canonical forms for linear differential algebraic equations with variable coefficients abstract we give a new set of local characterizing quantities for the treatment of linear differential algebraic equations with variable coefficients this leads to new global invariances under which we can find a generalization of the global or differentiation index and get a new existence and uniqueness theorem

some new results concerning index 3 differential algebraic equations on introduit la notion de tractabilite d indice 3 on donne un critere matriciel pour tester cette tractabilite pour l equation differentielle algebrique f x t x t t 0

singularly perturbed differential algebraic equations i asymptotic expansion of outer solutions abstract singularly perturbed nonlinear differential algebraic equations dae s are considered which are decomposed into two auxiliary problems called the outer and inner problems respectively the structure of solutions of the singularly perturbed dae s is determined by the outer and inner solutions both of which are proved to exist asymptotic expansions for outer and inner solutions are obtained and proved to be uniformly convergent this generalizes known results about asymptotic expansions of singularly perturbed ordinary differential equations the presentation of this work is separated into two parts because of the limitation of space the first part concerns the derivation of outer and inner problem and the existence and asymptotic expansion of outer solutions while the second part mainly focuses on the inner problem

multi multiplier ambient space formulations of constrained dynamical systems with an application to elastodynamics various formulations of the equations of motion for both finite and infinite dimensional constrained lagrangian dynamical systems are studied the different formulations correspond to different ways of enforcing constraints through multiplier fields all the formulations considered are posed on ambient spaces whose members are unrestricted by the need to satisfy constraint equations but each formulation is shown to possess an invariant set on which the constraint equations and physical balance laws are satisfied the stability properties of the invariant set within its ambient space are shown to be different in each case we use the specific model problem of linearized incompressible elastodynamics to compare properties of three different ambient space formulations we establish the well posedness of one formulation in the particular case of a homogeneous isotropic body subject to specified tractions on its boundary

differential algebraic systems as differential equations on manifolds based on the theory of differential equations on manifolds existence and uniqueness results are proved for a class of mixed systems of differential and algebraic equations as they occur in various applications both the autonomous and nonautonomous case are considered moreover a class of algebraically incomplete systems is introduced for which existence and uniqueness results only hold on certain lower dimensional manifolds this class includes systems for which the application of ode solvers is known to lead to difficulties finally some solution approach based on continuation techniques is outlined

a stability result for mean width of lp centroid bodies we give a different proof of a recent result of klartag b klartag a central limit theorem for convex sets invent math 168 1 2007 91 131 concerning the concentration of the volume of a convex body within a thin euclidean shell and proving a conjecture of anttila ball and perissinaki m anttila k ball i perissinaki the central limit problem for convex bodies trans amer math soc 355 12 2003 4723 4735 it is based on the study of the lp centroid bodies we prove an almost isometric reverse holder inequality for their mean width and a refined form of a stability result

numerical solution of differential algebraic equations for constrained mechanical motion the two most popular formulations of the equations of constrained mechanical motion thedescriptor andstate space forms each have severe practical limitations in this paper we discuss and relate some proposed reformulations of the equations which have improved numerical properties

a geometric treatment of implicit differential algebraic equations abstract a differential geometric approach for proving the existence and uniqueness of implicit differential algebraic equations is presented it provides for a significant improvement of an earlier theory developed by the authors as well as for a completely intrinsic definition of the index of such problems the differential algebraic equation is transformed into an explicit ordinary differential equation by a reduction process that can be abstractly defined for specific submanifolds of tangent bundles here called reducible submanifolds local existence and uniqueness results for differential algebraic equations then follow directly from the final stage of this reduction by means of an application of the standard theory of ordinary differential equations

review paper efficient numerical methods for the solution of stiff initial value problems and differential algebraic equations in recent years after a prolonged period of intense activity the study of numerical methods for solving stiff initial value problems for ordinary differential equations and differential algebraic equations has reached a certain maturity there now exist some excellent codes which are both efficient and reliable for solving these particular classes of problems in this paper we sketch some of the main theory which underpins stiff integration methods and we use this to describe and put into context some of the best codes currently available by referencing only codes which have been thoroughly tested and are widely available our aim is to direct users of numerical software to those codes which they should try initially if faced with the problem of solving ordinary differential equations of this type an additional feature is that the codes which we propose serve as benchmarks against which any new codes can be evaluated

approximation of spectral intervals and leading directions for differential algebraic equation via smooth singular value decompositions this paper is devoted to the numerical approximation of lyapunov and sacker sell spectral intervals for linear differential algebraic equations daes the spectral analysis for daes is improved and the concepts of leading directions and solution subspaces associated with spectral intervals are extended to daes numerical methods based on smooth singular value decompositions are introduced for computing all or only some spectral intervals and their associated leading directions the numerical algorithms as well as implementation issues are discussed in detail and numerical examples are presented to illustrate the theoretical results

random points and lattice points in convex bodies assume k subset of r d is a convex body and x is a large finite subset of k how many convex polytopes are there whose vertices belong to x is there a typical shape of such polytopes how well does the maximal such polytope which is actually the convex hull of x approximate k we are interested in these questions mainly in two cases the first is when x is a random sample of n uniform independent points from k in this case motivation comes from sylvester s famous four point problem and from the theory of random polytopes the second case is when x k boolean and z d where z d is the lattice of integer points in r d and the questions come from integer programming and geometry of numbers surprisingly or not so surprisingly the answers in the two cases are rather similar

mechanical systems subject to holonomic constraints differential algebraic formulations and conservative integration abstract the numerical integration in time of the equations of motion for mechanical systems subject to holonomic constraints is considered schemes are introduced for the direct treatment of a differential algebraic form of the equations that preserve the constraints the total energy and other integrals such as linear and angular momentum arising from affine symmetries moreover the schemes can be shown to preserve the property of time reversibility in an appropriate sense an example is given to illustrate various aspects of the proposed methods

solvability of general differential algebraic equations in the last few years there has been considerable research on differential algebraic equations daes f t x x 0 where f x is identically singular most of this effort has focused on computing a solution that is assumed to exist that is the dae is assumed solvable more recently there have been existence results developed using differential geometry for complex higher index systems these characterizations can be hard to verify in practice in this paper the computational verification of solvability is investigated this first requires developing an alternative set of sufficient conditions for solvability which are more amenable to computation verification of these conditions using readily available numerical and symbolic software is then discussed an example from robotics where classical graph theoretical approaches give an incorrect answer is worked to illustrate the usefulness of the sufficient condition and the computational approach

daes arising from traveling wave solutions of pdes ii abstract the study of traveling wave solutions of pdes sometimes leads to systems of differential algebraic equations daes this paper examines a family of daes that arise in this manner from the magnetohydrodynamics mhd equations these daes are of interest in their own right as a source of test problems for dae numerical integrators and because of their relationship to the mhd equations

numerical solution of hybrid systems of differential algebraic equations abstract we present a mathematical framework for general over and under determined hybrid switched systems of differential algebraic equations hdaes we give a systematic formulation of hdaes and discuss existence and uniqueness of solutions the treatment of the switch points and how to perform consistent initialization at switch points we show how numerical solution methods for daes can be adapted for hdaes and present a numerical results for these methods for the real world example of simulating an automatic gearbox

feedback design for regularizing descriptor systems this paper surveys numerical techniques for the regularization of descriptor generalized state space systems by proportional and derivative feedback we review generalizations of controllability and observability to descriptor systems along with definitions of regularity and index in terms of the weierstras canonical form three condensed forms display the controllability and observability properties of a descriptor system the condensed forms are obtained through orthogonal equivalence transformations and rank decisions so they may be computed by numerically stable algorithms in addition the condensed forms display whether a descriptor system is regularizable i e when the system pencil can be made to be regular by derivative and or proportional output feedback and if so what index can be achieved also included is a a new characterization of descriptor systems that can be made to be regular with index 1 by proportional and derivative output feedback

constraint preserving integrators for general nonlinear higher index daes in the last few years there has been considerable research on numerical methods for differential algebraic equations daes f x x t 0 where f x is identically singular the index provides one measure of the singularity of a dae most of the numerical analysis literature on daes to date has dealt with daes with indices no larger than three even in this case the systems were often assumed to have a special structure recently a numerical method was proposed that could in principle be used to integrate general unstructured higher index solvable daes however that method did not preserve constraints this paper will discuss a modification of that approach which can be used to design constraint preserving integrators for general nonlinear higher index daes

index determination and calculationof consistent initial values for daes the definition of the index of daes with properly stated leading term is based on a matrix sequence with suitably chosen projectors a possible way of determining of this matrix sequence is presented it includes the calculation of suitable projectors using generalized inverses with the help of the matrix sequence elements matrices projectors we calculate consistent initial values for daes up to index 2 by describing the hidden constraint examples are given

one step collocation methods for differential algebraic systems of index 1 abstract in this paper we propose one step collocation methods for linear differential algebraic equation dae systems of index 1 the proposed methods give a continuous approximate solution on the integration interval t0 tn we study the uniform convergence properties of the proposed collocation methods some of the methods show an order reduction phenomenon at the nodes similar to that observed for runge kutta methods petzold 1986 since dae systems may be considered as a kind of very stiff differential systems the collocation methods for dae systems have to verify strong stability properties for example collocation methods at radau points are particularly suitable for solving dae systems however the numerical results will also show that gauss legendre collocation methods may be applied successfully to dae systems also if these are only a stable methods

regular solutions of nonlinear differential algebraic equations and their numerical determination for a general class of nonlinear possibly higher index differential algebraic equations we show existence and uniqueness of solutions these solutions are regular in the sense that newton s method will converge locally and quadratically on the basis of the presented theoretical results numerical methods for the determination of consistent initial values and for the computation of regular solutions are developed several numerical examples are included

symmetric collocation for unstructured nonlinear differential algebraic equations of arbitrary index we examine a class of symmetric collocation schemes for the solution of nonlinear boundary value problems for unstructured nonlinear systems of differential algebraic equations with arbitrary index we show that these schemes converge with the same orders as one would expect for ordinary differential equations in particular we show superconvergence for a special choice of the collocation points we demonstrate the efficiency of the new approach with some numerical examples

punch stretching of sheet metal and differential equations on manifolds the goal of mathematical modelling of sheet metal forming processes is to provide predictive tools for use in the design of stamping processes and the selection of sheet materials most current approaches to finite element modelling of large deformation elastic plastic sheet metal forming problems use a rate form of the virtual work equilibrium equations and a single field finite element representation of the displacement components called the incremental method this approach does not produce approximations which satisfy the discrete equilibrium equations at all times and consequently it demands small time steps to insure stability and numerical accuracy this paper describes a variant of the mixed method in which displacements stresses effective strain and pressures are all given separate finite element representations the equilibrium equations in non rate form are discretized to produce a system of algebraic equations which are coupled with the constitutive equations and then integrated using state of the art numerical software when used to model rate sensitive sheet materials in hydrostatic bulging plane strain punch stretching and hemispherical punch stretching the new approach proved to be between 6 and 26 times as fast as the old incremental method

rosenbrook methods for differential algebraic equations this paper deals with the numerical solution of differential algebraic equations dae of index one it begins with the development of a general theory on the taylor expansion for the exact solutions of these problems which extends the well known theory of butcher for first order ordinary differential equations to dae s of index one as an application we obtain butcher type results for rosenbrock methods applied to dae s of index one we characterize numerical methods as applications of certain sets of trees we derive convergent embedded methods of order 4 3 which require 4 or 5 evaluations of the functions 1 evaluation of the jacobian and 1 lu factorization per step

one step and extrapolation methods for differential algebraic systems the paper analyzes one step methods for differential algebraic equations dae in terms of convergence order in view of extrapolation methods certain perturbed asymptotic expansions are shown to hold for the special dae extrapolation solver based on the semi implicit euler discretization the perturbed order pattern of the extrapolation tableau is derived in detail the theoretical results lead to modifications of the known code the efficiency of the modifications is illustrated by numerical comparisons over critical examples mainly from chemical combustion

manpak a set of algorithms for computations on implicitly defined manifolds abstract mathematical models often involve differentiable manifolds that are implicitly defined as the solution sets of systems of nonlinear equations the resulting computational tasks differ considerably from those arising for manifolds defined in parametric form here a collection of algorithms is presented for performing a range of essential tasks on general implicitly specified submanifolds of a finite dimensional space this includes algorithms for determining local parametrizations and their derivatives and for evaluating quantities related to the curvature with sensitivity measures the methods have been implemented as a fortran 77 package called manpak

on the discretization error of parametrized nonlinear equations many applications lead to nonlinear parameter dependent equations h y t y 0 where h y times t to y y 0 in operatorname rge h and the state space y is infinite dimensional while the parameter space t has finite dimension the case dim t 1 is of special interest in connection with continuation methods for this case a general theory is developed which provides for the existence of solution paths of a rather general class of such equations and of their finite dimensional approximations and which allows for an assessment of the error between these paths a principal tool in this analysis is the theory of nonlinear fredholm operators the results cover a more general class of operators than the mildly nonlinear mappings to which other approaches appear to be restricted

feedback stabilization of control systems described by a class of nonlinear differential algebraic equations a theoretical framework for the stabilization of con trol systems defined by a class of nonlinear differential alge braic equations is established assumptions are stated which guarantee that the nonlinear differential algebraic equations can be described by a nonlinear control system defined on a smooth manifold a procedure for obtaining a local state realization is developed conditions for local stabilization of a single equilibrium solution including one set of conditions which can easily be checked using standard computations are indicated

a nonlinear truck model and its treatment as a multibody system abstract a planar vertical truck model with nonlinear suspension and its multibody system formulation are presented the equations of motion of the model form a system of differential algebraic equations daes all equations are given explicitly including a complete set of parameter values consistent initial values and a sample road excitation thus the truck model allows various investigations of the specific dae effects and represents a test problem for algorithms in control theory mechanics of multibody systems and numerical analysis several numerical tests show the properties of the model

desingularization of implicit analytic differential equations the question of finding solutions to a given implicit differential equation ide is an important one in part because it appears very naturally in several problems in physics engineering and many other fields in this work we show how to reduce a given analytic ide to an analytic ide of locally constant rank this can be done by using some fundamental results on subanalytic subsets and desingularization of closed subanalytic subsets an example from nonholonomic mechanics is studied using these methods

automatic integration of euler lagrange equations with constraints abstract numerical difficulties in the integration of euler lagrange and similar equations are discussed a technique for reducing their index from three to two is introduced and it is shown that variable order variable step bdf methods converge for these index two problems the practical application of this reduction in a numerical setting is examined

on the analysis of laterally vibrating slender beams subject to various complicating effects the rayleigh ritz method with orthogonally generated polynomials as admissible functions is used to obtain a simple analysis of the free vibration problem of slender beams subject to various complicating effects these effects are the presence of concentrated masses and or concentrated moments of inertia intermediate supports axial loading and non uniformity of cross section such may exist separately or in any combination numerical results are presented for several example beams illustrating the accuracy and versatility of the approach

stability results for the volume of random simplices it is known that for a convex body k in r d of volume one the expected volume of random simplices in k is minimized if k is an ellipsoid and for d 2 maximized if k is a triangle here we provide corresponding stability estimates

nonstandard singular perturbation systems and higher index differential algebraic systems in order to obtain trajectory approximation results for a given singular perturbation system sps two systems are derived from it the slow and the fast one tikhonov s theorem gives sufficient conditions on them to ensure a good approximation for a standard sps i e its corresponding slow system is a differential algebraic system das of index 1 in this paper it is shown that a nonstandard sps with the parameter set to zero can be seen as a das of higher index this connection allows us to obtain a tikhonov s theorem when this das is of index 2

associating matrix pencils to generalized linear multivariable systems we consider quadruples of matrices e a b c representing generalized linear multivariable systems ex t ax t bu t y t cx t with e a square matrices and b c rectangular matrices we characterize equivalent quadruples by associating matrix pencils to them with respect to the equivalence relation corresponding to standard transformations basis changes for the state control and output spaces state feedback derivative feedback and output injection equivalent quadruples are those whose associated matrix pencils are simultaneously equivalent

a note on transverse vibrations of continuous beams subject to an axial force and carrying concentrated masses abstract polynomial co ordinate functions are used in order to obtain an approximate solution to the title problem the algorithmic procedure is quite simple leads to adequate engineering accuracy and can be easily extended to the case of several intermediate supports

the application of rosenbrock wanner type methods with stepsize control in differential algebraic equations two rosenbrock wanner type methods for the numerical treatment of differential algebraic equations are presented both methods possess a stepsize control and an index 1 monitor the first method dae34 is of order 3 4 and uses a full semi implicit rosenbrock wanner scheme the second method rkf4da is derived from the runge kutta fehlberg 4 5 pair where a semi implicit rosenbrock wanner method is embedded in order to solve the nonlinear equations the performance of both methods is discussed in artificial test problems and in technical applications

stability of the blaschke santalo and the affine isoperimetric inequality abstract a stability version of the blaschke santalo inequality and the affine isoperimetric inequality for convex bodies of dimension n 3 is proved the first step is the reduction to the case when the convex body is o symmetric and has axial rotational symmetry this step works for related inequalities compatible with steiner symmetrization secondly for these convex bodies a stability version of the characterization of ellipsoids by the fact that each hyperplane section is centrally symmetric is established

rosenbrock type methods adapted to differential algebraic systems we consider the numerical solution of differential algebraic systems of index one given in kronecker canonical form the methods described here are derived from the rosenbrock approach hence they do not require the solution of nonlinear systems of equations but one evaluation of the jacobian and one lu decomposition per step by construction the s stage method coincides with a solver for nonlinear equations of order s 1 if the stepsize is set to zero in this sense the adaptation to differential algebraic equations is performed the special structure of the method leads to simplified order conditions and to an easy implementation some particular methods up to order 4 are given especially an embedded 4 stage method of order 4 3 is derived

convergence results for a coordinate projection method applied to mechanical systems with algebraic constraints the equations of motion of mechanical multibody systems with algebraic constraints are of index 3 and therefore not directly solvable by standard ode or dae methods reducing the index by different

approximation methods for the consistent initialization of differential algebraic equations the algebraic constraints in a system of differential algebraic equations daes impose a consistency requirement on the initial values that can be difficult to satisfy in this thesis the consistency requirement is characterized by a system of equations an approximation method is introduced for these equations and the numerical solution of the resulting system is analyzed for certain important classes of daes finally some numerical experiments are described

completions of nonlinear dae flows based on index reduction techniques and their stabilization differential algebraic equations daes define a differential equation on a manifold a number of ways have been developed to numerically solve some classes of daes motivated by problems in control theory numerical simulation and the use of general purpose modeling environments recent research has considered the embedding of the dae solutions of a general dae into the solutions of an ode where the added dynamics have special properties this paper both provides new results on the linear time varying case and considers the important nonlinear case

least squares completions for nonlinear differential algebraic equations a method has been proposed for numerically solving lower dimensional nonlinear higher index differential algebraic equations for which more classical methods such as backward differentiation or implicit runge kutta may not be appropriate this method is based on solving nonlinear dae derivative arrays using nonlinear singular least squares methods the theoretical foundations generality and limitations of this approach remain to be determined this paper carefully examines several key aspects of this approach the emphasis is on general results rather than specific results based on the structure of various applications

vibration of beams with generally restrained boundary conditions using fourier series this work presents a method to find accurate vibration frequencies of beams with generally restrained boundary conditions using fourier series the suggested method is very convenient to find an accurate frequency parameter for beams with not only classical boundary conditions but also non classical boundary conditions restrained by rotational and translational springs numerical results for various degenerate cases are compared with existing results for natural frequency obtained by conventional analysis the results indicate that the present method can be used very effectively in the design of beams with various supporting conditions

linearly implicit extrapolation methods for differential algebraic systems we consider the numerical solution of implicit differential equations in which the solution derivative appears multiplied by a solution dependent singular matrix we study extrapolation methods based on two linearly implicit euler discretizations their error behaviour is explained by perturbed asymptotic expansions

multiple shooting for unstructured nonlinear differential algebraic equations of arbitrary index we study multiple shooting methods for the numerical solution of nonlinear boundary value problems for unstructured nonlinear systems of differential algebraic equations with arbitrary index we give a convergence analysis and demonstrate the results with some numerical examples

a new look at pencils of matrix valued functions abstract matrix pencils depending on a parameter and their canonical forms under equivalence are discussed the study of matrix pencils or generalized eigenvalue problems is often motivated by applications from linear differential algebraic equations daes based on the weierstrass kronecker canonical form of the underlying matrix pencil one gets existence and uniqueness results for linear constant coefficients daes in order to study the solution behavior of linear daes with variable coefficients one has to look at new types of equivalence transformations this then leads to new canonical forms and new invariances for pencils of matrix valued functions we give a survey of recent results for square pencils and extend these results to nonsquare pencils furthermore we partially extend the results for canonical forms of hermitian pencils and give new canonicalforms there too based on these results we obtain new existence and uniqueness theorems for differential algebraic systems which generalize the classical results of weierstrass and kronecker

regularization of linear descriptor systems with variable coefficients we study linear descriptor control systems with rectangular variable coefficient matrices we introduce condensed forms for such systems under equivalence transformations and use these forms to detect whether the system can be transformed to a uniquely solvable closed loop system via state or derivative feedback we show that under some mild assumptions every such system consists of an underlying square subsystem that behaves essentially like a standard state space system plus some solution components that are constrained to be zero

symmetric collocation methods for linear differential algebraic boundary value problems we present symmetric collocation methods for linear differential algebraic boundary value problems without restrictions on the index or the structure of the differential algebraic equation in particular we do not require a separation into differential and algebraic solution components instead we use the splitting into differential and algebraic equations which arises naturally by index reduction techniques and apply gaus type for the differential part and lobatto type for the algebraic part collocation schemes to obtain a symmetric method which guarantees consistent approximations at the mesh points under standard assumptions we show solvability and stability of the discrete problem and determine its order of convergence moreover we show superconvergence when using the combination of gaus and lobatto schemes and discuss the application of interpolation to reduce the number of function evaluations finally we present some numerical comparisons to show the reliability and efficiency of the new methods

backward differentiation approximations of nonlinear differential algebraic systems finite difference approximations of dynamical systems modeled by nonlinear semiexplicit differential algebraic equations are analyzed convergence for the backward differentiation method is proved for index two and index three problems when the numerical initial values obey certain constraints the appropriate asymptotic convergence rates and the leading error terms are determined

a new software package for linear differential algebraic equations we describe the new software package gelda for the numerical solution of linear differential algebraic equations with variable coefficients the implementation is based on the new discretization scheme introduced in p kunkel and v mehrmann siam j numer anal 33 1996 pp 1941 1961 it can deal with systems of arbitrary index and with systems that do not have unique solutions or inconsistencies in the initial values or the inhomogeneity the package includes a computation of all the local invariants of the system a regularization procedure and an index reduction scheme and it can be combined with every solution method for standard index 1 systems nonuniqueness and inconsistencies are treated in a least square sense we give a brief survey of the theoretical analysis of linear differential algebraic equations with variable coefficients and discuss the algorithms used in gelda furthermore we include a series of numerical examples as well as comparisons with results from other codes as far as this is possible

the additional dynamics of least squares completions for linear differential algebraic equations abstract several approaches have been proposed for numerically solving lower dimensional nonlinear higher index differential algebraic equations daes for which more classical numerical methods such as backward differentiation or implicit runge kutta may not be appropriate one of these approaches is called explicit integration ei this approach is based on solving nonlinear dae derivative arrays using nonlinear singular least squares methods this results in a computed ode called the least squares completion whose solutions contain those of the original dae this ode is then integrated by a classical numerical method the additional dynamics of the least squares completion can affect the numerical solution of the dae this paper begins the study of determining these extra dynamics

a note on bifurcation points for index 1 differential algebraic problems the arclength continuation method is proposed for index 1 daes with singular points in particular the cusp points the tangency points and equilibrium points are investigated the numerical iterative matrix is studied at a singular point the numerical examples are given to illustrate the robustness of the continuation method

an efficient approach for the numerical simulation of multibody systems the field of kinematics and dynamics of mechanical systems has progressed from a manual graphics art to a highly developed discipline in analytical geometry and dynamics various general purpose formulations for the dynamic analysis of constrained mechanical systems cms lead to mixed differential algebraic equations daes called the euler lagrange equations during the past fifteen years many contributions have been made to the theory of computational kinematics and dynamics of cms also called mutibody dynamics the recent advances in computer hardware and software have tremendously revolutionized the analysis of cms there are various numerical approaches for solving general vector fields in the previous paper appl math comput 92 1998 153 193 a complete and detailed analysis of various approaches for the numerical solution of vector fields is given in this paper we extend the algorithms presented in the above mentioned reference to solve the euler lagrange equations of motion for cms the numerical experiments suggest that perturbation approach performs better than the other approaches

symmetries and infinitesimal symmetries of singular differential equations the aim of this paper is to study symmetries of linearly singular differential equations namely equations that cannot be written in normal form because the derivatives are multiplied by a singular linear operator the concept of geometric symmetry of a linearly singular differential equation is introduced as a transformation that preserves the geometric data that define the problem it is proved that such symmetries are essentially equivalent to dynamic symmetries that is transformations mapping solutions into solutions similar results are given for infinitesimal symmetries to study the invariance of several objects under the flows of vector fields a careful study of infinitesimal variations is performed with a special emphasis on infinitesimal vector bundle automorphisms

forcing function diagnostics for nonlinear dynamics this article investigates the problem of model diagnostics for systems described by nonlinear ordinary differential equations odes i propose modeling lack of fit as a time varying correction to the right hand side of a proposed differential equation this correction can be described as being a set of additive forcing functions estimated from data representing lack of fit in this manner allows us to graphically investigate model inadequacies and to suggest model improvements i derive lack of fit tests based on estimated forcing functions model building in partially observed systems of odes is particularly difficult and i consider the problem of identification of forcing functions in these systems the methods are illustrated with examples from computational neuroscience

tracking in nonlinear differential algebraic control systems with applications to constrained robot systems we consider the design of a feedback control law for control systems described by a class of nonlinear differential algebraic equations so that certain desired outputs track given reference inputs the nonlinear differential algebraic control system being considered is not in state variable form assumptions are introduced and a procedure is developed such that an equivalent state realization of the control system described by nonlinear differential algebraic equations is expressed in a familiar normal form a nonlinear feedback control law is then proposed which ensures under appropriate assumptions that the tracking error in the closed loop differential algebraic system approaches zero exponentially applications to simultaneous contact force and position tracking in constrained robot systems with rigid joints constrained robot systems with joint flexibility and constrained robot systems with significant actuator dynamics are discussed

ode methods for the solution of differential algebraic systems in this paper we study the numerical solution of the differential algebraic systems f t y y 0 many of these systems can be solved conveniently and economically using a range of ode methods others can be solved only by a small subset of ode methods and still others present insurmountable difficulty for all current ode methods we examine the first two groups of problems and indicate which methods we believe to be best for them then we explore the properties of the third group which cause the methods to fail we describe a reduction technique which allows systems to be reduced to ones that can be solved it also provides a tool for the analytical study of the structure of systems

random polytopes in a convex polytope independence of shape and concentration of vertices write d for the set of all convex bodies convex compact sets with nonempty interior in d define o g l d as the set of those k e 5 b d with vol k 1 fix k e g i d and choose points x l x e k randomly independently and according to the uniform distribution on k then k c o n v x l xn is a random polytope in k write e k n for the expectation of the random variable v o l k k n e k n shows how well k approximates k in volume on the average groemer grl proved that among all convex bodies k e o g l d the ellipsoids are approximated worst i e

dependence of solutions of nonsmooth differential algebraic equations on parameters abstract the well posedness of nonsmooth differential algebraic equations daes is investigated more specifically semi explicit daes with caratheodory style assumptions on the differential right hand side functions and local lipschitz continuity assumptions on the algebraic equations the daes are classified as having differential index one in a generalized sense solution regularity is formulated in terms of projections of generalized clarke jacobians existence of solutions is derived under consistency and regularity of the initial data uniqueness of a solution is guaranteed under analogous caratheodory ordinary differential equation uniqueness assumptions the continuation of solutions is established and sufficient conditions for continuous and lipschitzian parametric dependence of solutions are also provided to accomplish these results a theoretical tool for analyzing nonsmooth daes is provided in the form of an extended nonsmooth implicit function theorem the findings here are a natural extension of classical results and lay the foundation for further theoretical and computational analyses of nonsmooth daes

the index of general nonlinear daes in the last few years there has been considerable research on differential algebraic equations daes f t y y 0 where f y is identically singular much of the mathematical effort has focused on computing a solution that is assumed to exist more recently there has been some discussion of solvability of daes there has historically been some imprecision in the use of the two key concepts of solvability and index for daes the index is also important in control and systems theory but with different terminology the consideration of increasingly complex nonlinear daes makes a clear and correct development necessary this paper will try to clarify several points concerning the index after establishing some new and more precise terminology that we need some inaccuracies in the literature will be corrected the two types of indices most frequently used the differentiation index and the perturbation index are defined with respect to solutions of unperturbed problems examples are given to show that these indices can be very different for the same problem we define new maximum indices which are the maxima of earlier indices in a neighborhood of the solution over a set of perturbations and show that these indices are simply related to each other these indices are also related to an index defined in terms of jacobians