curve shortening and the rendezvous problem for mobile autonomous robots if a smooth closed and embedded curve is deformed along its normal vector field at a rate proportional to its curvature it shrinks to a circular point this curve evolution is called euclidean curve shortening and the result is known as the gage hamilton grayson theorem motivated by the rendezvous problem for mobile autonomous robots we address the problem of creating a polygon shortening flow a linear scheme is proposed that exhibits several analogues to euclidean curve shortening the polygon shrinks to an elliptical point convex polygons remain convex and the perimeter of the polygon is monotonically decreasing

a common framework for complete and incomplete attitude synchronization in networks with switching topology in this paper we study attitude synchronization for elements in the unit sphere in mathbb r scriptscriptstyle scriptscriptstyle 3 and for elements in the three dimensional 3 d rotation group for a network with switching topology the agents angular velocities are assumed to be the control inputs and a switching control law for each agent is devised that guarantees synchronization provided that all elements are initially contained in a region which we identify later in the paper the control law is decentralized and it does not require a common orientation frame among all agents we refer to synchronization of unit vectors in mathbb r scriptscriptstyle 3 as incomplete synchronization and of 3 d rotation matrices as complete synchronization our main contribution lies in showing that these two problems can be analyzed under a common framework where all agents dynamics are transformed into unit vectors dynamics on a sphere of appropriate dimension

smooth approximation of potential fields from noisy scattered data geophysical potential field data generally are observed at scattered sampling points and are contaminated by measurement and preprocessing errors analysis and further treatment require a representation of the underlying potential fields on a regular grid in the presence of noise an approximation of the data is more appropriate than an exact interpolation analysis of the smoothness of potential fields indicates an approximation by band limited functions the resulting least squares problem can be formulated as a linear system of equations in which the system matrix is of block toeplitz type this special structure allows a computationally attractive and robust solution via the fast fourier transform and the conjugate gradient algorithm on this basis we present a gridding method that incorporates additional knowledge about the physical properties of potential fields and the statistics of the data experimental results for synthetic and real world gravity data demonstrate good performance for highly sc

circulant preconditioners for toeplitz matrices with piecewise continuous generating functions the authors consider the solution of n by n toeplitz systems t sub n x b by preconditioned conjugate gradient methods the preconditioner c sub n is the t chan circulant preconditioner which is defined to be the circulant matrix that minimizes parallel b sub n t sub n parallel sub f over all circulant matrices b sub n for toeplitz matrices generated by positive 2 pi periodic continuous functions they have shown earlier that the spectrum of the preconditioned system c sup minus 1 sub n t sub n is clustered around 1 and hence the convergence rate of the preconditioned system is superlinear however in this paper they show that if instead the generating function is only piecewise continuous then for all epsilon sufficiently small there are o log n eigenvalues of c sup minus 1 sub n t sub n that lie outside the interval 1 epsilon 1 epsilon in particular the spectrum of c sup minus 1 sub n t sub n cannot be clustered around 1 numerical examples are given to verify that the convergence rate of the method is no longer superlinear in general 20 refs

unified cooperative control of multiple agents on a sphere for different spherical patterns in this technical note we consider the cooperative control of multiple agents on a sphere provide the appropriate attraction repulsion interactions between agents with introduction of the state dependent repulsive coefficients and design the unified control laws for agents to achieve three fundamental spherical configurations or patterns rendezvous uniform deployment and formation using both the first and second order models we present the analysis of the stability scaling and geometric patterns of the models on the sphere under different attraction repulsion parameters with illustration of the geometrical patterns and the geometrical isomerism by simulations the closed loop dynamics of agents are invariant with respect to group rotation actions on the sphere

hybrid control for connectivity preserving flocking in this technical note we address the combined problem of motion and network topology control in a group of mobile agents with common objective the flocking behavior of the group instead of assuming network connectivity we enforce it by means of distributed topology control that decides on both deletion and creation of communication links between agents adapting the network to the group s spatial distribution with this protocol ensuring network connectivity a decentralized motion controller aligns agent velocity vectors and regulates inter agent distances to maintain existing network links the stability of the flocking controller is established in continuous time by means of an observability argument on a quadratic form of the graph laplacian that exploits the time delay between link deletion and creation caused by the topology control protocol which induces a dwell time between network switches

stabilization of the boussinesq equation via internal feedback controls in this paper we show that the steady state solutions to boussinesq equations in three dimensional domain are stabilizable by internal controllers with the supports in subsets 1 under some conditions on and 1

a unifying approach to some old and new theorems on distribution and clustering abstract a unifying approach is proposed to studying the distributions of eigenvalues and singular values of toeplitz matrices associated with a fourier series and multilevel toeplitz matrices associated with a multidimensional fourier series obtained are the extensions of the szego and avram parter theorems where the generating function is now required to belong to l 2 and not necessarily to l analogous extensions are given for multilevel toeplitz matrices in particular it is proved that if f x 1 x p l 2 then the p level complex toeplitz matrices allied with f have their singular values distributed as f x 1 x p the distribution results for the cesaro optimal circulants hold even if f l 1 also suggested are new theorems on clustering that have to do with the preconditioning of multilevel toeplitz matrices by multilevel circulants

on the rendezvous problem for multiple nonholonomic agents in this note a decentralized feedback control strategy that drives a system of multiple nonholonomic unicycles to a rendezvous point in terms of both position and orientation is introduced the proposed nonholonomic control law is discontinuous and time invariant and using tools from nonsmooth lyapunov theory and graph theory the stability of the overall system is examined similarly to the linear case the convergence of the multi agent system relies on the connectivity of the communication graph that represents the inter agent communication topology the control law is first defined in order to guarantee connectivity maintenance for an initially connected communication graph moreover the cases of static and dynamic communication topologies are treated as corollaries of the proposed framework

numerical analysis of the non uniform sampling problem we give an overview of recent developments in the problem of reconstructing a band limited signal from nonuniform sampling from a numerical analysis view point it is shown that the appropriate design of the finite dimensional model plays a key role in the numerical solution of the nonuniform sampling problem in the one approach often proposed in the literature the finite dimensional model leads to an ill posed problem even in very simple situations the other approach that we consider leads to a well posed problem that preserves important structural properties of the original infinite dimensional problem and gives rise to efficient numerical algorithms furthermore a fast multilevel algorithm is presented that can reconstruct signals of unknown bandwidth from noisy nonuniformly spaced samples we also discuss the design of efficient regularization methods for ill conditioned reconstruction problems numerical examples from spectroscopy and exploration geophysics demonstrate the performance of the proposed methods

equilibria and steering laws for planar formations abstract this paper presents a lie group setting for the problem of control of formations as a natural outcome of the analysis of a planar two vehicle formation control law the vehicle trajectories are described using the planar frenet serret equations of motion which capture the evolution of both the vehicle position and orientation for unit speed motion subject to curvature steering control the set of all possible relative equilibria for arbitrary g invariant curvature controls is described where g se 2 is a symmetry group for the control law and a global convergence result for the two vehicle control law is proved an n vehicle generalization of the two vehicle control law is also presented and the corresponding relative equilibria for the n vehicle problem are characterized work is on going to discover stability and convergence results for the n vehicle problem

the reconstruction of a band limited function and its fourier transform from a finite number of samples at arbitrary locations by singular value decomposition a method for the stable interpolation of a bandlimited function known at sample instants with arbitrary locations in the presence of noise is given singular value decomposition is used to provide a series expansion that in contrast to the method of sampling functions permits simple identification of vectors in the minimum norm space poorly represented in the sample values three methods miller regularization least squares estimation and maximum a posteriori estimation are given for obtaining regularized reconstructions when noise is present the singular value decomposition svd method is used to interrelate these methods examples illustrating the technique are given

the constrained rayleigh quotient with a general orthogonality constraint and an eigen balanced laplacian matrix the greatest lower bound and applications in cooperative control problems mathematically the rayleigh quotient is defined as the quadratic function of a symmetric matrix and a nonzero usually unconstrained variable vector in this paper we consider the constrained rayleigh quotient in which the variable vector has the orthogonality constraint i e it is constrained to be orthogonal to a nonzero vector this nonzero vector is called the orthogonality constraint vector or abbreviated as the oc vector for the variable vector the matrix for the rayleigh quotient is an eigen balanced eb laplacian matrix a tighter lower bound of the constrained rayleigh quotient has many implications in mathematics as well as in cooperative control problems the main contributions in this paper are as follows first we provide the greatest lower bound or the infimum of the constrained rayleigh quotient with respect to a general oc vector and the eb laplacian matrix whose results are novel and better than the existing results then we interpret the physical meaning of our results in an insightful geometric form and characterize the properties of the results finally as an example to illustrate the merit of the tighter lower bound of the constrained rayleigh quotient we consider the scale of the agents driving by a fundamental consensus protocol and show that the convergence rate of the agents scale is characterized by the greatest lower bound of the constrained rayleigh quotient

geometric analysis of the formation problem for autonomous robots in the formation control problem for autonomous robots a distributed control law steers the robots to the desired target formation a local stability result of the target formation can be derived by methods of linearization and center manifold theory or via a lyapunov based approach besides the target formation the closed loop dynamics of the robots feature various other undesired invariant sets such as nonrigid formations this note addresses a global stability analysis of the closed loop formation control dynamics we pursue a differential geometric approach and derive purely algebraic conditions for local stability of invariant embedded submanifolds these theoretical results are then applied to the well known example of a cyclic triangular formation and result in instability of all invariant sets other than the target formation

frames containing a riesz basis and approximation of the frame coefficients using finite dimensional methods abstract we consider frames which contain a riesz basis among those we find riesz frames i e frames with the property that every subfamily is a frame for its closed linear span with a common lower bound we point out the connection to the projection method which can be used to approximate the frame coefficients for a riesz frame using finitely many elements of the frame under different conditions we show that the projection method even approximates the frame coefficient in l 2 sense

coverage optimization and spatial load balancing by robotic sensor networks this technical note studies robotic sensor networks performing static coverage optimization with area constraints given a density function describing the probability of events happening and a performance function measuring the cost to service a location the objective is to position sensors in the environment so as to minimize the expected servicing cost moreover because of load balancing considerations the area of the region assigned to each robot is constrained to be a pre specified amount we characterize the optimal configurations as center generalized voronoi configurations the generalized voronoi partition depends on a set of weights one per robot assigned to the network we design a jacobi iterative algorithm to find the weight assignment whose corresponding generalized voronoi partition satisfies the area constraints this algorithm is distributed over the generalized delaunay graph we also design the move to center and compute weight strategy to steer the robotic network towards the set of center generalized voronoi configurations while monotonically optimizing coverage

a discrete theory of irregular sampling abstract we derive algorithms for the iterative reconstruction of discrete band limited signals from their irregular samples they converge at a geometric rate and all constants are computable explicitly we also treat the related problem of interpolation of trigonometric polynomials and give estimates for the condition number of certain positive definite toeplitz matrices

coordinated motion design on lie groups the present paper proposes a unified geometric framework for coordinated motion on lie groups it first gives a general problem formulation and analyzes ensuing conditions for coordinated motion then it introduces a precise method to design control laws in fully actuated and underactuated settings with simple integrator dynamics it thereby shows that coordination can be studied in a systematic way once the lie group geometry of the configuration space is well characterized applying the proposed general methodology to particular examples allows to retrieve control laws that have been proposed in the literature on intuitive grounds a link with brockett s double bracket flows is also made the concepts are illustrated on so 3 se 2 and se 3

intrinsic reduced attitude formation with ring inter agent graph this paper investigates the reduced attitude formation control problem for a group of rigid body agents using feedback based on relative attitude information under both undirected and directed cyc

solving ill conditioned and singular linear systems a tutorial on regularization it is shown that the basic regularization procedures for finding meaningful approximate solutions of ill conditioned or singular linear systems can be phrased and analyzed in terms of classical linear algebra that can be taught in any numerical analysis course apart from rewriting many known results in a more elementary form we also derive a new two parameter family of merit functions for the determination of the regularization parameter the traditional merit functions from generalized cross validation gcv and generalized maximum likelihood gml are recovered as special cases

control of a driftless bilinear vector field on n sphere in this paper we consider a multi input driftless bilinear system evolving on the n dimensional sphere s n we first provide examples drawn from rigid body mechanics that provide the motivation for the control of bilinear systems on s n for the general framework we establish the global controllability on s n and propose two linear control laws on s n that achieve asymptotic stabilization of an equilibrium point with an almost global domain of attraction further the asymptotically stable closed loop system trajectories are shown to be arcs on the geodesics of s n for a particular choice of the equilibrium point next we propose two linear time varying control laws to achieve trajectory tracking on s n and show the asymptotic stability of the tracking error a distributed control is designed for the consensus of multiagent bilinear systems on s n with an undirected tree as the communication graph the consensus manifold is shown to have an almost global domain of attraction

distributed geodesic control laws for flocking of nonholonomic agents we study the problem of flocking and velocity alignment in a group of kinematic nonholonomic agents in 2 and 3 dimensions by analyzing the velocity vectors of agents on a circle for planar motion or sphere for 3 d motion we develop a geodesic control law that minimizes a misalignment potential and results in velocity alignment and flocking the proposed control laws are distributed and will provably result in flocking when the underlying proximity graph which represents the neighborhood relation among agents is connected we further show that flocking is possible even when the topology of the proximity graph changes over time so long as a weaker notion of joint connectivity is preserved

exponential synchronization of the high dimensional kuramoto model with identical oscillators under digraphs abstract for the kuramoto model and its variations it is difficult to analyze the exponential synchronization under the general digraphs due to the lack of symmetry in this paper for the high dimensional kuramoto model of identical oscillators a matrix riccati differential equation mrde is proposed to describe the error dynamics based on the mrde the exponential synchronization is proved by constructing a total error function for the case of digraphs admitting directed spanning trees

irregular sampling toeplitz matrices and the approximation of entire functions of exponential type in many applications one seeks to recover an entire function of exponential type from its non uniformly spaced samples whereas the mathematical theory usually addresses the question of when such a function in l 2 r can be recovered numerical methods operate with a finite dimensional model the numerical reconstruction or approximation of the original function amounts to the solution of a large linear system we show that the solutions of a particularly efficient discrete model in which the data are fit by trigonometric polynomials converge to the solution of the original infinite dimensional reconstruction problem this legitimatizes the numerical computations and explains why the algorithms employed produce reasonable results the main mathematical result is a new type of approximation theorem for entire functions of exponential type from a finite number of values from another point of view our approach provides a new method for proving sampling theorems

high dimensional kuramoto models on stiefel manifolds synchronize complex networks almost globally abstract the kuramoto model of coupled phase oscillators is often used to describe synchronization phenomena in nature some applications e g quantum synchronization and rigid body attitude synchronization involve high dimensional kuramoto models where each oscillator lives on the n sphere or so n these manifolds are special cases of the compact real stiefel manifold st p n using tools from optimization and control theory we prove that the generalized kuramoto model on st p n converges to a synchronized state for any connected graph and from almost all initial conditions provided p n satisfies p 2 3 n 1 and all oscillator frequencies are equal this result could not have been predicted based on knowledge of the kuramoto model in complex networks over the circle in that case almost global synchronization is graph dependent it applies if the network is acyclic or sufficiently dense this paper hence identifies a property that distinguishes many high dimensional generalizations of the kuramoto models from the original model

consensus optimization on manifolds the present paper considers distributed consensus algorithms that involve n agents evolving on a connected compact homogeneous manifold the agents track no external reference and communicate their relative state according to a communication graph the consensus problem is formulated in terms of the extrema of a cost function this leads to efficient gradient algorithms to synchronize i e maximizing the consensus or balance i e minimizing the consensus the agents a convenient adaptation of the gradient algorithms is used when the communication graph is directed and time varying the cost function is linked to a specific centroid definition on manifolds introduced here as the induced arithmetic mean that is easily computable in closed form and may be of independent interest for a number of manifolds the special orthogonal group so n and the grassmann manifold text it grass p n are treated as original examples a link is also drawn with the many existing results on the circle

on exponential synchronization of kuramoto oscillators in this technical note we study the problem of exponential synchronization for one of the most popular models of coupled phase oscillators the kuramoto model we consider the special case of finite oscillators with distinct bounded natural frequencies our first result derives a lower bound on the coupling gain which is necessary for the onset of synchronization this bound improves the one derived by jadbabaie we then calculate a lower bound on the coupling gain that is sufficient to guarantee oscillator synchronization and derive further sufficient conditions to ensure exponential synchronization of the angular frequencies of all oscillators to the mean natural frequency of the group we also characterize the coupling gain that is sufficient for the oscillator phase differences to approach any desired compact set in finite time

synchronization and balancing on the n torus in this paper we study the behavior of a network of n agents each evolving on the circle we propose a novel algorithm that achieves synchronization or balancing in phase models under mild connectedness assumptions on the possibly time varying and unidirectional communication graphs the global convergence analysis on the n torus is a distinctive feature of the present work with respect to previous results that have focused on convergence in the euclidean space

the prolate matrix abstract the purpose of this note is to describe some properties of a particular positive definite toeplitz matrix of arbitrary order which we call the prolate matrix this matrix arises naturally in signal processing and it is extremely ill conditioned moreover its eigenvalues exhibit rather unusual behavior making it a very useful test matrix for computational algorithms