halanay type inequalities on time scales with applications a b s t r a c t this paper aims to introduce halanay type inequalities on time scales by means of these inequalities we derive new global stability conditions for nonlinear dynamic equations on time scales giving several examples we show that besides generalization and extension to q difference case our results also provide improvements for the existing theory regarding differential and difference inequalities which are the most important particular cases of dynamic inequalities on time scales

a continuum of relative equilibria in the five body problem abstract it is generally believed that the set of relative equilibria equivalence classes in the newtonian n body problem for a given set of positive masses is finite however the result has only been proven for n 3 and remains a difficult open question for n 4 a wintner princeton math ser 5 215 princeton university press princeton nj 1941 s smale mathematical intelligencer 20 1998 7 15 we demonstrate that the condition for the masses being positive is a necessary one by finding a continuum of relative equilibria in the five body problem which unfortunately includes one negative mass this family persists in similar potential functions including the logarithmic potential used to describe the motion of point vortices in a plane of fluid

on the characterization of sampled data controllers in the lifted domain abstract in this paper we address sampled data control problems on which the sampling and hold devices are not fixed before hand but are chosen as part of the design process the idea is to apply the lifting transformation to the plant and then characterize the set of generalized sampled data controllers i e those that can be implemented as the cascade of a generalized sampler a discrete time controller and a generalized hold we argue that our approach offers significant advantages over previous works and will illustrate this by considering three applications the input output stabilization by generalized sampled data controllers the solution of the h optimal control problem and the combination of analog and digital controllers

on the central configurations of the planar restricted four body problem abstract this article is devoted to answering several questions about the central configurations of the planar 3 1 body problem firstly we study bifurcations of central configurations proving the uniqueness of convex central configurations up to symmetry secondly we settle the finiteness problem in the case of two nonzero equal masses lastly we provide all the possibilities for the number of symmetrical central configurations and discuss their bifurcations and spectral stability our proofs are based on applications of rational parametrizations and computer algebra

super central configurations of the three body problem under the inverse integer power law in this paper we consider the problem of central configurations of the n body problem with the general homogenous potential 1 r where is a positive integer a configuration q q1 q2 qn is called a super central configuration if there exists a positive mass vector m m1 mn such that q is a central configuration for m with mi attached to qi and q is also a central configuration for m where m mandm is a permutation of m the main result in this paper is the existence and classifications of super central configurations in the rectilinear three body problem with general homogenous potential our results extend the previous work xie z j math phys 51 042902 2010 10 1063 1 3345125 from the case in which 2 to the case in which is a positive integer descartes rule of sign is extensively used in the proof of the main theorem

finiteness and bifurcations of some symmetrical classes of central configurations we consider here the classical question of finiteness see smale 13 and wintner 15 given n point masses is the corresponding number of central configurations finite we prove finiteness for a particular family of d dimensional symmetrical configurations of d 2 point masses also we study the bifurcations of these configurations and provide the exact number of central configurations when d 2 3 all our results stem from the application of a new method for studying symmetrical classes of central configurations which is presented in this work

ripple free tracking with robustness despite the previous efforts in deadbeat control a need still exists to offer a generic design methodology for both performance and robustness we propose a hybrid two degree of freedom 2dof controller for the fixed order constrained optimization problem addressing performance and robustness specifications the controller is given in terms of the solution of two diophantine equations simple parameterization of these diophantine equations allows for straight forward optimization and constraint satisfaction for both performance and robustness measures

periodic solutions for a neutral nonlinear dynamical equation on a time scale abstract let t be a periodic time scale we use a fixed point theorem due to krasnosel ski to show that the nonlinear neutral dynamic system with delay x t a t x t c t x t k q t x t x t k t t has a periodic solution we assume that k is a fixed constant if t r and is a multiple of the period of t if t r under a slightly more stringent inequality we show that the periodic solution is unique using the contraction mapping principle

an optimal algorithm for sampled data robust servomechanism controller using exponential hold a new structure of sampled data robust servomechanism controller using exponential hold is developed an optimal algorithm is also proposed for choosing the controller parameters of two important special classes the algorithm is derived by minimizing a square error performance index and the solution can be solved from a discrete time algebraic riccati equation

dziobek configurations of the restricted n 1 body problem with equal masses in this article we count the number of n 1 dimensional central configurations of the restricted n 1 body problem with equal masses for n 4 5 there are 25 56 central configurations for n 3 or n 6 the number of central configurations is 2n n 1

floquet theory for integral and integro differential equations we study the extension of hill s method of infinite determinants to the case of integro differential equations with periodic coefficients and kernels we develop the analytical theory of such methods and we obtain certain qualitative properties of the equations that determine the boundaries between regions of dynamic stability and dynamic instability

comments on ripple free deadbeat control of siso discrete systems by h r sirisena the commenter argues that the necessary and sufficient conditions given by m r sirisena ibid vol ac 30 p 168 170 feb 1985 for solving the ripple free deadbeat problem are incorrect

capabilities and limitations of multirate control schemes abstract a fundamental obstacle in control system design is the presence of plant zeros recently it has been suggested that in digital control systems such problems may be overcome by using multirate sampling in this paper we describe the capabilities of such multirate control schemes with regard to pole and zero assignment we also point out a limitation of the approach due to degradation in the intersample behavior in mita and chida 1988 proc 27th conf on dec and control pp 1883 1888 it is shown that with a two delay input control scheme it is possible to assign all the poles and zeros of the closed loop system using state feedback in this paper we generalize this result to the case of n delay input control using dynamic compensators in an n delay input control scheme the input to the continuous system is changed n times more often than the output is sampled using such a scheme we show how to design output feedback controllers for siso systems that ensure arbitrary placement of all the poles and zeros of the closed loop system corresponding results for the mimo regulator and the siso servomechanism are given we also show that a hidden cost associated with the use of n delay input control is a degradation in the intersample behavior we demonstrate this via simulation and present an analysis which explains why this degradation arises

generalization of the floquet theory abstract this paper introduces the concept of topological equivalence for the periodic systems necessary and sufficient conditions of topological equivalence are derived furthermore we generalize the floquet theory for a linear system with a periodic coefficient

second order q difference equations solvable by factorization method by solving an infinite nonlinear system of q difference equations one constructs a chain of q difference operators the eigenproblems for the chain are solved and some applications including the one related to q hahn orthogonal polynomials are discussed it is shown that in the limit q 1 the present method corresponds to the one developed by infeld and hull

a continuum of periodic solutions to the planar four body problem with two pairs of equal masses abstract in this paper we apply the variational method with structural prescribed boundary conditions spbc to prove the existence of periodic and quasi periodic solutions for the planar four body problem with two pairs of equal masses m 1 m 3 and m 2 m 4 a path q t on 0 t satisfies the spbc if the boundaries q 0 a and q t b where a and b are two structural configuration spaces in r 2 4 and they depend on a rotation angle 0 2 and the mass ratio m 2 m 1 r we show that there is a region 0 2 r such that there exists at least one local minimizer of the lagrangian action functional on the path space satisfying the spbc q t h 1 0 t r 2 4 q 0 a q t b for any the corresponding minimizing path of the minimizer can be extended to a non homographic periodic solution if is commensurable with or a quasi periodic solution if is not commensurable with in the variational method with the spbc we only impose constraints on the boundary and we do not impose any symmetry constraint on solutions instead we prove that our solutions that are extended from the initial minimizing paths possess certain symmetries the periodic solutions can be further classified as simple choreographic solutions double choreographic solutions and non choreographic solutions among the many stable simple choreographic orbits the most extraordinary one is the stable star pentagon choreographic solution when 4 5 1 remarkably the unequal mass variants of the stable star pentagon are just as stable as the equal mass choreographies

central configurations and total collisions for quasihomogeneous n body problems abstract we consider n body problems given by potentials of the form r a r b with a b constants 0 a b to analyze the dynamics of the problem we first prove some properties related to central configurations including a generalization of moulton s theorem then we obtain several qualitative properties for collision and near collision orbits in the manev type case a 1 at the end we point out some new relationships between central configurations relative equilibria and homothetic solutions

a unified floquet theory for discrete continuous and hybrid periodic linear systems abstract in this paper we study periodic linear systems on periodic time scales which include not only discrete and continuous dynamical systems but also systems with a mixture of discrete and continuous parts e g hybrid dynamical systems we develop a comprehensive floquet theory including lyapunov transformations and their various stability preserving properties a unified floquet theorem which establishes a canonical floquet decomposition on time scales in terms of the generalized exponential function and use these results to study homogeneous as well as nonhomogeneous periodic problems furthermore we explore the connection between floquet multipliers and floquet exponents via monodromy operators in this general setting and establish a spectral mapping theorem on time scales finally we show this unified floquet theory has the desirable property that stability characteristics of the original system can be determined via placement of an associated but time varying system s poles in the complex plane we include several examples to show the utility of this theory

central configurations of the 4 body problem with masses m 1 m 2 m 3 m 4 m 0 and m small in this paper we give a complete description of the families of central configurations of the planar 4 body problem with two pairs of equals masses and two equal masses sufficiently small in particular we give an analytical proof that this particular 4 body problem has exactly 34 different classes of central configurations moreover for this problem we prove the following two conjectures there is a unique convex planar central configuration of the 4 body problem for each ordering of the masses in the boundary of its convex hull which appears in albouy and fu 2007 3 we also prove the conjecture there is a unique convex planar central configuration having two pairs of equal masses located at the adjacent vertices of the configuration and it is an isosceles trapezoid finally the families of central configurations of this 4 body problem are numerically continued to the 4 body problem with four equal masses

exact output control for a family of linear plants with parameter uncertainties a family of linear continuous time plants parametrized by a vector of unknown parameters taking values in a known finite set is considered the problem investigated consists in finding a linear discrete time compensator that independently of the actual value of the parameter vector guarantees zero continuous time steady state output error with respect to a given continuous time external reference it is shown that the problem admits a solution in the class of periodic controllers

a numerical study of the large period limit of a zakharov shabat eigenvalue problem with periodic potentials deconinck and kutz 2006 j comput phys 219 296 321 developed an efficient algorithm for solving the zakharov shabat eigenvalue problem with periodic potentials numerically it is natural to use the same algorithm for solving the problem for non periodic potential decaying potentials defined over the whole real line using large periods in this paper we propose the use of a specific value of the floquet exponent our numerical results indicate that it can produce accurate results long before the period becomes large enough for the analytical convergence results of gardner 1997 j reine angew math 491 149 81 to be valid we also illustrate the rather complicated path to convergence of some nonlinear schr dinger potentials

numerical inverse scattering for the focusing and defocusing nonlinear schrodinger equations we solve the focusing and defocusing nonlinear schrodinger nls equations numerically by implementing the inverse scattering transform the computation of the scattering data and of the nls solution are both spectrally con vergent initial conditions in a suitable space are treated using the approach of biondini and bui 3 we numerically solve homogeneous robin boundary value problems on the half line finally using recent theoretical developments in the numerical approximation of riemann hilbert problems we prove that under mild assumptions our method of approximating solutions to the nls equations is uniformly accurate in their domain of definition

multirate robust servomechanism controllers of linear delay systems using a hybrid structure a multirate robust servomechanism controller design for a linear time delay system is presented the proposed controller is constructed by using a preparatory continuous internal model followed by a multirate error feedback compensator an optimal algorithm for choosing the controller parameters is also derived the algorithm is derived by minimizing a squared deviation error performance index and the practical optimal solution can simply be solved from a discrete time algebraic riccati equation a distinctive feature of the algorithm is that an optimal pulse compensation is also employed to improve the transient characteristics such a pulse compensation tends to eliminate the undesired influence arising from the initial deviation from the ultimate steady state in particular if the initial deviation from the ultimate steady state is known then the optimal pulse compensation can drive the system into the steady state in a time no larger than the input delay time plus the sampling period

convergence of hill s method for nonselfadjoint operators by the introduction of a generalized evans function defined by an appropriate 2 modified fredholm determinant we give a simple proof of convergence in location and multiplicity of hill s method for numerical approximation of spectra of periodic coefficient ordinary differential operators our results apply to operators of nondegenerate type under the condition that the principal coefficient matrix be symmetric positive definite automatically satisfied in the scalar case notably this includes a large class of non self adjoint operators which previously had not been treated in a simple way the case of general coefficients depends on an interesting operator theoretic question regarding properties of toeplitz matrices

a function space approach to sampled data control systems and tracking problems this paper presents a new framework for hybrid sampled data control systems instead of considering the state only at sampling instants this paper introduces a function piece during the sampling period as the state and gives an infinite dimensional model with such a state space this gives the advantage that sampled data systems with built in intersample behavior can be regarded as linear time invariant discrete time systems as a result the approach makes it possible to introduce such time invariant concepts as transfer functions poles and zeros to the sampled data systems even with the presence of the intersample behavior in particular tracking problems can be studied in this setting in a simple and unified way and ripples are completely characterized as a mismatch between the intersample reference signal and transmission zero directions this leads to the internal model principle for sampled data systems

convex central configurations for the n body problem abstract we give a simple proof of a classical result of macmillan and bartky trans amer math soc 34 1932 838 which states that for any four positive masses and any assigned order there is a convex planar central configuration moreover we show that the central configurations we find correspond to local minima of the potential function with fixed moment of inertia this allows us to show that there are at least six local minimum central configurations for the planar four body problem we also show that for any assigned order of five masses there is at least one convex spatial central configuration of local minimum type our method also applies to some other cases

super central configurations of the n body problem in this paper we consider the inverse problem of central configurations of the n body problem for a given q q1 q2 qn rd n let s q be the admissible set of masses by s q m m1 mn mi r q is a central configurationfor m for a given m s q let sm q be the permutational admissible set about m m1 m2 mn by sm q m m s q m m and m is apermutation of m here q is called a super central configuration if there exists m such that sm q is nonempty for any q in the planar four body problem q is not a super central configuration as an immediate consequence of a theorem proved by macmillan and bartky permanent configurations in the problem of four bodies trans am math soc 34 838 1932 the main discovery in this paper is the existence of super central configurations in the collinear three body problem we proved that for any q in the collinear three body problem and any m s q sm q has at most one element and the detailed classification of sm q is provided

robust tracking deadbeat control we derive the parametrization of all controllers that achieve both the robust tracking and the deadbeat control we also give the parametrization of all reference functions for which the robust tracking and designed deadbeat property is guaranteed by all robust tracking deadbeat controllers for one reference the solution to the robust tracking deadbeat problem for more than two reference functions is also given we use the h norm of the sensitivity function as the robustness index of the system by using two degree of freedom control systems we can achieve both minimal time deadbeat control and optimal robustness with only one controller

double objective optimal multivariable ripple free deadbeat control this paper presents novel results on optimal multivariable deadbeat control given a discrete time stable linear time invariant plant model we give a simple parameterization of all stabilizing ripple free deadbeat controllers of a given order the free parameter is then optimized in the sense that a quadratic index is kept minimal the optimality criterion has the advantage of accounting for both tracking performance and magnitude of the control effort the proposed design procedure is simple to use and allows the tuning of the controller with a scalar weighting factor simulation results are included to illustrate the effectiveness of the proposed design algorithm

deadbeat error control of discrete multivariable systemsdagger abstract the main objective of this paper is to delineate conditions under which deadbeat error control can be achieved between a plant output and a discrete reference input signal all facets of this problem are considered in the linear multivariable case i e when the reference is any non diminishing signal whether the plant is truly discrete or a discretized continuous one and whether the initial conditions are zero or unknown the results presented can also be used to resolve the case of known non zero initial conditions on the state of the plant as well as the compensator

minimal time minimal order deadbeat regulator with internal stability this paper is concerned with the output deadbeat finite settling time regulation with internal stability for linear discrete time multivariable systems the two cases are treated separately the one is the regulation by state feedback and the other is the regulation through function observers for both cases the basic solvability conditions the minimal settling time and the characterizations of the minimal time regulators are derived throughout the paper the well posedness condition plays a fundamental role a deterministic version of the so called separation theorem in lqg problem is derived namely the minimal time state feedback regulator combined with the minimal time minimal order function observer yields the overall minimal time minimal order regulator by output feedback

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

computing spectra of linear operators using the floquet fourier hill method in order to establish the stability of an equilibrium solution u of an infinite dimensional dynamical system u x u one is interested in the spectrum of the linear operator lu obtained by linearizing the dynamical system around u we use a spectrally accurate method for the computation of the spectrum of the maximal extension of the operator lu the method is particularly well suited to the case of periodic u although no periodic boundary conditions on the perturbations are imposed by incorporating the fundamentals of floquet theory an almost uniform approximation to the entire spectrum of the maximal extension is obtained as opposed to an approximation of a few selected elements the numerical component of the method is limited to i choosing the size of the matrices to be used and ii an eigenvalue solver such as the qr algorithm compared to often used finite difference approaches the method is an order of magnitude faster for comparable accuracy we illustrate that the method is efficiently extended to problems defined on the whole line

ripple free tracking problem in the deadbeat control of discrete plants originating from continuous ones it is important to have exact tracking not only at the sampling instants but also in between the instants this paper investigates such a problem in the general case of a reference signal that can vary within a pre specified class the design procedure proposed here is based on a polynomial algebra approach an internally stable closed loop system is derived by means of an algorithm where the transfer function of the controller originates from the solution of a set of linear equations algebraic conditions are given to determine the minimum time control scheme numerical examples are reported to show the feasibility of the method and the simplicity of including other specific requirements into the control law

elliptic solutions of the defocusing nls equation are stable the stability of the stationary periodic solutions of the integrable one dimensional cubic defocusing nonlinear schrodinger nls equation is reasonably well understood especially for solutions of small amplitude in this paper we exploit the integrability of the nls equation to establish the spectral stability of all such stationary solutions this time by explicitly computing the spectrum and the corresponding eigenfunctions associated with their linear stability problem an additional argument using an appropriate krein signature allows us to conclude the nonlinear orbital stability of all stationary solutions of the defocusing nls equation with respect to so called subharmonic perturbations perturbations that have period equal to an integer multiple of the period of the amplitude of the solution all results presented here are independent of the size of the amplitude of the solutions and apply equally to solutions with trivial and nontrivial phase profiles

finiteness of relative equilibria of the four body problem we show that the number of relative equilibria of the newtonian four body problem is finite up to symmetry in fact we show that this number is always between 32 and 8472 the proof is based on symbolic and exact integer computations which are carried out by computer

h sub spl infin type problem for sampled data control systems a solution via minimum energy characterization this paper aims at deriving a solution for h sub spl infin type problem for sampled data control systems the solution is given in terms of an equivalent discrete time h sub spl infin problem the reduction procedure is viewed and characterized from the viewpoint of minimum energy principle and j unitary transformations

symmetry of planar four body convex central configurations we study the relationship between the masses and the geometric properties of central configurations we prove that in the planar four body problem a convex central configuration is symmetric with respect to one diagonal if and only if the masses of the two particles on the other diagonal are equal if these two masses are unequal then the less massive one is closer to the former diagonal finally we extend these results to the case of non planar central configurations of five particles

some notes on four body co circular central configurations abstract there are two symmetric families of four body co circular central configurations namely the kite and isosceles trapezoid using mutual distances as coordinates we prove that if the four body central configuration is an isosceles trapezoid then the diagonals of the isosceles trapezoid cannot be perpendicular to each other furthermore we show that for any four body co circular central configuration the diagonals of the quadrilateral cannot be perpendicular except that the configuration is a kite

twisted angles for central configurations formed by two twisted regular polygons abstract in this paper we study the necessary conditions and sufficient conditions for the twisted angles of the central configurations formed by two twisted regular polygons in particular we prove that for the 2n body problem the twisted angles must be 0 or n

inherent design limitations for linear sampled data feedback systems there is a well developed theory describing inherent design limitations for linear time invariant feedback systems consisting of an analogue plant and analogue controller this theory describes limitations on achievable performance present when the plant has non minimum phase zeros unstable poles and or time delays the parallel theory for linear time invariant discrete time systems is less interesting because it describes system behaviour only at sampling instants this paper develops a theory of design limitations for sampled data feedback systems wherein the response of the analogue system output is considered this is done using the fact that the steady state response of a hybrid feedback system to a sinusoidal input consists of a fundamental component at the frequency of the input together with infinitely many harmonics at frequencies spaced integer multiples of the sampling frequency away from the fundamental this fact allows fundamental sensitivity and complementary sensitivity functions that re

ripple free deadbeat control for sampled data systems in this paper the ripple free deadbeat control problem for sampled data systems is considered this control objective is to settle the error to zero for all time after some finite settling time in other words to eliminate the ripples between the sampling instants in deadbeat control of sampled data systems the necessary and sufficient conditions to solve this problem are derived and related to the system type and a method of constructing the ripple free deadbeat control system is presented

approximations of strongly continuous families of unbounded self adjoint operators the problem of approximating the discrete spectra of families of self adjoint operators that are merely strongly continuous is addressed it is well known that the spectrum need not vary continuously as a set under strong perturbations however it is shown that under an additional compactness assumption the spectrum does vary continuously and a family of symmetric finite dimensional approximations is constructed an important feature of these approximations is that they are valid for the entire family uniformly an application of this result to the study of plasma instabilities is illustrated

parametrized solution of the periodic dead beat control problem the dead beat control problem for linear periodic discrete time systems in presence of unmeasurable disturbances and external reference signals is considered using a polynomial method a parametrization of all causal output feedback controllers solving the problem is provided

evans functions jost functions and fredholm determinants the principal results of this paper consist of an intrinsic definition of the evans function in terms of newly introduced generalized matrix valued jost solutions for general first order matrix valued differential equations on the real line and a proof of the fact that the evans function a finite dimensional determinant by construction coincides with a modified fredholm determinant associated with a birman schwinger type integral operator up to an explicitly computable nonvanishing factor

on the convergence of hill s method hill s method is a means to numerically approximate spectra of linear differential operators with periodic coefficients in this paper we address different issues related to the convergence of hill s method we show the method does not produce any spurious approximations and that for selfadjoint operators the method converges in a restricted sense furthermore assuming convergence of an eigenvalue we prove convergence of the associated eigenfunction approximation in the l 2 norm these results are not restricted to selfadjoint operators finally for certain selfadjoint operators we prove that the rate of convergence of hill s method to the least eigenvalue is faster than any polynomial power

conditions for the ripple free response of multirate systems using linear time invariant controllers abstract in this work necessary and sufficient conditions are derived for multirate systems using linear time invariant lti controllers to guarantee a ripple free response to step reference signals these conditions correspond to constraints on the coefficients of a finite impulse response pre filter the analysis is facilitated by the use of the lifting technique the modified z transform and up sampling and down sampling operators several examples of feasible filters are provided to show the existence of multiple solutions to the considered problem

optimal ripple free deadbeat controllers a ripple free deadbeat controller for a system exists if and only if there are no transmission zeros coinciding with the poles of the reference signal approaches to this problem often use the diophantine equation solution however solutions provided by the diophantine equation often exhibit extremely bad transient responses this approach gives a new affine parametrization of solutions of the diophantine equation based on this parametrization lmi conditions are used to provide optimal or constrained controllers for design quantities such as overshoot undershoot control amplitude slew rate as well as for norm bounds such as l1 l2 and l infinity

a continuous time recursive fixed lag smoother converging in finite time in this technical note we propose a new fixed lag smoother that estimates the fixed delayed state for a stochastic continuous time system the estimation error variance of the proposed smoother is minimized under the constraint that the estimated state converges to the real state exactly in finite time after noises or uncertainties disappear for numerical computing the proposed smoother is represented in a recursive form unlike other approaches any additional processes such as batch processing and sampling data through discrete time techniques are not required to achieve the finite time convergence a numerical example is presented to illustrate the finite time convergence of the proposed smoother in comparison with the asymptotic convergence of an optimal smoother

on central configurations this paper concerns an old problem which arose in celestial mechanics and has since taken on a life of its own the probem of central configurations of the n body problem

parametrization of all ripple free deadbeat controllers abstract this paper is concerned with deadbeat control in sampled data systems deadbeat control achieves finite time settling deadbeat settling at sampling instants but there may exist error called ripple between sampling instants even after the response is settled at sampling instants the objective of this paper is to give a parametrization of all ripple free deadbeat controllers controllers which achieve deadbeat settling without ripple in sampled data systems it is also shown that the following holds in general minimum time deadbeat control causes ripple when the pulse transfer function to be controlled has stable zeros

collinear central configurations and singular surfaces in the mass space for a given m m1 mn r n let p and q r3 n be two central configurations for m then we call p and qequivalent and write p q if they differ by an so 3 rotation followed by a scalar multiplication as well as by a permutation of bodies denote by l n m the set of equivalent classes of n body collinear central configurations in r3 for any given mass vector m m1 mn r n the main discovery in this paper is the existence of a union h3 of three non empty algebraic surfaces in the mass half space m1 m2 m1 m3 m2 r r2 besides the planes generated by equal masses which decreases the number of collinear central configurations the union h3 in r r2 is explicitly constructed by three 6 degree homogeneous polynomials in three variables such that for any mass vector m m1 m2 m3 r 3 l 3 m 3 if m1 m2 and m3 are mutually distinct and m1 m2 m1 m3 m2 h3 l 3 m 2 if m1 m2 and m3 are mutually distinct and m1 m2 m1 m3 m2 h3 l 3 m 2 if two of m1 m2 and m3 are equal but not the third l 3 m 1 if m1 m2 m3 we give also a sharp upper bound on l n m for any positive mass vector m r n

floquet theory based on new periodicity concept for hybrid systems involving q difference equations using the new periodicity concept based on shifts we construct a unified floquet theory for homogeneous and nonhomogeneous hybrid periodic systems on domains having continuous discrete or hybrid structure new periodicity concept based on shifts enables the construction of floquet theory on hybrid domains that are not necessarily additive periodic this makes periodicity and stability analysis of hybrid periodic systems possible on non additive domains in particular this new approach can be useful to know more about floquet theory for linear q difference systems defined on q z q n n z 0 where q 1 by constructing the solution of matrix exponential equation we establish a canonical floquet decomposition theorem determining the relation between floquet multipliers and floquet exponents we give a spectral mapping theorem on closed subsets of reals that are periodic in shifts finally we show how the constructed theory can be utilized for the stability analysis of dynamic systems on periodic time scales in shifts

nonlinear modulational stability of periodic traveling wave solutions of the generalized kuramoto sivashinsky equation abstract in this paper we consider the spectral and nonlinear stabilities of periodic traveling wave solutions of a generalized kuramoto sivashinsky equation in particular we resolve the long standing question of nonlinear modulational stability by demonstrating that spectrally stable waves are nonlinearly stable when subject to small localized integrable perturbations our analysis is based upon detailed estimates of the linearized solution operator which are complicated by the fact that the necessarily essential spectrum of the associated linearization intersects the imaginary axis at the origin we carry out a numerical evans function study of the spectral problem and find bands of spectrally stable periodic traveling waves in close agreement with previous numerical studies of frisch she thual bar nepomnyashchy chang demekhin kopelevich and others carried out by other techniques we also compare predictions of the associated whitham modulation equations which formally describe the dynamics of weak large scale perturbations of a periodic wave train with numerical time evolution studies demonstrating their effectiveness at a practical level for the reader s convenience we include in an appendix the corresponding treatment of the swift hohenberg equation a nonconservative counterpart of the generalized kuramoto sivashinsky equation for which the nonlinear stability analysis is considerably simpler together with numerical evans function analyses extending spectral stability analyses of mielke and schneider

ripple free robust output regulation and tracking for multirate sampled data control the problem of the robust continuous time asymptotic tracking and disturbance rejection for a multivariable multirate sampled data system is studied the necessity of a continuous time internal model of the exogenous signals except for the constant ones if any for obtaining a ripple free zero error response is proven the conditions are given for the existence of a hybrid control system including possibly a continuous time subcompensator for which the exponential stability and a continuous time convergence to zero of the tracking error are guaranteed at least in a neighbourhood of the nominal parameters this in particular both for when the dead beat convergence of the state free response and the continuous time error response is required at the nominal parameters and for when an exponential convergence with a prescribed rate of decay is required including design procedures of the continuous time and the discrete time subcompensators both for when the available description of the plant has a

parameterization of all multivariable ripple free deadbeat tracking controllers this paper presents an explicit parameterization of all ripple free deadbeat tracking controllers for multi input multi output mimo sampled data control systems the advantages over the previous results are the following first a useful necessary and sufficient condition to obtain a ripple free deadbeat tracking response for mimo sampled data control systems is presented second not only all ripple free mimo deadbeat tracking controllers but also all achievable deadbeat sequences and all control input ones are characterized by completely free parameters in an explicit form third the design method is simple and straightforward the computation consists of simple polynomial matrix equations without involving troublesome calculation such as the reduction to smith mcmillan form fourth the class of the reference inputs for which ripple free deadbeat tracking responses are achieved is generalized finally a numerical example demonstrates that the proposed parameterization makes it easy to clarify the r

the discrete linear time invariant time optimal control problem an overview a review of two decades of investigation into one of the fundamental problems of control theory the discrete linear time invariant time optimal control problem is presented two classes of multi input time optimal controller one based on system controllability and the other on eigenvalue assignment are considered in some detail also discussed are the dual state reconstruction problem the deadbeat control of linear systems with inaccessible state and other extensions the exposition is a unifying one and involves only that matrix algebra commonly encountered in control theory

four body central configurations with some equal masses we prove firstly that any convex non collinear central configuration of the planar 4 body problem with equal opposite masses 0 such that the diagonal corresponding to the mass is not shorter than that corresponding to the mass must possess a symmetry and therefore must be a kite then by a recent result of bernat llibre and perez chavela this kite is actually a rhombus secondly we prove that a convex non collinear planar 4 body central configuration with three equal masses must be a kite too we also prove that the concave central configuration with three equal masses forming a triangle and the fourth one with any given mass in the interior must be either an equilateral triangle with the fourth mass at its geometric center or an isosceles triangle with the fourth mass on the symmetry axis

optimal hold functions for sampled data regulation abstract this paper investigates the use of generalized sampled data hold function control gshf to optimize quadratic measures of performance in sampled data control loops the idea of gshf control is to use sampled data feedback but consider the hold function as a design parameter explicit solutions are given for the gshf versions of the optimal lq and lqg regulators the questions of existence uniqueness and computation of optimal hold functions are treated an example is presented

the golden ratio and super central configurations of the n body problem abstract in this paper we consider the problem of central configurations of the n body problem with the general homogeneous potential 1 r a configuration q q 1 q 2 q n is called a super central configuration if there exists a positive mass vector m m 1 m n such that q is a central configuration for m with m i attached to q i and q is also a central configuration for m where m m and m is a permutation of m the main discovery in this paper is that super central configurations of the n body problem have surprising connections with the golden ratio let r be the ratio q 3 q 2 q 2 q 1 of the collinear three body problem with the ordered positions q 1 q 2 q 3 on a line q is a super central configuration if and only if 1 r 1 r r 1 and r 1 where r 1 1 is a continuous function such that lim 0 r 1 the golden ratio the existence and classification of super central configurations are established in the collinear three body problem with general homogeneous potential 1 r super central configurations play an important role in counting the number of central configurations for a given mass vector which may decrease the number of central configurations under geometric equivalence

on the spectral stability of solitary wave solutions of the vector nonlinear schrodinger equation we consider a system of coupled cubic nonlinear schrodinger nls equations where the interaction coefficients jk are real the spectral stability of solitary wave solutions both bright and dark is examined both analytically and numerically our results build on preceding work by nguyen et al and others specifically we present closed form solitary wave solutions with trivial and nontrivial phase profiles their spectral stability is examined analytically by determining the locus of their essential spectrum their full stability spectrum is computed numerically using a large period limit of hill s method we find that all nontrivial phase solutions are unstable while some trivial phase solutions are spectrally stable to our knowledge this paper presents the first investigation of the stability of the solitary waves of the coupled cubic nls equation without the restriction that all components j are proportional to sech

a general framework for linear periodic systems with applications to h sup infinity sampled data control the authors present a framework for dealing with continuous time periodic systems the main tool is a lifting technique which provides a strong correspondence between continuous time periodic systems and certain types of discrete time time invariant systems with infinite dimensional input and output spaces despite the infinite dimensionality of the input and output spaces a lifted system has a finite dimensional state space if the original system does this fact permits rather constructive methods for analyzing these systems as a demonstration of the utility of this framework the authors use it to describe the continuous time i e intersample behavior of sampled data systems and to obtain a complete solution to the problem of parameterizing all controllers that constrain the l sup 2 induced norm of a sampled data system to within a certain bound

sampling theorem for periodic ripple problem this technical note presents a new sampling theorem it is the first sampling theorem derived in the framework of the study of sampled data control systems the sampling theorem can be a tool for the analysis and reduction of periodic intersample ripple

the symmetric central configurations of the 4 body problem with masses m1 m2 m3 m4 we characterize the planar central configurations of the 4 body problem with masses m 1 m 2 m 3 m 4 which have an axis of symmetry it is known that this problem has exactly two classes of convex central configurations one with the shape of a rhombus and the other with the shape of an isosceles trapezoid we show that this 4 body problem also has exactly two classes of concave central configurations with the shape of a kite this proof is assisted by computer

deadbeat ripple free tracking with disturbance rejection a polynomial approach the deadbeat ripple free tracking problem for sampled data systems is considered it is assumed that both the state and the output of the continuous time plant to be controlled are affected by external disturbances a polynomial approach is adopted and solvability conditions are given both in terms of the sampled system and of the original continuous time one

design of ripple free multivariable robust servomechanisms the application of standard robust servomechanism theory to sampled data systems guarantees asymptotic tracking at sampling instances and between sampling instants the output will normally contain ripple in this note the robust servomechanism theory is extended to sampled data systems and a technique is proposed for ripple free tracking of sinusoids and polynomials it is shown that a continuous internal model is necessary and sufficient to provide ripple free response

continuous time deadbeat control for sampled data systems in this paper the continuous time deadbeat control problem for the sampled data systems is considered we derived the class of all controllers that achieve the continuous time deadbeat control

deadbeat control and tracking of discrete time systems a new approach for forcing the state of a linear discrete time system to zero in a minimum number of steps is discussed the problem is formulated as the solution to a steady state optimal control problem with no cost on the control this problem in turn is set up as the solution to an associated eigenvalue problem no special assumption on the open loop system matrix and or the ratio of the number of states to controls is required stable numerical techniques are presented for solving for the feedback gain robust deadbeat tracking is also discussed

classification of four body central configurations with three equal masses it is known that a central configuration of the planar four body problem consisting of three particles of equal mass possesses a symmetry if the configuration is convex or is concave with the unequal mass in the interior we use analytic methods to show that besides the family of equilateral triangle configurations there are exactly one family of concave and one family of convex central configurations which completely classifies such central configurations

central configurations with a quasihomogeneous potential function for the classical n body problem if the masses are paced at rest in the configuration that is central then the masses will tend to the origin and result in a simultaneous collision a winter the analytical foundations of celestial mechanics 1st ed princeton university press princeton 1941 however if the potential is quasihomogeneous in nature that is the potential is of the form u v w where w i jamimjrij and v i jbmimjrij then we will dramatically get different results this paper determines the orbits that will result when the potential function is quasihomogeneous it also shows when there existences a relative equilibrium solution for the regular n gon configuration this is one half of perko s theorem for a homogeneous potential finally we show the existence of a central configuration which is not central for either w or v this configuration is the placement of two mutual equal pairs of masses placed at the vertices of a square this shows that the reverse implication of perk