pole placement by parametric output feedback abstract this note presents a new analytical solution to the problem of pole placement via constant output feedback under the condition m p n where n m and p are the number of states inputs and outputs respectively the approach is based upon parametric eigenstructure assignment of linear time invariant multivariable systems in combination with a special explicit formulation of the pole assignment equations thus the resulting analytical solution explicitly offers all remaining m p n degrees of freedom beyond eigenvalue assignment which can be used for additional design goals such as response shaping minimizing the norm of the feedback matrix and robust control respectively

almost sure well posedness of the cubic nonlinear schrodinger equation below l 2 mathbb t we consider the cauchy problem for the one dimensional periodic cubic non linear schrodinger equation nls with initial data below l 2 in particular we exhibit nonlinear smoothing when the initial data are randomized then we prove local well posedness of nls almost surely for the initial data in the support of the canonical gaussian measures on h s t for each s 1 and global well posedness for each s 1 12

a variational formula for the free energy of an interacting many particle system we consider n bosons in a box in r d with volume n rho under the influence of a mutually repellent pair potential the particle density p is an element of 0 infinity is kept fixed our main result is the identification of the limiting free energy f beta p at positive temperature 1 beta in terms of an explicit variational formula for any fixed rho if beta is sufficiently small and for any fixed beta if rho is sufficiently small the thermodynamic equilibrium is described by the symmetrized trace of e beta h n where h n denotes the corresponding hamilton operator the well known feynman kac formula reformulates this trace in terms of n interacting brownian bridges due to the symmetrization the bridges are organized in an ensemble of cycles of various lengths the novelty of our approach is a description in terms of a marked poisson point process whose marks are the cycles this allows for an asymptotic analysis of the system via a large deviations analysis of the stationary empirical field the resulting variational formula ranges over random shift invariant marked point fields and optimizes the sum of the interaction and the relative entropy with respect to the reference process in our proof of the lower bound for the free energy we drop all interaction involving infinitely long cycles and their possible presence is signalled by a loss of mass of the finitely long cycles in the variational formula in the proof of the upper bound we only keep the mass on the finitely long cycles we expect that the precise relationship between these two bounds lies at the heart of bose einstein condensation and intend to analyze it further in future

derivation of the gross pitaevskii hierarchy for the dynamics of bose einstein condensate consider a system of n bosons on the three dimensional unit torus interacting via a pair potential n2v n xi xj where x x1 xn denotes the positions of the particles suppose that the initial data n 0 satisfies the condition where hn is the hamiltonian of the bose system this condition is satisfied if n 0 wn n 0 where wn is an approximate ground state to hn and n 0 is regular let n t denote the solution to the schrodinger equation with hamiltonian hn gross and pitaevskii proposed to model the dynamics of such a system by a nonlinear schrodinger equation the gross pitaevskii gp equation the gp hierarchy is an infinite bbgky hierarchy of equations so that if ut solves the gp equation then the family of k particle density matrices k ut ut solves the gp hierarchy we prove that as n the limit points of the k particle density matrices of n t are solutions of the gp hierarchy our analysis requires that the n boson dynamics be described by a modified hamiltonian that cuts off the pair interactions whenever at least three particles come into a region with diameter much smaller than the typical interparticle distance our proof can be extended to a modified hamiltonian that only forbids at least n particles from coming close together for any fixed n 2006 wiley periodicals inc

grassmannian central projection and output feedback pole assignment of linear systems a sufficient rank condition for arbitrary pole assignability by static output feedback of m input p output linear systems of mcmillan degree n mp is given this condition is easy to check and is satisfied by the generic system an effective computational method to compute the feedback laws assigning any self conjugate set of closed loop poles is also given

unconditional uniqueness for the cubic gross pitaevskii hierarchy via quantum de finetti we present a new simpler proof of the unconditional uniqueness of solutions to the cubic gross pitaevskii hierarchy in one of the main tools in our analysis is the quantum de finetti theorem our uniqueness result is equivalent to the one established in the celebrated works of erd s schlein and yau 2015 wiley periodicals inc

periodic nonlinear schrodinger equation and invariant measures in this paper we continue some investigations on the periodic nlseiu u iu xx u u p 2 p 6 started in lrs we prove that the equation is globally wellposed for a set of data of full normalized gibbs measrue e beta h phi hd phi x h phi tfrac 1 2 int left phi right 2 tfrac 1 p int left phi right p after suitablel 2 truncation the set and the measure are invariant under the flow the proof of a similar result for the kdv and modified kdv equations is outlined the main ingredients used are some estimates from b1 on periodic nls and kdv type equations

on well posedness for general hierarchy equations of gross pitaevskii and hartree type gross pitaevskii and hartree hierarchies are infinite systems of coupled pdes emerging naturally from the mean field theory of bose gases their solutions are known to be related to initial value problems in particular the gross pitaevskii and hartree equations due to their physical and mathematical relevance the issues of well posedness and uniqueness for these hierarchies have recently been studied thoroughly using specific nonlinear and combinatorial techniques in this article we introduce a new approach for the study of such hierarchy equations by firstly establishing a duality between them and certain liouville equations and secondly solving the uniqueness and existence questions for the latter as an outcome we formulate a hierarchy equation starting from any initial value problem which is u 1 invariant and prove a general principle which can be stated formally as follows in particular several new well posedness results as well as a counterexample to uniqueness for the gross pitaevskii hierarchy equation are proved the novelty in our work lies in the aforementioned duality and the use of liouville equations with powerful transport techniques extended to infinite dimensional functional spaces

the quintic nls as the mean field limit of a boson gas with three body interactions we investigate the dynamics of a boson gas with three body interactions in dimensions d 1 2 we prove that in the limit of infinite particle number the bbgky hierarchy of k particle marginals converges to a limiting gross pitaevskii gp hierarchy for which we prove existence and uniqueness of solutions factorized solutions of the gp hierarchy are shown to be determined by solutions of a quintic nonlinear schrodinger equation our proof is based on and extends methods of erdos schlein yau klainerman machedon and kirkpatrick schlein staffilani

random data cauchy problem for the navier stokes equations on t3 in this paper we study the random data initial value problem of the navier stokes equations in l2 t3 by using the randomization approach recently developed by n burq and n tzvetkov we prove that for almost all where is the sample space of a probability space a p for the randomized initial data f h l 2 t3 the navier stokes equations have a global solution u c 0 l 2 t3 or more precisely u et f h c 0 h 12 t3

on the uniqueness of solutions to the periodic 3d gross pitaevskii hierarchy abstract in this paper we present a uniqueness result for solutions to the gross pitaevskii hierarchy on the three dimensional torus under the assumption of an a priori spacetime bound we show that this a priori bound is satisfied for factorized solutions to the hierarchy which come from solutions of the nonlinear schrodinger equation in this way we obtain a periodic analogue of the uniqueness result on r 3 previously proved by klainerman and machedon 75 except that in the periodic setting we need to assume additional regularity in particular we need to work in the sobolev class h for 1 by constructing a specific counterexample we show that on t 3 the existing techniques from the work of klainerman and machedon approach do not apply in the endpoint case 1 this is in contrast to the known results in the non periodic setting where these techniques are known to hold for all 1 in our analysis we give a detailed study of the crucial spacetime estimate associated to the free evolution operator in this step of the proof our methods rely on lattice point counting techniques based on the concept of the determinant of a lattice this method allows us to obtain improved bounds on the number of lattice points which lie in the intersection of a plane and a set of radius r depending on the number theoretic properties of the normal vector to the plane we are hence able to obtain a sharp range of admissible sobolev exponents for which the spacetime estimate holds

randomization and the gross pitaevskii hierarchy we study the gross pitaevskii hierarchy on the spatial domain mathbb t 3 by using an appropriate randomization of the fourier coefficients in the collision operator we prove an averaged form of the main estimate which is used in order to contract the duhamel terms that occur in the study of the hierarchy in the averaged estimate we do not need to integrate in the time variable an averaged spacetime estimate for this range of regularity exponents then follows as a direct corollary the range of regularity exponents that we obtain is alpha frac 3 4 it was shown in our previous joint work with gressman j funct anal 266 7 4705 4764 2014 that the range alpha 1 is sharp in the corresponding deterministic spacetime estimate this is in contrast to the non periodic setting which was studied by klainerman and machedon commun math phys 279 1 169 185 2008 where the spacetime estimate is known to hold whenever alpha geq 1 the goal of our paper is to extend the range of in this class of estimates in a probabilistic sense we use the new estimate and the ideas from its proof in order to study randomized forms of the gross pitaevskii hierarchy more precisely we consider hierarchies similar to the gross pitaevskii hierarchy but in which the collision operator has been randomized for these hierarchies we show convergence to zero in low regularity sobolev spaces of duhamel expansions of fixed deterministic density matrices we believe that the study of the randomized collision operators could be the first step in the understanding of a nonlinear form of randomization

embedding henselian fields into power series abstract every henselian field of residue characteristic 0 admits a truncation closed embedding in a field of generalised power series possibly with a factor set as corollaries we obtain the ax kochen ershov theorem and an extension of mourgues and ressayre s theorem every ordered field which is henselian in its natural valuation has an integer part we also give some results for the mixed and the finite characteristic cases

some remarks on real and complex output feedback we provide some new necessary and sufficient conditions which guarantee arbitrary pole placement of a particular linear system over the complex numbers we exhibit a nontrivial real linear system which is not controllable by real static output feedback and discuss a conjecture from algebraic geometry concerning the existence of real linear systems for which all static feedback laws are real

some results on the genericity of the pole placement problem abstract in this note the pole placement problem for a linear mimo systems with p outputs and m inputs is studied from the algebraic point of view a formulation is proposed that allows to analyze both theoretical and numerical aspects of the case min m p 2 with more sharpness moreover here it is shown that arbitrary pole placement by static output feedback of unitary rank is generically not possible even if m p n holds true

stabilization of discrete time linear systems by static output feedback this note proposes a systematic approach for the static output feedback control design for discrete time linear systems it is shown that if the open loop system satisfies some particular structural conditions a static output feedback gain can be calculated easily using a formula only involving the original system matrices among the conditions the system has to satisfy the stronger relies on a minimum phase argument square and nonsquare systems are considered

global wellposedness of defocusing critical nonlinear schrodinger equation in the radial case is globally wellposed in time more precisely we obtain a unique solution u u ch1 0 such that for all time u t depends continuously on the data in fact the dependence is even real analytic here moreover there is scattering for t the same statement holds for radial data h s 1 and proves in particular global existence of classical solutions in the radially symmetric case also this issue was open thus this is the analogue for nls of the result for the wave equation with quintic nonlinearity obtained by m struwe str in the radial case and by m grillakis gr s s in general in the case of the wave equation the proof is based on the following two different facts

least squares pole assignment by memoryless output feedback in this paper a pole assignment problem for a linear time invariant control system by memoryless output feedback is posed as a least squares optimisation problem and analysed the cost functions are appropriately modified so as to obtain convergence of the corresponding gradient flow and existence of the global minimum this approach is also extended to accommodate the output pole assignment problem with insensitivity against disturbances in system parameters the relation between the modified cost functions and the original pole assignment problem is revealed the proposed approach is compared with other existing approaches and illustrated by numerical results

on the uniqueness of solutions to the gross pitaevskii hierarchy the purpose of this note is to give a new proof of uniqueness of the gross pitaevskii hierarchy first established in 1 in a different space based on space time estimates similar in spirit to those of 2

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

mean field and classical limit of many body schrodinger dynamics for bosons we present a new proof of the convergence of the n particle schrodinger dynamics for bosons towards the dynamics generated by the hartree equation in the mean field limit for a restricted class of two body interactions we obtain convergence estimates uniform in hbar up to an exponentially small remainder for hbar 0 the classical dynamics in the mean field limit is given by the vlasov equation

rigorous derivation of the cubic nls in dimension one we derive rigorously the one dimensional cubic nonlinear schrodinger equation from a many body quantum dynamics the interaction potential is rescaled through a weak coupling limit together with a short range one we start from a factorized initial state and prove propagation of chaos with the usual two step procedure in the former step convergence of the solution of the bbgky hierarchy associated to the many body quantum system to a solution of the bbgky hierarchy obtained from the cubic nls by factorization is proven in the latter we show the uniqueness for the solution of the infinite bbgky hierarchy

on the cauchy problem for gross pitaevskii hierarchies the purpose of this paper is to investigate the cauchy problem for the gross pitaevskii infinite linear hierarchy of equations on rn n 1 we prove local existence and uniqueness of solutions in certain sobolev type spaces h of sequences of marginal density operators with n 2 in particular we give a clear discussion of all cases n 2 which covers the local well posedness problem for the gross pitaevskii hierarchy in this situation

on the propagation of weakly nonlinear random dispersive waves we study several basic dispersive models with random periodic initial data such that the different fourier modes are independent random variables motivated by the vast physics literature on related topics we then study how much the fourier modes of the solution at later times remain decorrelated our results are sensitive to the resonances associated with the dispersive relation and to the particular choice of the initial data

derivation of the gross pitaevskii equation for the dynamics of bose einstein condensate consider a system of n bosons in three dimensions interacting via a repulsive short range pair potential n 2 v n xi xj where x x1 xn denotes the positions of the particles let hn denote the hamiltonian of the system and let n t be the solution to the schrodinger equation suppose that the initial data n 0 satisfies the energy condition h n 0 h k n n 0i c k n k for k 1 2 we also assume that the k particle density matrices of the initial state are asymptotically factorized as n we prove that the k particle density matrices of n t are also asymptotically factorized and the one particle orbital wave function solves the gross pitaevskii equation a cubic non linear schrodinger equation with the coupling constant given by the scattering length of the potential v we also prove the same conclusion if the energy condition holds only for k 1 but the factorization of n 0 is assumed in a stronger sense ams subject classification number 81v70 81t18 35q55

computing dynamic output feedback laws the pole placement problem asks to find laws to feed the output of a plant governed by a linear system of differential equations back to the input of the plant so that the resulting closed loop system has a desired set of eigenvalues converting this problem into a question of enumerative geometry efficient numerical homotopy algorithms to solve this problem for general multiple input multiple output systems have been proposed recently despite the wider application range of dynamic feedback laws the realization of the output of the numerical homotopies as a machine to control the plant in the time domain has not been addressed before in this note we present symbolic numeric algorithms to turn the solution to the question of enumerative geometry into a useful control feedback machine we report on numerical experiments with our publicly available software phcpack and illustrate its application on various control problems from the literature

statistical mechanics of the nonlinear schrodinger equation we investigate the statistical mechanics of a complex fieldo whose dynamics is governed by the nonlinear schrodinger equation such fields describe in suitable idealizations langmuir waves in a plasma a propagating laser field in a nonlinear medium and other phenomena their hamiltonian h phi int omega frac 1 2 nabla phi 2 1 p phi p dx is unbounded below and the system will under certain conditions develop self focusing singularities in a finite time we show that when is the circle and thel 2 norm of the field which is conserved by the dynamics is bounded byn the gibbs measure obtained is absolutely continuous with respect to wiener measure and normalizable if and only ifp andn are such that classical solutions exist for all time no collapse of the solitons this measure is essentially the same as that of a one dimensional version of the more realisitc zakharov model of coupled langmuir and ion acoustic waves in a plasma we also obtain some properties of the gibbs state by both analytic and numerical methods asn and the temperature are varied

uniqueness of solutions to the 3d quintic gross pitaevskii hierarchy abstract in this paper we study solutions to the three dimensional quintic gross pitaevskii hierarchy we prove unconditional uniqueness among all small solutions in the critical space h 1 which corresponds to h 1 on the nls level with slight modifications to the proof we also prove unconditional uniqueness of solutions to the hartree hierarchy without a smallness condition our proof uses the quantum de finetti theorem and is an extension of the work by chen hainzl pavlovic and seiringer 9 and our previous work 32

bold diagrammatic monte carlo technique when the sign problem is welcome we introduce a monte carlo scheme for sampling a bold line diagrammatic series specifying an unknown function in terms of itself the range of convergence of this bold line diagrammatic monte carlo bmc technique is significantly broader than that of a simple iterative scheme for solving integral equations with the bmc technique a moderate sign problem turns out to be an advantage in terms of the convergence of the process for an illustrative purpose we solve the one particle s scattering problem as an important application we obtain the t matrix for a fermi polaron one spin down particle interacting with the spin up fermionic sea

quantum fluctuations and rate of convergence towards mean field dynamics the nonlinear hartree equation describes the macroscopic dynamics of initially factorized n boson states in the limit of large n in this paper we provide estimates on the rate of convergence of the microscopic quantum mechanical evolution towards the limiting hartree dynamics more precisely we prove bounds on the difference between the one particle density associated with the solution of the n body schrodinger equation and the orthogonal projection onto the solution of the hartree equation

a central limit theorem in many body quantum dynamics we study the many body quantum evolution of bosonic systems in the mean field limit the dynamics is known to be well approximated by the hartree equation so far the available results have the form of a law of large numbers in this paper we go one step further and we show that the fluctuations around the hartree evolution satisfy a central limit theorem interestingly the variance of the limiting gaussian distribution is determined by a time dependent bogoliubov transformation describing the dynamics of initial coherent states in a fock space representation of the system

the derivation of the mathbb t 3 t 3 energy critical nls from quantum many body dynamics we derive the 3d energy critical quintic nls from quantum many body dynamics with 3 body interaction in the mathbb t 3 periodic setting due to the known complexity of the energy critical setting previous progress was limited in comparison to the 2 body interaction case yielding energy subcritical cubic nls previously the only result for the 3d energy critical case was hong et al j funct anal 270 1 34 67 2016 which proved the uniqueness part of the argument in the case of small solutions in the main part of this paper we develop methods to prove the convergence of the bbgky hierarchy to the infinite gross pitaevskii gp hierarchy and separately the uniqueness of large gp solutions since the trace estimate used in the previous proofs of convergence is the false endpoint trace estimate in our setting we instead introduce a new frequency interaction analysis and apply the finite dimensional quantum de finetti theorem for the large solution uniqueness argument we discover the new hierarchical uniform frequency localization hufl property for the gp hierarchy and use it to prove a new type of uniqueness theorem the hufl property reduces to a new statement even for nls with the help of colliander et al ann math 2 167 3 767 865 2008 and ionescu and pausader duke math j 161 1581 1612 2012 which proved the global well posedness for the quintic nls this new uniqueness theorem establishes global uniqueness

rigorous derivation of the gross pitaevskii equation with a large interaction potential consider a system of n bosons in three dimensions interacting via a repulsive short range pair potential n 2 v n xi xj where x x1 xn denotes the positions of the particles let hn denote the hamiltonian of the system and let n t be the solution to the schrodinger equation suppose that the initial data n 0 satisfies the energy condition h n 0 hn n 0i cn and that the one particle density matrix converges to a projection as n then we prove that the k particle density matrices of n t factorize in the limit n moreover the one particle orbital wave function solves the time dependent gross pitaevskii equation a cubic non linear schrodinger equation with the coupling constant proportional to the scattering length of the potential v in 12 we proved the same statement under the condition that the interaction potential v is sufficiently small in the present work we develop a new approach that requires no restriction on the size of the potential

derivation of the time dependent gross pitaevskii equation without positivity condition on the interaction using a new method pickl in a simple derivation of mean field limits for quantum systems 2010 it is possible to derive mean field equations from the microscopic n body schrodinger evolution of interacting particles without using bbgky hierarchies in this paper we wish to analyze scalings which lead to the gross pitaevskii equation which is usually derived assuming positivity of the interaction erdos et al in commun pure appl math 59 12 1659 1741 2006 invent math 167 515 614 2007 the new method for dealing with mean field limits presented in pickl 2010 allows us to relax this condition the price we have to pay for this relaxation is however that we have to restrict the scaling behavior of the interaction and that we have to assume fast convergence of the reduced one particle marginal density matrix of the initial wave function mu psi 0 to a pure state 0 0

on the solvability of polynomial systems arising in control we consider a compactification of a general system of polynomial equations in weighted projective space and give some sufficient conditions about the solvability of a polynomial system over c and r we also prove that for generic linear subspaces the inverse eigenvalue problem with perturbations in the linear subspace always has n solutions

local existence of solutions to randomized gross pitaevskii hierarchies in this paper we study the local in time existence of solutions to randomized forms of the gross pitaevskii hierarchy on periodic domains in particular we study the independently randomized gross pitaevskii hierarchy and the dependently randomized gross pitaevskii hierarchy which were first introduced in the author s joint work with staffilani 2013 for these hierarchies we construct local in time low regularity solutions in spaces which contain a random component the constructed density matrices will solve the full randomized hierarchies thus extending the results from the author and staffilani s joint work where solutions solving arbitrarily long subhierarchies were given our analysis will be based on the truncation argument which was first used in the deterministic setting in the work of t chen and pavlovic 2013 the presence of randomization in the problem adds additional difficulties most notably to estimating the duhamel expansions that are crucial in the truncation argument these difficulties are overcome by a detailed analysis of the duhamel expansions in the independently randomized case we need to keep track of which randomization parameters appear in the duhamel terms whereas in the dependently randomized case we express the duhamel terms directly in terms of the initial data in both cases we can obtain stronger results with respect to the time variable if we assume additional regularity on the initial data

output feedback pole assignment for transfer functions with symmetries this paper studies the problem of pole assignment for symmetric and hamiltonian transfer functions a necessary and sufficient condition for pole assignment by complex symmetric output feedback transformations is given moreover in the case where the mcmillan degree coincides with the number of parameters appearing in the symmetric feedback transformations we derive an explicit combinatorial formula for the number of pole assigning symmetric feedback gains the proof uses intersection theory in projective space as well as a formula for the degree of the complex lagrangian grassmann manifold

random data cauchy theory for supercritical wave equations ii a global existence result we prove that the subquartic wave equation on the three dimensional ball with dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in bigcap s 1 2 h s theta times h s 1 theta we obtain this result as a consequence of a general random data cauchy theory for supercritical wave equations developed in our previous work 6 and invariant measure considerations inspired by earlier works by bourgain 2 3 on the non linear schrodinger equation which allow us to obtain also precise large time dynamical informations on our solutions

brief augmented gradient flows for on line robust pole assignment via state and output feedback this paper is concerned with robust pole assignment in synthesis of linear control systems via state and output feedbacks first both the pole assignment and robustness requirements are appropriately formulated as two optimization problems then gradient flow models are developed for the on line computation of feedback gain matrices that result in robust pole assignment by solving these two optimization problems a technique is introduced to facilitate the real time matrix inverse involved for realizing the gradient flow models the resulting augmented gradient flows have desired convergence properties simulation results are included to show the effectiveness of the proposed approach

a necessary condition for pole assignment by constant output feedback in this paper we shall investigate the pole assignment problem by constant output feedback using transfer function techniques in particular a necessary condition is given for the existence of constant output feedback which assigns the eigenvalues of closed loop system to any desired positions

l p properties for gaussian random series let c c n n n be an arbitrary sequence of l 2 n and let f c be a random series of the type f c n n g n c n e n where g n n n is a sequence of independent n c 0 1 gaussian random variables and e n n n an orthonormal basis of l 2 y the finite measure space y m being arbitrary by using the equivalence of gaussian moments and an integrability theorem due to fernique we show that a necessary and sufficient condition for f c to belong to l p y m p 2 oo for any c l 2 n almost surely is that sup n n e n l p y m oo one of the main motivations behind this result is the construction of a nontrivial gibbs measure invariant under the flow of the cubic defocusing nonlinear schrodinger equation posed on the open unit disc of r 2

invariant measures and long time behaviour for the benjamin ono equation iii we complete the program developed in our previous works aiming to construct an infinite sequence of invariant measures of gaussian type associated with the conservation laws of the benjamin ono equation

random data cauchy theory for supercritical wave equations i local theory we study the local existence of strong solutions for the cubic nonlinear wave equation with data in hs m s 1 2 where m is a three dimensional compact riemannian manifold this problem is supercritical and can be shown to be strongly ill posed in the hadamard sense however after a suitable randomization we are able to construct local strong solution for a large set of initial data in hs m where s 1 4 in the case of a boundary less manifold and s 8 21 in the case of a manifold with boundary

static output feedback pole placement via a trust region approach this technical note presents two closely related algorithms for the problem of pole placement via static output feedback the algorithms are based on two different trust region methods and utilize the derivatives of the closed loop poles extensive numerical experiments show the effectiveness of the algorithms in practice though convergence to a solution is not guaranteed for either algorithm while desired poles must be distinct strategies for dealing with repeated poles are also presented

derivation of the cubic non linear schrodinger equation from quantum dynamics of many body systems we prove rigorously that the one particle density matrix of three dimensional interacting bose systems with a short scale repulsive pair interaction converges to the solution of the cubic non linear schrodinger equation in a suitable scaling limit the result is extended to k particle density matrices for all positive integer k

a general formula for eigenvalue assignment by static output feedback with application to robust design in this note generalizing an idea of a previous paper we derive a new expression of the characteristic polynomial of a linear system subject to static output feedback from this expression an explicit form of the assignment equations in the unknown elements of the feedback matrix is obtained the equations are completely general and seem much simpler than many others proposed until now in special cases a closed or almost a closed form solution is obtained the assignment equations are then used within an algorithm for global optimization to solve the problem of exactly assigning the eigenvalues of the nominal plant while maximizing the norm of perturbation matrices for which the perturbed eigenvalues remain in specified regions of the complex plane

multivariable nyquist criteria root loci and pole placement a geometric viewpoint classical control theory is concerned with the topics in our title in the context of single input single output systems there is now a large and growing literature on the extension of these ideas to the multiinput multioutput case this development has posed certain difficulties some due to the intrinsic nature of the problem and some we would argue due to an inadequate reflection on what the multivariable problem calls for in this paper we describe what seems to us to be the natural multivariable analogs of these concepts from classical control theory a rather satisfactory generalization of the nyquist criterion will be described and a clear analog of the asymptotic properties of the root locus will be obtained in the multiparameter case however an example is given which illustrates the quite surprising fact that the root locus map is not always continuous at infinite gains this calls for a new ingredient a compactification of the space of gains and perhaps the most interesting new feature in this circle of ideas comes in the area of pole placement this problem is difficult in the multivariable case but by establishing a correspondence with a classical set of problems in geometry we are able to understand its main aspects and to derive results on pole placement by output feedback over the real field

mean field dynamics singular potentials and rate of convergence we consider the time evolution of a system of n identical bosons whose interaction potential is rescaled by n 1 we choose the initial wave function to describe a condensate in which all particles are in the same one particle state it is well known that in the mean field limit n the quantum n body dynamics is governed by the nonlinear hartree equation using a nonperturbative method we extend previous results on the mean field limit in two directions first we allow a large class of singular interaction potentials as well as strong possibly time dependent external potentials second we derive bounds on the rate of convergence of the quantum n body dynamics to the hartree dynamics

a tracial quantum central limit theorem we prove a central limit theorem for non commutative random variables in a von neumann algebra with a tracial state any non commutative polynomial of averages of i i d samples converges to a classical limit the proof is based on a central limit theorem for ordered joint distributions together with a commutator estimate related to the baker campbell hausdorff expansion the result can be considered a generalization of johansson s theorem on the limiting distribution of the shape of a random word in a fixed alphabet as its length goes to infinity math co 9906120 math pr 9909104

quantum dynamics with mean field interactions a new approach we propose a new approach for the study of the time evolution of a factorized n particle bosonic wave function with respect to a mean field dynamics with a bounded interaction potential the new technique which is based on the control of the growth of the correlations among the particles leads to quantitative bounds on the difference between the many particle schrodinger dynamics and the one particle nonlinear hartree dynamics in particular the one particle density matrix associated with the solution to the n particle schrodinger equation is shown to converge to the projection onto the one dimensional subspace spanned by the solution to the hartree equation with a speed of convergence of order 1 n for all fixed times

eigenvalue assignment by static output feedback on a new solvability condition and the computation of low gain feedback matrices in this article the static output feedback problem for linear time invariant systems is considered for arbitrary assignability of the roots of the characteristic polynomial by static output feedback a new necessary and sufficient condition is derived although the proof is based on simple analysis the known sufficient conditions derived by techniques of algebraic geometry are directly covered furthermore an algorithm for the calculation of feedback matrices assigning a desired set of eigenvalues is proposed this algorithm does not require the desired eigenvalues to be distinct and it explicitly exploits the available degrees of freedom for reducing the feedback gain the presented approach is illustrated on computational examples

derivation of the two dimensional nonlinear schrodinger equation from many body quantum dynamics we derive rigorously for both r 2 and l l 2 the cubic nonlinear schrodinger equation in a suitable scaling limit from the two dimensional many body bose systems with short scale repulsive pair interactions we first prove convergence of the solution of the bbgky hierarchy corresponding to the many body systems to a solution of the infinite gross pitaevskii hierarchy corresponding to the cubic nls and then we prove uniqueness for the infinite hierarchy which requires number theoretical techniques in the periodic case

the gross pitaevskii hierarchy on general rectangular tori in this work we study the gross pitaevskii hierarchy on general rational and irrational rectangular tori of dimensions two and three this is a system of infinitely many linear partial differential equations which arises in the rigorous derivation of the nonlinear schrodinger equation we prove a conditional uniqueness result for the hierarchy in two dimensions this result allows us to obtain a rigorous derivation of the defocusing cubic nonlinear schrodinger equation from the dynamics of many body quantum systems on irrational tori this question was posed as an open problem in the previous work of kirkpatrick schlein and staffilani

invariant gibbs measure evolution for the radial nonlinear wave equation on the 3d ball abstract we establish new global well posedness results along gibbs measure evolution for the nonlinear wave equation posed on the unit ball in r 3 via two distinct approaches the first approach invokes the method established in the works bourgain 1994 1996 3 4 5 based on a contraction mapping principle and applies to a certain range of nonlinearities the second approach allows to cover the full range of nonlinearities admissible to treatment by gibbs measure working instead with a delicate analysis of convergence properties of solutions the method of the second approach is quite general and we shall give applications to the nonlinear schrodinger equation on the unit ball in subsequent works bourgain and bulut 2013 10 11

on pole placement via eigenstructure assignment approach this note comes back to the hard problem of pole placement by static output feedback let a triplet of matrices a b c be given with n state variables m inputs and p ouputs find a matrix k such that the spectrum of a bkc equals a specified set when mp m p a simple noniterative technique based upon the notion of eigenstructure that in most cases assigns m p roots is proposed it therefore enables the designer to assign the whole of the desired spectrum when m p n mp

a quantum central limit theorem for sums of independent identically distributed random variables we formulate and prove a general central limit theorem for sums of independent identically distributed noncommutative random variables

a parametric solution to the pole assignment problem using dynamic output feedback a technique is presented for pole placement of linear time invariant systems using dynamic feedback a previously developed method for partial pole assignment using constant feedback is generalized to the dynamic output feedback case subject to a mild assumption on the number of complex conjugate poles to be assigned it is almost always possible to arbitrarily assign all the closed loop system poles using a compensator of order n spl phi max m l using this new method here n m and l are the order of the system and the number of inputs and outputs respectively and spl phi spl delta max m l max m l 2 max m l min m l where x denotes the nearest integer lower than or equal to x i e floor x and x denotes the nearest integer greater than or equal to x i e ceiling x an equivalent result is that using a compensator of order q it is almost always possible to arbitrarily assign min n q max m l 1 q spl phi closed loop system poles only the normal procedures of linear algebra are required to implement the technique note that spl phi spl ges l m 1 and therefore the result is stronger than previous exact pole assignment results since it does not involve iteration or any other numerical techniques it is possible to implement the method symbolically and therefore to obtain general parametric solutions to the pole assignment problem the freedom in this design approach can also often be used to guarantee the internal stability and or robustness of the resulting closed loop system

a sufficient condition for arbitrary eigenvalue assignment in linear descriptor systems by output feedback the problem of eigenvalue assignment in the linear descriptor system e mathdotx ax bu y cx via output feedback is considered it is shown that mp rank e m and p are respectively the numbers of inputs and outputs of the system is a sufficient condition for generic real output feedback eigenvalue assignability the result is the best possible for general m p rank e in general the assignment is achieved not in an exact sense but in arbitrarily small neighborhoods around a given set of eigenvalues

construction of a gibbs measure associated to the periodic benjamin ono equation we define a finite borel measure of gibbs type supported by the sobolev spaces of negative indexes on the circle the measure can be seen as a limit of finite dimensional measures these finite dimensional measures are invariant by the ode s which correspond to the projection of the benjamin ono equation posed on the circle on the first n n 1 modes in the trigonometric bases

gross pitaevskii equation as the mean field limit of weakly coupled bosons we consider the dynamics of n boson systems interacting through a pair potential n 1va xi xj where va x a 3v x a we denote the solution to the n particle schrodinger equation by n t recall that the gross pitaevskii gp equation is a nonlinear schrodinger equation and the gp hierarchy is an infinite bbgky hierarchy of equations so that if ut solves the gp equation then the family of k particle density matrices open image in new window solves the gp hierarchy under the assumption that a n for 0 3 5 we prove that as n the limit points of the k particle density matrices of n t are solutions of the gp hierarchy with the coupling constant in the nonlinear term of the gp equation given by v x dx the uniqueness of the solutions of this hierarchy remains an open question

invariant measures for the 2d defocusing nonlinear schrodinger equation consider the2d defocusing cubic nlsiu t u u u 2 0 with hamiltonian smallint left left nabla phi right 2 tfrac 1 2 left phi right 4 right it is shown that the gibbs measure constructed from the wick ordered hamiltonian i e replacing 4 by 4 is an invariant measure for the appropriately modified equationiu t u u u 2 2 u 2 dx u 0 there is a well defined flow on thesupport of the measure in fact it is shown that for almost all data the solutionu u 0 satisfiesu t e it c hs for somes 0 first a result local in time is established and next measure invariance considerations are used to extend the local result to a global one cf b2

output feedback pole placement with dynamic compensators derives a new rank condition which guarantees the arbitrary pole assignability of a given system by dynamic compensators of degree at most q by using this rank condition the authors establish several new sufficiency conditions which ensure the arbitrary pole assignability of a generic system the authors proofs also come with a concrete numerical procedure to construct a particular compensator which assigns a given set of closed loop poles