that the proposed methods provide better or at least the same results of the methods presented in the literature numerical results exemplify this fact these results are also used for fuzzy regulators and observers designs the nonlinear systems are represented by fuzzy models proposed by takagi and sugeno 1985 the stability analysis and the design of controllers are described by linear matrix inequalities that can be solved efficiently using convex programming techniques the specification of the decay rate constrains on control input and output are also discussed

network based robust passive control for fuzzy systems with randomly occurring uncertainties this paper investigates the problem of robust passive control for networked fuzzy systems where randomly occurring uncertainties variable sampling intervals and constant network induced delay are taken into account a discontinuous lyapunov functional is introduced for the closed loop systems which takes full advantage of the sawtooth structure of the time varying interval delay induced by sample and hold and signal transmission a sufficient condition is proposed to ensure the closed loop system to be robustly stochastically passive then the problem of robust passive control is solved two examples are utilized to illustrate the advantages and effectiveness of the results that are proposed in this paper

design of h control for fuzzy time delay systems abstract in this paper the design problems of h output feedback controllers for takagi sugeno fuzzy time delay systems are considered we first introduce takagi sugeno fuzzy time delay systems and the h performance norm for stable fuzzy time delay systems we consider systems with multiple time delays in state we give a sufficient condition for the norm being less than a given number we then consider the h problems with output feedback controllers both memoryless controller and controller with memory are considered in the design of output feedback controllers the selection of the premise variables in the if then rules is important as for the premise variable we consider three cases where the premise variable is a given function the output and the state of the underlying system in the first two cases we give a design method of h controllers and sufficient conditions which guarantee the required h performance of the closed loop system in the last case we give an h controller for an approximation of the original system two examples are given to illustrate the theory