system analysis via integral quadratic constraints this paper introduces a unified approach to robustness analysis with respect to nonlinearities time variations and uncertain parameters from an original idea by yakubovich 1967 the approach has been developed under a combination of influences from the western and russian traditions of control theory it is shown how a complex system can be described using integral quadratic constraints iqc for its elementary components a stability theorem for systems described by iqcs is presented that covers classical passivity dissipativity arguments but simplifies the use of multipliers and the treatment of causality a systematic computational approach is described and relations to other methods of stability analysis are discussed last but not least the paper contains a summarizing list of iqcs for important types of system components

recursion operators and non local symmetries a new formulation of recursion operators is presented which eliminates diffi culties associated with integro differential operators this interpretation treats recursion operators and their inverses on an equal footing efficient techniques for constructing non local symmetries of differential equations result

rational recursion operators for integrable differential difference equations in this paper we introduce the concept of prehamiltonian pairs of difference operators demonstrate their connections with nijenhuis operators and give a criteria for the existence of weakly nonlocal inverse recursion operators for differential difference equations we begin with a rigorous setup of the problem in terms of the skew field of rational pseudo difference operators over a difference field with a zero characteristic subfield of constants and the principal ideal ring of matrix rational pseudo difference operators in particular we give a criteria for a rational operator to be weakly nonlocal a difference operator is called prehamiltonian if its image is a lie subalgebra with respect to the lie bracket on the difference field two prehamiltonian operators form a prehamiltonian pair if any linear combination of them is prehamiltonian then we show that a prehamiltonian pair naturally leads to a nijenhuis operator and a nijenhuis operator can be represented in terms of a prehamiltonian pair this provides a systematic method to check whether a rational operator is nijenhuis as an application we construct a prehamiltonian pair and thus a nijenhuis recursion operator for the differential difference equation recently discovered by adler and postnikov the nijenhuis operator obtained is not weakly nonlocal we prove that it generates an infinite hierarchy of local commuting symmetries we also illustrate our theory on the well known examples including the toda the ablowitz ladik and the kaup newell differential difference equations

on a special class of compatible poisson structures of hydrodynamic type we consider the problem of classifying all compatible poisson brackets of hydrodynamic type one of which is linear in one of the coordinates and the other its derivative with respect to that coordinate

differential difference equations associated with the fractional lax operators we study integrable hierarchies associated with spectral problems of the form p q where p and q are difference operators the corresponding nonlinear differential difference equations can be viewed as inhomogeneous generalizations of the bogoyavlensky type lattices while the latter turn into the korteweg de vries equation under the continuous limit the lattices under consideration provide discrete analogs of the sawada kotera and kaup kupershmidt equations the r matrix formulation and several of the simplest explicit solutions are presented

why nonlocal recursion operators produce local symmetries new results and applications it is well known that integrable hierarchies in 1 1 dimensions are local while the recursion operators that generate these hierarchies usually contain nonlocal terms we resolve this apparent discrepancy by providing simple and universal sufficient conditions for a nonlocal recursion operator in 1 1 dimensions to generate a hierarchy of local symmetries these conditions are satisfied by virtually all recursion operators known today and are much easier to verify than those found in earlier work we also give explicit formulae for the nonlocal parts of higher recursion hamiltonian and symplectic operators of integrable systems in 1 1 dimensions using these two results we prove under some natural assumptions the maltsev novikov conjecture stating that higher hamiltonian symplectic and recursion operators of integrable systems in 1 1 dimensions are weakly nonlocal i e the coefficients of these operators are local and these operators contain at most one integration operator in each term

on the local systems hamiltonian in the weakly non local poission brackets we study in this work the important class of non local poisson brackets pb which we call weakly non local they appeared recently in some investigations in soliton theory however there was no theory of such brackets except for very special first order case even in this case the theory was not developed enough in particular we introduce the physical forms and find casimirs momentum and canonical forms for the most important hydrodynamic type pb of that kind and their dependence on the boundary conditions

integrable odes on associative algebras in this paper we give definitions of basic concepts such as symmetries first integrals hamiltonian and recursion operators suitable for ordinary differential equations on associative algebras and in particular for matrix differential equations we choose existence of hierarchies of first integrals and or symmetries as a criterion for integrability and justify it by examples using our componentless approach we have solved a number of classification problems for integrable equations on free associative algebras also in the simplest case we have listed all possible hamiltonian operators of low order

hydrodynamic chains and the classification of their poisson brackets infinite component poisson brackets of the dubrovin novikov type sov math dokl 27 665 669 1983 are considered the corresponding jacobi identity is significantly simplified in the liouville coordinates since the skew symmetry condition is automatically satisfied the concept of m poisson bracket connected with hydrodynamic chains is introduced then the jacobi identity is a nonlinear system of equations in partial derivatives which can be completely integrated in such a case a classification of infinite component poisson brackets of the dubrovin novikov type can be obtained two simplest examples m 0 and m 1 are considered also infinite component poisson brackets of the ferapontov am math soc transl 170 38 58 1995 type can be simplified in the liouville coordinates the jacobi identity for infinite component poisson brackets of the ferapontov mokhov type russ math surveys 45 218 219 1990 is presented in the liouville coordinates

a bi hamiltonian structure for the integrable discrete non linear schrodinger system abstract this paper shows that the al ablowitz ladik hierarchy of integrable equations can be explicitly viewed as a hierarchy of commuting flows which a are hamiltonian with respect to both a standard local poisson operator j and a new non local skew almost poisson operator k on the appropriate space b can be recursively generated from a recursion operator r k j 1 in addition the proof of these facts relies upon two new pivotal resolvent identities which suggest a general method for uncovering bi hamiltonian structures for other families of discrete integrable equations

some characterizations of symmetric inverse m matrices abstract we apply the previously proved equivalence of four models to present new characterizations of matrices whose inverse is a weakly diagonally dominant symmetric m matrix it is shown that the case of strictly ultrametric matrices of martinez michon and san martin is included as a corollary

mastersymmetries angle variables and recursion operator of the relativistic toda lattice conserved quantities bi hamiltonian formulation and recursive structure of the relativistic toda lattice rt are obtained in an algorithmic way without making use of the lax representation furthermore for the multisoliton solutions the gradients of the angle variables are described in terms of mastersymmetries a new hierarchy of completely integrable systems is discovered which turns out to correspond to the negative of the hierarchy of rt thus it is shown that the full algebra of time dependent symmetry group generators for each member of the rt hierarchy is isomorphic to the algebra of first order differential operators with laurent polynomials as coefficients the surprising phenomenon is revealed that the members of the rt hierarchy are connected to their negative counterparts by explicit backlund transformation

reciprocal transformations of hamiltonian operators of hydrodynamic type nonlocal hamiltonian formalism for linearly degenerate systems reciprocal transformations of hamiltonian operators of hydrodynamic type are investigated the transformed operators are generally nonlocal possessing a number of remarkable algebraic and differential geometric properties we apply our results to linearly degenerate semi hamiltonian systems in riemann invariants a typical example being rti m 1nrm ri rxi i 1 2 n since all such systems are linearizable by appropriate generalized reciprocal transformations our formulas provide an infinity of mutually compatible nonlocal hamiltonian structures explicitly parametrized by n arbitrary functions of one variable

h_ bm infty fuzzy control for systems with repeated scalar nonlinearities and random packet losses this paper is concerned with the h infin fuzzy control problem for a class of systems with repeated scalar nonlinearities and random packet losses a modified takagi sugeno t s fuzzy model is proposed in which the consequent parts are composed of a set of discrete time state equations containing a repeated scalar nonlinearity such a model can describe some well known nonlinear systems such as recurrent neural networks the measurement transmission between the plant and controller is assumed to be imperfect and a stochastic variable satisfying the bernoulli random binary distribution is utilized to represent the phenomenon of random packet losses attention is focused on the analysis and design of h infin fuzzy controllers with the same repeated scalar nonlinearities such that the closed loop t s fuzzy control system is stochastically stable and preserves a guaranteed h infin performance sufficient conditions are obtained for the existence of admissible controllers and the cone complementarity linearization procedure is employed to cast the controller design problem into a sequential minimization one subject to linear matrix inequalities which can be readily solved by using standard numerical software two examples are given to illustrate the effectiveness of the proposed design method

the initial value problem for the whitham averaged system we study the initial value problem for the whitham averaged system which is important in determining the kdv zero dispersion limit we use the hodograph method to show that for a generic non trivial monotone initial data the whitham averaged system has a solution within a region in thex t plane for all time bigger than a large time furthermore the whitham solution matches the burgers solution on the boundaries of the region for hump like initial data the hodograph method is modified to solve the non monotone inx solutions of the whitham averaged system in this way we show that for a hump like initial data the whitham averaged system has a solution within a cusp for a short time after the increasing and decreasing parts of the initial data beging to interact on the cusp the whitham and burgers solutions are matched

lenard scheme for two dimensional periodic volterra chain we prove that for compatible weakly nonlocal hamiltonian and symplectic operators hierarchies of infinitely many commuting local symmetries and conservation laws can be generated under some easily verified conditions no matter whether the generating nijenhuis operators are weakly nonlocal or not we construct a recursion operator of the two dimensional periodic volterra chain from its lax representation and prove that it is a nijenhuis operator furthermore we show that this system is a generalized bi hamiltonian system rather surprisingly the product of its weakly nonlocal hamiltonian and symplectic operators gives rise to the square of the recursion operator

whitham systems and deformations we consider the deformations of whitham systems including the dispersion terms and having the form of dubrovin zhang deformations of frobenius manifolds the procedure is connected with the b a dubrovin problem of deformations of frobenius manifolds corresponding to the whitham systems of integrable hierarchies under some nondegeneracy requirements we suggest a general scheme of the deformation of the hyperbolic whitham systems using the initial nonlinear system the general form of the deformed whitham system coincides with the form of the low dispersion asymptotic expansions used by b a dubrovin and y zhang in the theory of deformations of frobenius manifolds

output feedback control of markovian jump repeated scalar nonlinear systems this paper is concerned with the induced l2 dynamic output feedback controller dofc design problem for discrete time markovian jump repeated scalar nonlinear systems by employing both the switching sequence dependent lyapunov function approach and the positive definite diagonally dominant lyapunov function technique a sufficient condition is first established which guarantees the underlying system to be stochastically stable with an induced l2 disturbance attenuation performance then the desired full or reduced order dofcs are designed by using projection approach cone complementarity linearization procedure is employed to cast the nonconvex feasibility problem into a sequential minimization problem finally a numerical example is presented to show the effectiveness of the proposed methods

on the integrability of homogeneous scalar evolution equations abstract we determine the existence of infinitely many symmetries for equations of the formut uk f u uk 1 when they are homogeneous with respect to the scalinguk k with 0 algorithms are given to determine whether a system has a symmetry also independent oftandx if it has one generalized symmetry we prove it has infinitely many and these can be found using recursion operators or master symmetries the method of proof uses the symbolic method and results from diophantine approximation theory we list the 10 integrable hierarchies the methods can in principle be applied to the 0 case as we illustrate for one example with 0 which seems to be new in principle they can also be used for systems of evolution equations but so far this has only been demonstrated for one class of examples

further results for systems with repeated scalar nonlinearities a class of nonlinear systems described by a discrete time state equation containing a repeated scalar nonlinearity is considered this paper sharpens several previous results on the performance synthesis and model reduction for such systems extensions to the case that the nonlinearity is not an odd function are also discussed

oscillations of the zero dispersion limit of the korteweg de vries equation we attack the multiphase averaged systems for the zero dispersion limit of the kdv equation attention is paid to the most important case the single phase oscillations a scheme is developed to solve the whitham averaged system single phase averaged system this system under our scheme is transformed to a linear over determined system of euler poisson darboux type whose solution can be written down explicitly we show that for any smooth initial data which has only one hump or is a nontrivial monotone function the weak limit has single phase oscillations within a cusp in the x t plane for a short time after the breaking time for the corresponding burgers equation outside the cusp the limit satisfies the burgers equation more surprisingly we also show that the weak limit has global single phase oscillations within a cusp for any smooth nontrivial monotone initial data with only one inflection point 1993 john wiley sons inc

on complete integrability of the mikhailov novikov wang system in the present paper we consider a new two component fifth order integrable system recently found by mikhailov novikov and wang and show that this system possesses a hereditary recursion operator and infinitely many commuting symmetries and conservation laws as well as infinitely many compatible hamiltonian and symplectic structures and is therefore completely integrable the system in question admits a reduction to the kaup kupershmidt equation

an algorithm for determining if the inverse of a strictly diagonally dominant stieltjes matrix is strictly ultrametric it was recently shown that the inverse of a strictly ultrametric matrix is a strictly diagonally dominant stieltjes matrix on the other hand as it is well known that the inverse of a strictly diagonally dominant stieltjes matrix is a real symmetric matrix with nonnegative entries it is natural to ask conversely if every strictly diagonally dominant stieltjes matrix has a strictly ultrametric inverse examples show however that the converse is not true in general i e there are strictly diagonally dominant stieltjes matrices in n n for everyn 3 whose inverses are not strictly ultrametric matrices then the question naturally arises if one can determine which strictly diagonally dominant stieltjes matrices in n n n 3 have inverses which are strictly ultrametric here we develop an algorithm based on graph theory which determines if a given strictly diagonally dominant stieltjes matrixa has a strictly ultrametric inverse where the algorithm is applied toa and requires no computation of inverse moreover if this given strictly diagonally dominant stieltjes matrix has a strictly ultrametric inverse our algorithm uniquely determines this inverse as a special sum of rank one matrices

integrable systems and their recursion operators in this paper we discuss the structure of recursion operators we show that recursion operators of evolution equations have a nonlocal part that is determined by symmetries and cosymmetries this enables us to compute recursion operators more systematically under certain conditions which hold for all examples known to us nijenhuis operators are well defined i e they give rise to hierarchies of infinitely many commuting symmetries of the operator moreover the nonlocal part of a nijenhuis operator contains the candidates of roots and coroots

the zero dispersion limit of the korteweg de vries equation with periodic initial data on etudie le probleme aux valeurs initiales pour l equation de korteweg de vries u t 6uu x e 2 u xxx 0 dans la limite e 0 on calcul la limite faible de la solution pour des donnees initiales periodiques quand e 0

the generation of modulated wavetrains in the solution of the korteweg de vries equation on demontre une relation entre la methode des ondes modulees de whitham et la methode de lax et levermore contenant la limite de petite dispersion

the small dispersion limit of the korteweg de vries equation ii in part i the scattering transform method is used to study the weak limit of solutions to the initial value problem for the korteweg devries kdv equation as the dispersion tends to zero in that limit the associated schrodinger operator becomes semiclassical so the exact scattering data is replaced by its corresponding wkb expressions only nonpositive initial data are considered in that case the limiting reflection coefficient vanishes the explicit solution of kay and moses for the reflectionless inverse transform is then analyzed and the weak limit valid for all time is characterized by a quadratic minimum problem with constraints this minimum problem is reduced to a riemann hilbert problem in function theory in parts ii and iii we use function theoretical methods to solve the riemann hilbert problem in terms of solutions to an auxiliary initial value problem the weak limit satisfies the kdv equation with the dispersive term dropped until its derivatives become infinite up to that time the weak limit is a strong l2 limit at later times the weak limit is locally described by whitham s averaged equations or more generally by the equations found by flaschka et al using multiphase averaging for large times behavior of the weak limit is studied directly from the minimum problem

on construction of recursion operators from lax representation in this work we develop a general procedure for constructing the recursion operators for nonlinear integrable equations admitting lax representation several new examples are given in particular we find the recursion operators for some kdv type systems of integrable equations

reciprocal transformations and flat metrics on hurwitz spaces we consider hydrodynamic systems which possess a local hamiltonian structure to such a system there are also associated an infinite number of nonlocal hamiltonian structures we give the necessary and sufficient conditions so that after a nonlinear transformation of the independent variables the reciprocal system still possesses a local hamiltonian structure we show that under our hypotheses bi hamiltonicity is preserved by the reciprocal transformation finally we apply such results to the reciprocal systems of genus g whitham kdv modulation equations

jacobi structures of evolutionary partial differential equations abstract in this paper we introduce the notion of infinite dimensional jacobi structure to describe the geometrical structure of a class of nonlocal hamiltonian systems which appear naturally when applying reciprocal transformations to hamiltonian evolutionary pdes we prove that our class of infinite dimensional jacobi structures is invariant under the action of reciprocal transformations that only change the spatial variable the main technical tool is in a suitable generalization of the classical schouten nijenhuis bracket to the space of the so called quasi local multi vectors and a simple realization of this structure in the framework of supermanifolds these constructions are used to compute the lichnerowicz jacobi cohomologies and to prove a darboux theorem for jacobi structures with hydrodynamic leading terms we also introduce the notion of bi jacobi structures and consider the integrability of a system of evolutionary pdes that possesses a bi jacobi structure

nonfragile h_ infty fuzzy filtering with randomly occurring gain variations and channel fadings this paper is concerned with the nonfragile h infty filtering problem for a class of discrete time takagi sugeno t s fuzzy systems with both randomly occurring gain variations rogvs and channel fadings the phenomenon of the rogvs is introduced into the system model so as to account for the parameter fluctuations occurring during the filter implementation two sequences of random variables obeying the bernoulli distribution are employed to describe the phenomenon of the rogvs bounded by prescribed norms in addition the rice fading model is utilized to describe the phenomena of channel fadings where the occurrence probabilities of the random channel coefficients are allowed to time varying through stochastic analysis and lyapunov functional approach sufficient conditions are established under which the filtering error dynamics is exponentially mean square stable with a prespecified h infty performance the set of the desired nonfragile h infty filters is characterized by solving a convex optimization problem via the semidefinite programming method an illustrative example is given to show the usefulness and effectiveness of the proposed design method in this paper

artificial dissipation models for the euler equations various artificial dissipation models that are used with central difference algorithms for the euler equations are analyzed for their effect on accuracy stability and convergence rates in particular linear and nonlinear models are investigated using an implicit approximate factorization code arc2d for transonic airfoils fully implicit application of the dissipation models is shown to improve robustness and convergence rates the treatment of dissipation models at boundaries will be examined it will be shown that accurate error free solutions with sharp shocks can be obtained using a central difference algorithm coupled with an appropriate nonlinear artificial dissipation model i introduction t he solution of the euler equations using numerical techniques requires the use of either a differencing method with inherent dissipation or the addition of dissipation terms to a nondissipative scheme this is because the euler equations do not provide any natural dissipation mechanism such as viscosity in the navier stokes equations that would eliminate high frequencies which are caused by nonlinearitie s and especially shocks a variety of numerical algorithms and computer codes for the euler equations have been developed methods such as maccormack s1 explicit

model reduction of multidimensional and uncertain systems model reduction methods are presented for systems represented by a linear fractional transformation on a repeated scalar uncertainty structure these methods involve a complete generalization of balanced realizations balanced gramians and balanced truncation model reduction with guaranteed error bounds based on solutions to a pair of linear matrix inequalities which generalize lyapunov equations the resulting reduction methods immediately apply to uncertainty simplification and state order reduction in the case of uncertain systems but also may be interpreted as state order reduction for multidimensional systems

the zero dispersion limit of the korteweg de vries equation for initial potentials with non trivial reflection coefficient the inverse scattering method is used to determine the distribution limit as 0 of the solution u x t of the initial value problem ut 6uux 2uxxx 0 u x 0 v x where v x is a positive bump which decays sufficiently fast as x x the case v x 0 has been solved by peter d lax and c david levermore 8 9 10 the computation of the distribution limit of u x t as 0 is reduced to a quadratic maximization problem which is then solved

bounds of the induced norm and model reduction errors for systems with repeated scalar nonlinearities the class of nonlinear systems described by a discrete time state equation containing a repeated scalar nonlinearity as in recurrent neural networks is considered sufficient conditions are derived for the stability and induced norm of such systems using positive definite diagonally dominant lyapunov functions or storage functions satisfying appropriate linear matrix inequalities results are also presented for model reduction errors for such systems

nonstationary l2 l filtering for markov switching repeated scalar nonlinear systems with randomly occurring nonlinearities abstract this paper is concerned with nonstationary l 2 l filtering for markov switching repeated scalar nonlinear systems msrsnss with randomly occurring nonlinearities rons where measurement output is modeled by a mode dependent random variable that satisfying bernouli distribution the new relationship are proposed to depict multiple mutually independent markov chains between original mmsrsnss and nonstationary filters by constructing a proper lyapnov function the msrsnss is stochastically stable with l 2 l performance level is guaranteed accordingly the nonstationary filters are designed where filters are characterised by a two layer structure the paper provides a numerical example verifying the efficacy of established technique

efficient direct solvers for large scale computational fluid dynamics problems this paper presents some new results concerning the direct solution of large systems of equations resulting from the fully implicit coupled solution of the euler and navier stokes equations in three dimensions two methods for such direct solution have been investigated the first is based on the application of sparse matrix algorithms and the second on the implementation of a gauss elimination algorithm optimized for vector parallel processing on cray supercomputers results indicate that sparse matrix algorithms consistently deteriorate for rectangular i e n n m grids in three dimensions as m gets larger than n on the other hand gauss elimination seems to be ideally suited for processing on today s supercomputers achieving execution rates up to 83 of the cray ymp peak efficiency

the τ function of the universal whitham hierarchy matrix models and topological field theories the universal witham hierarchy is considered from the point of view of topological field theories the function for this hierarchy is defined it is proved that the algebraic orbits of whitham hierarchy can be identified with various topological matter models coupled with topological gravity

robust output feedback guaranteed cost control of nonlinear stochastic uncertain systems via an iqc approach this technical note presents a new approach to constructive output feedback robust nonlinear controller design based on the use of iqcs and minimax lqg control the approach involves a class of controllers which include copies of the nonlinearities in the controller the linear part of the controller is synthesized using minimax lqg control theory which is closely related to h infin control theory

compatible poisson brackets of hydrodynamic type some general properties of compatible poisson brackets of hydrodynamic type are discussed in particular a an invariant differential geometric criterion of the compatibility based on the nijenhuis tensor which is slightly different from those existing in the literature b the lax pair with a spectral parameter governing compatible poisson brackets in the diagonalizable case c the connection of this problem with the class of surfaces in euclidean space which possess non trivial deformations preserving the weingarten operator

the multi dimensional hamiltonian structures in the whitham method we consider the averaging of local field theoretic poisson brackets in the multi dimensional case as a result we construct a local poisson bracket for the regular whitham system in the multidimensional situation the procedure is based on the procedure of averaging of local conservation laws and follows the dubrovin novikov scheme of the bracket averaging suggested in one dimensional case however the features of the phase space of modulated parameters in higher dimensions lead to a different natural class of the averaged brackets in comparison with the one dimensional situation here we suggest a direct procedure of construction of the bracket for the whitham system for d 1 and discuss the conditions of applicability of the corresponding scheme at the end we discuss canonical forms of the averaged poisson bracket in the multidimensional case

the integrable kdv6 equations multiple soliton solutions and multiple singular soliton solutions abstract in this work the completely integrable sixth order nonlinear equations kdv6 are studied three distinct cases of these equations are selected for a reliable treatment multiple soliton solutions and multiple singular soliton solutions are formally derived for these equations the cole hopf transformation method combined with the hirota s bilinear sense are used to determine the two distinct structures of solutions the resonance relations for the three cases are investigated

the cole hopf transformation and improved tanh coth method applied to new integrable system kdv6 abstract in this paper we study a new coupled system kdv6 from the point of view of constructing exact solutions for it by using the cole hopf transformation and one improved tanh coth method traveling wave solution which includes periodic and soliton solutions are formally derived

extension of integrable equations in this communication we show that there is general construction to produce non evolutionary integrable equations from a given integrable evolutionary equation to support the main theorem a few examples are explicitly given

on hamiltonian perturbations of hyperbolic systems of conservation laws i quasi triviality of bi hamiltonian perturbations we study the general structure of formal perturbative solutions to the hamiltonian perturbations of spatially one dimensional systems of hyperbolic pdes vt v x 0 under certain genericity assumptions it is proved that any bi hamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on the derivatives the main tool is in constructing the so called quasimiura transformation of jet coordinates eliminating an arbitrary deformation of a semisimple bi hamiltonian structure of hydrodynamic type the quasi triviality theorem we also describe following 35 the invariants ofsuchbi hamiltonian structures with respect to the group of miura type transformations depending polynomially on the derivatives c

computational challenge euler solution for ellipses the present treatment of the inviscid flow past an ellipse via the numerical solution of the euler equations yields a lifting solution for any combination of grid and or angle of attack which is nonsymmetric in order to illustrate the cfd challenge posed by this unusual flow behavior the results obtained call into question the general capability and validity of numerical euler results in the realm of conventional difference methods specifically the mechanism generating lifting results is not understood and the problem s resolution is not yet in sight

stabilization and performance synthesis for systems with repeated scalar nonlinearities the class of nonlinear systems described by a discrete time state equation containing a repeated scalar nonlinearity as in recurrent neural networks is considered given a plant of this form sufficient conditions are derived for 1 the parametrization of all controllers of the same form such that the closed loop is stable in the sense of a diagonally dominant lyapunov function and 2 the synthesis of a controller of the same form so that the induced norm of the closed loop is under a prescribed level several of these conditions can be written into linear matrix inequalities

symmetries conserved quantities and hierarchies for some lattice systems with soliton structure basic invariants such as conserved quantities symmetries mastersymmetries and recursion operators are explicitly constructed for the following nonlinear lattice systems the modified korteweg de vries lattice the ablowitz ladik lattice the brusci ragnisco lattice the ragnisco tu lattice and some cases of the class of integrable systems introduced by bogoyavlensky the algorithmic basis for obtaining these quantities is described and the interrelation between the underlying mastersymmetry approach and the lax pair analysis is discussed by explicit presentation of the higher order members of the corresponding hierarchies new completely integrable lattice flows are found for all systems multi hamiltonian formulations are given

weakly nonlocal symplectic structures whitham method and weakly nonlocal symplectic structures of hydrodynamic type we consider the special type of field theoretical symplectic structures called weakly nonlocal the structures of this type are in particular very common for integrable systems such as kdv or nls we introduce here the special class of weakly nonlocal symplectic structures which we call weakly nonlocal symplectic structures of hydrodynamic type we investigate then the connection of such structures with the whitham averaging method and propose the procedure of averaging the weakly nonlocal symplectic structures the averaging procedure gives the weakly nonlocal symplectic structure of hydrodynamic type for the corresponding whitham system the procedure also gives action variables corresponding to the wave numbers of m phase solutions of the initial system which give the additional conservation laws for the whitham system

induced l sub 2 and generalized h sub 2 filtering for systems with repeated scalar nonlinearities this paper provides complete results on the filtering problem for a class of nonlinear systems described by a discrete time state equation containing a repeated scalar nonlinearity as in recurrent neural networks both induced l sub 2 and generalized h sub 2 indexes are introduced to evaluate the filtering performance for a given stable discrete time systems with repeated scalar nonlinearities our purpose is to design a stable full order or reduced order filter with the same repeated scalar nonlinearities such that the filtering error system is asymptotically stable and has a guaranteed induced l sub 2 or generalized h sub 2 performance sufficient conditions are obtained for the existence of admissible filters since these conditions involve matrix equalities the cone complementarity linearization procedure is employed to cast the nonconvex feasibility problem into a sequential minimization problem subject to linear matrix inequalities which can be readily solved by using standard numerical software if these conditions are feasible a desired filter can be easily constructed these filtering results are further extended to discrete time systems with both state delay and repeated scalar nonlinearities the techniques used in this paper are very different from those used for previous controller synthesis problems which enable us to circumvent the difficulty of dilating a positive diagonally dominant matrix a numerical example is provided to show the applicability of the proposed theories

using inverse nonlinearities in robust output feedback guaranteed cost control of nonlinear systems this paper extends the result of an earlier paper on constructive robust nonlinear output feedback controller design which is based on the use of iqcs and minimax lqg control in this paper in addition to copies of the nonlinearities we include copies of the inverses of the nonlinearities in the controller consequently more iqcs are exploited in the design procedure which gives a smaller bound on the closed loop cost function than the earlier result

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

integrable discretizations of derivative nonlinear schrodinger equations we propose integrable discretizations of derivative nonlinear schrodinger dnls equations such as the kaup newell equation the chen lee liu equation and the gerdjikov ivanov equation by constructing lax pairs the discrete dnls systems admit the reduction of complex conjugation between two dependent variables and possess bi hamiltonian structure through transformations of variables and reductions we obtain novel integrable discretizations of the nonlinear schrodinger nls modified kdv mkdv mixed nls matrix nls matrix kdv matrix mkdv coupled nls coupled hirota coupled sasa satsuma and burgers equations we also discuss integrable discretizations of the sine gordon equation the massive thirring model and their generalizations

a new scheme of integrability for bi hamiltonian pde we develop a new method for constructing integrable hamiltonian hierarchies of lax type equations which combines the fractional powers technique of gelfand and dickey and the classical hamiltonian reduction technique of drinfeld and sokolov the method is based on the notion of an adler type matrix pseudodifferential operator and the notion of a generalized quasideterminant we also introduce the notion of a dispersionless adler type series which is applied to the study of dispersionless hamiltonian equations non commutative hamiltonian equations are discussed in this framework as well

the generation propagation and extinction of multiphases in the kdv zero dispersion limit we study the multiphases in the kdv zero dispersion limit these phases are governed by the whitham equations which are 2g 1 quasi linear hyperbolic equations where g is the number of phases we are interested in both the interaction of two single phases and the breaking of a single phase for general initial data we analyze in detail how a double phase is generated from the interaction or breaking how it propagates in space time and how it collapses to a single phase in a finite time the whitham equations are known to be integrable via a hodograph transform the crucial step in our approach is to formulate the hodograph transform in terms of the euler poisson darboux solutions under our scheme the zeros of the jacobian of the transform are given by the zeros of the euler poisson darboux solution hence the problem of inverting the hodograph transform to give the whitham solution reduces to that of counting the zeros of the euler poisson darboux solution 2002 wiley periodicals inc

universal lie algebra extensions and their casimir invariants via commutative structures abstract we consider a special kind of lie algebra extension called universal extension introduced recently we show that there is one to one correspondence with commutative associative algebras we also investigate the casimir invariants for the universal extensions and consider in detail the so called coextension structures arising in the calculation of the casimir functions

hamiltonian structure of discrete soliton systems we describe an approach for investigating the hamiltonian structures of the lattice isospectral evolution equations associated with a general discrete spectral problem by using the so called implicit representations of the isospectral flows we demonstrate the existence of the recursion operator l which is a strong and hereditary symmetry of the flows it is then proven that every equation in the isospectral hierarchy possesses the hamiltonian structure if l has a skew symmetric factorization and the first equation ut k 0 in the hierarchy satisfies some simple condition we obtain related properties such as the implectic symplectic factorization of l the liouville complete integrability and the multi hamiltonian structures of the isospectral hierarchy four examples are given

unified approach to kdv modulations we develop a unified approach to integrating the whitham modulation equations our approach is based on the formulation of the initial value problem for the zero dispersion kdv as the steepest descent for the scalar riemann hilbert problem 6 and on the method of generating differentials for the kdv whitham hierarchy 9 by assuming the hyperbolicity of the zero dispersion limit for the kdv with general initial data we bypass the inverse scattering transform and produce the symmetric system of algebraic equations describing motion of the modulation parameters plus the system of inequalities determining the number the oscillating phases at any fixed point on the x t plane the resulting system effectively solves the zero dispersion kdv with an arbitrary initial datum c 2001 john wiley sons inc

reciprocal transformations and local hamiltonian structures of hydrodynamic type systems we start from a hyperbolic dubrovin and novikov dn hydrodynamic type system of dimension n which possesses riemann invariants and we settle the necessary conditions on the conservation laws in the reciprocal transformation so that after such a transformation of the independent variables one of the metrics associated with the initial system is flat we prove the following statement let n 3 in the case of reciprocal transformations of a single independent variable or n 5 in the case of transformations of both the independent variables then the reciprocal metric may be flat only if the conservation laws in the transformation are linear combinations of the canonical densities of conservation laws i e the casimirs the momentum and the hamiltonian densities associated with the hamiltonian operator for the initial metric then we restrict ourselves to the case in which the initial metric is either flat or of constant curvature and we classify the reciprocal transformations of one or both the independent variables so that the reciprocal metric is flat such characterization has an interesting geometric interpretation the hypersurfaces of two diagonalizable dn systems of dimension n 5 are lie equivalent if and only if the corresponding local hamiltonian structures are related by a canonical reciprocal transformation

virasoro symmetries of the extended toda hierarchy we prove that the extended toda hierarchy of 1 admits a nonabelian lie algebra of infinitesimal symmetries isomorphic to half of the virasoro algebra the generators lm m 1 of the lie algebra act by linear differential operators onto the tau function of the hierarchy we also prove that the tau function of a generic solution to the extended toda hierarchy is annihilated by a combination of the virasoro operators and the flows of the hierarchy as an application we show that the validity of the virasoro constraints for the cp1 gromov witten invariants and their descendents implies that their generating function is the logarithm of a particular tau function of the extended toda hierarchy

relativistic toda systems we present and study poincare invariant generalizations of the galilei invariant toda systems the classical nonperiodic systems are solved by means of an explicit action angle transformation

conservation laws and symmetries for a nonholonomic deformed volterra equation a nonholonomic deformed volterra equation is studied its lax representation infinitely many of conservation laws and generalized commutation symmetries are given a degenerate recursion operator to generate the generalized symmetries is proposed