relationships among three kinds of finite dimensional integrable hamiltonian systems abstract a new type of finite dimensional integrable hamiltonian systems containing many restricted soliton flows are considered their relations with the generalized classical gaudin models and the generalized classical gaudin models with boundary are established as an application the relationships among three different restricted mkdv flows are exhibited

the truncated complex moment problem let 2n denote a sequence of complex numbers 00 01 10 0 2n 2n 0 00 0 ij ji and let k denote a closed subset of the complex plane c the truncated complex k moment problem for entails determining whether there exists a positive borel measure on c such that ij zizj d 0 i j 2n and supp k for k kp a semi algebraic set determined by a collection of complex polynomials p pi z z mi 1 we characterize the existence of a finitely atomic representing measure with the fewest possible atoms in terms of positivity and extension properties of the moment matrix m n and the localizing matrices mpi we prove that there exists a rankm n atomic representing measure for 2n supported in kp if and only if m n 0 and there is some rankpreserving extension m n 1 for which mpi n ki 0 where deg pi 2ki or 2ki 1 1 i m

spectral theory for the periodic sine gordon equation a concrete viewpoint a summary of the spectral theory for quasiperiodic sine and sinh gordon equations is given analogies with whole line solitons and scattering theory motivates the discussion the relation between the ingredients in the inverse spectral solution of the periodic sine gordon equation and physical characteristics of sine gordon waves is emphasized the explicit topics covered are summarized in the table of contents in the introduction

middle convolution and harnad duality dettweiler reiter s additive description of katz s middle convolution can be interpreted in terms of the harnad duality using this interpretation together with mumford s geometric invariant theory we generalize the additive middle convolution to have a multi parameter and act on systems of linear ordinary differential equations with irregular singularities we show that the generalized operation holds basic properties of the original one and additionally show that katz s algorithm involving our generalization works well in generic case

backlund transformations for nonlinear sigma models with values in riemannian symmetric spaces this work deals with backlund transformations for the principal sl n sigma model together with all reduced models with values in riemannian symmetric spaces first the dressing method of zakharov mikhailov and shabat is shown for the case of a meromorphic dressing matrix to be equivalent to a backlund transformation for an associated linearly extended system comparison of this multi backlund transformation with the composition of ordinary ones leads to a new proof of the permutability theorem a new method of solution for such multi backlund transformations mbt is developed by the introduction of a soliton correlation matrix which satisfies a riccati system equivalent to the mbt using the geometric structure of this system a linearization is achieved leading to a nonlinear superposition formula expressing the solution explicitly in terms of solutions of a single backlund transformation through purely linear algebraic relations a systematic study of all reductions of the system by involutive automorphisms is made thereby defining the multi backlund transformations and their solution for all riemannian symmetric spaces

global optimization of rational functions a semidefinite programming approach we consider the problem of global minimization of rational functions on unconstrained case and on an open connected semi algebraic subset of or the partial closure of such a set constrained case we show that in the univariate case n 1 these problems have exact reformulations as semidefinite programming sdp problems by using reformulations introduced in the phd thesis of jibetean 16 this extends the analogous results by nesterov 13 for global minimization of univariate polynomials for the bivariate case n 2 we obtain a fully polynomial time approximation scheme fptas for the unconstrained problem if an a priori lower bound on the infimum is known by using results by de klerk and pasechnik 1 for the np hard multivariate case we discuss semidefinite programming based relaxations for obtaining lower bounds on the infimum by using results by parrilo 15 and lasserre 12

integrable decomposition of a hierarchy of soliton equations and integrable coupling system by semidirect sums of lie algebras abstract staring from a new spectral problem a hierarchy of the soliton equations is derived it is shown that the associated hierarchies are infinite dimensional integrable hamiltonian systems by the procedure of nonlinearization of the lax pairs the integrable decomposition of the whole soliton hierarchy is given further we construct two integrable coupling systems for the hierarchy by the conception of semidirect sums of lie algebras

consecutive rosochatius deformations of the neumann system consecutive rosochatius deformations of the neumann system are investigated it is first shown that different realizations of a classical sl 2 gaudin magnet model yield different integrable hamiltonian systems then an algorithm of constructing infinitely many symplectic realizations of sl 2 algebra from a known one is presented and thus the neumann system can be deformed consecutively the second rosochatius deformation of the neumann system is taken as an illustrative example to show that the deformed systems admit separations of variables and may be linearized on the jacobi variety

the c neumann problem as a completely integrable system on an adjoint orbit it is shown by purely lie algebraic methods that the c neumann problem the motion of a material point on a sphere under the influence of a quadratic potential is a completely integrable system of euler poisson equations on a minimal dimensional orbit of a semidirect product of lie algebras 1 the c neumann problem the motion of a point on the sphere s 1 under the influence of a quadratic potential u x ax x x e r a diag al an is a completely integrable hamiltonian system for n 3 this has been shown by c neumann in 1859 12 and for arbitrary n by k uhlenbeck 16 r devaney 3 j moser 10 11 m adler and p van moerbeke 2 in this paper we show how this problem fits naturally in the framework of euler poisson equations 41 5 14 17 proving that the c neumann problem is a hamiltonian system on a minimaldimensional adjoint orbit in a semidirect product of lie algebras thus its complete integrability will follow entirely from lie algebraic considerations the equations of motion are i ax1 axx i 1 n 1 1 where the lagrange multiplier a ax x 11x112 is chosen such that x e sn i during the motion set x y and get the equivalent system to 1 1 xi yi yi a1xi ax x 11y112 xi llxll 1 x y 0 1 2 the following crucial remark that motivated the present investigation is due to k uhlenbeck and can be verified without any difficulties lemma 1 1 put x xixj p yixj xiyj system 1 2 is equivalent to x p x p x a lixl 1 x y o 1 3 remark that if one replaces x and a by x id n and a tr a id n respectively where id is the n x n identity matrix equations 1 3 remain unchanged from now on we shall assume that in 1 3 this change has been made so that x p a e sl n the next section gives a lie algebraic interpretation to these equations received by the editors march 12 1980 1980 mathematics subject classification primary 58f07 70h05 secondary 53c15 17b99

the kowalewski and henon heiles motions as manakov geodesic flows on so 4 a two dimensional family of lax pairs the invariant surfaces for the kowalewski top the henon heiles system and the manakov geodesic flow onso 4 complete into abelian surfacesa by adjoining in each case a divisord of arithmetic genus 9 these divisors belong to the same linear system ona and they each define a polarization 2 4 therefore there are rational maps transforming the kowalewski top and the henon heiles system into manakov s geodesic flow onso 4 this paper deals with the precise geometric relationship between these three problems it is based on the splitting of the 8 dimensional space of sections ofd theta functions into an even and an odd part and also on a normal form for the six quadrics describinga as embedded in 7 as a byproduct we get a 2 dimensional family of lax pairs for both the kowalewski top and the henon heiles system

quadrics on real riemannian spaces of constant curvature separation of variables and connection with gaudin magnet the integrable systems are considered which are connected with separation of variables in real riemannian spaces of constant curvature an isomorphism is given for these systems with hyperbolic gaudin magnet using this isomorphism the complete classification of separable coordinate systems is provided by means of the corresponding l operators for the gaudin magnet

a tight small gain theorem for not necessarily iss systems a new version of the small gain theorem is presented for nonlinear finite dimensional systems the result provides conditions for global asymptotic stability under relaxed assumptions in particular the two interconnected subsystems need not be input to state stable in open loop

hyperelliptic quasi periodic and soliton solutions of the nonlinear schrodinger equation on considere la construction de solutions par des methodes de geometrie algebriques et en particulier par les fonctions theta hyperelliptiques

surfaces and the sklyanin bracket we discuss the lie poisson group structures associated to splittings of the loop group lgl n due to sklyanin concentrating on the finite dimensional leaves of the associated poisson structure we show that the geometry of the leaves is intimately related to a complex algebraic ruled surface with a invariant poisson structure in particular sklyanin s lie poisson structure admits a suitable abelianisation once one passes to an appropriate spectral curve the sklyanin structure is then equivalent to one considered by mukai tyurin and bottacin on a moduli space of sheaves on the poisson surface the abelianization procedure gives rise to natural darboux coordinates for these leaves as well as separation of variables for the integrable hamiltonian systems associated to invariant functions on the group

kac moody lie algebras and soliton equations iii stationary equations associated with a1 1 abstract the hamiltonian structure of stationary soliton equations associated with the akns eigenvalue problem is derived in two ways first it is shown to arise from the kostant kirillov symplectic structure on a coadjoint orbit in an infinite dimensional lie algebra second it is obtained as the restriction to a finite dimensional manifold of the infinite dimensional hamiltonian structure associated with a certain eigenvalue problem polynomial in the eigenvalue parameter

a generalised homogeneous solution for global stabilisation of a class of non smooth upper triangular systems this paper investigates the problem of using small controls to globally stabilise a class of upper triangular systems including non smooth systems a generalised definition of homogeneity with new weights is employed to relax the existing restrictions imposed on the nonlinearities and enable the design of non smooth stabilisers by developing a more delicate design approach small state feedback controllers are recursively constructed in a bottom to top fashion to globally stabilise the upper triangular systems in addition a number of typical examples are discussed in order to illustrate that the proposed result not only encompasses several existing works under the same topic but also solves the global stabilisation problem of more general upper triangular systems using simpler controller structures

a nonconfocal generator of involutive systems and three associated soliton hierarchies quite a few new finite dimensional integrable systems have been obtained by the nonlinearization procedure of eigenvalue problems suggested by the author almost all of their associated involutive systems are generated from a confocal generator k nj 1 k j 1 b2kj bkj pkqj pjqk in the present paper a nonconfocal generator k nj 1 2k 2j 1 k j 2a2kj e k j 2 b2kj akj pkpj eqkqj is found from which some new integrable systems and soliton hierarchies are obtained

quotients on the sato grassmannian and the moduli of vector bundles it is shown that there exists a geometric quotient of the subscheme of stable points of gr c z r under the action of sl r c the consequences in terms of vector bundles on an algebraic curve are studied

quasi periodic and periodic solutions for coupled nonlinear schrodinger equations of manakov type we consider travelling periodic and quasi periodic wave solutions in coupled nonlinear schrodinger equations in fibre optics these equations can be used to model single mode fibres with strong birefringence and two mode optical fibres recently these equations appear as a model describing pulse pulse interactions in wavelength division multiplexed channels of optical fibre transmission systems in some cases this model reduces to the integrable manakov system ims two phase quasi periodic solutions for the ims are given in terms of two dimensional kleinian functions the reduction of quasi periodic solutions to elliptic functions is discussed new solutions are found in terms of generalized hermite polynomials which are associated with two gap treibich verdier potentials

on computing rational gauss chebyshev quadrature formulas we provide an algorithm to compute the nodes and weights for gauss chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary real poles outside 1 1 contrary to existing rational quadrature formulas the computational effort is very low even for extremely high degrees and under certain conditions on the poles it can be shown that the complexity is of order o n this method is based on the derivation of explicit expressions for chebyshev orthogonal rational functions which are thus far the only examples of explicitly known orthogonal rational functions on 1 1 with arbitrary real poles outside this interval

a loop group approach to the c neumann problem and moser veselov factorization a geometrical description of continuous and discrete versions of the neumann oscillator in terms of a loop group framework is investigated it is shown that the continuous neumann oscillator can be integrated by the riemann hilbert factorization on the twisted loop group of o 3 lo 3 the solution of the problem is given in terms of a special class of flows on the quotient space lo 3 lo 3 where lo 3 is a subgroup of positive loops in lo 3 it is also shown that the moser veselov algorithm for integrating a discrete version of the neumann oscillator heisenberg xyz chain is induced by a discrete flow in this space the flow can be explicitly integrated by solving the matrix riccati equation in both cases discrete and continuous conservation laws are derived from time invariance of a relation that holds for coefficients of the fourier expansion of the flows

the camassa holm hierarchy n dimensional integrable systems and algebro geometric solution on a symplectic submanifold this paper shows that the camassa holm ch spectral problem yields two different integrable hierarchies of nonlinear evolution equations nlees one is of negative order ch hierachy while the other one is of positive order ch hierarchy the two ch hierarchies possess the zero curvature representations through solving a key matrix equation we see that the well known ch equation is included in the negative order ch hierarchy while the dym type equation is included in the positive order ch hierarchy furthermore under two constraint conditions between the potentials and the eigenfunctions the ch spectral problem is cast in 1 a new neumann like n dimensional system when it is restricted into a symplectic submanifold of 2n which is proven to be integrable by using the dirac poisson bracket and the r matrix process and 2 a new bargmann like n dimensional system when it is considered in the whole 2n which is proven to be integrable by using the standard poisson bracket and the r matrix process

isospectral hamiltonian flows in finite and infinite dimensions the approach to isospectral hamiltonian flow introduced in part i is further developed to include integration of flows with singular spectral curves the flow on finite dimensional ad invariant poisson submanifolds of the dual widetilde gl r of the positive part of the loop algebra widetilde gl r is obtained through a generalization of the standard method of linearization on the jacobi variety of the invariant spectral curves these curves are embedded in the total space of a line bundlet p1 c allowing an explicit analysis of singularities arising from the structure of the image of a moment map tilde j r m n r times m n r to widetilde gl r from the space of rank r deformations of a fixedn n matrixa it is shown how the linear flow of line bundles e t to tilde s over a suitably desingularized curve tilde s may be used to determine both the flow of matricial polynomialsl and the hamiltonian flow in the spacemn r mn r in terms of functions the resulting flows are proved to be completely integrable the reductions to subalgebras developed in part i are shown to correspond to invariance of the spectral curves and line bundles e t to tilde s under certain linear or anti linear involutions the integration of two examples from part i is given to illustrate the method the rosochatius system and the cnls coupled non linear schrodinger equation

integrable rosochatius deformations of the restricted soliton flows a method to construct integrable rosochatius deformations of the restricted soliton flows in the setup of lax formulation is presented the integrable rosochatius deformations of the restricted soliton flows such as the restricted ablowitz kaup newell segur flow the restricted tu meng flow the restricted tu flow with neumann type constraints and the restricted modified korteweg de vries flow together with their lax representations are presented in addition a lax representation of the jacobi rosochatius system is obtained

homogeneity with incremental degrees and global stabilisation of a class of high order upper triangular systems the problem of global asymptotic stabilisation is investigated for a class of upper triangular systems with uncontrollable linearisation a notion of generalised homogeneity is introduced in this article providing a new insight and a deeper perspective on how to categorise nonlinear terms into the higher order and equal order terms in a non traditional way the delicate analysis and new categorisation of the vector field make it possible to encompass more general nonlinearities a globally stabilising controller is recursively constructed by a bottom up step by step procedure containing the equal order terms as its indispensable components it is illustrated that the results obtained in this article incorporate and extend existing results for the global stabilisation of upper triangular systems

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

robust stabilization via saturated feedback in this paper we deal with the problem of stabilization of uncertain systems in the presence of input constraint first algebraic conditions are derived for input to state stability of linear system with saturated linear feedback of low dimension then a recursive design procedure is derived for robust stabilization of block upper triangular nonlinear systems with feedforward structure

generalizations of the neumann system a curve theoretical approach part iii order n systems the neumann system is a well known algebraically completely integrable hamiltonian system its geometry has roots in hyperelliptic curve theory and the isospectral deformation theory of hill s operator in this paper generalizations of the neumann system are found for n sheeted riemann surfaces and the isospectral deformation theory of operators of order n trace formulas lax pairs and constants of motion are found the new systems are shown to be algebraically completely integrable

an extension of the classical theory of algebraic invariants to pseudo riemannian geometry and hamiltonian mechanics we develop a new approach to the study of killing tensors defined in pseudo riemannian spaces of constant curvature that is ideologically close to the classical theory of invariants the main idea which provides the foundation of the new approach is to treat a killing tensor as an algebraic object determined by a set of parameters of the corresponding vector space of killing tensors under the action of the isometry group the spaces of group invariants and conformal group invariants of valence two killing tensors defined in the minkowski plane are described the group invariants which are the generators of the space of invariants are applied to the problem of classification of orthogonally separable hamiltonian systems defined in the minkowski plane transformation formulas to separable coordinates expressed in terms of the parameters of the corresponding space of killing tensors are presented the results are applied to the problem of orthogonal separability of the drach superintegrable potentials

a simplified small gain theorem for time varying nonlinear systems the small gain theorem as established in two previous papers has been given in terms of a quite general interconnection of two nonlinear subsystems this note gives a simplified version and a more straightforward proof for the small gain theorem for a simpler interconnection of two nonlinear subsystems by using a different characterization of the input to state stable iss concept the resulting small gain condition features the familiar contraction mapping inequality for time invariant systems moreover we have also incorporated some type of model uncertainties into the system description thus making this special version directly applicable to the global robust stabilization and output regulation problems

integrable rosochatius deformations of higher order constrained flows and the soliton hierarchy with self consistent sources we propose a systematic method for generalizing the integrable rosochatius deformations for finite dimensional integrable hamiltonian systems to integrable rosochatius deformations for infinite dimensional integrable equations an infinite number of the integrable rosochatius deformed higher order constrained flows of some soliton hierarchies which includes the generalized integrable henon heiles system and the integrable rosochatius deformations of the kdv hierarchy with self consistent sources of the akns hierarchy with self consistent sources and of the mkdv hierarchy with self consistent sources as well as their lax representations are presented

integrable evolution systems based on gerdjikov ivanov equations bi hamiltonian structure finite dimensional integrable systems and n fold darboux transformation a spectral problem and the associated gerdjikov ivanov gi hierarchy of nonlinear evolution equations is presented as a reduction the well known gi equation of derivative nonlinear schrodinger equations is obtained it is shown that the gi hierarchy is integrable in a liouville sense and possesses bi hamiltonian structure moreover the spectral problem can be nonlinearized as a finite dimensional completely integrable system under the bargmann constraint between the potentials and the eigenfunctions in particular an explicit n fold darboux transformation for the gi equation is constructed with the help of a gauge transformation of spectral problems and a reduction technique some explicit solitonlike solutions of the gi equation are given by applying its darboux transformation

linear r matrix algebra for classical separable systems we consider a hierarchy of the natural type hamiltonian systems of n degrees of freedom with polynomial potentials separable in general ellipsoidal and general paraboloidal coordinates we give a lax representation in terms of 2 2 matrices for the whole hierarchy and construct the associated linear r matrix algebra with the r matrix dependent on the dynamical variables a yang baxter equation of dynamical type is proposed using the method of variable separation we provide the integration of the systems in classical mechanics constructing the separation equations and hence the explicit form of action variables the quantization problem is discussed with the help of the separation variables

the deduction of the lax representation for constrained flows from the adjoint representation for x and tn finite dimensional hamiltonian systems obtained from the decompositions of zero curvature equations it is shown that their lax representations can be deduced directly from the adjoint representations of the auxiliary linear problems as a consequence the zero curvature representation for soliton hierarchy with source is presented

separation of variables in the classical integrable sl 3 magnetic chain there are two fundamental problems studied by the theory of hamiltonian integrable systems integration of equations of motion and construction of action angle variables a third problem however should be added to the list separation of variables though much simpler thant the two others it has important relations to quantum integrability separation of variables is constructed for thesl 3 magnetic chain an example of an integrable model associated to a nonhyperelliptic algebraic curve

the moment problem on compact semi algebraic sets in this chapter we begin the study of the multidimensional moment problem the passage to dimensions d 2 brings new difficulties and unexpected phenomena in sect 3 2 we derived solvability criteria of the moment problem on intervals in terms of positivity conditions it seems to be natural to look for similar characterizations in higher dimensions as well this leads us immediately into the realm of real algebraic geometry and to descriptions of positive polynomials on semi algebraic sets in this chapter we treat this approach for basic closed compact semi algebraic subsets of mathbb r d it turns out that for such sets there is a close interaction between the moment problem and positivstellensatze for strictly positive polynomials

r matrix construction of electromagnetic models for the painleve transcendents the painleve transcendents pi pv and their representations as isomonodromic deformation equations are derived as nonautonomous hamiltonian systems from the classical r matrix poisson bracket structure on the dual space slr 2 of the loop algebra slr 2 the hamiltonians are obtained by composing elements of the poisson commuting ring of spectral invariant functions on slr 2 with a time dependent family of poisson maps whose images are four dimensional rational coadjoint orbits in slr 2 each system may be interpreted as describing a particle moving on a surface of zero curvature in the presence of a time varying electromagnetic field the painleve equations follow from reduction of these systems by the hamiltonian flow generated by a second commuting element in the ring of spectral invariants 1995 american institute of physics

the k moment problem with densities given a compact basic semi algebraic set mathbf k subset mathbb r n a rational fraction f mathbb r n to mathbb r and a sequence of scalars y y we investigate when y alpha int mathbf k x alpha f d mu holds for all alpha in mathbb n n i e when y is the moment sequence of some measure fd supported on k this yields a set of convex linear matrix inequalities lmi we also use semidefinite programming to develop a converging approximation scheme to evaluate the integral fd when the moments of are known and f is a given rational fraction numerical expreriments are also provided we finally provide again lmi conditions on the moments of two measures nu mu with support contained in k to have d nu f d mu for some rational fraction f

robust global stabilization of a class of uncertain feedforward nonlinear systems abstract the problem of global asymptotically stabilizing a certain class of uncertain feedforward nonlinear systems is considered the control law is obtained by nesting saturation functions whose amplitude can be rendered arbitrarily small with respect to previous works on the subject the design procedure is able to deal with uncertain possibly time varying parameters ranging within the prescribed compact sets which can affect also the linear approximation of the system the small gain theorem for nonlinear systems which are input to state stable with restrictions is shown to be a key tool for designing a state feedback saturated control law

neumann type integrable reduction for nonlinear evolution equations in 1 1 and 2 1 dimensions a family of new neumann type systems is given in view of the nonlinearization technique realizing the variable separation of the modified jaulent miodek hierarchy and a new coupled modified kadomtsev petviashvili equation on the symplectic submanifold tsn 1 by two casimir functions and a special solution of the lenard eigenvalue equation we deduce the lax moser matrix of the neumann type systems that yields integrals of motion and the constrained hamiltonians whose vector fields are tangent to tsn 1 based on the dirac poisson bracket and a lax equation on tsn 1 a new systematic way is proposed to prove the liouville integrability of a family of neumann type systems synchronously the dirac poisson bracket and the generating function are used to reveal the explicit relation between infinite dimensional integrable systems and neumann type systems and then we point out that compatible solutions of neumann type systems yield the finite parametric solutions of 1 1 and 2 1 dimensional integrable nonlinear

isospectral hamiltonian flows in finite and infinite dimensions i generalized moser systems and moment maps into loop algebras a moment map jr jia gl r is constructed from the poisson manifold j a of rank r perturbations of a fixed n xn matrix a to the dual r f e positive part of the formal loop algebra i r g r c 1 the adler kostant symes theorem is used to give hamiltonians which generate commutative isospectral flows on g r the pull back of these hamiltonians by the moment map gives rise to commutative isospectral hamiltonian flows in j a the latter may be identified with flows on finite dimensional coadjoint orbits in g and linearized on the jacobi variety of an invariant spectral curve xr which generically is an r sheeted riemann surface reductions of j a are derived corresponding to subalgebras of gl r c and sl r c determined as the fixed point set of automorphism groupes generated by involutions i e jdl the classical algebras as well as reductions to twisted subalgebras of s r c the theory is illustrated by a number of examples of finite dimensional isospectral flows defining integrable hamiltonian systems and their embeddings as finite gap solutions to integrable systems of pde s

lax representation r matrix method and separation of variables for the neumann type restricted flow the r matrix method and the method of separation of variables for restricted flows with neumann constraints are considered restricted jaulent miodek flows are taken as illustrative examples lax representations are obtained from the adjoint representation of the corresponding soliton hierarchy the r matrix formula based on this lax representation is derived and the separation of variables for the hamilton jacobi equation are performed

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

a nonlinear small gain theorem for the analysis of control systems with saturation a nonlinear small gain theorem is presented that provides a formalism for analyzing the behavior of certain control systems that contain or utilize saturation the theorem is used to show that an iterative procedure can be derived for controlling systems in a general nonlinear feedforward form this result in turn is applied to the control of 1 linear systems stable and unstable with inputs subject to magnitude and rate saturation and time delays 2 the cascade of globally asymptotically stable nonlinear systems with certain linear systems those that are stabilizable right invertible and such that all of their invariant zeros have nonpositive real part 3 the inverted pendulum on a cart and 4 the planar vertical takeoff and landing aircraft

some algebro geometric solutions for the coupled modified kadomtsev petviashvili equations arising from the neumann type systems a treatment is described for getting some algebro geometric solutions of the coupled modified kadomtsev petviashvili cmkp equations and a hierarchy of 1 1 dimensional integrable nonlinear evolution equations inlees by using the neumann type systems through three steps i according to the nonlinearization of lax pair the cmkp equations and the 1 1 dimensional inlees are reduced to a family of compatible neumann type systems on a symplectic submanifold whose involutive solutions give rise to the finite parametric solutions of the inlees in both 2 1 and 1 1 dimensions ii from the holomorphic differentials and the abel map on a hyperelliptic curve of riemman surface the abel jacobi variables are introduced to straighten out the neumann type flows the 1 1 and 2 1 dimensional flows giving the abel jacobi solutions iii based on the riemann theorem and the trace formulas the jacobi inversion is applied to the straightened flows for getting some new algebro geometric solutions of the cmkp equations and the 1 1 dimensional integrable hierarchy

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

finite gap solutions of 2 1 dimensional integrable nonlinear evolution equations generated by the neumann systems each soliton equation in the korteweg de vries kdv hierarchy the 2 1 dimensional breaking soliton equation and the 2 1 dimensional caudrey dodd gibbon kotera sawada cdgks equation are reduced to two or three neumann systems on the tangent bundle tsn 1 of the unit sphere sn 1 the lax moser matrix for the neumann systems of degree n 1 is deduced in view of the mckean trubowitz identity and a bilinear generating function whose favorite characteristic accounts for the problem of the genus of riemann surface matching to the number of elliptic variables from the lax moser matrix the constrained hamiltonians in the sense of dirac poisson bracket for all the neumann systems are written down in a uniform recursively determined by integrals of motion the involution of integrals of motion and constrained hamiltonians is completed on tsn 1 by using a lax equation and their functional independence is displayed over a dense open subset of tsn 1 by a direct calculation which contribute to the liouville integ

the soliton correlation matrix and the reduction problem for integrable systems integrable 1 1 dimensional systems associated to linear first order matrix equations meromorphic in a complex parameter as formulated by zakharov mikhailov and shabat 1 3 zms are analyzed by a new method based upon the soliton correlation matrix m matrix the multi backlund transformation which is equivalent to the introduction of an arbitrary number of poles in the zms dressing matrix is expressed by a pair of matrix riccati equations for them matrix through a geometrical interpretation based upon group actions on grassman manifolds the solution of this system is explicitly determined in terms of the solutions to the zms linear system reductions of the system corresponding to invariance under finite groups of automorphisms are also solved by reducing them matrix suitably so as to preserve the class of invariant solutions

the finite band solution of the jaulent miodek equation a method to construct the finite band solution to the soliton equation is presented we take the jaulent miodek equation as an example with the use of the nonlinearization of lax pair the jaulent miodek equation is decomposed into two finite dimensional integrable systems whose properties are studied from the view of an r matrix then through solving these finite dimensional integrable systems the finite band solution of the jaulent miodek equation is obtained

multi hamiltonian structures for r matrix systems for the rational elliptic and trigonometric r matrices we exhibit the links between three levels of poisson spaces a some finite dimensional spaces of matrix valued meromorphic functions on the complex line b spaces of spectral curves and sheaves supported on them and c symmetric products of a surface we have at each level a linear space of compatible poisson structures and the maps relating the levels are poisson this leads in a natural way to the nijenhuis coordinates for these spaces at level b there are hamiltonian systems on these spaces which are integrable for each poisson structure in the family and which are such that the lagrangian leaves are the intersections of the symplectic leaves over the poisson structures in the family specific examples include many of the well known integrable systems

discrete time analogues of some nonlinear oscillators in the inverse square potential we propose a family of symplectic maps providing finite difference approximations for the wojciechowski and rosochatius systems the lax pair representations are found and complete integrability is proved

a hierarchy of garnier rosochatius systems a hierarchy of hamiltonian systems including the garnier rosochatius system as the first member is introduced this hierarchy of hamiltonian systems is proved to be completely integrable and the corresponding flows are linearized on the jacobi variety of the associated hyperelliptic curve in addition a relation between these flows and the kdv equation is found and the connection with finite gap solution of kdv equation is shown

hamiltonians and zero curvature equations for integrable partial differential equations we discuss the relationship between two approaches to integrable partial differential equations one using formal affine lie algebras and the other banach lie groups in the first approach integrability of the equations follows from commutativity of hamiltonian flows on the lie algebra while in the second it follows from commutativity of certain flows induced by an action on the banach lie group we show that these two methods are essentially equivalent since one can calculate one type of the flows from the other the classes of solutions encompassed by the two methods however vary significantly we demonstrate this relationship specifically with the nonlinear schrodinger equation

disturbance attenuation of feedforward systems with dynamic uncertainty this paper studies the disturbance attenuation problem of a class of nonlinear systems in feedforward form that is subject to both dynamic uncertainty and disturbance when the disturbance vanishes the equilibrium of the closed loop system is globally asymptotically stable two versions of small gain theorem with restrictions are employed to establish the global attractiveness and local stability of the closed loop system at the equilibrium respectively

kac moody lie algebras and soliton equations ii lax equations associated with a1 1 the soliton equations associated with sl 2 eigenvalue problems polynomial in the eigenvalue parameter are given a unified treatment they are shown to be generated by a single family of commuting hamiltonians on a subalgebra of the loop algebra of sl 2 the conserved densities and fluxes of the usual anks hierarchy are identified with conserved densities and fluxes for the polynomial eigenvalue problems the hamiltonian structures of the soliton equations associated with the polynomial eigenvalue problems are given a unified treatment

abelian integrals and the reduction method for an integrable hamiltonian system a classical finite dimensional integrable hamiltonian system corresponding to the motion of a particle constrained to an n dimensional sphere n 0 x2 1 with the hamiltonian h 1 2 y2 u2 x2 e x2 where u and e are constants and y are the momenta conjugate to x is integrated using several different methods these are the following 1 the projection of geodesic free flow on a larger space namely the sphere s2n 1 for e 0 the flow is obtained in terms of elementary functions 2 separation of variables in the hamilton jacobi equation in elliptic coordinates or alternatively the use of a complete set of integrals of motion in involution to reduce hamilton s equations to quadratures the flow is obtained in terms of abelian integrals which are then inverted in terms of generalized functions the relation between the different methods and results is clarified using methods of algebraic geometry in particular the geometry of quadrics

on liouville integrability of zero curvature equations and the yang hierarchy sufficient conditions for a zero curvature equation ut vx u v 0 being liouville integrable are investigated in the case that the equation is integrable an explicit formula of the poisson bracket h lambda h mu for hamiltonians h is proposed the yang hierarchy is derived and shown to be liouville integrable

on the integrability of stationary and restricted flows of the kdv hierarchy a bi hamiltonian formulation for stationary flows of the kdv hierarchy is derived in an extended phase space a map between stationary flows and restricted flows is constructed in one case it connects an integrable henon heiles system and the garnier system moreover a new integrability scheme for hamiltonian systems is proposed that holds in the standard phase space

algebraic integrability of the confluent neumann system in this paper we study the neumann system which describes the harmonic oscillator of arbitrary dimension constrained to the sphere in particular we will consider the confluent case where two eigenvalues of the potential coincide which implies that the system has s1 symmetry we will prove complete algebraic integrability of the confluent neumann system and show that its flow can be linearized on the generalized jacobian torus of some singular algebraic curve the symplectic reduction of s1 action will be described and we will show that the general rosochatius system is a symplectic quotient of the confluent neumann system where all the eigenvalues of the potential are double