external boundary conditions for three dimensional problems of computational aerodynamics we consider an unbounded steady state flow of viscous fluid over a three dimensional finite body or configuration of bodies for the purpose of solving this flow numerically we discretize the governing equations navier stokes on a finite difference grid prior to the discretization we obviously need to truncate the original unbounded domain by introducing an artificial computational boundary at a finite distance from the body otherwise the number of discrete variables will not be finite this artificial boundary is typically the external boundary of the domain covered by the grid the flow problem both continuous and discretized formulated on the finite computational domain is clearly subdefinite unless supplemented by some artificial boundary conditions abcs at the external computational boundary in this paper we present an innovative approach to constructing highly accurate abcs for three dimensional flow computations the approach extends our previous technique developed for the two dimensional case it employs the finite difference counterparts to calderon s pseudodifferential boundary projections calculated in the framework of the difference potentials method dpm of ryaben kii the resulting abcs appear spatially nonlocal but are particularly easy to implement along with the existing flow solvers the new boundary conditions have been successfully combined with the nasa developed production code tlns3d and used for the analysis of wing shaped configurations in subsonic and transonic flow regimes as demonstrated by the computational experiments and comparison with the standard local methods the dpm based abcs allow one to greatly reduce the size of the computational domain while still maintaining high accuracy of the numerical solution moreover they may provide for a noticeable speedup of multigrid convergence

shape estimation from shading defocus and correspondence using light field angular coherence light field cameras are quickly becoming commodity items with consumer and industrial applications they capture many nearby views simultaneously using a single image with a micro lens array thereby providing a wealth of cues for depth recovery defocus correspondence and shading in particular apart from conventional image shading one can refocus images after acquisition and shift one s viewpoint within the sub apertures of the main lens effectively obtaining multiple views we present a principled algorithm for dense depth estimation that combines defocus and correspondence metrics we then extend our analysis to the additional cue of shading using it to refine fine details in the shape by exploiting an all in focus image in which pixels are expected to exhibit angular coherence we define an optimization framework that integrates photo consistency depth consistency and shading consistency we show that combining all three sources of information defocus correspondence and shading outperforms state of the art light field depth estimation algorithms in multiple scenarios

local and global controllability for nonlinear systems a technique to study local and global controllability properties for a wide class of nonlinear systems is presented in addition to an extensive study of systems on r2 we propose as an application of our method a new criterion for global controllability of systems with polynomial drift term defined on rn in the case of linear systems the criterion reduces to the classical kalman condition a study of local controllability at the equilibrium point of the drift term of systems defined on r2 enables us not only to rederive a local controllability result derived by hermes 4 5 but also to extend this result when no assumptions are made on the boundedness of the controls

a local algorithm for the computation of image velocity via constructive interference of global fourier components a novel fourier based technique for local motion detection from image sequences is proposed in this method the instantaneous velocities of local image points are inferred directly from the global 3d fourier components of the image sequence this is done by selecting those velocities for which the superposition of the corresponding fourier gratings leads to constructive interference at the image point hence image velocities can be assigned locally even though position is computed from the phases and amplitudes of global fourier components spanning the whole image sequence that have been filtered based on the motion constraint equation reducing certain aperture effects typically arising from windowing in other methods regularization is introduced for sequences having smooth flow fields aperture effects and their effect on optic flow regularization are investigated in this context the algorithm is tested on both synthetic and real image sequences and the results are compared to those of other local methods finally we show that other motion features i e motion direction can be computed using the same algorithmic framework without requiring an intermediate representation of local velocity which is an important characteristic of the proposed method

classic and inverse compositional gauss newton in global dic today effective implementations of digital image correlation dic are based on iterative algorithms with constant linear operators a relevant idea of the classic finite element or more generally global dic fe dic solver consists in replacing the gradient of the deformed state image with that of the reference image so as to obtain a constant operator different arguments small strains small deformations equality of the two gradients close to the solution have been given in the literature to justify this approximation but none of them are fully accurate indeed the convergence of the optimization algorithm has to be investigated from its ability to produce descent directions through such a study this paper attempts to explain why this approximation works and what is its domain of validity then an inverse compositional gauss newton icgn implementation of fe dic is proposed as a cost effective and mathematically sound alternative to this approximation

necessary and sufficient condition for global controllability of planar affine nonlinear systems in this correspondence we will investigate the global controllability of general planar affine nonlinear systems and present a simple necessary and sufficient condition for global controllability the analysis is carried out in detail which is based on some basic facts in planar topology such as the jordan curve theorem and in the geometrical theory of ordinary differential equations including the poincare bendixson theorem furthermore we will also discuss the global controllability for higher dimensional affine nonlinear system with a triangular like structure

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

two dimensional transonic aerodynamic design method this paper demonstrates the capabilities of a new transonic two dimensional design method based on the simultaneous solution of multiple streamtubes coupled through the position of and pressure at the streamline interfaces this allows the specification of either the airfoil shape direct analysis mode or the airfoil surface pressure distribution inverse design mode the nonlinear system of equations is formulated in a conservative manner which guarantees the correct treatment of shocks and is solved by a rapid newton solution method viscous effects can also be included through a coupled integral boundary layer analysis the first set of results shows the effect of different far field treatments demonstrating the improvement in accuracy obtained by including the second order doublet terms in addition to the usual first order vortex term the results are also compared to those obtained by specifying straight far field streamlines corresponding to solid wall wind tunnel experiments or constant far field pressure corresponding to freejet experiments to show the sensitivity to the farfield distance in the second set of results the design method is used to design a transonic airfoil with c 1 000 at a oo 0 70 it is shown that the off design performance is improved by specifying a surface pressure distribution with a very weak shock

on the global controllability of nonlinear systems fountains recurrence and applications to hamiltonian systems in this paper a form of open local accessibility for nonlinear control systems is introduced called the continuous fountain condition subject to the conditions that i the states of a system are continuous fountains and ii one of various recurrence conditions holds it is established that the system state space is globally controllable it is shown that these controllability results imply the controllability of certain subsets of the state space of hamiltonian control systems called energy slices the relations between the notion of a fountain and i local accessibility and ii the full lie algebra rank condition for control affine systems are investigated finally the results in this paper are shown to have application to hierarchical hybrid control theory in that they give conditions for a finite analytic partition to satisfy the so called in block controllability property this is illustrated with a linear mass spring system example

domain decomposition methods for large linearly elliptic three dimensional problems abstract the idea of solving large problems using domain decomposition technique appears particularly attractive on present day large scale parallel computers but the performance of such techniques used on a parallel computer depends on both the numerical efficiency of the proposed algorithm and the efficiency of its parallel implementation the approach proposed herein splits the computational domain in unstructured subdomains of arbitrary shape and solves for unknown on the interface using the associated trace operator the steklov poincare operator on the continuous level or the schur complement matrix after a finite element discretization and a preconditioned conjugate gradient method this algorithm involves the solution of dirichlet and of neumann problems defined on each subdomain and which can be solved in parallel this method has been implemented on a cray 2 computer using multitasking and on an intel hypercube it was tested on a large scale industrial ill conditioned three dimensional linear elasticity problem which gives a fair indication of its performance in a real life environment in such situations the proposed method appears operational and competitive on both machines compared to standard techniques it yields faster results with far less memory requirements

on image gradients in digital image correlation summary in digital image correlation dic the unknown displacement field is typically identified by minimizing the linearized form of the brightness conservation equation while the minimization scheme also involves a linearization yielding a two step linearization with four implicit assumptions these assumptions become apparent by minimizing the non linear brightness conservation equation in a consistent mathematical setting yielding a one step linearization allowing a thorough study of the dic tangent operator through this analysis eight different image gradient operators are defined and the impact of these alternative image gradients on the accuracy efficiency and initial guess robustness is discussed on the basis of a number of academic examples and representative test cases the main conclusion is that for most cases the image gradient most common in literature is recommended except for cases with 1 large rotations 2 initial guess instabilities and 3 costly iterations due to other reasons e g integrated dic where a large deformation corrected mixed gradient is recommended instead copyright 2015 john wiley sons ltd

a fully integrated noise robust strategy for the identification of constitutive laws from digital images constitutive parameter identification has been greatly improved by the achievement of full field measurements in this context noise sensitivity has been shown to be of great importance it is crucial to incorporate noise sensitivity minimization in the design of robust identification procedures in this paper we investigate noise sensitivity reduction techniques for constitutive parameter identification based on finite element model updating after examining the existing strategies we propose a single step algorithm based on a mixed optical mechanical cost function the key point of this novel procedure is that no boundary conditions are needed a first example on a real case illustrates the advantages of the proposed methodology in terms of noise sensitivity a second example shows its capabilities to identify a non linear consitutive law copyright 2010 john wiley sons ltd

asymptotic stabilization of a class of smooth two dimensional systems this paper studies the asymptotic stabilizability of two dimensional control systems the class under consideration includes c infty systems that satisfy a certain genericity assumption and all real analytic systems necessary and sufficient conditions for feedback stabilization using continuous feedback and a sufficient condition for c 1 feedback stabilization are given this latter condition is given in terms of an inequality involving two indices if the direction of the inequality is changed an obstruction to c infty feedback stabilizability is obtained a subclass of polynomial systems is also studied and given complete necessary and sufficient conditions for global asymptotic stabilization using c 1 feedback

a 3d domain decomposition approach for the identification of spatially varying elastic material parameters summary the post treatment of 3d displacement fields for the identification of spatially varying elastic material parameters is a large inverse problem that remains out of reach for massive 3d structures we explore here the potential of the constitutive compatibility method for tackling such an inverse problem provided an appropriate domain decomposition technique is introduced in the method described here the statically admissible stress field that can be related through the known constitutive symmetry to the kinematic observations is sought through minimization of an objective function which measures the violation of constitutive compatibility after this stress reconstruction the local material parameters are identified with the given kinematic observations using the constitutive equation here we first adapt this method to solve 3d identification problems and then implement it within a domain decomposition framework which allows for reduced computational load when handling larger problems copyright 2015 john wiley sons ltd

global external linearization of nonlinear systems via feedback this note presents necessary conditions and sufficient conditions for an affine nonlinear system to be globally feedback equivalent to a controllable linear system over an open subset v of rn when v equals rn necessary and sufficient conditions are obtained

global 2d digital image correlation for motion estimation in a finite element framework a variational formulation and a regularized pyramidal multi grid implementation in this study the inverse problem of reconstructing the in plane 2d displacements of a monitored surface through a sequence of two dimensional digital images severely ill posed in hadamard s sense is deeply investigated a novel variational formulation is presented for the continuum 2d digital image correlation problem and critical issues such as semi coercivity and solution multiplicity are discussed by functional analysis tools in the framework of a galerkin finite element discretization of the displacement field a robust implementation for 2d digital image correlation is outlined aiming to attenuate the spurious oscillations which corrupt the deformation scenario especially when very fine meshes are adopted recourse is made to a hierarchical family of grids linked by suitable restriction and prolongation operators and defined over an image pyramid multi grid cycles are performed ascending and descending along the pyramid with only one newton iteration per level irrespective of the tolerance satisfaction as if the problem were linear at each level the conventional least square matching functional is herein enriched by a tychonoff regularization provision preserving the solution against an unstable response the algorithm is assessed on the basis of both synthetic and truly experimental image pairs copyright 2013 john wiley sons ltd

coupled problem of the inverse design and constraint optimization two dimensional inviscid compressible flow solver has been implemented validated and extended for completing inverse design based optimization problems euler equations are used for modelling basic physics the standard cell centred finite volume method has been applied with roe s approximated riemann solver muscl monotone upstream schemes for conservation laws approach and mulder limiter stratford s separation prediction method with constrained sqp sequential quadratic programming optimization is implemented to evaluate the optimum pressure distribution at strong adverse pressure gradient flow conditions a particular inverse design method is developed to complete wall surface modification till the optimum target pressure distribution is reached by means of inverse wall modification and direct algorithms the direct solver is validated and the coupled system of the inverse design method with constraint optimization is tested over the naca 65 410 airfoil

robust non local tv l 1 optical flow estimation with occlusion detection in this paper we propose a robust non local tv l 1 optical flow method with occlusion detection to address the problem of weak robustness of optical flow estimation with motion occlusion first a tv l 1 form for flow estimation is defined using a combination of the brightness constancy and gradient constancy assumptions in the data term and by varying the weight under the charbonnier function in the smoothing term second to handle the potential risk of the outlier in the flow field a general non local term is added in the tv l 1 optical flow model to engender the typical non local tv l 1 form third an occlusion detection method based on triangulation is presented to detect the occlusion regions of the sequence the proposed non local tv l 1 optical flow model is performed in a linearizing iterative scheme using improved median filtering and a coarse to fine computing strategy the results of the complex experiment indicate that the proposed method can overcome the significant influence of non rigid motion motion occlusion and large displacement motion results of experiments comparing the proposed method and existing state of the art methods by respectively using middlebury and mpi sintel database test sequences show that the proposed method has higher accuracy and better robustness

a domain coupling method for finite element digital image correlation with mechanical regularization application to multiscale measurements and parallel computing a general method is proposed to couple two sub regions analysed with finite element digital image correlation fe dic even when using a mechanical regularization r dic a lagrange multiplier is introduced to stitch both displacements fields in order to recover continuity over the full region of interest another interface unknown is introduced to ensure additionally the equilibrium of the mechanical models used for regularization as a first application the method is used to perform a single measurement from images at two different resolutions secondly the method is also extended to parallel computing in r dic the problem is formulated at the interface and solved with gmres a dedicated preconditioner is proposed to significantly accelerate convergence the resulting method is a good candidate for the analysis of large data sets copyright

further results on global controllability of planar nonlinear systems the global controllability of a class of planar affine nonlinear systems with perturbation is investigated it has been proven that the systems can still hold their global controllability under the perturbation which can be dominated by a function of state and vanishes when control input is zero

local enrichment of nurbs patches using a non intrusive coupling strategy geometric details local refinement inclusion fracture in this work we apply a non intrusive global local coupling strategy for the modelling of local phenomena in a nurbs patch the idea is to consider the nurbs patch to be enriched as the global model this results in a simple flexible strategy first the global nurbs patch remains unchanged which completely eliminates the need for costly re parametrization procedures even if the local domain is expected to evolve then easy merging of a linear nurbs code with any other existing robust codes suitable for the modelling of complex local behaviour is possible the price to pay is the number of iterations of the non intrusive solver but we show that this can be strongly reduced by means of acceleration techniques the main development for nurbs is to be able to handle non conforming geometries only slight changes in the implementation process including the setting up of suitable quadrature rules for the evaluation of the interface reaction forces are made in response to this issue a range of numerical examples in two dimensional linear elasticity are given to demonstrate the performance of the proposed methodology and its significant potential to treat any case of local enrichment in a nurbs patch simply

technical communique global nonlinear feedback stabilization and nonpeaking conditions in this paper some new results of global feedback stabilization for nonlinear control systems are obtained our results deal with the case where the system does not satisfy the nonpeaking conditions given by byrnes and isidori as an implication of the results we also illustrate that as far as feedback stabilization is concerned global exponentially minimum phase is not necessarily a stronger hypothesis than global asymptotically critically minimum phase

n dimensional controllability with n 1 controls let m be a connected real analytic n dimensional manifold f g 1 g n 1 be complete real analytic vector fields on m which are linearly independent at some point of m and u 1 u n 1 be real valued controls consider the controllability of the system dot x t f x t sum min i 1 max n 1 u i t g i x t x 0 x 0 in m necessary and sufficient conditions are given so that this system is controllable on any simply connected domain d contained in m on which g 1 g n 1 are linearly independent these conditions depend on the computation of lie brackets at those points where f g 1 g n 1 are linearly dependent

a correlation relaxation labeling framework for computing optical flow template matching from a new perspective optical flow estimation is discussed based on a model for time varying images more general than that implied by horn and schunk 1981 the emphasis is on applications where low contrast imagery nonrigid or evolving object patterns movement as well as large interframe displacements are encountered template matching is identified as having advantages over point correspondence and the gradient based approach in dealing with such applications the two fundamental uncertainties in feature matching whether template matching or feature point correspondences are discussed correlation template matching procedures are established based on likelihood measurement a method for determining optical flow is developed by combining template matching and relaxation labeling a number of candidate displacements for each template and their respective likelihood measures are determined then relaxation labeling is employed to iteratively update each candidate s likelihood by requiring smoothness within a motion field real cloud images from satellites are used to test the method

determining optical flow optical flow cannot be computed locally since only one independent measurement is available from the image sequence at a point while the flow velocity has two components a second constraint is needed a method for finding the optical flow pattern is presented which assumes that the apparent velocity of the brightness pattern varies smoothly almost everywhere in the image an iterative implementation is shown which successfully computes the optical flow for a number of synthetic image sequences the algorithm is robust in that it can handle image sequences that are quantized rather coarsely in space and time it is also insensitive to quantization of brightness levels and additive noise examples are included where the assumption of smoothness is violated at singular points or along lines in the image

nonuniform image motion estimation using the maximum a posteriori principle an iterative scheme for frame to frame motion estimation from a pair of noisy images is established the algorithm is developed by assuming that the karhunen loeve coefficients of the motion vector waveform are zero mean and gaussian random variables following the derivation of the generalized maximum likelihood gml algorithm and invoking the maximum a posteriori map criterion an iterative motion estimator is developed a linear analysis of the algorithm is presented and the convergence of the algorithm is discussed simulation experiments are performed and comparisons are made with the gml algorithm the algorithm reported by a n netravali and j d robbins 1979 and the scheme developed by k p g horn and g g schunck 1981

quantifying time varying coordination of multimodal speech signals using correlation map analysis this paper demonstrates an algorithm for computing the instantaneous correlation coefficient between two signals the algorithm is the computational engine for analyzing the time varying coordination between signals which is called correlation map analysis cma correlation is computed around any pair of points in the two input signals thus coordination can be assessed across a continuous range of temporal offsets and be detected even when changing over time due to temporal fluctuations the correlation algorithm has two major features i it is structurally similar to a tunable filter requiring only one parameter to set its cutoff frequency and sensitivity ii it can be applied either uni directionally computing correlation based only on previous samples or bi directionally computing correlation based on both previous and future samples computing instantaneous correlation for a range of time offsets between two signals produces a 2d correlation map in which correlation is characterized as

on global controllability of planar affine nonlinear systems with a singularity this paper is concerned with the global controllability of planar affine nonlinear systems with a singular point by using some recent developed methods a necessary and sufficient condition is obtained which can also be viewed as a generalization of the corresponding result in the book by nikitin s nikitin global controllability and stabilization of nonlinear systems world scientific publishing co pte ltd singapore 1994

gd aided iol input output linearisation controller for handling affine form nonlinear system with loose condition on relative degree abstractinput output linearisation iol may encounter a singularity problem when applied to the tracking control of affine form nonlinear system afns which may not have a well defined relative degree the singularity problem has occurred in the area of control for decades in this paper we incorporate the gradient dynamics gd into iol which leads to the gd aided iol method to solve the singularity problem with the proposition of the loose condition on relative degree moreover detailed theoretical analyses on tracking error bound and convergence performance of the corresponding gd aided iol controller are presented simulations and comparisons substantiate that the proposed gd aided iol method is capable of completing the tracking control task and conquering the singularity encountered in the afns

sensitivity analysis of unsteady aerodynamic loads in cascades a method for computing the effect perturbations in the shape of airfoils in a cascade have on the steady and unsteady flow through the cascade is presented first the full potential equation is used to describe the behavior of the nonlinear steady flow and the small disturbance unsteady flow through the cascade the steady flow and small disturbance unsteady flow versions of the full potential equation are then discretized on a computational grid of quadrilateral cells using a variational finite element technique the resulting discretized equations describing the nonlinear steady flow are solved using newton iteration with lower upper decomposition at each iteration similarly the discretized unsteady small disturbance equations which are linear are solved using a single lower upper decomposition next a sensitivity analysis is performed to determine the effect small changes in cascade and airfoil geometry have on the steady and unsteady flowfields the sensitivity analysis makes use of the nominal steady and unsteady flow lower upper decompositions so that no additional matrices need to be factored hence the present method is computationally very efficient

euler based inverse method for turbomachine blades part 2 three dimensional flows the aerodynamic design method for turbomachine blades reported in part 1 is extended to three dimensions in this inverse method the blade pressure loading i e pressure difference between blade upper and lower surfaces and the blade thickness distributions are prescribed and the corresponding blade camber surface is sought the inverse problem is formulated using a pressure loading boundary condition across the blade surfaces and modification of the blade geometry is achieved using the flow tangency conditions along the blade surfaces the method is demonstrated for the design of axial flow machines ranging from the subsonic to the supersonic flow regimes

linear reachability versus global stabilization in this paper the problem of whether or not linear reachability implies global stabilizability of nonlinear control systems is studied although it is not hard to imagine that the pointwise reachability does not imply the stabilizability of a nonlinear system constructing counter examples is not as trivial as one might at first think in this paper four examples are used to show why in general the reachability of a x b x everywhere does not necessarily imply the global stabilizability of the nonlinear system in any conventional sense and what additional hypotheses may be needed

improved design of subcritical and supercritical cascades using complex characteristics and boundary layer correction the method of complex characteristics and hodograph transformation for the design of shockless airfoils was extended to design supercritical cascades with high solidities and large inlet angles this capability was achieved by introducing a conformal mapping of the hodograph domain onto an ellipse and expanding the solution in terms of tchebycheff polynomials a computer code was developed based on this idea a number of airfoils designed with the code are presented various supercritical and subcritical compressor turbine and propeller sections are shown the lag entrainment method for the calculation of a turbulent boundary layer was incorporated to the inviscid design code the results of this calculation are shown for the airfoils described the elliptic conformal transformation developed to map the hodograph domain onto an ellipse can be used to generate a conformal grid in the physical domain of a cascade of airfoils with open trailing edges with a single transformation a grid generated with this transformation is shown for the korn airfoil previously announced in star as n83 24474

viscous inviscid analysis of transonic and low reynolds number airfoils a method of accurately calculating transonic and low reynolds number airfoil flows implemented in the viscous inviscid design analysis code ises is presented the euler equations are discretized on a conservative streamline grid and are strongly coupled to a two equation integral boundary layer formulation using the displacement thickness concept a transition prediction formulation of the e type is derived and incorporated into the viscous formulation the entire discrete equation set including the viscous and transition formulations is solved as a fully coupled nonlinear system by a global newton method this is a rapid and reliable method for dealing with strong viscous inviscid interactions which invariably occur in transonic and low reynolds number airfoil flows the results presented demonstrate the ability of the ises code to predict transitioning separation bubbles and their associated losses the rapid airfoil performance degradation with decreasing reynolds number is thus accurately predicted also presented is a transonic airfoil calculation involving shock induced separation showing the robustness of the global newton solution procedure good agreement with experiment is obtained further demonstrating the performance of the present integral boundary layer formulation

euler based inverse method for turbomachine blades i two dimensional cascades a three dimensio nal inverse method for the aerodynamic design of turbomachine blades using robust timemarching algorithms for the numerical solutions of the euler equations is proposed in this inverse method the circumferential mass averaged tangential velocity or the blade loading is the primary specified flow quantity and the corresponding blade geometry is sought after the presence of the blades is represented by a periodic array of discrete body forces which is included in the equations of motion a four stage runge kutta time stepping scheme is used to march a finite volume formulation of the unsteady euler equations to a steady state solution modification of the blade geometry during this time marching process is achieved using the flow tangency conditions along the blade surfaces in this paper the method is demonstrated for the design of two dimensional infinitely thin cascaded blades ranging from the subsonic to the supersonic flow regimes including cases with rotational flows and complex shock structures in the flow passage

linear controllability versus global controllability in this note the relationship between linear controllability and global controllability for a class of two dimensional nonlinear control systems x dot ax b x u has been discussed in detail it is shown that the linear controllability can imply the global controllability under some certain conditions furthermore a counterexample is constructed to answer an open problem in the paper x m hu and c martin linear reachability versus global stabilization ieee transactions on automatic control vol 44 no 6 pp 1303 1305 1999