semi global symplectic invariants of the spherical pendulum abstract we compute the semi global symplectic invariants near the focus focus point of the spherical pendulum a modified birkhoff normal form procedure is presented to compute the expansion of the hamiltonian near the focus focus point in eliasson variables combining this with explicit formulas for the action we find the semi global symplectic invariants near the focus focus point introduced by v ng c 2003 32 we show that the birkhoff normal form at the focus focus point is the inverse of a complete elliptic integral over a vanishing cycle we close with some remarks about the pendulum for which the invariants can be related to theta functions in a beautiful way

constructing integrable systems of semitoric type let m be a connected symplectic 4 manifold a semitoric integrable system on m essentially consists of a pair of independent real valued smooth functions j and h on m for which j generates a hamiltonian circle action under which h is invariant in this paper we give a general method to construct starting from a collection of five ingredients a symplectic 4 manifold equipped a semitoric integrable system then we show that every semitoric integrable system on a symplectic 4 manifold is obtained in this fashion in conjunction with the uniqueness theorem proved recently by the authors this gives a classification of semitoric integrable systems on 4 manifolds in terms of five invariants

repetitive control system a new type servo system for periodic exogenous signals a control scheme called repetitive control is proposed in which the controlled variables follow periodic reference commands a high accuracy asymptotic tracking property is achieved by implementing a model that generates the periodic signals of period l into the closed loop system sufficient conditions for the stability of repetitive control systems and modified repetitive control systems are derived by applying the small gain theorem and the stability theorem for time lag systems synthesis algorithms are presented by both the state space approach and the factorization approach in the former approach the technique of the kalman filter and perfect regulation is utilized while coprime factorization over the matrix ring of proper stable rational functions and the solution of the hankel norm approximation are used in the latter one

inverse spectral theory for semiclassical jaynes cummings systems quantum semitoric systems form a large class of quantum hamiltonian integrable systems with circular symmetry which has received great attention in the past decade they include systems of high interest to physicists and mathematicians such as the jaynes cummings model 1963 which describes a two level atom interacting with a quantized mode of an optical cavity and more generally the so called systems of jaynes cummings type in this paper we consider the joint spectrum of a pair of commuting semiclassical operators forming a quantum integrable system of jaynes cummings type we prove assuming the bohr sommerfeld rules hold that if the joint spectrum of two of these systems coincide up to mathcal o hbar 2 then the systems are isomorphic

two sided tangential interpolation with real rational units in h the problem of computing units in rh that satisfy ball gohberg rodman two sided tangential interpolation conditions over the open right half complex plane is considered we first show that every unit in rh can be written up to left multiplication by an orthogonal matrix as a finite product of positive real functions in rh with invertible value at this factorization is used to recast interpolation conditions on units as coupled positive real interpolation problems for the factors an algebraic necessary and sufficient condition for the existence of solutions and a parameterization of all solutions when one exists are given we introduce a multivariable parity interlacing property for left interpolation data and present an algorithm to compute units the algorithm involves a line search and is guaranteed to terminate in a finite number of operations when the data set has parity interlacing property as an application strong stabilization of lti plants is discussed a procedure to construct a stable stabilizing controller is given the results are illustrated with a simple mimo example a necessary and sufficient condition characterizing all real rational units that satisfy tangential interpolation conditions is given an algorithm to compute units for left interpolation is presented

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

stable controllers for robust stabilization of systems with infinitely many unstable poles abstract this paper studies the problem of robust stabilization by a stable controller for a linear time invariant single input single output infinite dimensional system we consider a class of plants having finitely many simple unstable zeros but possibly infinitely many unstable poles first we show that the problem can be reduced to an interpolation minimization by a unit element next by the modified nevanlinna pick interpolation we obtain both lower and upper bounds on the multiplicative perturbation under which the plant can be stabilized by a stable controller in addition we find stable controllers to provide robust stability we also present a numerical example to illustrate the results and apply the proposed method to a repetitive control system

sensitivity reduction by stable controllers for mimo infinite dimensional systems via the tangential nevanlinna pick interpolation we study the problem of finding a stable stabilizing controller that satisfies a desired sensitivity level for an mimo infinite dimensional system the systems we consider have finitely many simple transmission zeros in c but they are allowed to possess infinitely many poles in c we compute both upper and lower bounds of the minimum sensitivity achievable by a stable controller via the tangential nevanlinna pick interpolation we also obtain stable controllers attaining such an upper bound to illustrate the results we discuss a repetitive control system as an application of the proposed method

semitoric integrable systems on symplectic 4 manifolds let m be a symplectic 4 manifold a semitoric integrable system on m is a pair of smooth functions j h c m r for which j generates a hamiltonian s 1 action and the poisson brackets j h vanish we shall introduce new global symplec tic invariants for these systems some of these invariants encode topological or geometric aspects while others encode analytical information about the singularities and how they stand with respect to the system our goal is to prove that a semitoric system is completely determined by the invariants we introduce

on a general structure of the stabilizing controllers based on stable range in this paper we prove that some stabilizing controllers of a plant which admits a left right coprime factorization have a special form where their stable and unstable parts are separated the dimension of the unstable part depends on the algebraic concept of stable range of the ring a of siso stable plants moreover we prove that if the stable range of a is equal to 1 then every plant defined by a transfer matrix with entries in the quotient field of a and admitting a left right coprime factorization can be stabilized by a stable controller strong stabilization in particular using a result of treil proving that the stable range of h infty mathbb d is equal to 1 we show that every stabilizable plant defined by a transfer matrix with entries in the quotient field of h infty mathbb d or h infty mathbb c is strongly stabilizable and equivalently every couple of stabilizable plants can be simultaneously stabilized by a controller simultaneous stabilization finally using the fact that the topological stable range of h infty mathbb d is equal to 2 a result due to suarez we show that every unstabilizable siso plant defined by a transfer function with entries in the quotient field of h infty mathbb d is as close as we want to a stabilizable plant in the product topology

sensitivity reduction by strongly stabilizing controllers for mimo distributed parameter systems this note investigates a sensitivity reduction problem by stable stabilizing controllers for a linear time invariant multi input multioutput distributed parameter system the plant we consider has finitely many unstable zeros which are simple and blocking but can possess infinitely many unstable poles we obtain a necessary condition and a sufficient condition for the solvability of the problem using the matrix nevanlinna pick interpolation with boundary conditions we also develop a necessary and sufficient condition for the solvability of the interpolation problem and show an algorithm to obtain the solutions our method to solve the interpolation problem is based on the schur nevanlinna algorithm

structured systems within the transfer matrix approach application to decoupling abstract we consider a new approach to structured systems within the transfer matrix framework we associate with such a structured system a graph which concentrates both knowledge on structural properties of independent subsystems and on their interconnections the analysis of this associated graph allows to conclude on the decouplability of structured systems

sensitivity minimization by strongly stabilizing controllers for a class of unstable time delay systems weighted sensitivity minimization is studied within the framework of strongly stabilizing stable hinfin controller design for a class of infinite dimensional systems this problem has been solved by ganesh and pearson for finite dimensional plants using nevanlinna pick interpolation we extend their technique to a class of unstable time delay systems moreover we illustrate suboptimal solutions and their robust implementation

relationships between internal and external stability for infinite dimensional systems with applications to a servo problem a study is made of the relationships among various stability motions for a class of infinite dimensional systems which contains a class of systems not covered by existing methods e g those having infinitely many unstable poles it is proved that internal l sup 2 stability and exponential stability are equivalent and internal stability implies h sup infinity stability several necessary and sufficient conditions for internal stability are derived in particular under certain conditions a canonical realization is internally stable if it is externally stable these results are applied to the servo problem involving this class of systems it is shown that i an internal model is necessary for tracking ii an internal model along with closed loop stability implies tracking a typical example called a repetitive control system is discussed to illustrate the results

semiclassical quantization and spectral limits of ħ pseudodifferential and berezin toeplitz operators we introduce a minimalistic notion of semiclassical quantization and use it to prove that the convex hull of the semiclassical spectrum of a quantum system given by a collection of commuting operators converges to the convex hull of the spectrum of the associated classical system this gives a quick alternative solution to the isospectrality problem for quantum toric systems if the operators are uniformly bounded the convergence is uniform analogous results hold for non commuting operators

moment polytopes for symplectic manifolds with monodromy abstract a natural way of generalising hamiltonian toric manifolds is to permit the presence of generic isolated singularities for the moment map for a class of such almost toric 4 manifolds which admits a hamiltonian s 1 action we show that one can associate a group of convex polygons that generalise the celebrated moment polytopes of atiyah guillemin sternberg as an application we derive a duistermaat heckman formula demonstrating a strong effect of the possible monodromy of the underlying integrable system

correction to inverse spectral theory for semiclassical jaynes cummings systems we explain why theorem b in the original article does not follow from the main result of this paper theorem a while we conjecture that theorem b should nevertheless be true in this erratum we prove a slightly weaker version of it

convexity properties of the moment mapping re examined abstract consider a hamiltonian action of a compact lie group on a compact symplectic manifold a theorem of kirwan s says that the image of the momentum mapping intersects the positive weyl chamber in a convex polytope i present a new proof of kirwan s theorem which gives explicit information on how the vertices of the polytope come about and on how the shape of the polytope near any point can be read off from infinitesimal data on the manifold it also applies to some interesting classes of noncompact or singular hamiltonian spaces such as cotangent bundles and complex affine varieties