the fractional representation approach to synthesis problems an algebraic analysis viewpoint part i weakly doubly coprime factorizations in this paper we show how to reformulate the fractional representation approach to analysis and synthesis problems within an algebraic analysis framework in terms of modules we give necessary and sufficient conditions so that a system admits weakly left right doubly coprime factorizations moreover we explicitly characterize the integral domains a such that every plant defined by means of a transfer matrix whose entries belong to the quotient field of a admits weakly doubly coprime factorizations finally we show that this algebraic analysis approach allows us to recover on the one hand the approach developed in m c smith ieee trans automat control 34 1989 pp 1005 1007 and on the other hand the ones developed in k mori and k abe siam j control optim 39 2001 pp 1952 1973 v r sule siam j control optim 32 1994 pp 1675 1695 and 36 1998 pp 2194 2195 m vidyasagar h schneider and b a francis ieee trans automat control 27 1982 pp 880 894 m vidyasagar control system synthesis a factorization approach mit press cambridge ma 1985

on the existence of caratheodory solutions in mechanical systems with friction it is shown that the differential equation with discontinuous right hand side which describes the dynamic behavior of a mechanical system with friction has a caratheodory solution this result supports theoretically the prior work on the control of mechanical systems with friction which assumes the existence of caratheodory solutions without proof

transcendental and interpolation methods in simultaneous stabilization and simultaneous partial pole placement problems in this paper we investigate the existence of a compensator which simultaneously renders a given r tuple of multiinput multioutput p times m linear dynamical systems internally stable in particular we parametrize a set of simultaneously stabilizable r tuples of plants and show that provided r leqq max m p the above set is semialgebraic and dense in the space sigma of r tuples of plants furthermore we also consider an extension of the classical pole placement and stabilization problems and investigate the simultaneous partial pole placement problem consequently we consider a suitable topology in and obtain a necessary condition and a sufficient condition for the generic partial pole placement problem surprisingly enough the problem of simultaneously stabilizing a triplet of m times m plants chosen generically is shown equivalent to the problem of partially pole placing a single m times m plant by a stable minimum phase compensator on the other hand the problem of simultaneous

error propagation when approximating multi solitons the kdv equation as a case study abstract the paper studies the influence of the time discretizations when simulating some phenomena involving more than one solitary wave taking the kdv equation as a case study we obtain some conditions on the numerical method in order to get a more correct simulation of multi soliton solutions they are related to the evolution of the conserved quantities of the problem through the numerical integration it is shown that when approximating n solitons a method that preserves n invariants of the problem shows a better time propagation of the error than that of a general scheme as a consequence of this the simulation of some physical parameters that characterize the waves is more suitable when using conservative integrators we also show how these results can be extended to the approximation to multi solitons of any equation of the kdv hierarchy and more generally other integrable equations

shadow hybrid monte carlo an efficient propagator in phase space of macromolecules shadow hybrid monte carlo shmc is a new method for sampling the phase space of large molecules particularly biological molecules it improves sampling of hybrid monte carlo hmc by allowing larger time steps and system sizes in the molecular dynamics md step the acceptance rate of hmc decreases exponentially with increasing system size n or time step t this is due to discretization errors introduced by the numerical integrator shmc achieves an asymptotic o n1 4 speedup over hmc by sampling from all of phase space using high order approximations to a shadow or modified hamiltonian exactly integrated by a symplectic md integrator shmc satisfies microscopic reversibility and is a rigorous sampling method shmc requires extra storage modest computational overhead and a reweighting step to obtain averages from the canonical ensemble this is validated by numerical experiments that compute observables for different molecules ranging from a small n alkane butane with four united atoms to a larger solvated protein with 14 281 atoms in these experiments shmc achieves an order magnitude speedup in sampling efficiency for medium sized proteins sampling efficiency is measured by monitoring the rate at which different conformations of the molecules dihedral angles are visited and by computing ergodic measures of some observables

construction of multisymplectic schemes of any finite order for modified wave equations in this paper we analyze several symplectic discretizations of linear wave equations with periodic boundary conditions and show the discretizations are multisymplectic of arbitrary finite order for wave equations when the periodic boundary condition vanishes

error growth in the numerical integration of periodic orbits with application to hamiltonian and reversible systems we analyze in detail the growth with time of the coefficients of the asymptotic expansion of the error in the numerical integration with one step methods of periodic solutions of systems of ordinary differential equations variable stepsizes are allowed we successively consider general hamiltonian and reversible problems for hamiltonian and reversible systems and under fairly general hypotheses on the orbit being integrated numerical methods with relevant geometric properties symplecticness energy conservation reversibility are proved to have better error growth than general methods

what do multistep methods approximate standard analysis of multistep methods for ode s assumes the application of an initialization routine that generates the starting points here ak step method is considered directly as a mappingr kn r n it is shown to approximate a mapping which is expressible directly in terms of the flow of the vector field some useful properties of that mapping are shown and for strictly stable methods these are applied to the question of invariant circles near a hyperbolic periodic solution

simultaneous controller design for time varying linear systems abstract in this paper we establish the parametrization of all simultaneously stabilizing controllers for two time varying linear systems a new approach to the simultaneous stabilization of three time varying linear systems is also provided

algebraic design techniques for reliable stabilization in this paper we study two problems in feedback stabilization the first is the simultaneous stabilization problem which can be stated as follows given plants g 0 g 1 g l does there exist a single compensator c that stabilizes all of them the second is that of stabilization by a stable compensator or more generally a least unstable compensator given a plant g we would like to know whether or not there exists a stable compensator c that stabilizes g if not what is the smallest number of right half place poles counted according to their mcmillan degree that any stabilizing compensator must have we show that the two problems are equivalent in the following sense the problem of simultaneously stabilizing l 1 plants can be reduced to the problem of simultaneously stabilizing l plants using a stable compensator which in turn can be stated as the following purely algebraic problem given 2l matrices a 1 a l b 1 b l where a i b i are right coprime for all i does there exist a matrix m such that a i mb i is unimodular for all i conversely the problem of simultaneously stabilizing l plants using a stable compensator can be formulated as one of simultaneously stabilizing l 1 plants the problem of determining whether or not there exists an m such that a bm is unimodular given a right coprime pair a b turns out to be a special case of a question concerning a matrix division algorithm in a proper euclidean domain we give an answer to this question and we believe this result might be of some independent interest we show that given two n times m plants g 0 and g 1 we can generically stabilize them simultaneously provided either n or m is greater than one in contrast simultaneous stabilizability of two single input single output plants g 0 and g 1 is not generic

the nose poincare method for constant temperature molecular dynamics we present a new extended phase space method for constant temperature canonical ensemble molecular dynamics our starting point is the hamiltonian introduced by nose to generate trajectories corresponding to configurations in the canonical ensemble using a poincare time transformation we construct a hamiltonian system with the correct intrinsic timescale and show that it generates trajectories in the canonical ensemble our approach corrects a serious deficiency of the standard change of variables nose hoover dynamics which yields a time reversible system but simultaneously destroys the hamiltonian structure a symplectic discretization method is presented for solving the nose poincare equations the method is explicit and preserves the time reversal symmetry in numerical experiments it is shown that the new method exhibits enhanced stability when the temperature fluctuation is large extensions are presented for nose chains holonomic constraints and rigid bodies

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

brief a phase plane approach to time optimal control of single dof mechanical systems with friction in this paper we attempt to solve the time optimal control problem for single degree of freedom dof mechanical systems with friction while taking into account not only the velocity dependent control input constraint but also the state constraint direct application of the pontryagin s maximum principle pmp leads to a sixth order nonlinear two point boundary value problem tpbvp which is very difficult to solve numerically in this context we take a phase plane analysis without resorting to the pmp thereby the exact time optimal solution can be obtained simply by solving a set of first order differential equations with continuous right hand sides we also present some simulation results to demonstrate the practical use of the time optimal solution

a time reversible variable stepsize integrator for constrained dynamics this article considers the design and implementation of variable timestep methods for simulating holonomically constrained mechanical systems symplectic variable stepsizes are briefly discussed and we consider time reparameterization techniques employing a time reversible symmetric integration method to solve the equations of motion we give several numerical examples including a simulation of an elastic inextensible unshearable rod undergoing large deformations and collisions with the sides of a bounding box numerical experiments indicate that adaptive stepping can significantly smooth the numerical energy and improve the overall efficiency of the simulation

on higher order semi explicit symplectic partitioned runge kutta methods for constrained hamiltonian systems in this paper we generalize the class of explicit partitioned runge kutta prk methods for separable hamiltonian systems to systems with holonomic constraints for a convenient analysis of such schemes we first generalize the backward error analysis for systems in bbb r m to systems on manifolds embedded in bbb r m by applying this analysis to constrained prk methods we prove that such methods will in general suffer from order reduction as well known for higher index differential algebraic equations however this order reduction can be avoided by a proper modification of the standard prk methods this modification increases the number of projection steps onto the constraint manifold but leaves the number of force evaluations constant we also give a numerical comparison of several second fourth and sixth order methods

computing global orbits of the forced spherical pendulum abstract orbits of the spherical pendulum with time periodic forcing are considered a numerical framework is developed which allows orbits to explore the entire globe the spherical pendulum is considered as an invariant manifold in an ambient six dimensional euclidean space the numerical integrator is the second order stormer verlet method coupled with the shake rattle algorithm the algorithm preserves numerically the phase space of the sphere which is a manifold to machine accuracy poincare sections restricted to the configuration space are used to illustrate the transition from oscillatory behavior to chaotic solutions as the amplitude of the pivot motion is changed the qualitative change in the poincare sections from regular to chaotic behavior is in excellent qualitative agreement with corresponding computations of the lyapunov exponents les the les are also computed using a novel variant of the shake rattle algorithm the results show that irregular behavior can explore the entire sphere even at low forcing amplitudes and therefore local methods which parameterize only part of the sphere are inadequate in general and may lead to spurious dynamics the numerical framework provides a tool for detailed investigation of the symmetric chaos of the forced spherical pendulum

error propagation in numerical approximations near relative equilibria we study the propagation of errors in the numerical integration of perturbations of relative equilibrium solutions of hamiltonian differential equations with symmetries first it is shown that taking an initial perturbation of a relative equilibrium the corresponding solution is related in a first approximation to another relative equilibrium with the parameters perturbed from the original then this is used to prove that for stable relative equilibria error growth with respect to the perturbed solution is in general quadratic but only linear for schemes that preserve the invariant quantities of the problem in this sense the conclusion is similar to the one obtained when integrating unperturbed relative equilibria numerical experiments illustrate the results

various closeness concepts in numerical ode s abstract we present several results pointing out that appropriate concepts of closeness between exact solutions and their discretization approximation depend on the inner structure of the underlying differential equations in many cases especially if function is not bounded the exact solution segment t x r m t 0 x is suggested to be compared witha h invariant i e overflowing invariant with respect to the one step discretization mapping h curve starting at some point j x r m properties like continuity surjectivity bijectivity and nearness to identity of the assignment x j h x are also considered

stabilizability and existence of system representations for discrete time time varying systems in this paper right and left representations as an alternate but equivalent framework to coprime factorizations of operators are developed the main theorem of the paper establishes that a linear time varying discrete time plant is stabilizable if and only if its graph can be represented as the range respectively kernel of a causal bounded operator which is left respectively right invertible the proof relies on certain factorization theorems of arveson for nest algebras the paper extends the youla parametrization of all stabilizing compensators to this framework also it is proven that a time invariant plant that is not stabilizable by a time invariant compensator is not stabilizable with a time varying compensator an example of a time varying plant of feintuch is considered and shown to be not stabilizable finally the continuous time case is examined and the problems encountered in extending the proof are discussed however it is shown that a stabilizable plant must have a closed graph

on invariant closed curves for one step methods we show that a one step method as applied to a dynamical system with a hyperbolic periodic orbit exhibits an invariant closed curve for sufficiently small step size this invariant curve converges to the periodic orbit with the order of the method and it inherits the stability of the periodic orbit the dynamics of the one step method on the invariant curve can be described by the rotation number for which we derive an asymptotic expression our results complement those of 2 3 where one step methods were shown to create invariant curves if the dynamical system has a periodic orbit which is stable in either time direction or if the system undergoes a hopf bifurcation

on the discretization of a partial differential equation in the neighborhood of a periodic orbit this article is concerned with the comparison of the dynamic of a partial differential equation and its time discretization we restrict our attention to the neighborhood of a hyperbolic periodic orbit we show that the discretization possesses an invariant closed curve near the periodic orbit and that the trajectories of the semigroups defined by the partial differential equations and its approximation are close in a sense to be precised provided that different data are allowed this answers partly an open problem posed in 4 examples of application to dissipative partial equations are provided

an averaging technique for highly oscillatory hamiltonian problems in this paper we are concerned with the numerical solution of highly oscillatory hamiltonian systems with a stiff linear part we construct an averaged system whose solution remains close to the exact one over bounded time intervals possesses the same adiabatic and hamiltonian invariants as the original system and is nonstiff we then investigate its numerical approximation through a method which combines a symplectic integration scheme and an acceleration technique for the evaluation of time averages developed in e cances et al numer math 100 2005 pp 211 232 eventually we demonstrate the efficiency of our approach on two test problems with one or several frequencies

existence of carathe spl acute odory solutions in nonlinear systems with discontinuous switching feedback controllers in this note we consider the existence of a carathe spl acute odory solution in a single input single output nonlinear system with a discontinuous switching feedback controller the main contribution is to show that if the nonlinear system can be transformed into a global normal form then we can specify the value of the discontinuous switching feedback controller on the switching hypersurface so that the closed loop system has a carathe spl acute odory solution

aspects of backward error analysis of numerical odes abstract in this paper some preliminary cases are studied to see how the idea of backward error analysis carries over to numerical solution of ordinary differential equations the interest here is in one step methods and infinite time intervals

the life span of backward error analysis for numerical integrators backward error analysis is a useful tool for the study of numerical approximations to ordinary differential equations the numerical solution is formally interpreted as the exact solution of a perturbed differential equation given as a formal and usually divergent series in powers of the step size for a rigorous analysis this series has to be truncated in this article we study the influence of this truncation to the difference between the numerical solution and the exact solution of the perturbed differential equation results on the long time behaviour of numerical solutions are obtained in this way we present applications to the numerical phase portrait near hyperbolic equilibrium points to asymptotically stable periodic orbits and hopf bifurcation and to energy conservation and approximation of invariant tori in hamiltonian systems

conservation of energy momentum and actions in numerical discretizations of non linear wave equations for classes of symplectic and symmetric time stepping methods trigonometric integrators and the stormer verlet or leapfrog method applied to spectral semi discretizations of semilinear wave equations in a weakly non linear setting it is shown that energy momentum and all harmonic actions are approximately preserved over long times for the case of interest where the cfl number is not a small parameter such results are outside the reach of standard backward error analysis here they are instead obtained via a modulated fourier expansion in time

approximate momentum conservation for spatial semidiscretizations of semilinear wave equations we prove that a standard second order finite difference uniform space discretization of the semilinear wave equation with periodic boundary conditions analytic nonlinearity and analytic initial data conserves momentum up to an error which is exponentially small in the stepsize our estimates are valid for as long as the trajectories of the full semilinear wave equation remain real analytic the method of proof is that of backward error analysis whereby we construct a modified equation which is itself lagrangian and translation invariant and therefore also conserves momentum this modified equation interpolates the semidiscrete system for all time and we prove that it remains exponentially close to the trigonometric interpolation of the semidiscrete system these properties directly imply approximate momentum conservation for the semidiscrete system we also consider discretizations that are not variational as well as discretizations on non uniform grids through numerical example as well as arguments from geometric mechanics and perturbation theory we show that such methods generically do not approximately preserve momentum

numerical integration of lie poisson systems while preserving coadjoint orbits and energy in this paper we apply geometric integrators of the rkmk type to the problem of integrating lie poisson systems numerically by using the coadjoint action of the lie group g on the dual lie algebra mbox normalsize mathfrak g to advance the numerical flow we devise methods of arbitrary order that automatically stay on the coadjoint orbits first integrals known as casimirs are retained to machine accuracy by the numerical algorithm within the proposed class of methods we find integrators that also conserve the energy these schemes are implicit and of second order nonlinear iteration in the lie algebra and linear error growth of the global error are discussed numerical experiments with the rigid body and a finite dimensional truncation of the euler equations for a two dimensional 2d incompressible fluid are used to illustrate the properties of the algorithm

on manifolds of connecting orbits in discretizations of dynamical systems it is shown that one step methods when applied to a one parametric dynamical system with a homoclinic orbit exhibit a closed loop of discrete homoclinic orbits on this loop the parameter varies periodically while the orbit shifts its index after one revolution we show that at least two homoclinic tangencies occur on this loop our approach works for systems with finite smoothness and also applies to general connecting orbits it provides an alternative to the interpolation approach by fiedler and scheurle mem amer math soc 119 570 1996 and it allows to recover some of their results on exponentially small splittings of separatrices by using some recent backward error analysis for the analytic case

two concepts for numerical periodic solutions of ode s in computing a numerical approximation for a hyperbolic periodic orbit of an ordinary differential equation one can look either for periodic points of the mapping corresponding the numerical method or for an invariant set of that mapping here the one step methods are considered it is shown that both of these concepts for numerical approximation of periodic solutions are meaningful in the sense that the corresponding sets exist and are c k circles provided the method is accurate enough also both of these are accurate to the order of the method further some relations of these concepts are studied

characterization of controllers in simultaneous stabilization this paper gives a convenient parametrization for the class of all stabilizing controllers for two or more plants the result represent a generalization of the youla parametrization of the class of all stabilizing controllers in terms of an arbitrary stable proper transfer function although as expected the additional constraints for simultaneous stabilization are not readily incorporated into h h2 optimization procedures as in the standard case there is immediate application of the theory for reduction of controllers which simultaneously stabilize two or more plants

discretization errors in molecular dynamics simulations with deterministic and stochastic thermostats we investigate the influence of numerical discretization errors on computed averages in a molecular dynamics simulation of tip4p liquid water at 300k coupled to different deterministic nose hoover and nose poincare and stochastic langevin thermostats we propose a couple of simple practical approaches to estimating such errors and taking them into account when computing the averages we show that it is possible to obtain accurate measurements of various system quantities using step sizes of up to 70 of the stability threshold of the integrator which for the system of tip4p liquid water at 300k corresponds to the step size of about 7fs

backward error analysis for numerical integrators backward error analysis has become an important tool for understanding the long time behavior of numerical integration methods this is true in particular for the integration of hamiltonian systems where backward error analysis can be used to show that a symplectic method will conserve energy over exponentially long periods of time such results are typically based on two aspects of backward error analysis i it can be shown that the modified vector fields have some qualitative properties which they share with the given problem and ii an estimate is given for the difference between the best interpolating vector field and the numerical method these aspects have been investigated recently for example by benettin and giorgilli in j statist phys 74 1994 pp 1117 1143 by hairer in ann numer math 1 1994 pp 107 132 and by hairer and lubich in numer math 76 1997 pp 441 462 in this paper we aim at providing a unifying framework and a simplification of the existing results and corresponding proofs our approach to backward error analysis is based on a simple recursive definition of the modified vector fields that does not require explicit taylor series expansion of the numerical method and the corresponding flow maps as in the above cited works as an application we discuss the long time integration of chaotic hamiltonian systems and the approximation of time averages along numerically computed trajectories

reversible long term integration with variable stepsizes the numerical integration of reversible dynamical systems is considered a backward analysis for variable stepsize one step methods is developed and it is shown that the numerical solution of a symmetric one step method implemented with a reversible stepsize strategy is formally equal to the exact solution of a perturbed differential equation which again is reversible this explains geometrical properties of the numerical flow such as the nearby preservation of invariants in a second part the efficiency of symmetric implicit runge kutta methods linear error growth when applied to integrable systems is compared with explicit nonsymmetric integrators quadratic error growth

pid control in the presence of static friction a comparison of algebraic and describing function analysis pid position control in the presence minimal coulomb friction and non zero static friction is shown to lead to a frictional limit cycle for all stabilizing combinations of p i and d parameters an idealized coulomb static friction model dimensional analysis and the study of all possible solutions of the equations of motion are used to achieve rigorous and general results for this nonlinear problem the algebraic prediction of stick slip for this system is contrasted with predictions of the describing function the predictions of the describing function are shown to be unreliable for this problem of friction and control

discretizing dynamical systems with generalized hopf bifurcations we consider the discretizations of parameter dependent continuous time dynamical systems we show that the general one step methods shift a generalized hopf bifurcation and turn it into a generalized neimark sacker point we analyze the effect of discretization methods on the emanating hopf curve in particular we obtain estimates for the eigenvalues of the discretized system along this curve a detailed analysis of the discretized first lyapunov coefficient is also given the results are illustrated by a numerical example dynamical consequences are discussed

some conservation issues for the dynamical cores of nwp and climate models the rationale for designing atmospheric numerical model dynamical cores with certain conservation properties is reviewed the conceptual difficulties associated with the multiscale nature of realistic atmospheric flow and its lack of time reversibility are highlighted a distinction is made between robust invariants which are conserved or nearly conserved in the adiabatic and frictionless limit and non robust invariants which are not conserved in the limit even though they are conserved by exactly adiabatic frictionless flow for non robust invariants a further distinction is made between processes that directly transfer some quantity from large to small scales and processes involving a cascade through a continuous range of scales such cascades may either be explicitly parameterized or handled implicitly by the dynamical core numerics accepting the implied non conservation an attempt is made to estimate the relative importance of different conservation laws it is argued that satisfactory model performance requires spurious sources of a conservable quantity to be much smaller than any true physical sources for several conservable quantities the magnitudes of the physical sources are estimated in order to provide benchmarks against which any spurious sources may be measured

unitary integration with operator splitting for weakly dissipative systems unitary integration is a numerical method that preserves the structure of the quantum liouville equation by evolving the density via unitary transformations unitary integrators preserve the kinematic invariants cj tr j j 1 n to all orders in the time step here we extend unitary integration to weakly dissipative systems we apply the technique of operator splitting using a unitary integrator for the hamiltonian evolution and a conventional integrator for the dissipative piece in this way we guarantee that all dissipation and decoherence variation of the cj is due to the new non hamiltonian terms and not to any numerical artifacts we illustrate the method with examples

the stabilization problem for discrete time varying linear systems in this paper we consider the strong stabilization problem for a time varying linear system which admits a coprime factorization we also study the simultaneous and robust stabilization problems for this class of systems

simultaneous stabilization of three or more plants conditions on the positive real axis do not suffice the problem of the simultaneous stabilizability of a finite family of single input single output time invariant systems by a time invariant controller is studied the link between stabilization and avoidance is shown and is used to derive necessary conditions for the simultaneous stabilization of k plants these necessary conditions are proved to be in general not sufficient this result also disproves a long standing conjecture on the stabilizability condition of a single plant with a stable minimum phase controller the main result is to show that unlike the case of two plants the existence of a simultaneous stabilizing controller for more than two plants is not guaranteed by the existence of a controller such that the closed loops have no real unstable poles

simultaneously stabilising controllers for time varying linear systems within the framework of nest algebras abstractfrom the perspective of strong transitivity a controller design method is provided to simultaneously stabilise a collection of time varying linear systems within the framework of nest algebras in particular all simultaneously stabilising controllers for a class of linear plants are characterised based on the doubly coprime factorisations these results hold as well in the time invariant case an illustrative example is given to demonstrate the validity of the method

estimates in discretizing normally hyperbolic compact invariant manifolds of ordinary differential equations abstract discretization methods in the vicinity of normally hyperbolic compact invariant manifolds of ordinary differential equations are consisered the existence of a discretized normally hyperboic compact invariant manifold is proved and in various metric topologies the distance between the original manifold and its discretized counterpart is estimated proofs are based on reconsidering the hirsch 1 approach and supplying the necessary estimates

on the stability of some algorithms for computing the action of the matrix exponential abstract this paper deals with the stability of some one step methods for computing the action of a matrix exponential on a vector i e e a b where a is an n by n matrix and b is a vector of dimension n the methods are based on certain assumptions made for the case that a is a matrix having sufficiently small norm if the norm of a is larger a fixed step size is used to perform a scaling of a an analysis of the roundoff error of the computed e a b shows that these methods are backward stable if a is hermitian and forward stable if for example a is skew hermitian or if a is real and essentially nonnegative and b is real and nonnegative in addition an upper bound on the forward error of the computed e a b is obtained for all a and b this bound which is expressed in terms of the condition number is evaluated for several examples of the data the results of this paper partly apply to the algorithm proposed by al mohy and higham 2011 for computing e a b

new sufficient conditions for strong simultaneous stabilization abstract we present sufficient conditions for the existence of a single stable controller to stabilize a set of n siso plants p 1 p 2 p n strong simultaneous stabilization as is well known this is equivalent to the existence of a single controller not necessarily stable to stabilize n 1 plants simultaneous stabilization the basic assumption required in this paper is that all the plants have the same unstable zeros a practical example of where such an assumption holds is when each plant has the same pure delay which is approximated by a non minimum phase rational function the results here are applied to examples of this type

a sufficient condition for simultaneous stabilization the condition under which it is possible to find a single controller that stabilizes k single input single output linear time invariant systems p sub i s i 1 k is investigated the concept of avoidance in the complex plane is introduced and used to derive a sufficient condition for k systems to be simultaneously stabilizable a method for constructing a simultaneous stabilizing controller is also provided and is illustrated by an example

what good are numerical simulations of chaotic dynamical systems abstract numerical simulations of mathematical models can suggest that the models are chaotic for example one can compute an orbit and its associated finite time lyapunov exponents and these computed exponents can be positive it is not clear how far these suggestions can be trusted because as is well known numerical methods can introduce spurious chaos or even suppress actual chaos this focused review examines the fidelity of numerical methods we look at the didactic example of the gauss map from the theory of continued fractions which allows a simple examination of backward error analysis for discrete dynamical systems and gives a clear picture of the effects of floating point arithmetic a similar use of backward error analysis in the form of defect control gives a useful understanding in the case of continuous dynamical systems finally we discuss limitations of this backward point of view

on the existence and uniqueness of solutions in adaptive control systems the issue of existence and uniqueness of solutions which arises in the analysis of adaptive control systems is investigated sufficient conditions that guarantee existence and uniqueness in positive time of solutions to differential equations describing a wide class of adaptive schemes are given the concept of filippov s solution is introduced as a method of dealing with discontinuous adaptive systems

on structural stability of ordinary differential equations with respect to discretization methods on compact m dimensional discs morse smale differential systems having no periodic orbits are considered the main result is that they are correctly reproduced by one step discretization methods for methods of order p and stepsize h sufficiently small the time h map of the induced local flow and the h discretized system are joined by a conjugacy o h p near to the identity the paper fits well in the rapidly growing list of results stating that hyperbolic transversal structures are preserved by discretization the proof relies heavily on techniques elaborated by robbin 1971 in establishing his structural stability theorem on self diffeomorphisms of compact c infty manifolds

the fractional representation approach to synthesis problems an algebraic analysis viewpoint part ii internal stabilization in this second part of the paper a quadrat siam j control optim 40 2003 pp 266 299 we show how to reformulate the fractional representation approach to synthesis problems within an algebraic analysis framework in terms of modules we give necessary and sufficient conditions for internal stabilizability moreover we characterize all the integral domains a of siso stable plants such that every mimo plant defined by means of a transfer matrix whose entries belong to the quotient field k q a of a is internally stabilizable finally we show that this algebraic analysis approach allows us to recover on the one hand the approach developed in m c smith ieee trans automat control 34 1989 pp 1005 1007 and on the other hand the ones developed in k mori and k abe siam j control optim 39 2001 pp 1952 1973 s shankar and v r sule siam j control optim 30 1992 pp 11 30 v r sule siam j control optim 32 1994 pp 1675 1695 m vidyasagar h schneider and b a francis ieee trans automat control 27 1982 pp 880 894 m vidyasagar control system synthesis a factorization approach mit press cambridge ma 1985

on the hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms we reconsider the problem of the hamiltonian interpolation of symplectic mappings following moser s scheme we prove that for any mapping e analytic and e close to the identity there exists an analytic autonomous hamiltonian system he such that its time one mapping he differs from e by a quantity exponentially small in 1 e this result is applied in particular to the problem of numerical integration of hamiltonian systems by symplectic algorithms it turns out that when using an analytic symplectic algorithm of orders to integrate a hamiltonian systemk one actually follows exactly namely within the computer roundoff error the trajectories of the interpolating hamiltonian he or equivalently of the rescaled hamiltonian ke e 1he which differs fromk but turns out to be e5 close to it special attention is devoted to numerical integration for scattering problems

on spurious fixed points of runge kutta methods in this paper we investigate the onset of spurious fixed points when runge kutta methods are applied to study the dynamics of differential equations it is shown computationally that the spurious equilibria of griffithset al 14 are connected at infinity with fixed points inherited from the differential equation we introduce and study the concept of b regularity which is in connection to the concept of regularity introduced by iserles principium cuius hinc nobis exordia sumet nullam rem e nilo gigni diuinitus umquam nam si de nilo fierent ex omnibus rebus omne genus nasci posset nil semine egeret de rerum natura liber primus 149 lucrecio

gshmc an efficient method for molecular simulation the hybrid monte carlo hmc method is a popular and rigorous method for sampling from a canonical ensemble the hmc method is based on classical molecular dynamics simulations combined with a metropolis acceptance criterion and a momentum resampling step while the hmc method completely resamples the momentum after each monte carlo step the generalized hybrid monte carlo ghmc method can be implemented with a partial momentum refreshment step this property seems desirable for keeping some of the dynamic information throughout the sampling process similar to stochastic langevin and brownian dynamics simulations it is however ultimate to the success of the ghmc method that the rejection rate in the molecular dynamics part is kept at a minimum otherwise an undesirable zitterbewegung in the monte carlo samples is observed in this paper we describe a method to achieve very low rejection rates by using a modified energy which is preserved to high order along molecular dynamics trajectories the modified energy is based on backward error results for symplectic time stepping methods the proposed generalized shadow hybrid monte carlo gshmc method is applicable to nvt as well as npt ensemble simulations

the transitivity in simultaneous stabilization this paper deals with the transmission problem in simultaneous stabilization of time varying linear systems we establish two criteria for the transitivity in simultaneous stabilization in particular we give a necessary and sufficient condition for certain linear systems to satisfy a strengthened version of transitivity

fractional representation algebraic geometry and the simultaneous stabilization problem an explicit relationship between the fractional representation approach feedback system design and the algebro geometric approach to system theory is formulated and used to derive a global solution to the feedback system problem these techniques are then applied to the simultaneous stabilization problem yielding a natural geometric criterion for a set of plants to be simultaneously stabilized by a single compensator

lyapunov stability theory of nonsmooth systems this paper develops nonsmooth lyapunov stability theory and lasalle s invariance principle for a class of nonsmooth lipschitz continuous lyapunov functions and absolutely continuous state trajectories computable tests based on filipov s differential inclusion and clarke s generalized gradient are derived the primary use of these results is in analyzing the stability of equilibria of differential equations with discontinuous right hand side such as in nonsmooth dynamic systems or variable structure control