generalized invariant subspaces and parameter insensitive disturbance rejection problems with static output feedback in the geometric approach some generalized invariant subspaces for uncertain linear systems are investigated and then various parameter insensitive disturbance rejection problems with static output feedback are formulated and necessary and sufficient conditions for the problems to be solvable are presented further some sufficient conditions which are useful to check the solvability conditions are presented using the notions of generalized invariant subspaces

disturbance localization with dead beat control for linear periodic discrete time systems in this paper the classical notion of a controlled invariant subspace together with its main properties is extended to the case of linear periodic discrete time systems and used for deriving necessary and sufficient solvability conditions for the disturbance localization problem and a synthesis procedure for the solution moreover the notions of outer and inner controllable subspaces are introduced and studied for the same class of system thus allowing the derivation of necessary and sufficient solvability conditions for the disturbance localization problem with output or state dead beat control and to give synthesis procedures for the solutions

survey paper static output feedback a survey this paper reviews the static output feedback problem in the control of linear time invariant lti systems it includes analytical and computational methods and presents in a unified fashion the knowledge gained in the decades of research into this important open problem the paper shows that although many approaches and techniques exist to approach different versions of the problem no efficient algorithmic solutions are available

backlund transformations and loop group actions we construct a local action of the group of rational maps from 2 to gl n on local solutions of flows of the zs akns sl n hierarchy we show that the actions of simple elements linear fractional transformations give local backlund transformations and we derive a permutability formula from different factorizations of a quadratic element we prove that the action of simple elements on the vacuum may give either global smooth solutions or solutions with singularities however the action of the subgroup of the rational maps that satisfy the u n reality condition on the space of global rapidly decaying solutions of the flows in the u n hierarchy is global and the action of a simple element gives a global backlund transformation the actions of certain elements in the rational loop group on the vacuum give rise to explicit time periodic multisolitons multibreathers we show that this theory generalizes the classical backlund theory of the sine gordon equation the group structures of backlund transformations for various hierarchies are determined by their reality conditions we identify the reality conditions the group structures for the sl n u k n k kdv kupershmidt wilson and gel fand dikii hierarchies the actions of linear fractional transformations that satisfy a reality condition modulo the center of the group of rational maps give backlund and darboux transformations for the hierarchy defined by the reality condition since the factorization cannot always be carried out under this reality condition the action is again local and backlund transformations only generate local solutions for these hierarchies unless singular solutions are allowed 2000 john wiley sons inc

solitons from dressing in an algebraic approach to the constrained kp heirachy the algebraic matrix hierarchy approach based on affine lie sl n algebras leads to a variety of 1 1 soliton equations by varying the rank of the underlying sl n algebra as well as its gradation in the affine setting one encompasses the set of the soliton equations of the constrained kp hierarchy the soliton solutions are then obtained as elements of the orbits of the dressing transformations constructed in terms of representations of the vertex operators of the affine sl n algebras realized in the unconventional gradations such soliton solutions exhibit non trivial dependence on the kdv odd time flows and kp odd and even time flows which distinguishes them from the conventional structure of the darboux backlund wronskian solutions of the constrained kp hierarchy

generalized c a b pairs and parameter insensitive disturbance rejection problems with dynamic compensator in the so called geometric approach the notion of the generalized c a b pairs is introduced for uncertain linear systems and its properties are investigated further the parameter insensitive disturbance rejection problem with dynamic compensator is formulated and some sufficient conditions for the problem to be solvable are presented

regular conjugacy classes in the weyl group and integrable hierarchies generalized kdv hierarchies associated by drinfeld sokolov reduction with grade 1 regular semisimple elements from non equivalent heisenberg subalgebras of a loop algebra g x c lambda lambda 1 are studied the graded heisenberg subalgebras containing such elements are labelled by the regular conjugacy classes in the weyl group w g of the simple lie algebra g a representative w epsilon w g of a regular conjugacy class can be lifted to an inner automorphism of g given by w exp 2i pi ad i0 m where i0 is the defining vector of an sl2 subalgebra of g the grading is then defined by the operator d m i0 m lambda d d lambda ad i0 and any grade 1 regular element lambda from the heisenberg subalgebra associated with w takes the form lambda c lambda c where i0 c m 1 c and c is included in an sl2 subalgebra containing i0 the largest eigenvalue of adi0 is m 1 except for some cases in f4 e6 7 8 we explain how these lie algebraic results follow from known results and apply them to construct integrable systems if the largest ad i0 eigenvalue is m 1 then using any grade 1 regular element from the heisenberg subalgebra associated with w we can construct a kdv system possessing the standard w algebra defined by i0 as its second poisson bracket algebra for g a classical lie algebra we derive pseudo differential lax operators for those non principal kdv systems that can be obtained as discrete reductions of kdv systems related to gln non abelian toda systems are also considered

coupling integrable couplings and bi hamiltonian structure associated with the boiti pempinelli tu hierarchy a kind of new lie algebra is introduced whose corresponding loop algebra is followed to define from which two isospectral problems are established whose compatibility condition gives rise to a coupling integrable coupling of the boiti pempinelli tu bpt hierarchy by making use of the quadratic form identity its bi hamiltonian structure is obtained finally we reduce the coupling integrable coupling to obtain two integrable couplings of the bpt hierarchy furthermore the well known mkdv equation and a new coupled equation with two dependent variables are produced respectively

extensions of the matrix gelfand dickey hierarchy from generalized drinfeld sokolov reduction thep p matrix version of ther kdv hierarchy has been recently treated as the reduced system arising in a drinfeld sokolov type hamiltonian symmetry reduction applied to a poisson submanifold in the dual of the lie algebra widehat gl pr 1 here a series of extensions of this matrix gelfand dickey system is derived by means of a generalized drinfeld sokolov reduction defined for the lie algebra widehat gl pr s 1 using the natural embeddinggl pr glpr s fors any positive integer the hierarchies obtained admit a description in terms of ap p matrix pseudo differential operator comprising anr kdv type positive part and a non trivial negative part this system has been investigated previously in thep 1 case as a constrained kp system in this paper the previous results are considerably extended and a systematic study is presented on the basis of the drinfeld sokolov approach that has the advantage that it leads to local poisson brackets and makes clear the conformal w algebra structures related to the kdv type hierarchies

robust disturbance rejection problems for linear spl omega periodic discrete time systems two robust disturbance rejection problems with state feedback and with incomplete state feedback for linear spl omega periodic discrete time systems are studied in the framework of the so called geometric approach also some necessary conditions and or sufficient conditions for the problems to be solvable are presented

the generalised zakharov shabat system and the gauge group action the generalized zakharov shabat systems with complex valued non regular cartan elements and the systems studied by caudrey beals and coifman cbc systems and their gauge equivalent are studied this study includes the properties of fundamental analytical solutions for the gauge equivalent to cbc systems and the minimal set of scattering data the description of the class of nonlinear evolutionary equations solvable by the inverse scattering method and the recursion operator related to such systems the hierarchies of hamiltonian structures the results are illustrated in the example of the multi component nonlinear schrodinger equations and the corresponding gauge equivalent multi component heisenberg ferromagnetic type models related to so 5 c algebra

a necessary and sufficient condition for parameter insensitive disturbance rejection problem with state feedback in this paper a necessary and sufficient condition for a parameter insensitive disturbance rejection problem with state feedback which was pointed out as an open problem by bhattacharyya to be solvable is proved a constructive algorithm of simultaneously a b invariant subspaces for a finite number of linear systems and a relationship between simultaneously a b invariant subspaces and generalized a b invariant subspaces play an important role to prove the main result

local and non local conserved quantities for generalized non linear schrodinger equations it is shown how to construct infinitely many conserved quantities for the classical non linear schrodinger equation associated with an arbitrary hermitian symmetric spaceg k these quantities are non local in general but include a series of local quantities as a special case their poisson bracket algebra is studied and is found to be a realization of the half kac moody algebrakr consisting of polynomials in positive powers of a complex parameter which have coefficients in the compact real form ofk the lie algebra ofk

vortex filament on symmetric lie algebras and generalized bi schrodinger flows in this article we display an evolving model on symmetric lie algebras from a purely geometric way by using the hamiltonian or para hamiltonian gradient flow of a fourth order functional called generalized bi schrodinger flows which corresponds to the fukumoto moffatt s model in the theory of moving curves or the vortex filament in physical words in mathbb r 3 the theory of vortex filament in mathbb r 3 or mathbb r 2 1 up to the third order approximation is shown to be generalized to symmetric lie algebras in a unified way

two dimensional generalized toda lattice the zero curvature representation is obtained for the two dimensional generalized toda lattices connected with semisimple lie algebras the reduction group and conservation laws are found and the mass spectrum is calculated

bi hamiltonian structures of the coupled akns hierarchy and the coupled yajima oikawa hierarchy the hamiltonian theory for the two component akns hierarchy and yajima oikawa hierarchy is considered from the viewpoint of reduction we show that the second hamiltonian structures of the former is a dirac reduction of the sl 3 current algebra while the latter is related to the classical w 3 4 algebra

group theoretical symmetries and generalized backlund transformations for integrable systems a notion of symmetry for 1 1 dimensional integrable systems is presented which is consistent with their group theoretic description it is shown how a group symmetry may be used together with a dynamical reduction to produce new generalizations of the backlund transformation for the korteweg de vries equation to its sl n c generalization an additional application to the relativistic invariance of the leznov saveliev systems is given

gauge covariant theory of the generating operator i a gauge covariant formulation of the generating operator operator theory for the zakharov shabat system is proposed the operator tilde lambda corresponding to the gauge equivalent system in the pole gauge is explicitly calculated thus the unified approach to the nonlinear schrodinger type equations based on is automatically reformulated with the help of tilde lambda for the heisenberg ferromagnet type equations consequently it is established that the conserved densities for the heisenberg ferromagnet type equations are polynomial ins x and itsx derivatives special attention is paid to the interrelation between the hierarchies of symplectic structures corresponding to the above mentioned families of gauge equivalent equations it is shown that the geometrical properties of the conjugated operator are gaugeindependent

the matrix nonlinear schrodinger equation in dimension 2 in this paper we study the existence of global solutions to the cauchy problem for the matrix nonlinear schrodinger equation mnls in 2 space dimensions a sharp condition for the global existence is obtained for this equation this condition is in terms of an exact stationary solution of a semilinear elliptic equation in the scalar case the mnls reduces to the well known cubic nonlinear schrodinger equation for which existence of solutions has been studied by many authors

a variational approach to the macroscopic electrodynamics of anisotropic hard superconductors we consider bean s critical state model for anisotropic superconductors a variational problem solved by the quasi static evolution of the internal magnetic field is obtained as the limit of functionals arising from maxwell s equations combined with a power law for the dissipation moreover the quasi static approximation of the internal electric field is recovered using a first order necessary condition if the sample is a long cylinder subjected to an axial uniform external field the macroscopic electrodynamics is explicitly determined

aks hierarchy and bi hamiltonian geometry of gelfand zakharevich type a bi hamiltonian system is a system of differential equations which can be written in hamiltonian form in two distinct ways the applications of gelfand zakharevich bi hamiltonian structure which is an extension of a poisson nijenhuis structure on phase space has been extensively explored by falqui magri and pedroni in the context of separation of variables it is well known that the integrable hamiltonian systems defined by the adler kostant symes aks scheme contains bi hamiltonian structure in this paper we unveil the connection between adler kostant symes formalism applied to loop algebra and the gelfand zakharevich bi hamiltonian structure by superposition the results of fordy and kulish in the aks scheme we also study the commuting flows of the aks hierachy and its connection to the zakharov shabat hierarchy

riemann equations in bidifferential calculus we consider equations that formally resemble a matrix riemann or hopf equation in the framework of bidifferential calculus with different choices of a first order bidifferential calculus we obtain a variety of equations including a semi discrete and a fully discrete version of the matrix riemann equation a corresponding universal solution generating method then either yields a continuous or discrete cole hopf transformation or leaves us with the problem of solving riemann equations hence an application of the hodograph method if the bidifferential calculus extends to second order solutions of a system of riemann equations are also solutions of an equation that arises on the universal level of bidifferential calculus as an integrability condition depending on the choice of bidifferential calculus the latter can represent a number of prominent integrable equations like self dual yang mills as well as matrix versions of the two dimensional toda lattice hirota s bilinear difference equation 2 1 dimensional nonlinear schrodinger nls kadomtsev petviashvili kp equation and davey stewartson equations for all of them a recent non isospectral binary darboux transformation result in bidifferential calculus applies which can be specialized to generate solutions of the associated riemann equations for the latter we clarify the relation between these specialized binary darboux transformations and the aforementioned solution generating method from arbitrary size matrix versions of the riemann equations associated with an integrable equation possessing a bidifferential calculus formulation multi soliton type solutions of the latter can be generated this includes breaking multi soliton type solutions of the self dual yang mills and the 2 1 dimensional nls equation which are parametrized by solutions of riemann equations

t duality in 2d integrable models the non conformal analogue of abelian t duality transformations relating pairs of axial and vector integrable models from the non abelian affine toda family is constructed and studied in detail

reductions of n wave interactions related to low rank simple lie algebras i z2 reductions the analysis and the classification of all reductions for the nonlinear evolution equations solvable by the inverse scattering method is an interesting and still open problem we show how the second order reductions of the n wave interactions related to low rank simple lie algebras g can be embedded also in the weyl group of g this allows us to display along with the well known ones a number of new types of integrable n wave systems some of the reduced systems find applications to nonlinear optics

generalized controlled and conditioned invariances for linear ω periodic discrete time systems abstract in this paper the three notions of generalized controlled a b invariance generalized conditioned c a invariance and generalized a b c invariance for uncertain linear periodic discrete time systems are studied and then the parameter insensitive disturbance rejection problems with static output feedback and or with state feedback are formulated and their solvability conditions are presented

vertex operators and soliton solutions of affine toda model with u 2 symmetry the symmetry structure of the non abelian affine toda model based on the coset sl 3 sl 2 u 1 is studied it is shown that the model possess non abelian noether symmetry closing into a q deformed sl 2 u 1 algebra specific two vertex soliton solutions are constructed

integrable evolution equations on associative algebras this paper surveys the classification of integrable evolution equations whose field variables take values in an associative algebra which includes matrix clifford and group algebra valued systems a variety of new examples of integrable systems possessing higher order symmetries are presented symmetry reductions lead to an associative algebra valued version of the painleve transcendent equations the basic theory of hamiltonian structures for associative algebra valued systems is developed and the bihamiltonian structures for several examples are found

integrable dynamical systems with quadratic and cubic nonlinearities related to graded lie algebras practical implementation of two important methods of constructing integrable dynamical systems the lax pair method and the adler konstant symes theorem is studied general considerations are illustrated by several examples that contain multiwave interaction systems and interacting garnier systems as particular cases

nonautonomous integrable nonlinear schrodinger equations with generalized external potentials inhomogeneous nonlinear schrodinger equations are constructed systematically using the covariance with respect to the darboux transformation the darboux invariants and the spectral parameter are functions of time and space which result in equations describing nonautonomous solitons moving in generalized external potentials all the known inhomogeneous nonlinear schrodinger equations as well as new ones of a more generalized form can be derived from the constructed equation one and two soliton solutions are explicitly constructed using the darboux transformation

four lie algebras associated with r6 and their applications the first part in the paper reads that a three dimensional lie algebra is first introduced whose corresponding loop algebra is constructed for which isospectral problems are established by employing zero curvature equations a modified kaup newell mkn soliton hierarchy of evolution equations is obtained the corresponding hereditary operator and hamiltonian structure are worked out respectively then two types of enlarging semisimple lie algebras isomorphic to the linear space r6 are followed to construct one of them is a complex lie algebra their corresponding loop algebras are also given so that two types of new isospectral problems are introduced to generate two kinds of integrable couplings of the above mkn hierarchy the hereditary operators hamiltonian structures of the hierarchies are produced again respectively the exact computing formulas of the constant appearing in the trace identity and the variational identity are derived under the semisimple algebras the second part of this pap

the geometric approach for linear periodic discrete time systems abstract recently the disturbance localization problem and the problems of designing disturbance decoupled observers and of giving a geometric characterization of invariant zeros were solved for linear periodic discrete time systems through an extension of the geometric approach based on the notions of controlled invariant and conditionally invariant subspaces to periodic ones here the existing periodic geometric theory is supplemented with the notions of outer reachable subspace and controllability subspace and with some further results on the previously introduced notions of inner reachable controllable subspace and outer controllable subspace in addition it is shown that any such periodic geometric theory can be restated in an equivalent time invariant form which is just the same theory written for a suitable time invariant system having an extended state space

dressing method for the vector sine gordon equation and its soliton interactions in this paper we develop the dressing method to study the exact solutions for the vector sine gordon equation the explicit formulas for one kink and one breather are derived the method can be used to construct multi soliton solutions two soliton interactions are also studied the formulas for position shift of the kink and position and phase shifts of the breather are given these quantities only depend on the pole positions of the dressing matrices

a new family of integrable models in 2 1 dimensions associated with hermitian symmetric spaces in a series of papers fordy and his collaborators have studied families of integrable models in 1 1 dimensions associated with hermitian symmetric spaces these models were also generalized to 2 1 dimensions however the generalization used is not unique and in this paper a different generalization is considered resulting in new families of integrable models in 2 1 dimensions

a discretization of the matrix nonlinear schrodinger equation in this paper we shall show that a class of solutions to the discrete coupled matrix nonlinear schrodinger equation dcmnlse is gauge equivalent to the discrete equation of the schrodinger flow of maps into the grassmannian and the realizing gauge transformation is only the discretization of a classical gauge transformation between the matrix nonlinear schrodinger equation mnlse and the schrodinger flow of maps into the grassmannian in other words from the viewpoint of gauge equivalence a class of solutions of the dcmnlse is a correct discretization of the mnlse

matrix nonlinear schrodinger equations and moment maps into loop algebras it is shown how darboux coordinates on a reduced symplectic vector space may be used to parametrize the phase space on which the finite gap solutions of matrix nonlinear schrodinger equations are realized as isospectral hamiltonian flows the parametrization follows from a moment map embedding of the symplectic vector space reduced by suitable group actions into the dual g of the algebra g of positive frequency loops in a lie algebra g the resulting phase space is identified with a poisson subspace of g consisting of elements that are rational in the loop parameter reduced coordinates associated to the various hermitian symmetric lie algebras g k corresponding to the classical lie algebras are obtained

a powerful approach to generate new integrable systems in this paper an effective algorithm to generate integrable systems is given as a result many new integrable equations are derived in a systematic way

vector nonlinear schrodinger equation on the half line we investigate the manakov model or more generally the vector nonlinear schrodinger equation on the half line using a backlund transformation method two classes of integrable boundary conditions are derived mixed neumann dirichlet and robin boundary conditions integrability is shown by constructing a generating function for the conserved quantities we apply a nonlinear mirror image technique to construct the inverse scattering method with these boundary conditions the important feature in the reconstruction formula for the fields is the symmetry property of the scattering data emerging from the presence of the boundary particular attention is paid to the discrete spectrum an interesting phenomenon of transmission between the components of a vector soliton interacting with the boundary is demonstrated this is specific to the vector nature of the model and is absent in the scalar case for one soliton solutions we show that the boundary can be used to make certain components of the incoming soliton vanishingly small this is reminiscent of the phenomenon of light polarization by reflection

functional representations of integrable hierarchies we consider a general framework for integrable hierarchies in lax form and derive certain universal equations from which functional representations of particular hierarchies such as kp discrete kp mkp akns i e formulations in terms of functional equations are systematically and quite easily obtained the formalism genuinely applies to hierarchies where the dependent variables live in a noncommutative typically matrix algebra the obtained functional representations can be understood as noncommutative analogues of fay identities for the kp hierarchy

disturbance decoupling with quadratic stability for switched linear systems in this paper disturbance decoupling problems without stability and with quadratic stability for switched linear systems are formulated in the framework of the so called geometric approach firstly necessary and sufficient conditions for the problem without stability to be solvable are given secondly sufficient conditions for the problem with quadratic stability to be solvable are given further for switched linear systems composed of two subsystems necessary and sufficient conditions for the problem with quadratic stability to be solvable are also investigated finally an illustrative example is shown

sit nls solitons in hermitian symmetric spaces soliton solutions of multi component sit nls systems associated with hermitian symmetric spaces are obtained using the backlund transformation projectors of special form are introduced and solved to satisfy specific conditions to obtain one soliton solutions the non abelian permutability theorem is developed and used to obtain two soliton solutions

nonlinear schrodinger equations and simple lie algebras we associate a system of integrable generalised nonlinear schrodinger nls equations with each hermitian symmetric space these nls equations are considered as reductions of more general systems this time associated with a reductive homogeneous space the nonlinear terms are related to the curvature and torsion tensors of the appropriate geometrical space the hamiltonian structure is investigated using r matrix techniques and shown to be canonical for all these equations throughout the reduction procedure this hamiltonian structure does not degenerate each of the above systems of equations is gauge equivalent to a generalised ferromagnet reductions of the latter are discussed in terms of the corresponding nls type equations