a natural approach to the numerical integration of riccati differential equations this paper introduces a new class of methods which we call mobius schemes for the numerical solution of matrix riccati differential equations the approach is based on viewing the riccati equation in its natural geometric setting as a flow on the grassmannian of m dimensional subspaces of an n m dimensional vector space since the grassmannians are compact differentiable manifolds and the coefficients of the equation are assumed continuous there are no singularities or intrinsic instabilities in the associated flow the presence of singularities and numerical instabilities is an artifact of the coordinate system but since mobius schemes are based on the natural geometry they are able to deal with numerical instability and pass accurately through the singularities a number of examples are given to demonstrate these properties

stability analysis of continuous time periodic systems via the harmonic analysis asymptotic stability of finite dimensional linear continuous time periodic fdlcp systems is studied by harmonic analysis it is first shown that stability can be examined with what we call the harmonic lyapunov equation another necessary and sufficient stability criterion is developed via this generalized lyapunov equation which reduces the stability test into that of an approximate fdlcp model whose transition matrix can be determined explicitly by extending the gerschgorin theorem to linear operators on the linear space l sub 2 yet another disc group criterion is derived which is only sufficient stability of the lossy mathieu equation is analyzed as a numerical example to illustrate the results

boundary optimal lq control of coupled hyperbolic pdes and odes this contribution addresses the development of a linear quadratic lq regulator for a set of hyperbolic pdes coupled with a set of odes through the boundary the approach is based on an infinite dimensional hilbert state space description of the system and the well known operator riccati equation ore in order to solve the optimal control problem the ore is converted to a set of matrix riccati equations the feedback operator is found by solving the resulting matrix riccati equations the performance of the designed control policy is assessed by applying it to a system of interconnected continuous stirred tank reactor cstr and a plug flow reactor pfr through a numerical simulation

some analysis methods for rotating systems with periodic coefficients two of the more common procedures for analyzing the stability and forced response of equations with periodic coefficients are reviewed the use of floquet methods and the use of multiblade coordinate and harmonic balance methods the analysis procedures of these periodic coefficient systems are compared with those of the more familiar constant coefficient systems previously announced in star as n82 23702

h sampled data synthesis and related numerical issues in this paper we present a new compact derivation of state space formulae for the so called discretisation based solution of the h sampled data control problem our approach is based on the established technique of continuous time lifting which is used to isometrically map the continuous time linear periodically time varying sampled data problem to a discretetime linear time invariant problem state space formulae are derived for the equivalent discrete time problem by solving a set of two point boundary value problems the formulae accommodate a direct feed through term from the disturbance inputs to the controlled outputs of the original plant and are simple requiring the computation of only a single matrix exponential it is also shown that the resultant formulae can be easily re structured to give a numerically robust algorithm for computing the state space matrices

the recursive reduced order numerical solution of the singularly perturbed matrix differential riccati equation under stability observability conditions imposed on a singularly perturbed system an efficient numerical method for solving the corresponding matrix differential riccati equation is obtained in terms of the reduced order problems the order reduction is achieved via the use of the chang transformation applied to the hamiltonian matrix of a singularly perturbed linear quadratic control problem an efficient numerical recursive algorithm with a quadratic rate of convergence is developed for solving the algebraic equations comprising the chang transformation

solving the matrix differential riccati equation a lyapunov equation approach in this technical note we investigate a solution of the matrix differential riccati equation that plays an important role in the linear quadratic optimal control problem unlike many methods in the literature the approach that we propose employs the negative definite anti stabilizing solution of the matrix algebraic riccati equation and the solution of the matrix differential lyapunov equation an illustrative numerical example is provided to show the efficiency of our approach

hopf bifurcation with cubic symmetry and instability of abc flow we examine the dynamics of generic hopf bifurcation in a system that is symmetric under the action of the rotational symmetries of the cube we classify the generic branches of periodic solutions at bifurcation there are generically 27 branches corresponding to maximal symmetries organized into five symmetry types there are also up to 22 periodic solution branches of two other symmetry types these results are found by examination of the normal form with s 1 normal form symmetry for the bifurcation truncated at the third order in addition to the periodic branches whose branching and stability we find there are several branches of tori homoclinic bifurcations and chaotic attractors in the dynamics of the third order normal form since many of these features are not amenable to analysis we give some numerical examples on breaking the normal form symmetry there may be breakup of the branches of tori but the predictions for the periodic solutions will be reliable for the navier stokes equations with a particular forcing an abc flow is a dynamically stable solution for small reynolds numbers r for the most symmetric case a b c 1 the first instability of this system is a hopf bifurcation at r 13 04 with rotational symmetry of the cube we use our normal form analysis to explain the observed behaviour of solutions at this primary instability numerical simulations show that there is supercritical branching to rotating waves that alternate between the three axes which undergo secondary hopf bifurcation to a 2 torus at approximately r 13 09 the eight symmetrically related tori break up and then merge to form a chaotic attractor with full symmetry we can explain all these features by use of the generic third order normal form and s 1 normal form symmetry breaking terms

the control of linear time periodic systems using floquet lyapunov theory in this paper we use floquet lyapunov theory to derive the floquet factors of the state transition matrix of a given linear time periodic system we show how the periodicity of one of the factors can be determined a priori using a constant matrix which we call the yakubovich matrix based upon the signs of the eigenvalues of the monodromy matrix we then describe a method for the numerical computation of the floquet factors relying upon a boundary value problem formulation and the yakubovich matrix further we show how the invertibility of the controllability gramian and a specific form for the feedback gain matrix can be used to derive a control law for the closed loop system the controller can be full state or observer based it also allows the engineer to assign all the invariants of the system i e the full monodromy matrix deriving the feedback matrix requires solving a matrix integral equation for the periodic floquet factor of the new state transition matrix of the closed loop system this is

best approximation of positive power series this paper extends work of fiedler jurkat and the present author to series of the form la x where a is a moment sequence and 0 and f ty 1 ja jri in this paper we consider series of the form e anx with 0 x 1 where an is as above the results obtained here are similar to those in 4 we show the best approximation explicitly for the cases in which it can be calculated in particular in the notation of 4 we show that the error of the best l approximation is given by

reduced order solution to the finite time optimal control problems of linear weakly coupled systems the optimal solution to the finite time optimal control problem of weakly coupled linear systems is found in terms of completely decoupled reduced order differential equations for both closed loop and open loop control this is achieved through the use of the decoupling transformation that block diagonalizes the hamiltonian matrix of the weakly coupled linear quadratic control problem the convergence to the optimal solution is pretty rapid the proposed technique is very well suited for parallel computations

static duality and a stationary action application abstract conservative dynamical systems propagate as stationary points of the action functional using this representation it has previously been demonstrated that one may obtain fundamental solutions for two point boundary value problems for some classes of conservative systems via solution of an associated dynamic program further such a fundamental solution may be represented as a set of solutions of differential riccati equations dres where the solutions may need to be propagated past escape times notions of static duality and stat quad duality are developed where the relationship between the two is loosely analogous to that between convex and semiconvex duality static duality is useful for smooth functionals where one may not be guaranteed of convexity or concavity some simple properties of this duality are examined particularly commutativity application to stationary action is considered which leads to propagation of dres past escape times via propagation of stat quad dual dres

efficient matrix valued algorithms for solving stiff riccati differential equations in the time varying case a classical approach which has been widely used to compute the solution of the riccati matrix equation of for example size n n is to unroll the matrices into vectors and integrate the resulting system of n sup 2 vector differential equations directly if the system of vectorized differential equations is stiff the cost computation time and storage requirements of applying the popular backward differentiation formulas bdfs to the stiff equations will be very high for large n because a linear system of algebraic equations of size n sup 2 n sup 2 must be solved at each time step new matrix valued algorithms based on a matrix generalization of the bdfs are proposed for solving stiff riccatti differential equations the amount of work required to compute the solution per time step is only o n sup 3 flops by using the matrix valued algorithms whereas the classical approach requires o n sup 6 flops per time step

brief h2 and h norm computations of linear continuous time periodic systems via the skew analysis of frequency response operators the h 2 and h norm computations of finite dimensional linear continuous time periodic fdlcp systems through the frequency response operators defined by steady state analysis are discussed by the skew truncation the h 2 norm can be reached to any degree of accuracy by that of an asymptotically equivalent linear time invariant lti continuous time system the h norm can be approximated by the maximum singular value of the frequency response of an asymptotically equivalent lti continuous time system over a certain frequency range via the modified skew truncation by the latter result a hamiltonian test is proved for fdlcp systems in an lti fashion based on which a modified bisection algorithm is developed

a riccati transformation method for solving linear bvps ii computational aspects in part i of this paper siamj numer anal 25 1988 pp 1055 1073 a riccati transformation method for solving linear boundary value problems for ordinary differential equations is presented here the implementation aspects of this method are discussed in particular a double sweep integration process which includes a strategy for solving a matrix riccati differential equation and forming a continuous representation of its solution is described the efficacy of the method is demonstrated on a number of singular perturbation problems particular attention is paid to the choice of scaling for the variables and its effect on the solution method the computational experience while not based upon a refined implementation indicates that the riccati method may well prove to be competitive with the other general methods

computing the l2 induced norm of a compression operator abstract a new computationally viable approach is derived for computing the induced norm of a state space compression operator that is an integral operator defined on the finite length space l2 0 h determining this norm is a crucial component in the control analysis of both delay systems and sampled data systems the technique developed may have application to a wider variety of integral operators

a lifting technique for linear periodic systems with applications to sampled data control abstract a lifting technique is developed for periodic linear systems and applied to the h and h 2 sampled data control problems

quasi genericity of bifurcations to high dimensional invariant tori for maps we consider a family of maps in a banach spacee near the situation when the derivative at the fixed point has two pairs of complex eigenvalues lying on the unit circle the other part of the spectrum being strictly inside the unit disc we focus our attention on the region of the parameter space where the truncated normal form of the maps shows a bifurcation of a family of invariantt 1 circles into a family of invariantt 2 tori we show that this problem needs a 3 dimensional parameter unfolding and that for the complete maps bifurcation occurs at points where is the rotation number on the non normally hyperbolict 1 circle ande 2i are the eigenvalues of the constant matrix conjugated to the non contracting part of the linearization on the normal fiber bundle overt 1 making some non resonance and diophantine assumptions on leading to a positive measure cantor set int 2 we show that in paraboloidal regions of the 3 dim parameter space we have clean bifurcations as for the truncated normal form the complement of these regions forms a set of bubbles such as the ones obtained by chenciner in chen for a codimension 2 problem for maps in 2 the main tool here is a generalization for a matrix function ont 1 close to a constant of the quasi conjugacy to a constant modulo a minimum of additional parameters moved quasi conjugacy for the infinite dimensional case we use ac decoupling result on the angular dependent linear parts into a contraction still angular dependent and another part quasi conjugated to a constant matrix this type of analysis applies for a wide range of problems where truncated normal forms of the maps give bifurcations fromt n tot n 1 tori and this needs a n 1 dimensional parameter unfolding

brief paper computing the l2 gain for linear periodic continuous time systems a method to compute the l 2 gain is developed for the class of linear periodic continuous time systems that admit a finite dimensional state space realisation a bisection search for the smallest upper bound on the gain is employed where at each step an equivalent discrete time problem is considered via the well known technique of time domain lifting the equivalent problem involves testing a bound on the gain of a linear shift invariant discrete time system with the same state dimension as the periodic continuous time system it is shown that a state space realisation of the discrete time system can be constructed from point solutions to a linear differential equation and two differential riccati equations all subject to only single point boundary conditions these are well behaved over the corresponding one period intervals of integration and as such the required point solutions can be computed via standard methods for ordinary differential equations a numerical example is presented and comparisons made with alternative techniques

spectral properties and their applications of frequency response operators in linear continuous time periodic systems article history received 25 july 2008 available online 11 december 2008 submitted by h zwart spectral properties related to the frequency response operators of finite dimensional linear continuous time periodic fdlcp systems are examined rigorously and thoroughly in the paper as applications of the spectral features positive realness of fdlcp systems is scrutinized and then a harmonic hamiltonian criterion is derived for the h norm of fdlcp systems in particular positive realness of fdlcp systems is interpreted in term of toeplitz operators for the first time in this study together with a testing algorithm deriving the harmonic hamiltonian criterion with a frequency domain approach bridges the spectra of the frequency response operators in fdlcp systems with their time domain behaviors

a gmres based bdf method for solving differential riccati equations abstract differential riccati equations play a fundamental role in control theory for example optimal control filtering and estimation decoupling and order reduction etc the most popular codes to solve stiff differential riccati equations use backward differentiation formula bdf methods in this paper a new approach to solve differential riccati equations by means of a bdf method is described in each step of these methods an algebraic riccati equation is obtained which is solved by means of newton s method in the standard approach this system is transformed into a sylvester equation which could be solved by means of the well known bartels stewart method in our code we obtain a system of linear equations defined from a kronecker product of matrices related to coefficient matrices of the differential riccati equation that is solved by means of the iterative generalized minimum residual gmres method we have also implemented an efficient matrix vector product in order to reduce the computational and storage cost of the gmres method the above approach has been applied in the development of an algorithm to solve differential riccati equations the accuracy and efficiency of this algorithm has been compared with the bdf algorithm that uses the bartels stewart method experimental results show the advantages of the new algorithm

numerical integration of the differential riccati equation and some related issues abstract in this paper the problem of direct numerical integration of differential riccati equations dres and some related issues are considered the dre is an expression of a particular change of variables for a linear system of ordinary differential equations the error that an approximate solution of the dre induces on the original variables of the system is considered and it is related to geometrical properties of the system itself sharp bounds on the global error for the computed solution are also given in terms of local errors and geometrical properties of the original system nonstiff and stiff dres of unsymmetric and symmetric type are considered a useful matrix interpretation is given for many integration schemes such as the backward differentiation formulas bdf when applied to the dre this allows the matrix structure of the problem to be exploited in particular for stiff dres the resulting strategy allows for a saving of three orders of magnitude with respect to the standard reform

existence conditions and properties of the frequency response operators of continuous time periodic systems the definition of the frequency response operator via the steady state analysis in finite dimensional linear continuous time periodic fdlcp systems is revisited it is shown that the frequency response operator is guaranteed to be well defined only densely on the linear space l2 which is different from the usual understanding fortunately however it turns out that this frequency response operator can have an extension onto l2 so that the equivalence between the time domain h2 norm respectively the l2 induced norm and the frequency domain h2 norm respectively the h infty norm of the frequency response operator is recovered under some stronger assumptions it is also shown that the frequency response operator can be viewed as a bounded operator from l1 to l1 which can also be established via the steady state analysis

spatially periodic steady solutions to the three dimensional navier stokes equation with the abc force abstract for an appropriately scaled abc forcing the abc flow is a steady space periodic solution to the three dimensional navier stokes equation for an arbitrary reynolds number r we are investigating both numerically and analytically different branches of steady solutions in the case a b c when the equation has a group of symmetries isomorphic to the rotation group of the cube two families of steady flows each comprised of three mutually symmetric branches were detected numerically one was shown to persist for 7 9 r 2000 and another for 149 r 1000 branches of the first family intersect twice with the abc flow in a bifurcation generic to a system with symmetry group d3 other possible bifurcations of the abc flow and the possibility of existence of branches of steady flows with other symmetries are considered

survey invariant representations of discrete time periodic systems the correspondence between periodic and time invariant systems can take a variety of forms the purpose of this survey paper is to present a complete and clear picture on time invariant representations for discrete time periodic systems the core of the treatment evolves around the concept of periodic transfer function

rational approximate solutions and error bounds for the nonsymmetric riccati matrix differential equation in this paper we determine an interval where the exact solution of the nonsymmetric riccati matrix differential equation is defined rational approximate solutions are whose error is smaller than a prefixed number e 0 are constructed in terms of the data

linear quadratic optimal control of hereditary differential systems infinite dimensional riccati equations and numerical approximations recent theory of infinite dimensional riccati equations is applied to the linear quadratic optimal control problem for hereditary differential systems and it is shown that for most such problems the operator solutions of the riccati equations are of trace class i e nuclear with special attention to trace norm convergence an abstract approximation theory is developed and applied to a particular approximation scheme numerical examples are given problems on both finite and infinite time intervals are studied for both the hereditary system and the approximating systems in the infinite time problem characteristic equations are derived for the closed loop eigenvalues and formulas for the corresponding eigenvectors are given

frequency response of sampled data systems this paper introduces the concept of frequency response for sampled data systems and explores some basic properties as well as its computational procedures it is shown that 1 by making use of the lifting technique the notion of frequency response can be naturally introduced to sampled data systems in spite of their time varying characteristics 2 it represents a frequency domain steady state behaviour and 3 it is also closely related to the original transfer function representation via an integral formula it is shown that the computation of the frequency response can be reduced to a finite dimensional eigenvalue problem and some examples are presented to illustrate the results

a shishkin mesh for a singularly perturbed riccati equation in this paper a singularly perturbed riccati initial value problem is examined parameter explicit bounds on the solution and its derivatives are given a numerical method composed of an implicit difference operator and a piecewise uniform shishkin mesh is constructed a theoretical parameter independently bound on the errors in the numerical approximations is established numerical results are presented which are in agreement with the theoretical error bound

rigorous bounds on the fast dynamo growth rate involving topological entropy the fast dynamo growth rate for ack 1 map or flow is bounded above by topological entropy plus a 1 k correction the proof uses techniques of random maps combined with a result of yomdin relating curve growth to topological entropy this upper bound implies the following anti dynamo theorem inc systems fast dynamo action is not possible without the presence of chaos in addition topological entropy is used to construct a lower bound for the fast dynamo growth rate in the caserm

a survey of recursive algorithms for the solution of the discrete time riccati equation abstract the riccati equation plays a fundamental role in many fields of mathematics science and engineering its solution constitutes an integral prerequisite to the solution of important problems in the above fields due to the importance of the riccati equation there exists considerable literature on its algebraic as well as algorithmic solution a very large number of those studies are devoted to the continuous time riccati equation in this paper we present a survey of classical as well as more recent recursive algorithms that solve the discrete time riccati equatbn emanating from the kaiman filter as well as from the lainiotis filter equations either using per step calculations or the doubling principle it is established that these algorithms converge fast and are numerically stable

bifurcations in phase oscillator networks with a central element abstract a system of phase oscillators with identical natural frequencies and the star like architecture of connections is considered interaction functions are described by two terms of fourier expansion bifurcation analysis of small systems containing 3 or 4 oscillators has been performed the results are summarized in bifurcation diagrams that provide a full description of the boundaries between regions with different dynamics and the types of bifurcations that lead to the changes in the topology of phase space the bifurcations include changes of fixed point stability and formation destruction of limit and heteroclinic cycles for the system with 4 oscillators chaotic behaviour has been investigated the results can be useful to control system dynamics through an appropriate choice and variation of parameter values the generalization of the results to the systems with an arbitrary number of oscillators and application of the results in computational neuroscience are discussed

symmetry increasing bifurcation of chaotic attractors abstract bifurcation in symmetric is typically associated with spontaneous symmetry breaking that is bifurcation is associated with new solution having less symmetry in this paper we show that symmetry increasing bifurcation in the discrete dynamics of symmetric mappings is possible and is perhaps generic the reason for these bifurcations may be understood as follows the existence of one attractor in a system with symmetry gives rise to a family of conjugate attractors all related by symmetry typically in computer experiments what we see is a sequence of symmetry breaking bifurcations leading to the existence of conjugate chaotic attractors as the bifurcation parameter is varied these attractors grow in size and merge leading to a single attractor having greater symmetry we prove a theorem suggesting why this new attractor should have greater symmetry and present a number of striking examples of the symmetric patterns that can be formed by iterating the simplest mappings on the plane with the symmetry of the regular m gon in the last section we discuss period doubling in the presence of symmetry

modelling instability of abc flow using a mode interaction between steady and hopf bifurcations with rotational symmetries of the cube abstract we consider hydrodynamic stability of symmetric abc flow with a b c 1 regarded as a steady state of the three dimensional navier stokes equation with appropriate forcing numerical investigations have shown that its first instability on increasing the reynolds number is a hopf bifurcation at r 13 044 and that at this bifurcation the second dominant eigenvalue is real motivated by this we study generic interaction of steady state and hopf bifurcations in systems with rotational symmetry of the cube o with an eight dimensional normal form the generic branching and bifurcation behavior of the third order truncated normal form is investigated this is used to analyse a range of the bifurcations for particular values of the coefficients obtained by center manifold reduction from the hydrodynamic system the normal form system shows a sequence of bifurcations and attractors that closely follows the sequence observed for the original hydrodynamic system up to about r 13 91 this includes a torus breakdown to a chaotic attractor and a crisis of attractors that leads to a change in symmetry we discuss numerical simulations of the hydrodynamic system for larger values of r finally we present evidence that the system has robust heteroclinic cycles between fully symmetric abc flow and six steady states with broken symmetry for a range of parameter values near r 15

a family of anadromic numerical methods for matrix riccati differential equations matrix riccati differential equations mrdes x a21 xa11 a22x xa12x x 0 x0 where aij aij t appear frequently throughout applied mathematics science and engineering naturally the existing conventional runge kutta methods and linear multi step methods can be adapted to solve mrdes numerically indeed they have been adapted there are a few unconventional numerical methods too but they are suited more for time invariant mrdes than time varying ones for stiff mrdes existing implicit methods which are preferred to explicit ones require solving nonlinear systems of equations of possibly much higher dimensions than the original problem itself of for example implicit runge kutta methods and thus they can pose implementation difficulties and also be expensive in the past the property of an mrde which has been most preserved is the symmetry property for a symmetric mrde many other crucial properties have been discarded besides the symmetry property our proposed methods also preserve two other important properties bilinear rational dependence on the initial value and a generalized inverse relation between an mrde and its complementary mrde by preserving the generalized inverse relation our methods are accurately able to integrate an mrde whose solution has singularities by preserving the property of bilinear dependence on the initial value our methods also conserve the rank of change to the initial value and a solution s monotonicity property our methods are anadromic 1meaning if an mrde is integrated by one of our methods from t to and then integrated backward from t to using the same method the value at t is recovered in the absence of rounding errors this implies that our methods are necessarily of even order of convergence for time invariant mrdes methods of any even order of convergence are established while for time varying mrdes methods of order as high as 10 are established but only methods of order up to 6 are stated in detail our methods are semi implicit in the sense that there are no nonlinear systems of matrix equations to solve only linear ones unlike any pre existing implicit method given the availability of high quality codes for linear matrix equations our methods can easily be implemented and embedded into any application software package that needs a robust mrde solver numerical examples are presented to support our claims 1we coined anadromic from the greek roots back up and o o act of running of anadromous which describes fish that must return from the ocean to spawn in the same streams whence they hatched after we had tried to describe our numerical methods with words such as reflexive 27 28 symmetric which is still used today in the literature e g 22 and reversible which turned out to be overused received by the editor received august 25 2009 and in revised form october 30 2010 2010 mathematics subject classification primary 65l05

the 1 2 mode interaction and heteroclinic networks in boussinesq convection abstract methods of equivariant bifurcation theory are applied to boussinesq convection in a plane layer with stress free horizontal boundaries and an imposed square lattice periodicity in the horizontal directions we consider the problem near the onset of instability of the uniform conducting state where spatial roll patterns with two different wavelengths in the ratio 1 2 become simultaneously unstable at a mode interaction center manifold reduction yields a normal form on c 4 with very rich dynamical behavior for a fixed prandtl number p the mode interaction occurs at an isolated point in the parameter plane r l where r is the rayleigh number and l the length of the horizontal periodicities and acts as an organizing center for many nearby bifurcations the normal form predicts the appearance of many steady states and travelling waves which are classified by their symmetries it also predicts the appearance of robust heteroclinic networks involving steady states with several different symmetries and robust attractors of generalized heteroclinic type that include connections from equilibria to subcycles this is the first example of a heteroclinic network in a fluid dynamical system that has depth greater than one the normal form dynamics is in good correspondence both quantitatively and qualitatively with direct numerical simulations of the full convection equations

implementing the hamiltonian test for the h norm in linear continuous time periodic systems by the floquet similarity transformations of finite dimensional linear continuous time periodic fdlcp systems the time domain hamiltonian test for the h norm is first interpreted in terms of toeplitz operators of the system matrices based on this novel frequency domain interpretation implementing the time domain hamiltonian test can be accomplished by working on its frequency domain counterpart via truncations this gives us a bisection algorithm for evaluating the h norm of a class of fdlcp systems through finite dimensional linear time invariant continuous time models the finite dimensional hamiltonian test is necessary and sufficient in the asymptotic sense and claimed only via fourier coefficients of the system matrices without the transition matrix of the fdlcp system concerned there are numerical examples to illustrate the suggested algorithm

eigenfactor solution of the matrix riccati equation a continuous square root algorithm this paper introduces a new algorithm for solving the matrix riccati equation differential equations for the eigenvalues and eigenvectors of the solution matrix are developed in which their derivatives are expressed in terms of the eigenvalues and eigenvectors themselves and not as functions of the solution matrix the solution of these equations yields then the time behavior of the eigenvalues and eigenvectors of the solution matrix a reconstruction of the matrix itself at any desired time is immediately obtained through a trivial similarity transformation this algorithm serves two purposes first being a square root solution it entails all the advantages of square root algorithms such as nonnegative definiteness and accuracy secondly it furnishes the eigenvalues and eigenvectors of the solution matrix continuously without resorting to the complicated route of solving the equation directly and then decomposing the solution matrix into its eigenvalues and eigenvectors the algorithm which handles cases of distinct as well as multiple eigenvalues is tested on several examples through these examples it is seen that the algorithm is indeed more accurate than the ordinary one moreover it is seen that the algorithm works in cases where the ordinary algorithm fails and even in cases where the closed form solution cannot be computed as a result of numerical difficulties

on the benefits of the ldlt factorization for large scale differential matrix equation solvers abstract we propose efficient algorithms for solving large scale matrix differential equations in particular we deal with the differential riccati equations dre and state the applicability to the differential lyapunov equations dle we focus on methods based on standard versions of ordinary differential equations in the matrix setting the application of these methods yields algebraic lyapunov equations ales with a certain structure to be solved in every step the alternating direction implicit adi algorithm and krylov subspace based methods allow to exploit this special structure however a direct application of classic low rank formulations requires the use of complex arithmetic using an l d l t type decomposition of both the right hand side and the solution of the equation we avoid this problem thus the proposed methods are a more practical alternative for large scale problems arising in applications also they make the application of higher order methods feasible the numerical results show the better performance of the proposed methods compared to earlier formulations

invariants equivariants and characters in symmetric bifurcation theory in the analysis of stability in bifurcation problems it is often assumed that the appropriate reduced equations are in normal form in the presence of symmetry the truncated normal form is an equivariant polynomial map therefore the determination of invariants and equivariants of the group of symmetries of the problem is an important step in general these are hard problems of invariant theory and in most cases they are tractable only through symbolic computer programs nevertheless it is desirable to obtain some of the information about invariants and equivariants without actually computing them for example the number of linearly independent homogeneous invariants or equivariants of a certain degree generating functions for these dimensions are generally known as molien functions we obtain formulae for the number of linearly independent homogeneous invariants or equivariants for hopf bifurcation in terms of characters we also show how to construct molien functions for invariants and equivariants for hopf bifurcation our results are then applied to the computation of the number of invariants and equivariants for hopf bifurcation for several finite groups and the continuous group o 3

numerical comparisons of nonlinear convergence accelerators as part of a continuing program of numerical tests of convergence accelerators we have compared the iterated aitken s a2 method wynn s algorithm brezinski s 0 algorithm and levin s u transform on a broad range of test problems linearly convergence alternating monotone and irregular sign series logarithmically convergent series power method and bernoulli method sequences alternating and monotone asymptotic series and some perturba tion series arising in applications in each category either the algorithm or the u transform gives the best results of the four methods tested in some cases differences among methods are slight and in others they are quite striking 1 introduction we reported in 19 on a numerical test program that compared eleven different scalar convergence accelerators on a broad range of infinite series problems in an attempt to find an across the board accelerator that would handle both linear and logarithmic convergence and both alternating and monotone series or sequences the best candidates to emerge from that study were levin s u transform 14 and brezinski s 0 algorithm 6 we also provided some theoretical support for that conclusion in terms of the wide range of series summed exactly by these methods and by proving that 02 and the levin transforms all accelerate linear convergence the other levin transforms t and v produced essentially equivalent numerical results to u except that t fails on logarithmic convergence we have continued the test program started with 19 to include monotone linearly convergent series series with irregular sign patterns specifically fourier series and sequences whose terms do not depend explicitly on the previous term and or the index e g bernoulli s method since convergence accelerators are often known to transform divergent series to convergent ones we have also included in our test program divergent power series asymptotic series and a few perturbation series the methods considered in this paper are the u and 0 algorithms noted above and

technical communique approximation of frequency response for sampled data control systems this paper proves that the frequency response gains of fast sample fast hold approximations of a sampled data system converge to that of the original system as the sampling rate gets faster while this may appear to hold trivially there is a serious technical difficulty and the proof is indeed nontrivial it is also guaranteed that this convergence is uniform on the total frequency range the latter property is necessary to guarantee that a single approximant can be used for frequency response computation for the overall frequency range

on the non linear stability of the 1 1 1 abc flow abstract abc flows which can be considered as prototypes for the study of the onset of three dimensional spatio temporal turbulence are known both analytically and numerically to be linearly unstable we analyze the nonlinear evolution of the abc flow a 1 with a b c 1 and with characteristic wavenumber k0 1 in the interval of reynolds number 13 r 50 we solve numerically the forced navier stokes equations with periodic boundary conditions for up to 9 9 104 eddy turnover times bifurcations towards progressively more complex flows obtain with a relaminarization window loss of symmetries and chaotic oscillations probably revealing an underlying heteroclinic structure in the chaotic regime only three steady solutions emerge besides a1 they consist of a perturbed abc flow a 2 with a b c plus cyclic permutations at 23 r 50 an unstructured temporal chaos is observed with the flow still dominated by the largest scales

on monodromy matrix computation abstract we present a study on the critical time step for the numerical integration based on the runge kutta method of the monodromy matrix the fundamental matrix solution associated with a set of n first order linear ordinary differential equations with periodic coefficients by applying the lyapunov schmidt method for any dimension n and systems which are perturbations of autonomous systems we give an approximation to the critical time step which involves the autonomous part as well as the periodic perturbations

frequency response of sampled data systems this paper develops a frequency domain theory that provides a method to analyze and design sampled data control systems including their intersample behaviors the key idea is to consider the signal space x x t x t n x xn exp j t jn st n x xn 2 where s is the sampling angular frequency it is shown that a stable sampled data system equipped with a strictly proper pre filter before the sampler maps x into x l2 in the steady state that mapping is denoted by q j and is referred to as an fr operator it is proved that the norm of the sampled data system as an operator from l2 to l2 is given by max q j 2 2 where 12 s 12 s a set of equations relating outputs of a closed loop system to its inputs in the frequency domain is derived and their solution is given in an explicit form based on that solution the sensitivity fr operator y j and the complementary sensitivity fr operator t j are defined for feedback control systems and it is shown that y j gives the improvement of the sensitivity of the transfer characteristics from the reference to the controlled output and also represents the ability of rejecting disturbances and that t j represents the degree of robust stability and at the same time gives the effect of detection noises it is also shown that y j t j i and thus a frequency domain paradigm for the design of sampled data control systems which is exactly parallel to the continuous time case is established

almost analytic representation for the solution of the differential matrix riccati equation an almost analytic representation is presented for the solution of the nonhomogeneous and homogeneous time invariant and time variant differential matrix riccati equation this representation gives a new insight into the behavior of the optimal estimator on a finite time interval for the time invariant case this solution is a basis for an exact true iterative algorithm for the time invariant case an approximate iterative algorithm is derived

gip integrators for matrix riccati differential equations matrix riccati differential equations mrdes are initial value problems of the form x a 2 1 xa 1 1 a 2 2x xa 1 2x x 0 x 0 these equations arise frequently throughout applied mathematics science and engineering it can happen that even when the a i j are smooth functions of t or constant the solution x may have a singularity or even infinitely many singularities this paper shows several classes of numerical algorithms which we call gip integrators that can solve for x past its singularities furthermore none of the algorithms require knowledge of the placement or even existence of singularities in x also it is shown how embedded runge kutta methods can be used to construct gip integrators to not only approximate x past singularities but also provide for error estimation to allow efficient time stepping finally several examples are shown to validate the theory

numerical integration of the differential matrix riccati equation two new bernoulli substitution methods for solving the riccati differential equation are tested numerically against direct integration of the riccati equation the chandrasekhar algorithm and the davison maki method on a large set of problems taken from the literature the first of these new methods was developed for the time invariant case and uses the matrix analog of completing the square to transform the problem to a bisymmetric riecati equation whose solution can be given explicitly in terms of a matrix exponential of order n this method is fast and accurate when the extremal solutions of the associated algebraic riccati equation are well separated the second new method was developed as a means of eliminating the instabilities associated with the davison maki algorithm by using reinitialization at each time step the davison maki algorithm can be recast as a recursion which is over three times faster than the original method and is easily shown to be stable for both time invariant and time dependent problems from the results of our study we conclude that the modified davison maki method gives superior performance except for those problems where the number of observers and controllers is small relative to the number of states in which ease the chandrasekhar algorithm is better

a new approach to the discretization of continuous time controllers a new method for discretizing continuous time controllers is derived using principles of controller approximation it focuses on the closed loop use of the discrete time controller the resulting approximation criterion is a measure for stability of the control system provides an upper bound on the sampling time for which stability can be guaranteed and because it is based on continuous time controller approximation it indicates the cost of discretization in terms of performance degradation the discrete time controller is obtained through minimization of this criterion which can be performed with standard software used in h sub infinity controller design

h norm computation of linear continuous time periodic systems by a structure preserving algorithm a new reliable structure preserving algorithm for computing h norm of linear continuous time periodic systems is proposed in this paper in the computation of the h norm no riccati differential equations are needed to solve and only eigenvalues of a monodromy matrix of the associated periodic hamiltonian system will be evaluated first the monodromy matrix is expressed as the product of state transition matrices of the hamiltonian system second these state transition matrices which have been proved to be symplectic matrices are evaluated by a structure preserving magnus series method then in order to preserve the standard symplectic form of the monodromy matrix the structure preserving matrices obtained by state transition matrices are employed to compute the monodromy matrix at last the effectiveness and the high accuracy of the proposed structure preserving algorithm are demonstrated by numerical examples

max plus representation for the fundamental solution of the time varying differential riccati equation using the tools of optimal control semiconvex duality and max plus algebra this work derives a unifying representation of the solution for the matrix differential riccati equation dre with time varying coefficients it is based upon a special case of the max plus fundamental solution first proposed in fleming and mceneaney 2000 such a fundamental solution can extend a particular solution of certain bivariate dres into the general solution and the dres can be analytically solved from any initial condition this paper also shows that under a fixed duality kernel the semiconvex dual of a dre solution satisfies another dual dre whose coefficients satisfy the matrix compatibility conditions involving hamiltonian and certain symplectic matrices for the time invariant dre this allows us to make dual dre linear and thereby solve the primal dre analytically this paper also derives various kernel duality relationships between the primal and time shifted dual dres which lead to an array of dre solution methods time invariant analogue of one of these methods was first proposed in mceneaney 2008

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

accurate numerical integration of stiff differential riccati equations abstract in this paper accurate numerical integration methods for four times continuously differentiable coefficient stiff riccati matrix differential equations are proposed first a discrete numerical solution is constructed at a net of points of the predetermined existence interval using implicit one step matrix methods finally using linear b spline matrix functions a continuous numerical solution whose error is uniformly upper bounded by a prefixed error in the existence domain is constructed

an argument principle based framework for structural and spectral characteristics in linear dynamical systems abstractby means of the argument principle in complex analysis the paper proposes a novel complex geometric framework about structural features in terms of controllability observability and stability stabilisability and spectral ones in terms of bounded positive realness for transfer functions in linear dynamical systems the proposed criteria render alternative complex frequency domain interpretations about the structural and spectral essentials moreover the criteria are implementable graphically with locus plotting independent of singularities distribution locus orientation specification singularity related locus encirclement counting and prior frequency sweeping the criteria are also utilisable numerically via complex argument increment integration without locus plotting independent of iterative algorithms in either ways the criteria are numerically tractable and opt for robustness analysis and synthesis numerical examples are included to illustrate the main results

a new fundamental solution for differential riccati equations arising in control the matrix differential riccati equation dre is ubiquitous in control and systems theory the presence of the quadratic term implies that a simple linear systems fundamental solution does not exist of course it is well known that the bernoulli substitution may be applied to obtain a linear system of doubled size here however tools from max plus analysis and semiconvex duality are brought to bear on the dre we consider the dre as a finite dimensional solution to a deterministic linear quadratic control problem taking the semiconvex dual of the associated semigroup one obtains the solution operator as a max plus integral operator with quadratic kernel the kernel is equivalently represented as a matrix using the semigroup property of the dual operator one obtains a matrix operation whereby the kernel matrix propagates as a semigroup the propagation forward is through some simple matrix operations this time indexed family of matrices forms a new fundamental solution for the dre solution for any initial condition is obtained by a few matrix operations on the fundamental solution and the initial condition in analogy with standard algebra linear systems the fundamental solution can be viewed as an exponential form over a certain idempotent semiring this fundamental solution has a particularly nice control interpretation and might lead to improved dre solution speeds

a riccati transformation method for solving linear bvps i theoretical aspects a method of solving linear two point boundary value problems bvps for first order systems of ordinary differential equations odes is considered of particular interest is the class of singularly perturbed bvps the method attempts to split the integration of the rapidly increasing decreasing solution components typical for such problems into different initial value problems ivps the case of separated boundary conditions bcs is considered and their ability to detect solution components that eventually increase decrease is exploited by using the riccati transformation the original bvp is then reformulated as three ivps one of them being a matrix riccati equation the riccati equation is nonlinear and a multiple embedding setting must be used the entire method is then fit theoretically into a multiple shooting like framework and the original bvp as a chain of subproblems is studied in this way a generalized decoupling result is presented which enables the study of stability to be brough