symplectic integrators for index one constraints we show that symplectic runge kutta methods provide effective symplectic integrators for hamiltonian systems with index one constraints these include the hamiltonian description of variational problems subject to position and velocity constraints nondegenerate in the velocities such as those arising in sub riemannian geometry and control theory

geometric description of vakonomic and nonholonomic dynamics comparison of solutions we treat the vakonomic dynamics with general constraints within a new geometric framework which can be useful in the study of optimal control problems we compare our formulation with the one of vershik and gershkovich in the case of linear constraints we show how nonholonomic mechanics also admits a new geometrical description which allows us to develop an algorithm of comparison between the solutions of both dynamics examples illustrating the theory are treated

nonholonomic versus vakonomic dynamics on a riemann cartan manifold for the chaplygin s nonholonomic constrained systems the constraint manifold can be endowed with riemann cartan geometric structure by nonholonomic mapping into a riemann manifold the two kinds of existing dynamics nonholonomic dynamics and vakonomic dynamics are compared in the framework of riemann cartan geometry it is proved that the equations of motion for nonholonomic and vakonomic dynamics are described by the equations of autoparallel and geodesic trajectories on the riemann cartan constraint manifold respectively if the metricity condition of riemann cartan connection is satisfied the torsion contorsion of the riemann cartan manifold characterizes the difference between the autoparallel and geodesic trajectories as well as the distinction between the nonholonomic and vakonomic equations

discrete vakonomic mechanics discretizations of vakonomic mechanics are developed by using the methodology of variational integrators the resulting algorithms are symplectic in some sense that we will explain and preserve the momentum map associated with a lie group of symmetries a specific example illustrating the theory is tested numerically

subdifferential regularities of perturbed distance functions outside the target set in banach spaces abstract this paper is devoted to study the frechet and proximal regularities at points outside the target set of perturbed distance functions d s j determined by a closed subset s and a lipschitz function j also we provide some important results on the frechet proximal and clarke subdifferentials of d s j at those points in banach spaces

d alembert lagrange analytical dynamics for nonholonomic systems the d alembert lagrange principle dlp is designed primarily for dynamical systems under ideal geometric constraints although it can also cover linear velocity constraints its application to nonlinear kinematic constraints has so far remained elusive mainly because there is no clear method whereby the set of linear conditions that restrict the virtual displacements can be easily extracted from the equations of constraint on recognition that the commutation rule traditionally accepted for velocity displacements in lagrangian dynamics implies displaced states that do not satisfy the kinematic constraints we show how the property of possible displaced states can be utilized ab initio so as to provide an appropriate set of linear auxiliary conditions on the displacements which can be adjoined via lagrange s multipliers to the d alembert lagrange equation to yield the equations of state and also new transpositional relations for nonholonomic systems the equations of state so obtained for systems under general nonlinear velocity and acceleration constraints are shown to be identical with those derived in appendix a from the quite different gauss principle the present advance therefore solves a long outstanding problem on the application of dlp to ideal nonholonomic systems and as an aside provides validity to axioms as the chetaev rule previously left theoretically unjustified a more general transpositional form of the boltzmann hamel equation is also obtained

on the geometry of non holonomic lagrangian systems we present a geometric framework for non holonomic lagrangian systems in terms of distributions on the configuration manifold if the constrained system is regular an almost product structure on the phase space of velocities is constructed such that the constrained dynamics is obtained by projecting the free dynamics if the constrained system is singular we develop a constraint algorithm which is very similar to that developed by dirac and bergmann and later globalized by gotay and nester special attention to the case of constrained systems given by connections is paid in particular we extend the results of koiller for caplygin systems an application to the so called non holonomic geometry is given

generic solutions for some perturbed optimization problem in non reflexive banach spaces abstract let z be a closed boundedly relatively weakly compact nonempty subset of a banach space x and j z r a lower semicontinuous function bounded from below if x 0 is a convex subset in x and x 0 has approximatively z property k then the set of all points x in x 0 z for which there exists z 0 z such that j z 0 x z 0 x and every sequence z n z satisfying lim n j z n x z n x for x contains a subsequence strongly convergent to an element of z is a dense g subset of x 0 z moreover under the assumption that x 0 is approximatively z strictly convex we show more namely that the set of all points x in x 0 z for which there exists a unique point z 0 z such that j z 0 x z 0 x and every sequence z n z satisfying lim n j z n x z n x for x converges strongly to z 0 is a dense g subset of x 0 z here x inf j z x z z z these extend s cobzas s result j math anal appl 243 2000 344 356

on the k uniform rotund and the fully convex banach spaces let x be a real banach space if x is uniformly convex then it is known that x is 2 uniformly rotund s and fully 2 convex 2 furthermore if a banach space x is either k uniformly rotund or fully k convex for some k then x is reflexive in this paper we show that if x is strictly convex and kuniformly rotund then x is fully k 1 convex however there exists a superreflexive space which is fully 2 convex but is not 2 uniformly rotund and for each k 2 there exists a strictly convex space which is k uniformly rotund but is not fully k convex thus for each k 2 there exist fully kconvex banach space which is not fully k 1 convex for x x2 xk in x let

well posedness of a class of perturbed optimization problems in banach spaces abstract let x be a banach space and z a nonempty subset of x let j z r be a lower semicontinuous function bounded from below and p 1 this paper is concerned with the perturbed optimization problem of finding z 0 z such that x z 0 p j z 0 inf z z x z p j z which is denoted by min j x z the notions of the j strictly convex with respect to z and of the kadec with respect to z are introduced and used in the present paper it is proved that if x is a kadec banach space with respect to z and z is a closed relatively boundedly weakly compact subset then the set of all x x for which every minimizing sequence of the problem min j x z has a converging subsequence is a dense g subset of x z 0 where z 0 is the set of all points z z such that z is a solution of the problem min j z z if additionally p 1 and x is j strictly convex with respect to z then the set of all x x for which the problem min j x z is well posed is a dense g subset of x z 0

equivalence of the dynamics of nonholonomic and variational nonholonomic systems for certain initial data in this paper we discuss the necessary and sufficient conditions for the equivalence of the dynamics of nonholonomic mechanics and variational nonholonomic vakonomic dynamics for certain initial conditions we derive a priori results for identifying equivalence and specializing to abelian chaplygin systems prove that equivalence results if and only if the constrained nonholonomic equations are lagrangian we eliminate the need to solve the variational nonholonomic problem when checking equivalence by obtaining explicit formulae for the system s multipliers and then derive conditions under which the multiplier free lagrangian gives equivalence of the dynamics we consider nonholonomic systems possessing invariant measures showing when equivalence and hamiltonization are the same we also derive conditions under which measure preserving systems exhibit equivalence we apply the results to many of the known nonholonomic systems

on combining feasibility descent and superlinear convergence in inequality constrained optimization extension of quasi newton techniques from unconstrained to constrained optimization via sequential quadratic programming sqp presents several difficulties among these are the possible inconsistency away from the solution of first order approximations to the constraints resulting in infeasibility of the quadratic programs and the task of selecting a suitable merit function to induce global convergence in ths case of inequality constrained optimization both of these difficulties disappear if the algorithm is forced to generate iterates that all satisfy the constraints and that yield monotonically decreasing objective function values feasibility of the successive iterates is in fact required in many contexts such as in real time applications or when the objective function is not well defined outside the feasible set it has been recently shown that this can be achieved while preserving local two step superlinear convergence in this note the essential ingredients for an sqp based method exhibiting the desired properties are highlighted correspondingly a class of such algorithms is described and analyzed tests performed with an efficient implementation are discussed

geometric theory of the equivalence of lagrangians for constrained systems the lagrangian description of a system is analysed from a geometric viewpoint in order to find a concept for equivalence of singular lagrangians generalising that of the regular case geometric and gauge equivalence of singular lagrangians are studied and the authors also give some conditions in which second order differential equations exist satisfying the dynamical equation on the final constraint submanifold

comments on the presymplectic formalism and the theory of regular lagrangians with constraints the lagrangian formalism for systems with constraints is developed using a singular lagrangian defined in the extended tangent bundle t q r the dynamics defined by the new extended lagrangian that incorporates the constraints is studied using the formalism of the presymplectic geometry a comparative study with other geometric approaches is presented

limiting subdifferentials of perturbed distance functions in banach spaces abstract we explore in arbitrary banach spaces the frechet type e subdifferentials and the limiting subdifferentials for the perturbed distance function d s j determined by a closed subset s and a lower semicontinuous function j defined on s in particular upper and lower estimates for the frechet type e subdifferentials and for the limiting subdifferentials are provided in terms of the corresponding subdifferentials of the sum of the associated functions j and s

newton s law and integrability of nonholonomic systems in this paper we consider nonholonomic control systems on riemannian manifolds such systems evolve on subbundles of tangent bundles defined by the nonholonomic constraints this paper promotes the view of such systems as the restriction to the nonholonomic subbundle of newton law type problems on the entire tangent bundle defined by in general non riemannian connections these connections should be related to specific geometric properties of the nonholonomic system we introduce a particular class of connections and demonstrate the richness of the class through four examples the rolling ball the constrained particle the rolling penny and the generalized rolling ball this class of connections is strongly related to questions of integrability of the original nonholonomic system this in turn provides additional insight into the relation between nonholonomic control systems formulated as kinematic equations and those that are formulated as the full dynamic equations

poisson structures for reduced non holonomic systems borisov mamaev and kilin have recently found certain poisson structures with respect to which the reduced and rescaled systems of certain non holonomic problems involving rolling bodies without slipping become hamiltonian the hamiltonian function being the reduced energy we study further the algebraic origin of these poisson structures showing that they are of rank 2 and therefore the mentioned rescaling is not necessary we show that they are determined up to a non vanishing factor function by the existence of a system of first order differential equations providing two integrals of motion we generalize the form of the poisson structures and extend their domain of definition we apply the theory to the rolling disc the routh s sphere the ball rolling on a surface of revolution and its special case of a ball rolling inside a cylinder

porosity of perturbed optimization problems in banach spaces let x be a banach space and z a nonempty closed subset of x l etj z r be a lower semicontinuous function bounded from below this paper is concerned with the perturbed optimization problem infz z j z x z denoted by x j inf for x x in the case when x is compactly fully 2 convex it is proved in the present paper that the set of all points x in x for which there does not exist z0 z such that j z0 x z0 infz z j z x z is a porous set in x f urthermore ifx is assumed additionally to be compactly locally uniformly convex we verify that the set of all points x x z 0 such that the problem x j inf fails to be approximately compact is a porous set in x z 0 w herez 0 denotes the set of all z z such that z pz z moreover a counterexample to which some results of ni r x ni generic solutions for some perturbed optimization problem in nonreflexive banach space j math anal appl 302 2005 417 424 fail is provided

least squares estimation techniques for position tracking of radioactive sources this paper describes least squares estimation algorithms used for tracking the physical location of radioactive sources in real time as they are moved around in a facility we present both recursive and moving horizon nonlinear least squares estimation algorithms that consider both the change in the source location and the deviation between measurements and model predictions the measurements used to estimate position consist of four count rates reported by four different gamma ray detectors there is an uncertainty in the source location due to the large variance of the detected count rate and the uncertainty in the background count rate this work represents part of a suite of tools which will partially automate security and safety assessments allow some assessments to be done remotely and provide additional sensor modalities with which to make assessments

a geometric approach to the gibbs appell equations in lagrangian mechanics we present a geometrical derivation of the gibbs appell equations in constrained lagrangian mechanics using the affine structures existing in the spaces of velocities and accelerations the theoretical basis is the gauss principle of least constraint and the necessary conditions for the minimum expressed in terms of the gibbs function give rise to the gibbs appell equations we give a second order formulation of the constraint equation and using the concept of moore penrose inverse we provide an explicit intrinsic construction of the projector on the space of the effective or non ideal forces the gibbs appell equations are expressed in an intrinsic way in terms of the fibre derivative of the gibbs function and the effective part of the applied force

reduction of some classical non holonomic systems with symmetry two types of nonholonomic systems with symmetry are treated i the configuration space is a total space of a g principal bundle and the constraints are given by a connection ii the configuration space is g itself and the constraints are given by left invariant forms the proofs are based on the method of quasicoordinates in passing a derivation of the maurer cartan equations for lie groups is obtained simple examples are given to illustrate the algorithmical character of the main results

reduction of constrained mechanical systems and stability of relative equilibria a mechanical system with perfect constraints can be described under some mild assumptions as a constrained hamiltonian system m h d w m thephase space is a symplectic manifold h thehamiltonian a smooth function onm d theconstraint submanifold a submanifold ofm andw theprojection bundle a vector sub bundle oft d m the reduced tangent bundle alongd we prove that when these data satisfy some suitable conditions the time evolution of the system is governed by a well defined differential equation ond we define constrained hamiltonian systems with symmetry and prove a reduction theorem application of that theorem is illustrated on the example of a convex heavy body rolling without slipping on a horizontal plane two other simple examples show that constrained mechanical systems with symmetry may have an attractive or repulsive set of relative equilibria

nonholonomic mechanical systems with symmetry this work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of lagrangian mechanics and with a view to control theoretical applications the basic methodology is that of geometric mechanics applied to the formulation of lagrange d alembert generalizing the use of connections and momentum maps associated with a given symmetry group to this case we begin by formulating the mechanics of nonholonomic systems using an ehresmann connection to model the constraints and show how the curvature of this connection enters into lagrange s equations unlike the situation with standard configuration space constraints the presence of symmetries in the nonholonomic case may or may not lead to conservation laws however the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables this momentum equation which plays an important role in control problems involves parallel transport operators and is computed explicitly in coordinates an alternative description using a body reference frame relates part of the momentum equation to the components of the euler poincar e equations along those symmetry directions consistent with the constraints one of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not an example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard a modified version of the skateboard which uses momentum coupling for locomotion generation we construct a synthesis of the mechanical connection and the ehresmann connection defining the constraints obtaining an important new object we call the nonholonomic connection when the nonholonomic connection is a principal connection for the given symmetry group we show how to perform lagrangian reduction in the presence of nonholonomic constraints generalizing previous results which only held in special cases several detailed examples are given to illustrate the theory september 1994 revised march 1995 revised june 1995

quantizing conditionally variational nonholonomic systems in this paper we present a quantization procedure for a class of nonholonomic systems briefly mechanical systems subject to nonintegrable constraints on the velocities whose reduced mechanics is hamiltonian we illustrate the theory developed through several examples one of which demonstrates how the classical nonholonomic constraints are sometimes manifested quantum mechanically as a shift in the ground state energy of the system

radiological source detection and localisation using bayesian techniques the problem considered in this paper is detection and estimation of multiple radiation sources using a time series of radiation counts from a collection of sensors a bayesian framework is adopted source detection is approached as a model selection problem in which competing models are compared using partial bayes factors given the number of sources the posterior mean is the minimum mean square error estimator of the source parameters exact calculation of the partial bayes factors and the posterior mean is not possible due to the presence of intractable integrals importance sampling using progressive correction is proposed as a computationally efficient method for approximating these integrals previously proposed algorithms have been restricted to one or two sources a simulation analysis shows that the proposed methods can detect and accurately estimate the parameters of four sources with reasonable computational expense

optimal control and geodesic flows in this paper we analyze and generalize from the point of view of the maximum principle a class of nonlinear optimal control problems originally introduced in brockett 1994 the optimal control problems discussed here lie on the adjoint orbits of compact lie groups or on grassmann manifolds and the structure of the equations that arise depends on the metric used the so called normal metric we consider some special cases of the general problem and their meaning from the point of view of riemannian geometry and hamiltonian mechanics

existence and porosity for a class of perturbed optimization problems in banach spaces abstract let x be a banach space and z a nonempty closed subset of x let j z r be an upper semicontinuous function bounded from above this paper is concerned with the perturbed optimization problem sup z z j z x z which is denoted by x j sup we shall prove in the present paper that if z is a closed boundedly relatively weakly compact nonempty subset then the set of all x x for which the problem x j sup has a solution is a dense g subset of x in the case when x is uniformly convex and j is bounded we will show that the set of all points x in x for which there does not exist z 0 z such that j z 0 x z 0 sup z z j z x z is a porous subset of x and the set of all points x x z 0 such that there exists a maximizing sequence of the problem x j sup which has no convergent subsequence is a porous subset of x z 0 where z 0 denotes the set of all z z such that z is in the solution set of z j sup

the geometrical theory of constraints applied to the dynamics of vakonomic mechanical systems the vakonomic bracket a vakonomic mechanical system can be alternatively described by an extended lagrangian using the lagrange multipliers as new variables since this extended lagrangian is singular the constraint algorithm can be applied and a dirac bracket giving the evolution of the observables can be constructed

reduction of constrained systems with symmetries a general model is proposed for constrained dynamical systems on a symplectic manifold which covers among others the description of lagrangian and hamiltonian systems with nonholonomic constraints and the canonical description of mechanical systems with a singular lagrangian the reduction properties of these systems in the presence of symmetry are investigated within this general framework

generic existence of solutions for some perturbed optimization problems abstract let x be a real banach space z a closed nonvoid subset of x and j z r a lower semicontinuous function bounded from below if x is reflexive and has the kadets property then the set of all x x for which there exists z0 z such that j z0 x z0 inf j z x z z z is g and dense in x this extends a result of k s lau 1978 indiana univ math j 27 791 795 on nearest points the case j 0