bounds on the trace of a solution to the lyapunov equation with a general stable matrix abstract some new estimates for the eigenvalue decay rate of the lyapunov equation ax xa t b with a low rank right hand side b are derived the new bounds show that the right hand side b can greatly influence the eigenvalue decay rate of the solution this suggests a new choice of the adi parameters for the iterative solution the advantage of these new parameters is illustrated on second order damped systems with a low rank damping matrix

is gauss quadrature better than clenshaw curtis we compare the convergence behavior of gauss quadrature with that of its younger brother clenshaw curtis seven line matlab codes are presented that implement both methods and experiments show that the supposed factor of 2 advantage of gauss quadrature is rarely realized theorems are given to explain this effect first following o hara and smith in the 1960s the phenomenon is explained as a consequence of aliasing of coefficients in chebyshev expansions then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log z 1 z 1 in the complex plane gauss quadrature corresponds to pade approximation at z infty clenshaw curtis quadrature corresponds to an approximation whose order of accuracy at z infty is only half as high but which is nevertheless equally accurate near 1 1

decay estimates of functions through singular extensions of vector valued laplace transforms abstract let x be a banach space and let f l r x whose laplace transform extends analytically to some region containing i r 0 possibly having a pole at the origin in this paper we give estimates of the decay of certain slight suitable modification of f in terms of the growth of its laplace transform along the imaginary axis this technique is applied to obtain decay estimates of smooth orbits of bounded c 0 semigroups whose infinitesimal generators have an arbitrary finite boundary spectrum these results are close to those given recently by c j k batty and t duyckaerts

3 d attitude representation of human joints a standardization proposal abstract in view of the singularities asymmetries and other adverse properties of existing three dimensional definitions for joint and segment angles the present paper proposes a new convention for unambiguous and easily interpretable 3 d joint angles based on the concept of the attitude vector as derived from euler s theorem the suggested standard can be easily explained to non mathematically trained clinicians is readily implemented in software and can be simply related to classical cardanic eulerian angles for planar rotations about a coordinate system s axes the proposed convention coincides with the cardanic convention the attitude vector dispenses with the gimbal lock and non orthogonality disadvantages of cardanic eulerian conventions therefore its components have better metrical properties and they are less sensitive to measurement errors and to coordinate system uncertainties than cardanic eulerian angles a sensitivity analysis and a physical interpretation of the proposed standard are given and some experimental results that demonstrate its advantages

decay of hankel singular values of analytic control systems we show that control systems with an analytic semigroup and control and observation operators that are not too unbounded have a hankel operator that belongs to the schatten class sp for all positive p this implies that the hankel singular values converge to zero faster than any polynomial rate this in turn implies fast convergence of balanced truncations as a corollary decay rates for the eigenvalues of the controllability and observability gramians are also provided applications to the heat equation and a plate equation are given

partial models and logics programming abstract three extensions of the standard prolog fixpoint semantics are presented called sat strong and weak using partial models models which may fail to assign truth values to all formulas each of these semantics takes negation and quantification into account all thee are conservative they agree with the conventional semantics on pure horn clause programs the sat and the strong semantics incorporate the domain closure assumption but differ on whether to assign a truth value to a classically valid formula some part of which lacks a truth value the weak semantics is similar to the strong semantics but abandons the domain closure condition and consequently all programs give rise to continuous operators in this semantics for the weak semantics a sound and complete proof procedure is given based on semantics tableaus or equivalently gentzen sequents

absolute stability results in infinite dimensions we derive absolute stability results of popov and circle criterion type for infinite dimensional systems in an input output setting our results apply to feedback systems in which the linear part is the series interconnection of an input output stable linear system and an integrator and the nonlinearity satisfies a sector condition which in particular allows for saturation and deadzone effects we use the input output theory developed to derive state space results on absolute stability applying to feedback systems in which the linear part is the series interconnection of an exponentially stable well posed infinite dimensional system and an integrator

tauberian theorems and stability of one parameter semigroups the main result is the following stability theorem let t t t tzo be a bounded co semigroup on a reflexive space x denote by a the generator of t and by a a the spectrum of a if a a n ir is countable and no eigenvalue of a lies on the imaginary axis then limt boo t t x o for all x e x

a new iterative method for solving large scale lyapunov matrix equations in this paper we propose a new projection method to solve large scale continuous time lyapunov matrix equations the new approach projects the problem onto a much smaller approximation space generated as a combination of krylov subspaces in a and a 1 the reduced problem is then solved by means of a direct lyapunov scheme based on matrix factorizations the reported numerical results show the competitiveness of the new method compared to a state of the art approach based on the factorized alternating direction implicit iteration

model reduction by balanced truncation for systems with nuclear hankel operators we prove the h infinity error bounds for lyapunov balanced truncation and for optimal hankel norm approximation under the assumption that the hankel operator is nuclear this is an improvement of the result from glover curtain and partington siam j control optim 26 1998 pp 863 898 where additional assumptions were made the proof is based on convergence of the schmidt pairs of the hankel operator in a sobolev space we also give an application of this convergence theory to a numerical algorithm for model reduction by balanced truncation

a class of programs for which sldnf resolution and naf rule are complete in this paper we prove completeness of sldnf resolution and naf rule for the class of allowed and locally stratified programs satisfying the further condition of well behavedness well behavedness imposes a computational restriction on programs whose aim is to ensure that the 3 and 2 valued consequences of clark s completion of a program belonging to the above mentioned class do coincide then one can apply kunen s 3 valued completeness theorem in order to get a 2 valued completeness result

decay of singular values for infinite dimensional systems with gevrey regularity abstract we consider the decay rate of the singular values of the input map the output map and the hankel operator for a class of infinite dimensional systems this class is characterized by the control operator or the observation operator having a smoothing effect we capture this in the definition of a gevrey operator which generalizes the known concept of a gevrey vector in applications to pdes this abstract assumption on the control operator is typically satisfied when the input is multiplied by a function which is a compactly supported gevrey function in the spatial variable using the theory of polynomial approximation in particular truncated chebyshev expansions we obtain that the singular values decay exponentially in a root of the approximation dimension the power of the root depends on the order of the gevrey operator and on whether the underlying semigroup is nilpotent exponentially stable or polynomially stable

nuclearity of hankel operators for ultradifferentiable control systems nuclearity of the hankel operator is a known sufficient condition for convergence of lyapunov balanced truncations we show how a previous result on nuclearity of hankel operators of systems with an analytic semigroup can be extended to systems with a semigroup of class dp with p 1 the case p 1 being the analytic case for semigroups that are generated by a dunford schwartz spectral operator we prove that being of class dp is equivalent to being gevrey ultradifferentiable of order p we illustrate how for certain partial differential equations our results lead to an easy way of showing nuclearity of the hankel operator for a wide range of control and observation operators by considering several examples of damped beams

on the eigenvalue decay of solutions to operator lyapunov equations this paper is concerned with the eigenvalue decay of the solution to operator lyapunov equations with right hand sides of finite rank we show that the kth generalized eigenvalue decays exponentially in root k provided that the involved operator a generates an exponentially stable analytic semigroup and a is either self adjoint or diagonalizable with its eigenvalues contained in a strip around the real axis numerical experiments with discretizations of 1d and 2d pde control problems confirm this decay c 2014 elsevier b v all rights reserved

the linear quadratic control problem for infinite dimensional systems with unbounded input and output operators this paper establishes a general semigroup framework for solving quadratic control problems with infinite dimensional state space and unbounded input and output operators

on low rank approximability of solutions to high dimensional operator equations and eigenvalue problems low rank tensor approximation techniques attempt to mitigate the overwhelming complexity of linear algebra tasks arising from high dimensional applications in this work we study the low rank approximability of solutions to linear systems and eigenvalue problems on hilbert spaces although this question is central to the success of all existing solvers based on low rank tensor techniques very few of the results available so far allow to draw meaningful conclusions for higher dimensions in this work we develop a constructive framework to study low rank approximability one major assumption is that the involved linear operator admits a low rank representation with respect to the chosen tensor format a property that is known to hold in a number of applications simple conditions which are shown to hold for a fairly general problem class guarantee that our derived low rank truncation error estimates do not deteriorate as the dimensionality increases c 2015 elsevier inc all rights reserved

realisation and approximation of linear infinite dimensional systems with error bounds the class of linear infinite dimensional systems with finite dimensional inputs and outputs whose impulse response h satisfies h in l 1 cap l 2 0 infty mathbb c p times m and induces a nuclear hankel operator is said to be of nuclear type for this class of systems it is shown that balanced or output normal realisations always exist and their truncations converge to the original system in various topologies furthermore explicit l infty bounds on the transfer function errors l 1 and l 2 bounds on the impulse response errors and hilbert schmidt and nuclear bounds on the hankel operator errors are obtained these truncations also generate an approximating sequence to the optimal hankel norm approximations to the original system and various error bounds of these approximants are deduced

linear quadratic optimal actuator location in control of vibrations diffusion and many other problems governed by partial differential equations there is freedom in the choice of actuator location the actuator location should be chosen to optimize performance objectives in this paper we consider linear quadratic performance two types of cost are considered the choice depends on whether the response to the worst initial condition is to be minimized or whether the initial condition is regarded as random in practice approximations are used in controller design and thus in selection of the actuator locations the optimal cost and location of the approximating sequence should converge to the exact optimal cost and location in this work conditions for this convergence are given in the case of linear quadratic control examples are provided to illustrate that convergence may fail when these conditions are not satisfied

b convexity the analytic radon nikodym property and individual stability ofc0 semigroups abstract lett t t t 0be ac0 semigroup on a banach spacex with generatoraand growth bound assume thatx0 xis such that the local resolvent r a x0admits a bounded holomorphic extension to the right half plane re 0 we prove the following results i ifhas fourier type 1 2 then lim 0 for all 1 and ii ifhas the analytic rnp then lim 0 for all 1 and iii ifis arbitrary then weak lim 0 for all 1 and as an application we prove a tauberian theorem for the laplace transform of functions with values in ab convex banach space

the solution of the kato square root problem for second order elliptic operators on rn we prove the kato conjecture for elliptic operators on jfin more precisely we establish that the domain of the square root of a uniformly complex elliptic operator l div av with bounded measurable coefficients in ietn is the

accuracy of a ct based bone contour registration method to measure relative bone motions in the hindfoot abstract for measuring the in vivo range of motion of the hindfoot a ct based bone contour registration method ct bcm was developed to determine the three dimensional position and orientation of bones to validate this technique we hypothesized that the range of motion in the hindfoot is equally accurately measured by roentgen stereophotogrammetric analysis rsa as by the ct bcm technique tantalum bone markers were placed in the distal tibia talus and calcaneus of one cadaver specimen with a fixed lower leg the cadaveric foot was held in neutral and subsequently loaded in eight extreme positions immediately after acquiring a ct scan with the foot in a position rsa radiographs were made bone contour registration and rsa was performed helical axis parameters were calculated for talocrural and subtalar joint motion from neutral to extreme positions and between opposite extreme positions differences between ct bcm and rsa were calculated compared with rsa the ct bcm data registered an overall root mean square difference rmsd of 0 21 for rotation about the helical axis and 0 20 mm translation along the helical axis for the talocrural and subtalar joint and for all motions combined the rmsd of the position and direction of the helical axes was 3 3 mm and 2 4 respectively the latter errors were larger with smaller helical rotations the differences are similar to those reported for validated rsa and thus are not clinically relevant concluding ct bcm is an accurate and accessible alternative for studying joint motion as it does not have the risk of infection and overlapping bone markers

finite centroid and helical axis estimation from noisy landmark measurements in the study of human joint kinematics abstract recent work on joint kinematics indicates that the finite centroid centre of rotation and the finite helical axis axis of rotation screw axis twist axis are highly susceptible to measurement errors when they are experimentally determined from landmark position data this paper presents an analytical model to describe these effects under isotropic conditions for the measurement errors and for the spatial landmark distribution it appears that the position and direction errors are inversely proportional to the rotation magnitude and that they are much more error prone than the relatively well determined rotation and translation magnitudes furthermore the direction and rotation magnitude errors are inversely proportional to the landmark distribution radius and the position and translation magnitude errors are minimal if the mean position of the landmarks coincides with the centroid or helical axis for the planar centroid the use of rigid body constraints results in considerable precision improvement relative to the classical finite reuleaux method for centroid reconstruction these analytical results can be used to define suitable measurement configurations and they are used in this paper to explain experimental results on rontgenphotogrammetrically acquired in vitro wrist joint movement

the hilbert schmidt property of feedback operators in this article we prove new sufficient conditions under which the feedback operator associated with the linear quadratic regulator control design for distributed parameter systems is nuclear or hilbert schmidt examples illustrating the main results are also given

tauberian theorems for the laplace stieltjes transform let a 0 oo c be a function of locally bounded variation with a 0 0 whose laplace stieltjes transform g z fo e z da t is absolutely convergent for re z 0 let e be the singular set of g in ir and suppose that 0 e various estimates for lim supt la t g o l are obtained in particular a t g 0 as t oo if i e is null ii sup sup e aysda s oo yee t o o iii lim lim sup sup a s a t 0 610 t oo t 6 s t this result contains tauberian theorems for laplace transforms power series and dirichlet series

a note on a system theoretic approach to a conjecture by peller khrushchev based on the construction of infinite dimensional balanced realizations an alternative solution to the following inverse spectral problem is presented given a monotonically decreasing sequence of positive numbers n n 1 does there exist a hankel operator whose sequence of singular values is n n 1

riccati equations and normalized coprime factorizations for strongly stabilizable infinite dimensional systems the first part of the paper concerns the existence of strongly stabilizing solutions to the standard algebraic riccati equation for a class of infinite dimensional systems of the form a b s 1 2b d where a is dissipative and all the other operators are bounded these systems are not exponentially stabilizable and so the standard theory is not applicable the second part uses the riccati equation results to give formulas for normalized coprime factorizations over h for positive real transfer functions of the form d s 1 2b author a 1 b

finite rank adi iteration for operator lyapunov equations we give an algorithmic approach to the approximative solution of operator lyapunov equations for controllability motivated by the successfully applied alternating direction implicit adi iteration for matrix lyapunov equations we consider this method for the determination of gramian operators of infinite dimensional control systems in the case where the input space is finite dimensional this method provides approximative solutions of finite rank under the assumption of infinite time admissibility and boundedness of the semigroup we analyze convergence in several operator norms we show that under a mild assumption on the shift parameters convergence to the gramian is obtained particular emphasis is placed on systems governed by a heat equation with boundary control we present that adi iteration for the heat equation consists of solving a sequence of helmholtz equations two numerical examples are presented the first showing the benefit of adaptive finite elements and the second illustrating con

polynomial stability of the rao nakra beam with a single internal viscous damping abstract in this paper we consider the stability of the rao nakra sandwich beam equation with various boundary conditions which consists of two wave equations for the longitudinal displacements of the top and bottom layers and one euler bernoulli beam equation for the transversal displacement polynomial stability of certain orders are obtained when there is only one viscous damping acting either on the beam equation or one of the wave equations for a few special cases optimal orders are confirmed we also study the synchronization of the model with viscous damping on the transversal displacement our results reveal that the order of the polynomial decay rate is sensitive to various boundary conditions and to the damping locations

l approximation and nuclearity of delay systems abstract the l convergence of partial fraction decompositions of single input single output delay systems is studied conditions for the nuclearity of such systems are derived and examples given

balanced pod for model reduction of linear pde systems convergence theory we consider convergence analysis for a model reduction algorithm for a class of linear infinite dimensional systems the algorithm computes an approximate balanced truncation of the system using solution snapshots of specific linear infinite dimensional differential equations the algorithm is related to the proper orthogonal decomposition and it was first proposed for systems of ordinary differential equations by rowley int j bifurc chaos appl sci eng 15 3 997 1013 2005 for the convergence analysis we consider the algorithm in terms of the hankel operator of the system rather than the product of the system gramians as originally proposed by rowley for exponentially stable systems with bounded finite rank input and output operators we prove that the balanced realization can be expressed in terms of balancing modes which are related to the hankel operator the balancing modes are required to be smooth and this can cause computational difficulties for pde systems we show how this smoothness requirement can be lessened for parabolic systems and we also propose a variation of the algorithm that avoids the smoothness requirement for general systems we prove entrywise convergence of the matrices in the approximate reduced order models in both cases and present numerical results for two example pde systems

frequency domain approach for the polynomial stability of a system of partially damped wave equations in this paper we study the stability of a system of wave equations which are weakly coupled and partially damped using a frequency domain approach based on the growth of the resolvent on the imaginary axis we establish the polynomial energy decay rate for smooth initial data we show that the behavior of the system is sensitive to the arithmetic property of the ratio of the wave propagation speeds of the two equations

hankel norm approximation for well posed linear systems the sub optimal hankel norm approximation problem is solved for a well posed linear system with generating operators a b c and transfer function g satisfying some mild assumptions in the special case of the sub optimal nehari problem an explicit parameterization of all solutions is obtained in terms of the system parameters a b c and g 0

operators l1 r x and the norm continuity problem for semigroups we present a new method for constructing c0 semigroups for which properties of the resolvent of the generator and continuity properties of the semigroup in the operator norm topology are controlled simultaneously it allows us to show that a there exists a c0 semigroup which is continuous in the operator norm topology for no t 0 1 such that the resolvent of its generator has a logarithmic decay at infinity along vertical lines b there exists a c0 semigroup which is continuous in the operator norm topology for no t r such that the resolvent of its generator has a decay along vertical lines arbitrarily close to a logarithmic one these examples rule out any possibility of characterizing norm continuity of semigroups on arbitrary banach spaces in terms of resolvent norm decay on vertical lines

on the decay rate of hankel singular values and related issues this paper investigates the decay rate of the hankel singular values of linear dynamical systems this issue is of considerable interest in model reduction by means of balanced truncation for instance since the sum of the neglected singular values provides an upper bound for an appropriate norm of the approximation error the decay rate involves a new set of invariants associated with a linear system which are obtained by evaluating a modified transfer function at the poles of the system these considerations are equivalent to studying the decay rate of the eigenvalues of the product of the solutions of two lyapunov equations the related problem of determining the decay rate of the eigenvalues of the solution to one lyapunov equation will also be addressed very often these eigenvalues like the hankel singular values are rapidly decaying this fact has motivated the development of several algorithms for computing low rank approximate solutions to lyapunov equations however until now conditions assuring rapid decay have not been well understood such conditions are derived here by relating the solution to a numerically low rank cauchy matrix determined by the poles of the system bounds explaining rapid decay rates are obtained under some mild conditions

a least squares algorithm for the equiform transformation from spatial marker co ordinates the present paper describes an algorithm for estimating the translation vector and the rotation matrix of a moving body from noisy measurements on the spatial co ordinates of at least three non collinear markers a sensitivity analysis of the estimated parameters and of the helical axis is presented in terms of characteristics of the marker distribution the implementation of the proposed algorithm in a fortran subroutine is appended

on controllability of diagonal systems with one dimensional input space abstract this paper deals with diagonal systems on a hilbert state space with a one dimensional input space and a possibly unbounded control operator a priori it is not assumed that the input operator is admissible necessary and sufficient conditions for different notions of controllability such as null controllability exact controllability and approximate controllability are presented these conditions which are given in terms of the eigenvalues of the diagonal operator and in terms of the control operator are linked with the theory of interpolation in hardy spaces

infinite dimensional linear systems with unbounded control and observation a functional analytic approach abstract the object of these notes is to develop a unifying framework for the functional analytic representation of infinite dimensional linear systems with unbounded input and output operators on the basis of the general approach new results are derived on the wellposedness of feedback systems and on the linear quadratic control problem the implications of the theory for large classes of functional and partial differential equations are discussed in detail keywords representation of infinite dimensional systems semigroups boundary control feedback linear quadratic control

optimal polynomial decay of functions and operator semigroups we characterize the polynomial decay of orbits of hilbert space c 0 semigroups in resolvent terms we also show that results of the same type for general banach space semigroups and functions obtained recently in batty and duyckaerts j evol eq 8 765 780 2008 are sharp this settles a conjecture posed in batty and duyckaerts 2008

a study on damping rates and boundary feedback control in distributed parameter vibrating systems abstract the study of damping rates for distributed and boundary feedback control systems is important in the design of control mechanisms for distributed parameter systems despite the fact that many papers have been published regarding the determination of spectra and also possibly the sharpest systemwide uniform rates of energy decay many problems remain open in this paper we first mention a few examples of distributed and boundary damping then proceed to examine the simplest such possible case of boundary feedback control of the one dimensional wave equation using the wave propagation method one can see that the analysis of the spectra is quite sophisticated indeed for most cases with seemingly the best efforts one can only pin down the rates of damping in some narrow range several open questions are posed at the end of the paper which along with higher dimensional problems we hope will further motivate more studies in such decay spectral analysis

characteristic conditions of the generation of c0 semigroups in a hilbert space abstract in this paper new characteristic conditions in terms of a and the first order resolvent of a and a which assure that a generates a c 0 semigroup in a hilbert space are proposed and proved the conditions can be used to investigate the well posedness problem of non dissipative systems an example is also given to show how to use them

compactness and nuclearity of the hankel operator and internal stability of infinite dimensional state linear systems we examine the relationships between the exponential or strong stability of certain classes of regular linear systems and the compactness and nuclearity properties of the hankel operator new sufficient conditions for nuclearity are given for exponentially stable regular linear systems with an analytic semigroup

the suboptimal nehari problem for well posed linear systems we solve the suboptimal nehari problem for a transfer function that has a state space realization as a system stable input output and input output stable well posed linear system we obtain an explicit solution in terms of the state space parameters

eigenvalue decay bounds for solutions of lyapunov equations the symmetric case abstract we present two new bounds for the eigenvalues of the solutions to a class of continuous and discrete time lyapunov equations these bounds hold for lyapunov equations with symmetric coefficient matrices and right hand side matrices of low rank they reflect the fast decay of the nonincreasingly ordered eigenvalues of the solution matrix