the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

nilpotent and high order approximations of vector field systems this paper is concerned with the study of structurally stable properties associated with systems described by one or several real analytic vector fields via approximating systems which retain inf

contribution of leaky modes in the modal analysis of unbounded problems with perfectly matched layers the modal analysis of wave problems of unbounded type involves a continuous sum of radiation modes this continuum is difficult to handle mathematically and physically it can be approximated by a discrete set of leaky modes corresponding to improper modes growing to infinity perfectly matched layers pmls have been widely applied in numerical methods to efficiently simulate infinite media most often without considering a modal approach this letter aims to bring insight into the modal basis computed with pmls pmls actually enable to reveal of the contribution of leaky modes by redefining the continua two for elastodynamics discretized after pml truncation

an averaging theorem for time periodic degree zero homogeneous differential equations abstract this paper considers the stability of the differential equation dot x ex t x e x r n where x t x e is a time periodic degree zero homogeneous vector field and e 0 is a parameter it is shown that asymptotic stability of the time averaged equation implies asymptotic stability of the original system for e sufficiently small

exponential control of a rotational motion of a rigid body using quaternions a new developed technique is used to study the problem of the exponential stabilization of the rotational motion of a rigid body about a fixed axis in the body with the help of rotors in this technique the exponential asymptotic stability of this motion is proved the control moments leading to the rotational motion are obtained the conditions that ensure the exponential asymptotic stabilization of this motion are used to obtain the stabilizing control law as non linear functions of the phase coordinates of the system finally many special cases and analyzing the results of the studied problem are investigated

a krylov stability corrected coordinate stretching method to simulate wave propagation in unbounded domains the krylov subspace projection approach is a well established tool for the reducedorder modeling of dynamical systems in the time domain in this paper we address the main issues obstructing the application of this powerful approach to the time domain solution of exterior wave problems we use frequency independent perfectly matched layers to simulate the extension to infinity pure imaginary stretching functions based on zolotarev s optimal rational approximation of the square root are implemented leading to perfectly matched layers with a controlled accuracy over a complete spectral interval of interest a new krylov based solution method via stabilitycorrected operator exponents is presented which allows us to construct reduced order models roms that respect the delicate spectral properties of the original scattering problem the roms are unconditionally stable and are based on a renormalized bi lanczos algorithm we give a theoretical foundation of our method and illustrate its performance through a number of numerical examples in which we simulate two dimensional electromagnetic wave propagation in unbounded domains including a photonic waveguide example the new algorithm outperforms the conventional finitedifference time domain method for problems on large time intervals

global feedback stabilization of the angular velocity of a symmetric rigid body abstract in this paper we prove that the angular velocity equations of a symmetric rigid body can be globally stabilized by means of linear feedback when two control torques act on the body

a parallel domain decomposition algorithm of mixed element equation for second order elliptic dirichlet boundary value problem a parallel domain decomposition algorithm of schwarz type is introduced to solve the mixed finite element equation for the dirichlet boundary value problem of a second order partial differential equation of elliptic type and convergence of this algorithm is analyzed

trajectory based local approximations of ordinary differential equations the present paper introduces a new definition of local approximation for ordinary differential equations locally around an equilibrium point this definition generalizes the well known linear and homogeneous approximations the approach is based on approximating trajectories near the origin this concept of local approximation is applied to the study of local uniform asymptotic stability leading to alternative proofs for and extensions of several existing stability results

control of underactuated spacecraft with bounded inputs bounded feedback control laws are designed for stabilization and tracking of underactuated spacecraft the flat outputs of the system are computed and are used to generate reference trajectories for the tracking problem

analysis of a finite element pml approximation for the three dimensional time harmonic maxwell problem in our paper math comp 76 2007 597 614 we considered the acoustic and electromagnetic scattering problems in three spatial dimensions in particular we studied a perfectly matched layer pml approximation to an electromagnetic scattering problem we demonstrated both the solvability of the continuous pml approximations and the exponential convergence of the resulting solution to the solution of the original acoustic or electromagnetic problem as the layer increased in this paper we consider finite element approximation of the truncated pml electromagnetic scattering problem specifically we consider approximations which result from the use of nedelec edge finite elements we show that the resulting finite element problem is stable and gives rise to quasi optimal convergence when the mesh size is sufficiently small

spin axis stabilisation of underactuated rigid spacecraft under sinusoidal disturbance spin axis stabilisation of spacecraft is a problem of partial stabilisation for non linear dynamical systems in this article the analysis of spin axis stabilisation of underactuated rigid spacecraft in the presence of sinusoidal disturbances is presented by using the euler poisson form to describe the equations of motion and assuming the disturbances in three axes are decoupled with known frequencies the paper first studies the problem of the underactuated rigid axisymmetric spacecraft by applying the internal modal principle to eliminate the sinusoidal disturbance then the paper turns to the more complicated asymmetric spacecraft where the boundedness of the angular velocity for the underactuated axis is analysed in detail the paper also proves the global asymptotic stability of the closed loop systems for both axisymmetric spacecraft and asymmetric spacecraft by combining the lyapunov direct method with the lasalle s theorem the simulation results show that the proposed control law is effective in the presence of sinusoidal disturbance

new convergence estimates for multigrid algorithms in this paper new convergence estimates are proved for both symmetric and nonsymmetric multigrid algorithms applied to symmetric positive definite problems our theory relates the convergence of multigrid algorithms to a regularity and approximation parameter cap alpha epsilon 0 1 and the number of relaxations m we show that for the symmetric and nonsymmetric nu cycles the multigrid iteration converges for any positive m at a rate which deteriorates no worse than 1 cj sup 1 sup cap alpha sup sup cap alpha where j is the number of grid levels we then define a generalized nu cycle algorithm which involves exponentially increasing for example doubling the number of smoothings on successively coarser grids we show that the resulting symmetric and nonsymmetric multigrid iterations converge for any cap alpha with rates that are independent of the mesh size the theory is presented in an abstract setting which can be applied to finite element multigrid and finite difference multigrid methods

lyapunov design of stabilizing controllers for cascaded systems the design of a state feedback law for an affine nonlinear system to render a as small as possible compact neighborhood of the equilibrium of interest globally attractive is discussed following z artstein s theorem 1983 the problem can be solved by designing a so called control lyapunov function for systems which are in a cascade form a lyapunov function meeting artstein s conditions is designed assuming the knowledge of a control law stabilizing the equilibrium of the head nonlinear subsystem in particular for planar systems this gives sufficient and necessary conditions for a compact neighborhood of the equilibrium to be stabilized

feasible methods for nonconvex nonsmooth problems with applications in green communications we propose a general feasible method for nonsmooth nonconvex constrained optimization problems the algorithm is based on the inexact solution of a sequence of strongly convex optimization subproblems followed by a step size procedure key features of the scheme are i it preserves feasibility of the iterates for nonconvex problems with nonconvex constraints ii it can handle nonsmooth problems and iii it naturally leads to parallel distributed implementations we illustrate the application of the method to an open problem in green communications whereby the energy consumption in mimo multiuser interference networks is minimized subject to nonconvex quality of service constraints

comments on the stabilization of the angular velocity of a rigid body abstract this paper discusses the stabilizability by smooth feedback of the angular velocity of a rigid body first we show how to asymptotically stabilize the system with a single control then it is shown that a single control aligned with a principal axis cannot asymptotically stabilize the system finally we present some new results on stabilization by means of scalar feedback control

configuration flatness of lagrangian systems underactuated by one control lagrangian control systems that are differentially flat with flat outputs that depend only on configuration variables are said to be configuration flat we provide a complete characterization of configuration flatness for systems with n degrees of freedom and n 1 controls whose range of control forces only depends on configuration and whose lagrangian has the form of kinetic energy minus potential the method presented allows us to determine if such a system is configuration flat and if so provides a constructive method for finding all possible configuration flat outputs our characterization relates configuration flatness to riemannian geometry we illustrate the method with two examples

rate of convergence for some constraint decomposition methods for nonlinear variational inequalities some general subspace correction algorithms are proposed for a convex optimization problem over a convex constraint subset one of the nontrivial applications of the algorithms is the solving of some obstacle problems by multilevel domain decomposition and multigrid methods for domain decomposition and multigrid methods the rate of convergence for the algorithms for obstacle problems is of the same order as the rate of convergence for jump coefficient linear elliptic problems in order to analyse the convergence rate we need to decompose a finite element function into a sum of functions from the subspaces and also satisfying some constraints a special nonlinear interpolation operator is introduced for decomposing the functions

numerical linear algebra and solvability of partial differential equations it was observed long ago that the obstruction to the accurate computation of eigenvalues of large non self adjoint matrices is inherent in the problem the basic idea is that the resolvent of a highly non normal operator can be very large far away from the spectrum this leads to an easily observable fact that algorithms for locating eigenvalues will typically find some false eigenvalues these false eigenvalues also explain one of the most surprising phenomena in linear pdes namely the fact discovered by hans lewy in 1957 in berkeley that one cannot always locally solve the pde p u f almost immediately after that discovery hormander provided an explanation of lewy s example showing that almost all operators with non constanct complex valued coefficients are not locally solvable in modern language that was done by considering the essentially dual problem of existence of non propagating singularities the purpose of this article is to review this work in the context of almost eigenvalues and from the point of view of semi classical analysis

homogeneous lyapunov function for homogeneous continuous vector field the goal of this article is to provide a construction of a homogeneous lyapunov function p associated with a system of differential equations j f x x r n 1 under the hypotheses 1 f c r n vanishes at x 0 and is homogeneous 2 the zero solution of this system is locally asymptotically stable moreover the lyapunov function v x tends to infinity with 1 x i and belongs to c r 0 r n cp with p e as large as wanted as application to the theory of homogeneous systems we present two well known results of robustness in a slightly extended form and with simpler proofs

a geometric approach to feedback stabilization of nonlinear systems with drift the paper presents an approach to the construction of stabilizing feedback for strongly nonlinear systems the class of systems of interest includes systems with drift which are affine in control and which cannot be stabilized by continuous state feedback the approach is independent of the selection of a lyapunov type function but requires the solution of a nonlinear programming satisficing problem stated in terms of the logarithmic coordinates of flows as opposed to other approaches point to point steering is not required to achieve asymptotic stability instead the flow of the controlled system is required to intersect periodically a certain reachable set in the space of the logarithmic coordinates

on spurious solutions in finite element approximations of resonances in open systems in this paper we discuss problems arising when computing resonances with a finite element method in the pre asymptotic regime we detect for the one dimensional case spurious solutions in finite

the computation of resonances in open systems using a perfectly matched layer in this paper we consider the problem of computing resonances in open systems we first characterize resonances in terms of improper eigen functions of the helmholtz operator on an unbounded domain the perfectly matched layer pml technique has been successfully applied to the computation of scattering problems we shall see that the application of pml converts the resonance problem to a standard eigenvalue problem still on an infinite domain this new eigenvalue problem involves an operator which resembles the original helmholtz equation transformed by a complex shift in the coordinate system our goal will be to approximate the shifted operator first by replacing the infinite domain by a finite computational domain with a convenient boundary condition and second by applying finite elements on the computational domain we shall prove that the first of these steps leads to eigenvalue convergence to the desired resonance values which is free from spurious computational eigenvalues provided that the size of computational domain is sufficiently large the analysis of the second step is classical finally we illustrate the behavior of the method applied to numerical experiments in one and two spatial dimensions

a dual method for minimizing a nonsmooth objective over one smooth inequality constraint we consider the class of nondifferentiable convex problems which minimizes a nonsmooth convex objective over a single smooth constraint exploiting the smoothness of the feasible set and using duality we introduce a simple first order algorithm proven to globally converge to an optimal solution with a mathcal o 1 varepsilon o 1 efficiency estimate the performance of the algorithm is demonstrated by solving large instances of the convex sparse recovery problem

preconditioning in h div and applications we consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator i grad div the natural setting for such problems is in the hilbert space h div and the variational formulation is based on the inner product in h div we show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques these preconditioners are shown to be spectrally equivalent to the inverse of the operator as a consequence they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations with the number independent of the mesh discretization we describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems

pseudospectra of linear operators if a matrix or linear operator a is far from normal its eigenvalues or more generally its spectrum may have little to do with its behavior as measured by quantities such as an or exp ta more may be learned by examining the sets in the complex plane known as the pseudospectra of a defined by level curves of the norm of the resolvent zi a 1 five years ago the author published a paper that presented computed pseudospectra of thirteen highly nonnormal matrices arising in various applications since that time analogous computations have been carried out for differential and integral operators this paper a companion to the earlier one presents ten examples each chosen to illustrate one or more mathematical or physical principles

controllability and motion algorithms for underactuated lagrangian systems on lie groups we provide controllability tests and motion control algorithms for underactuated mechanical control systems on lie groups with lagrangian equal to kinetic energy examples include satellite and underwater vehicle control systems with the number of control inputs less than the dimension of the configuration space local controllability properties of these systems are characterized and two algebraic tests are derived in terms of the symmetric product and the lie bracket of the input vector fields perturbation theory is applied to compute approximate solutions for the system under small amplitude forcing in phase signals play a crucial role in achieving motion along symmetric product directions motion control algorithms are then designed to solve problems of point to point reconfiguration static interpolation and exponential stabilization we illustrate the theoretical results and the algorithms with applications to models of planar rigid bodies satellites and underwater vehicles

further comments on the stabilizability of the angular velocity a rigid body abstract we prove that the angular velocity equations can be smoothly stabilized with a single torque controller for bodies having an axis of symmetry this complements a recent result of aeyels and szafranski

boundedness properties for time varying nonlinear systems a liapunov theorem guaranteeing uniform boundedness and uniform ultimate boundedness for a time varying nonlinear system dot x t f x t t has been established the study of uniform boundedness and uniform ultimate boundedness of particular classes of time varying nonlinear systems dot x t f x t t is reduced to the study of the corresponding time invariant frozen systems dot x t f x t sigma for all sigma in mathbb r this approach is illustrated for time varying homogeneous systems with a positive order for particular classes of time varying nonhomogeneous systems and for time varying lotka volterra equations

rigid body attitude stabilization with attitude and angular rate constraints abstract in this paper a solution to the problem of rest to rest three axis attitude reorientation of a fully actuated rigid body under multiple attitude constraint zones and angular velocity limits is presented based on the unit quaternion parameterized attitude constrained zones a quadratic potential function is developed with a global minimum locating at the desired attitude and high potential closing to the constrained zones in addition to limit the magnitude of the angular velocity another logarithmic potential function is also designed using these two potential functions and sliding mode control technique a nonlinear attitude control law is obtained to guarantee asymptotic convergence of the closed loop system with consideration of attitude and angular rate constraints and external disturbances the effectiveness of the constrained attitude control method is demonstrated through numerical simulation

time varying exponential stabilization of a rigid spacecraft with two control torques rigid body models with two controls cannot be locally asymptotically stabilized by continuous feedbacks which are functions of the state only this impossibility no longer holds when the feedback is also a function of time and time varying asymptotically stabilizing feedbacks have already been proposed however due to the smoothness of the feedbacks the convergence rate is only polynomial in this paper exponential convergence is obtained by considering time varying feedbacks which are only continuous

decentralized formation control of mobile agents a unified framework abstract this paper studies the formation control problem for systems consisting of multiple mobile agents that are described by first order differential equations and second order differential equations respectively the following issues are investigated bounded input disturbance and time delay in derivation of the main results a blend of graph theoretic and system theoretic tools is employed where local potential functions and lasalle s invariant principle play central roles finally numerical examples which support the analytical results very well are also included

flatness and defect of non linear systems introductory theory and examples we introduce flat systems which are equivalent to linear ones via a special type of feedback called endogenous their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of kalman s controllability the distance to flatness is measured by a non negative integer the defect we utilize differential algebra where flatness and defect are best defined without distinguishing between input state output and other variables many realistic classes of examples are flat we treat two popular ones the crane and the car with n trailers the motion planning of which is obtained via elementary properties of plane curves the three non flat examples the simple double and variable length pendulums are borrowed from non linear physics a high frequency control strategy is proposed such that the averaged systems become flat

stabilizability of the angular velocity of a rigid body revisited abstract we prove that the jurdjevic quinn techniques for smooth stabilization can be applied to the equation of the angular velocity of a rigid body an explicit polynomial control is given for the symmetric case

convergence estimates for product iterative methods with applications to domain decomposition in this paper we consider iterative methods for the solution of symmetric positive definite problems on a space which are defined in terms of products of operators defined with respect to a number of subspaces the simplest algorithm of this sort has an error reducing operator which is the product of orthogonal projections onto the complement of the subspaces new normreduction estimates for these iterative techniques will be presented in an abstract setting applications are given for overlapping schwarz algorithms with many subregions for finite element approximation of second order elliptic problems

further remarks on strict input to state stable lyapunov functions for time varying systems we study the stability properties of a class of time varying non linear systems we assume that non strict input to state stable iss lyapunov functions for our systems are given and posit a mild persistency of excitation condition on our given lyapunov functions which guarantee the existence of strict iss lyapunov functions for our systems next we provide simple direct constructions of explicit strict iss lyapunov functions for our systems by applying an integral smoothing method we illustrate our constructions using a tracking problem for a rotating rigid body

on the stabilization in finite time of locally controllable systems by means of continuous time varying feedback law it is proven that if for any positive time t there exists an open loop control u a t depending continuously on the initial data a vanishing for a 0 and steering a small neighborhood of 0 into 0 in time t then the control system can be locally stabilized in small time by means of continuous time varying feedback law provided that the dimension of the state space is at least 4 and the strong accessibility rank condition holds

sliding mode control of underactuated multibody systems and its application to shape change control in this article we introduce an approach based on sliding mode control to design full state feedback controllers for stabilisation of underactuated non linear multibody systems we define first order sliding surfaces as a linear combination of actuated and unactuated coordinate tracking errors lyapunov stability analysis guarantees that all system trajectories reach and remain on the sliding surfaces however stability of the sliding surfaces depends on the equilibrium manifold if the system has isolated equilibrium points it is linearly controllable and asymptotic stability can be guaranteed under certain conditions otherwise the control system fails brockett s necessary condition for existence of a smooth stabilising feedback in the latter case if the total momentum is conserved the closed loop control system will be marginally stable consequently a procedure is proposed to achieve an asymptotically stable discontinuous control law through sliding surface redefinition and shape changes it i

a moving balls approximation method for a class of smooth constrained minimization problems we introduce a new algorithm for a class of smooth constrained minimization problems which is an iterative scheme that generates a sequence of feasible points that approximates the constraints set by a sequence of balls and is accordingly called the moving balls approximation algorithm mba the computational simplicity of mba which uses first order data information makes it suitable for large scale problems theoretical and computational properties of mba in its primal and dual forms are studied and convergence and global rate of convergence results are established for nonconvex and convex problems we then introduce a variant of mba that includes an active set technique and is particularly suitable for problems with a large number of constraints this variant is as simple as mba and is proven to preserve the convergence properties of mba extension of mba is also developed for a class of variational inequalities initial numerical experiments on quadratically constrained problems demonstrate the viability and performance of our methods when compared to some existing state of the art optimization methods software such as a sequential quadratic programming solver from the imsl library and the cvxopt software package for convex optimization

three dimensional perfectly matched layer for the absorption of electromagnetic waves the perfectly matched layer is a technique of free space simulation developed for solving unbounded electromagnetic problems with the finite difference time domain method referred to as pml this technique has been described in a previous paper for two dimensional problems the present paper is devoted to the three dimensional case the theory of the perfectly matched layer is generalized to three dimensions and some numerical experiments are shown to illustrate the efficiency of this new method of free space simulation

majorization minimization procedures and convergence of sqp methods for semi algebraic and tame programs in view of solving nonsmooth and nonconvex problems involving complex constraints like standard nlp problems we study general maximization minimization procedures produced by families of strongly convex subproblems using techniques from semi algebraic geometry and variational analysis in particular ojasiewicz inequality we establish the convergence of sequences generated by these types of schemes to critical points the broad applicability of this process is illustrated in the context of nlp in that case critical points coincide with kkt points when the data are semi algebraic or real analytic our method applies for instance to the study of various sequential quadratic programming sqp schemes the moving balls method the penalized sqp method and the extended sqp method under standard qualification conditions this provides to the best of our knowledge the first general convergence results for general nonlinear programming problems we emphasize the fact that unlike most works on this subjec

multigrid convergence for second order elliptic problems with smooth complex coefficients the finite element method when applied to a second order partial differential equation in divergence form can generate operators that are neither hermitian nor definite when the coefficient function is complex valued for such problems under a uniqueness assumption we prove the continuous dependence of the exact solution and its finite element approximations on data provided that the coefficients are smooth and uniformly bounded away from zero then we show that a multigrid algorithm converges once the coarse mesh size is smaller than some fixed number providing an efficient solver for computing discrete approximations numerical experiments while confirming the theory also reveal pronounced sensitivity of gauss seidel iterations on the ordering of the unknowns for certain problems

adding an integrator for the stabilization problem abstract we study the relationship between the following two properties p1 the system x f x y y v is locally asymptotically stabilizable and p2 the system x f x u is locally asymptotically stabilizable where x r n y r dayawansa martin and knowles have proved that these properties are equivalent if the dimension n 1 here using the so called control lyapunov function approach a we propose another more constructive and somewhat simpler proof of dayawansa martin and knowles s result b we show that in general p1 does not imply p2 for dimensions n larger than 1 c we prove that p2 implies p1 if some extra assumptions are added like homogeneity of the system by using the latter result recursively we obtain a sufficient condition for the local asymptotic stabilizability of systems in a triangular form

output feedback control for stabilization on se 3 this paper addresses the problem of stabilizing systems that evolve on se 3 the proposed solution consists of an output feedback controller that guarantees almost global asymptotic stability of the desired equilibrium point in the sense that the equilibrium point is stable and we have convergence for all initial conditions except for those in a nowhere dense set of measure zero the output vector is formed by the position coordinates of a collection of landmarks fixed in the environment the resulting closed loop system exhibits the following properties i the position and orientation subsystems are decoupled ii the position error is globally exponentially stable and iii the orientation error is almost globally exponentially stable results are also provided that allow one to select landmark configurations so as to control how the position and orientation of the rigid body converge to their final equilibrium values

uniform convergence of multigrid v cycle iterations for indefinite and nonsymmetric problems in this paper an analysis of a multigrid method for nonsymmetric and or indefinite elliptic problems is presented in this multigrid method various types of smoothers may be used one type of smoother considered is defined in terms of an associated symmetric problem and includes point and line jacobi and gauss seidel iterations smoothers based entirely on the original operator are also considered one smoother is based on the normal form that is the product of the operator and its transpose other smoothers studied include point and line jacobi and gauss seidel it is shown that the uniform estimates of j h bramble and j e pasciak math comp 60 1993 pp 447 471 for symmetric positive definite problems carry over to these algorithms more precisely the multigrid iteration for the nonsymmetric and or indefinite problem is shown to converge at a uniform rate provided that the coarsest grid in the multilevel iteration is sufficiently fine but not dependent on the number of multigrid levels

on the rational stability of autonomous dynamical systems applications to control chained systems in this paper several sufficient conditions for rational stability are provided we applied our conditions to show that all chained systems are rationally stabilized by holderian feedback laws in partial sense in addition we show that the backstepping techniques can be extended to rational stabilizability theory

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

global solution for the optimal feedback control of the underactuated heisenberg system we present a global solution for an optimal feedback controller of the underactuated heisenberg system or nonholonomic integrator employing a recently developed technique based on generating functions appearing in the hamilton jacobi theory we circumvent a singularity caused by under actuation to develop a nonlinear optimal feedback control in an implicitly analytical form the systematic procedure to deal with under actuation indicates that generating functions should be effective tools for solving general underactuated optimal control problems

dynamics and control of a class of underactuated mechanical systems this paper presents a theoretical framework for the dynamics and control of underactuated mechanical systems defined as systems with fewer inputs than degrees of freedom control system formulation of underactuated mechanical systems is addressed and a class of underactuated systems characterized by nonintegrable dynamics relations is identified controllability and stabilizability results are derived for this class of underactuated systems examples are included to illustrate the results these examples are of underactuated mechanical systems that are not linearly controllable or smoothly stabilizable

spectral element computation of high frequency leaky modes in three dimensional solid waveguides a numerical method is proposed to compute high frequency low leakage modes in structural waveguides surrounded by infinite solid media in order to model arbitrary shape structures a waveguide formulation is used which consists of applying to the elastodynamic equilibrium equations a space fourier transform along the waveguide axis and then a discretization method to the cross section coordinates however several numerical issues must be faced related to the unbounded nature of the cross section the number of degrees of freedom required to achieve an acceptable error in the high frequency regime as well as the number of modes to compute in this paper these issues are circumvented by applying perfectly matched layers pml along the cross section directions a high order spectral element method for the discretization of the cross section and an eigensolver shift suited for the computation of low leakage modes first computations are performed for an embedded cylindrical bar for which literature results are available the proposed pml waveguide formulation yields good agreement with literature results even in the case of weak impedance contrast its performance with high order spectral elements is assessed in terms of convergence and accuracy and compared to traditional low order finite elements then computations are performed for an embedded square bar dispersion curves exhibit strong similarities with cylinders these results show that the properties of low leakage modes observed in cylindrical bars can also occur in other types of geometry

a comparison of two multilevel iterative methods for nonsymmetric and indefinite elliptic finite element equations we describe and analyze two multilevel iterative procedures for solving linear systems arising from certain finite element discretizations of nonself adjoint and indefinite elliptic partial differential equations we prove convergence and optimal work estimate for both schemes

linearization of attitude control error dynamics some useful properties of direction cosine and quaternion attitude formulations that include specification of attitude error and the inversion of rigid body rotational dynamics are reviewed the inversion procedure is then applied to a measure of attitude error to realize a new model follower control system that exhibits linear attitude error dynamics error analyses and simulation results for spacecraft attitude control systems are presented to demonstrate the more robust performance obtainable from an exact linear error formulation over that obtained from either direction cosine or quaternion formulations with simple linear feedback control laws

the attitude control problem a general framework for the analysis of the attitude tracking control problem for a rigid body is presented a large family of globally stable control laws is obtained by using the globally nonsingular unit quaternion representation in a lyapunov function candidate whose form is motivated by the consideration of the total energy of the rigid body the controllers share the common structure of a proportional derivative feedback plus some feedforward which can be zero the model independent case the coriolis torque compensation or an adaptive compensation these controller structures are compared in terms of the requirement on the a priori model information guaranteed transient performance and robustness the global stability of the luh walker paul robot end effector controller is also analyzed in this framework

analysis of a multigrid algorithm for time harmonic maxwell equations this paper considers a multigrid algorithm suitable for efficient solution of indefinite linear systems arising from finite element discretization of time harmonic maxwell equations in particular a backslash multigrid cycle is proven to converge at rates independent of refinement level if certain indefinite block smoothers are used the method of analysis involves comparing the multigrid error reduction operator with that of a related positive definite multigrid operator this idea has previously been used in multigrid analysis of indefinite second order elliptic problems however the maxwell application involves a nonelliptic indefinite operator with the help of a few new estimates the earlier ideas can still be applied some numerical experiments with lowest order nedelec elements are also reported

orientation control on so 3 with piecewise sinusoids this paper addresses rigid body orientation control for the case that angular velocity is restricted to a class of piecewise sinusoidal signals the orientation kinematics are represented by the group of rotation matrices so 3 we present a new closed form solution of the rotation matrix kinematics and we use this solution to prove controllability we then present kinematic level orientation feedback controllers for setpoint tracking and command following

on the orientation of a rigid body using point masses in this paper we consider a new approach to study the problem of the orientation of a rigid body containing moving masses in this approach the exponential asymptotic stability of the equilibrium positions of a rigid body with the help of moving masses are investigated a non linear feedback control law that ensuring asymptotic stability of these positions is obtained as functions of the phase coordinates of the system the control forces ensuring exponential stability of these positions are obtained the general solution of the equations of controlled system is obtained and analyzed

an optimal order process for solving finite element equations a k level iterative procedure for solving the algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed the work estimate for this procedure is proportional to the number of unknowns an optimal order result general geometry is permitted for the underlying domain but the shape plays a role in the rate of convergence through elliptic regularity finally a short discussion of the use of this method for parabolic problems is presented

feedback stabilization of strongly non linear systems using the cbh formula this paper presents an approach to the construction of stabilizing feedback for strongly non linear systems the class of systems of interest includes systems with drift which are affine in the controls and which cannot be stabilized by continuous state feedback the approach is independent of the selection of a lyapunov function but requires the on line solution of a non linear programming satisficing problem stated in terms of expressions resulting from the composition of flows via the campbell baker hausdorff formula

analysis of a parallel schwarz algorithm for elliptic problems a parallel algorithm which is derived directly from the schwarz alternating algorithm with an overlapping domain decomposition for solving boundary value problems of elliptic partial differential equations is analyzed a new general analysis framework different from lions ones is established to describe the recursive relations of error functions between approximate solutions and the exact solution at the different iterative steps some abstract theories on convergence and convergent rate of the synchronously parallel algorithm are established under different conditions

a very simple sqcqp method for a class of smooth convex constrained minimization problems with nice convergence results we introduce a new and very simple algorithm for a class of smooth convex constrained minimization problems which is an iterative scheme related to sequential quadratically constrained quadratic programming methods called sequential simple quadratic method ssqm the computational simplicity of ssqm which uses first order information makes it suitable for large scale problems theoretical results under standard assumptions are given proving that the whole sequence built by the algorithm converges to a solution and becomes feasible after a finite number of iterations when in addition the objective function is strongly convex then asymptotic linear rate of convergence is established

a lie backlund approach to equivalence and flatness of nonlinear systems a new system equivalence relation using the framework of differential geometry of jets and prolongations of infinite order is studied in this setting two systems are said to be equivalent if any variable of one system may be expressed as a function of the variables of the other system and of a finite number of their time derivatives this is a lie backlund isomorphism the authors prove that although the state dimension is not preserved the number of input channels is kept fixed they also prove that a lie backlund isomorphism can be realized by an endogenous feedback the differentially flat nonlinear systems introduced by the authors 1992 via differential algebraic techniques are generalized and the new notion of orbitally flat systems is defined they correspond to systems which are equivalent to a trivial one with time preservation or not the endogenous linearizing feedback is explicitly computed in the case of the vtol aircraft to track given reference trajectories with stability

analysis of a cartesian pml approximation to acoustic scattering problems in r2 abstract in this paper we consider a cartesian pml approximation to solutions of acoustic scattering problems on an unbounded domain in r 2 the perfectly matched layer pml technique in a curvilinear coordinate system has been researched for acoustic scattering applications both in theory and computation our goal will be to extend the results of spherical cylindrical pml to pml in cartesian coordinates that is the well posedness of cartesian pml approximation on both the unbounded and truncated domains the exponential convergence of approximate solutions as a function of domain size is also shown we note that once the stability and convergence of the continuous truncated problem has been achieved the analysis of the resulting finite element approximations is then classical finally the results of numerical computations illustrating the theory and efficiency of the cartesian pml approach will be given

the analysis of pmhss multigrid methods for elliptic problems with smooth complex coefficients in this paper we investigate a class of multigrid methods with hss smoothers applied to second order elliptic problems whose dominant coefficient is complex valued these smoothers are based on hermitian skew hermitian splitting hss modified hss mhss and preconditioned mhss pmhss iterations it is shown that the abstract theory for multigrid method can be extended and applied to elliptic problems with complex coefficient under some assumption conditions we prove that multigrid method with pmhss smoother is convergent and optimal numerical results are reported to confirm the theoretical analysis