pulsating flows of the 2d euler poisson equations abstract it is well known that solutions of gaseous or fluid dynamical systems can easily blowup or develop shock waves in finite time in this paper we show the existence of a class of radially symmetric rotational solutions to the two dimensional pressureless euler poisson equations the flows are global i e exist for all t 0 have compact support at all times and pulsate periodically the method of construction is novel it comprises the piecing together of suitable shells of moving particles in a delicate manner with careful choice of initial data each shell is the solution of a member of a continuum of ordinary differential equations detailed analysis of the equations is carried out to ensure that neighboring shells can be chosen to pulsate with the same period we also show that any such solution can be extended to a larger flow by adding annuli of pulsating flows our result exhibits another example of rotation preventing the blowup of solutions

control of a three coleader formation in the plane this paper considers a formation of three point agents moving in the plane where the agents have a cyclic ordering with each one required to maintain a nominated distance from its neighbour further each agent is allowed to determine its movement strategy using local knowledge only of the direction of its neighbour and the current and desired distance from its neighbour the motion of the formation is studied when distances are initially incorrect a convergence result is established to the effect that provided agents never become collinear the correct distances will be approached exponentially fast and the formation as a whole will rotate by a finite angle and translate by a finite distance

necessary and sufficient graphical conditions for formation control of unicycles the feasibility problem is studied of achieving a specified formation among a group of autonomous unicycles by local distributed control the directed graph defined by the information flow plays a key role it is proved that formation stabilization to a point is feasible if and only if the sensor digraph has a globally reachable node a similar result is given for formation stabilization to a line and to more general geometric arrangements

distributed formation control of multi agent systems using complex laplacian the paper concentrates on the fundamental coordination problem that requires a network of agents to achieve a specific but arbitrary formation shape a new technique based on complex laplacian is introduced to address the problems of which formation shapes specified by inter agent relative positions can be formed and how they can be achieved with distributed control ensuring global stability concerning the first question we show that all similar formations subject to only shape constraints are those that lie in the null space of a complex laplacian satisfying certain rank condition and that a formation shape can be realized almost surely if and only if the graph modeling the inter agent specification of the formation shape is 2 rooted concerning the second question a distributed and linear control law is developed based on the complex laplacian specifying the target formation shape and provable existence conditions of stabilizing gains to assign the eigenvalues of the closed loop system at desired locations are given moreover we show how the formation shape control law is extended to achieve a rigid formation if a subset of knowledgable agents knowing the desired formation size scales the formation while the rest agents do not need to re design and change their control laws

ratio of distance rigidity theory with application to similar formation control this paper develops a ratio of distance rod rigidity theory to study when a framework can be uniquely determined by a set of rod constraints up to similar transformations translation rotation scaling and reflection in particular a framework is specified by a set of rod constraints the rod of a pair of edges joining a common vertex instead of distance bearing and angle constraints assumed in existing literature its relations to three existing rigidity theories distance bearing and angle rigidity theories are established the proposed rod rigidity theory is further applied to the rod based similar formation stabilization problem where the desired formation shape is expressed as a set of rod constraints finally numerical simulations are presented to illustrate the effectiveness of the theoretical results

three and higher dimensional autonomous formations rigidity persistence and structural persistence in this paper we generalize the notion of persistence which has been originally introduced for two dimensional formations to r d for d 3 seeking to provide a theoretical framework for real world applications which often are in three dimensional space as opposed to the plane persistence captures the desirable property that a formation moves as a cohesive whole when certain agents maintain their distances from certain other agents we verify that many of the properties of rigid and or persistent formations established in r 2 are also valid for higher dimensions analysing the closed subgraphs and directed paths in persistent graphs we derive some further properties of persistent formations we also provide an easily checkable necessary condition for persistence we then turn our attention to consider some practical issues raised in multi agent formation control in three dimensional space we display a new phenomenon not present in r 2 whereby subsets of agents can behave in a problematic way when this behaviour is precluded we say that the graph depicting the multi agent formation has structural persistence in real deployment of controlled multi agent systems formations with underlying structurally persistent graphs are of interest we analyse the characteristics of structurally persistent graphs and provide a streamlined test for structural persistence we study the connections between the allocation of degrees of freedom dofs across agents and the characteristics of persistence and or structural persistence of a directed graph we also show how to transfer dofs among agents when the formation changes with new agent s added to preserve persistence and or structural persistence

harmonic analysis with respect to heat kernel measure this paper surveys developments over the last decade in harmonic analysis on lie groups relative to a heat kernel measure these include analogs of the hermite expansion the segal bargmann transform and the taylor ex pansion some of the results can be understood from the standpoint of geo metric quantization others are intimately related to stochastic analysis

brief paper formation control using range only measurements this paper proposes algorithms to coordinate a formation of mobile agents when the agents are not able to measure the relative positions of their neighbors but only the distances to their respective neighbors in this sense less information is available to agents than is normally assumed in formation stabilization or station keeping problems to control the shape of the formation the solution advanced in the paper involves subsets of non neighbor agents cyclically localizing the relative positions of their respective neighbor agents while these are held stationary and then moving to reduce the value of a cost function which is nonnegative and assumes the zero value precisely when the formation has correct distances the movement schedule is obtained by a novel vertex coloring algorithm whose computation time is linear in the number of agents when implemented on the graphs of minimally rigid formations since in some formations it may be that an agent is never allowed to be stationary e g it is a fixed wing aircraft or because formations may be required to move as a whole in a certain direction the results are extended to allow for cyclic localization of agents in this case the tool used is the cayley menger determinant

bearing ratio of distance rigidity theory with application to directly similar formation control abstract this paper develops a bearing ratio of distance b rod rigidity theory to study when a framework can be uniquely determined by a set of b rod constraints up to directly similar transformation translation positive scaling in particular a framework is specified by a set of b rod constraints i e mixed bearing and rod measurements the ratio of distances of a pair of edges joining a common vertex instead of distance bearing and angle constraints assumed in existing literature the proposed b rod rigidity theory is further applied to the b rod based directly similar formation stabilization problem where the desired formation shape is expressed as a set of b rod constraints finally numerical simulations are presented to illustrate the effectiveness of the theoretical results

distributed self localization for relative position sensing networks in 2d space this paper studies the 2d localization problem of a sensor network given anchor node positions in a common global coordinate frame and relative position measurements in local coordinate frames between node pairs it is assumed that the local coordinate frames of different sensors have different orientations and the orientation difference with respect to the global coordinate frame are not known in terms of graph connectivity a necessary and sufficient condition is obtained for self localizability that leads to a fully distributed localization algorithm moreover a distributed verification algorithm is developed to check the graph connectivity condition which can terminate successfully when the sensor network is self localizable finally a fully distributed linear and iterative algorithm based on the complex valued laplacian associated with the sensor network is proposed which converges globally and gives the correct localization result

representations of lie superalgebras and generalized boson fermion equivalence in quantum stochastic calculus the boson fermion equivalence scheme of 5 can be generalized ton dimensions withr boson andn r fermion creation and annihilation fields the same stochastic integral prescription replaces then 2 generalized number processes ai j which form representations of the lie algebragl n by processes forming representations of the lie superalgebragl n r

comments on global stabilization of rigid formations in the plane automatica 49 2013 14361441 this paper shows that the modified gradient control law proposed in tian and wang 2013 does not globally stabilize rigid formation shapes also further analysis on the control law and a numerical example showing that the control law cannot drive the agents out from a stable incorrect equilibrium of the corresponding gradient control system are provided

a graph laplacian approach to coordinate free formation stabilization for directed networks this paper concentrates on coordinate free formation control for directed networks for which the dynamic motion of each agent is assumed to be governed only by a local control we develop a graph laplacian approach to solve the global and exponential formation stablization problem using merely relative position measurements between neighbors first to capture the sensing and control architectures that are needed to maintain the shape of a formation a necessary and sufficient topological condition is proposed second a laplacian based control law is developed for the stablization problem of a group of mobile agents to a desired formation shape under both fixed and switching topologies due to temporal node failures simulation results are provided to demonstrate that our laplacian based formation control strategy is inherently fault tolerant and robust to node failures

formation tracking of the second order multi agent systems using position only information via impulsive control with input delays in this paper the formation tracking problem for the second order multi agent system with and without input delay are investigated respectively the objective is to design the formation tracking algorithm such that a certain follower follows the trajectory of the leader while also maintains a certain desired geometric formation with other agents simultaneously the impulsive algorithms are designed by using only the relative position information of the neighbors for both the cases with and without the input delay by using properties of the laplacian matrix and combining the stability theory of impulsive systems necessary and sufficient conditions are derived to achieve the formation tracking of the second order multi agent system with and without the input delay respectively the numerical examples are given to illustrate the effectiveness of our theoretical results

stable protocol admissible synchronizability for high order singular complex networks with switching topologies abstract stable protocol admissible synchronization analysis and design problems for singular complex networks with both connected switching topologies and jointly connected switching topologies are investigated firstly a singular observer type synchronization protocol is introduced where the errors of protocol states among neighboring agents are applied to make complex networks satisfy a specific separation principle which can simplify synchronization analysis and design problems then by the orthonormal projection operators between the synchronization subspace and the complement synchronization subspace stable protocol admissible synchronization problems are transformed into admissible ones of multiple subsystems with lower dimensions and an approach is given to determine the synchronization function furthermore stable protocol admissible synchronizability criteria for connected switching topology cases and jointly connected switching topology cases are proposed respectively finally two numerical examples are given to demonstrate the effectiveness of theoretical results

decentralized control of vehicle formations this paper investigates a method for decentralized stabilization of vehicle formations using techniques from algebraic graph theory the vehicles exchange information according to a pre specified communication digraph g a feedback control is designed using relative information between a vehicle and its in neighbors in g we prove that a necessary and sufficient condition for an appropriate decentralized linear stabilizing feedback to exist is that g has a rooted directed spanning tree we show the direct relationship between the rate of convergence to formation and the eigenvalues of the directed laplacian of g various special situations are discussed including symmetric communication graphs and formations with leaders several numerical simulations are used to illustrate the results

matrix weighted consensus and its applications abstract this paper proposes the matrix weighted consensus algorithm which is a generalization of the consensus algorithm given a networked dynamical system where the interconnections between agents are weighted by nonnegative definite matrices it is shown that consensus and clustering phenomena naturally exist we examine algebraic and algebraic graph conditions for achieving a consensus

exact similarity solutions of ideal mhd equations for plane motions for the non linear two dimensional ideal mhd equations we use the known lie point symmetries and for the case of adiabatic exponent 2 we give local lie backlund symmetries we find exact solutions using similarity analysis among the calculated similarity solutions describing plasma expansion into vacuum shock waves and spiral flows with plasma sources

leader follower formation via complex laplacian the paper introduces complex valued laplacians for graphs whose edges are attributed with complex weights and studies the leader follower formation problem based on complex laplacians the main goal is to control the shape of a planar formation of point agents in the plane using simple and linear interaction rules related to complex laplacians we present a characterization of complex laplacians that preserve a specific planar formation as an equilibrium solution for both single integrator kinematics and double integrator dynamics planar formations under study are subject to translation rotation and scaling in the plane but can be determined by two co leaders in leader follower networks furthermore when a complex laplacian does not result in an asymptotically stable behavior of the multi agent system we show that a stabilizing matrix which updates the complex weights exists to asymptotically stabilize the system while preserving the equilibrium formation also algorithms are provided to find stabilizing matrices finally simulations are presented to illustrate our results

global stabilization of triangulated formations formation control deals with the design of decentralized control laws that stabilize mobile autonomous agents at prescribed distances from each other we call any configuration of the agents a tar

the two parameter free unitary segal bargmann transform and its biane gross malliavin identification abstract motivated by a conditional expectation interpretation of the segal bargmann transform we derive the integral kernel for the large n limit of the two parameter segal bargmann hall transform over the unitary group u n and explore its limiting behavior we also extend the notion of circular systems to more general elliptic systems in order to give an alternate construction of our new two parameter free unitary segal bargmann hall transform via a biane gross malliavin type theorem

short time behavior of hermite functions on compact lie groups abstract let p t x be the gaussian heat kernel on r n at time t the classical hermite polynomials at time t may be defined by a rodriguez formula given by h x t p t x p t x where is a constant coefficient differential operator on r n recent work of gross 1993 and hijab 1994 has led to the study of a new class of functions on a general compact lie group g in analogy with the r n case these hermite functions on g are obtained by the same formula wherein p t x is now the heat kernel on the group x is replaced by x 1 and is a right invariant differential operator let g be the lie algebra of g composing a hermite function on g with the exponential map produces a family of functions on g we prove that these functions scaled appropriately in t approach the classical hermite polynomials at time 1 as t tends to 0 both uniformly on compact subsets of g and in l p g where 1 p is a gaussian measure on g similar theorems are established when g is replaced by g k where k is some closed connected subgroup of g

brief paper collective motions and formations under pursuit strategies on directed acyclic graphs a novel pursuit based approach is presented to investigate collective motions and formations of a large number of agents with both single integrator kinematics and double integrator dynamics on directed acyclic graphs dags each agent pursues its neighbors according to a directed acyclic graph in which the agents without neighbors are leaders based on signal flow graph analysis and mason s rule necessary and sufficient conditions are derived for bibo stability of resulting pursuit systems moreover achievable collective motions and formations are analyzed by adjusting a set of control parameters when leaders keep stationary perform uniform rectilinear motions and perform uniform circular motions finally simulations are provided for achieving a static formation and mimicking several complex collective behaviors observed in nature such as v formation vortex motions and tornado motions

finite time stabilisation of cyclic formations using bearing only measurements this paper studies decentralised formation control of multiple vehicles in the plane when each vehicle can only measure the local bearings of their neighbours by using bearing only sensors since the inter vehicle distance cannot be measured the target formation involves no distance constraints more specifically the target formation considered in this paper is an angle constrained cyclic formation where each vehicle has exactly two neighbours and the angle at each vehicle subtended by its two neighbours is pre specified to stabilise the target formation we propose a discontinuous control law that only requires the sign information of the angle errors due to the discontinuity of the proposed control law the stability of the closed loop system is analysed by employing a locally lipschitz lyapunov function and nonsmooth analysis tools we prove that the target formation is locally finite time stable with collision avoidance guaranteed

phase space bounds for quantum mechanics on a compact lie group let k be a compact connected lie group and k bbb c its complexification i consider the hilbert space cal h l 2 left k bbb c nu t right of holomorphic functions introduced in h1 where the parameter t is to be interpreted as planck s constant in light of l s the complex group k bbb c may be identified canonically with the cotangent bundle of k using this identification i associate to each f in cal h l 2 left k bbb c nu t right a phase space probability density the main result of this paper is theorem 1 which provides an upper bound on this density which holds uniformly over all f and all points in phase space specifically the phase space probability density is at most a t left 2 pi t right n where n dim k and a t is a constant which tends to one exponentially fast as t tends to zero at least for small t this bound cannot be significantly improved

coherent state transforms and the weyl equation in clifford analysis we study a transform inspired by coherent state transforms from the hilbert space of clifford algebra valued square integrable functions l2 m dx m to a hilbert space of solutions of the weyl equation on m 1 m namely to the hilbert space l2 m 1 d of m valued monogenic functions on m 1 which are l2 with respect to an appropriate measure d we prove that this transform is a unitary isomorphism of hilbert spaces and that it is therefore an analog of the segal bargmann transform for clifford analysis as a corollary we obtain an orthonormal basis of monogenic functions on m 1 we also study the case when m is replaced by the m torus x1d54b m quantum mechanically this extension establishes the unitary equivalence of the schrodinger representation on m for m m and m x1d54b m with a representation on the hilbert space l2 m d of solutions of the weyl equation on the space time m

the taylor map on complex path groups let w g denote the path group of an arbitrary complex connected lie group the existence of a heat kernel measure t on w g has been shown in m cecil b k driver heat kernel measure on loop and path groups preprint http www math uconn edu cecil papers p2 pdf infin dimens anal quantum probab relat top submitted for publication the present work establishes an isometric map the taylor map from the space of l2 t holomorphic functions on w g to a subspace of the dual of the universal enveloping algebra of lie h g where h g is the lie subgroup of finite energy paths this map is shown to be surjective in the case where g is a simply connected graded lie group

the segal bargmann transform on classical matrix lie groups abstract we study the complex time segal bargmann transform b s k n on a compact type lie group k n where k n is one of the following classical matrix lie groups the special orthogonal group so n r the special unitary group su n or the compact symplectic group sp n our work complements and extends the results of driver hall and kemp on the segal bargman transform for the unitary group u n we provide an effective method of computing the action of the segal bargmann transform on trace polynomials which comprise a subspace of smooth functions on k n extending the polynomial functional calculus using these results we show that as n the finite dimensional transform b s k n has a meaningful limit g s which can be identified as an operator on the space of complex laurent polynomials

a new tabular form for determining root distribution of a complex polynomial with respect to the imaginary axis the original routh table dealing with real polynomials is further investigated for complex polynomials a tabular form for determining root distribution of a complex polynomial with respect to the imaginary axis is developed and modified procedures for directly treating singularities in the array are proposed also procedures are developed for determining the respective orders of simple and or repeated roots lying on the imaginary axis conditional stability and instability can be distinguished from each other by the criteria developed here

collective motion from consensus with cartesian coordinate coupling collective motions including rendezvous circular patterns and logarithmic spiral patterns can be achieved by introducing cartesian coordinate coupling to existing consensus algorithms we study the collective motions of a team of vehicles in 3d by introducing a rotation matrix to an existing consensus algorithm for double integrator dynamics it is shown that the network topology the damping gain and the value of the euler angle all affect the resulting collective motions we show that when the nonsymmetric laplacian matrix has certain properties the damping gain is above a certain bound and the euler angle is below equal or above a critical value the vehicles will eventually rendezvous move on circular orbits or follow logarithmic spiral curves lying on a plane perpendicular to the euler axis in particular when the vehicles eventually move on circular orbits the relative radii of the orbits respectively the relative phases of the vehicles on their orbits are equal to the relative magnitudes respectively the relative phases of the components of a right eigenvector associated with a critical eigenvalue of the nonsymmetric laplacian matrix simulation results are presented to demonstrate the theoretical results

necessary and sufficient graphical conditions for affine formation control this paper introduces a new multi agent control problem called an affine formation control problem with the objective of asymptotically reaching a configuration that preserves collinearity and ratios of distances with respect to a target configuration suppose each agent updates its own state using a weighted sum of its neighbor s relative states with possibly negative weights then the affine control problem can be solved for either undirected or directed interaction graphs it is shown in this paper that an affine formation is stabilizable over an undirected graph if and only if the undirected graph is universally rigid while an affine formation is stabilizable over a directed graph in the d dimensional space if and only if the directed graph is d 1 rooted rigorous analysis is provided mainly relying on laplacian associated with the interaction graph which contain both positive and negative weights

exponential stability for formation control systems with generalized controllers a unified approach this paper discusses generalized controllers for distance based rigid formation shape stabilization and aims to provide a unified approach for the convergence analysis we consider two types of formation control systems according to different characterizations of target formations minimally rigid target formation and non minimally rigid target formation for the former case we firstly prove the local exponential stability for rigid formation systems when using a general form of shape controllers with certain properties from this viewpoint different formation controllers proposed in previous literature can be included in a unified framework we then extend the result to the case that the target formation is non minimally rigid and show that exponential stability of the formation system is still guaranteed with generalized controllers

the complex time segal bargmann transform abstract we introduce a new form of the segal bargmann transform for a connected lie group k of compact type we show that the heat kernel t x t 0 x k has a space time analytic continuation to a holomorphic function c z re 0 z k c where k c is the complexification of k the new transform is defined by the integral b f z k c z k 1 f k d k z k c if s 0 and d s s the disk of radius s centered at s this integral defines a holomorphic function on k c for each f l 2 k s we construct a heat kernel density s on k c such that for all s as above b s b l 2 k s is an isometric isomorphism from l 2 k s onto the space of holomorphic functions in l 2 k c s when t s the transform b t t coincides with the one introduced by the second author for compact groups and extended by the first author to groups of compact type when t 0 2 s the transform b s t coincides with the one introduced by the first two authors

control of minimally persistent formations in the plane this paper studies the problem of controlling the shape of a formation of point agents in the plane a model is considered where the distance between certain agent pairs is maintained by one of the agents making up the pair if enough appropriately chosen distances are maintained with the number growing linearly with the number of agents then the shape of the formation will be maintained the detailed question examined in the paper is how one may construct decentralized nonlinear control laws to be operated at each agent that will restore the shape of the formation in the presence of small distortions from the nominal shape using the theory of rigid and persistent graphs the question is answered as it turns out a certain submatrix of a matrix known as the rigidity matrix can be proved to have nonzero leading principal minors which allows the determination of a stabilizing control law

laplace and segal bargmann transforms on hermitian symmetric spaces and orthogonal polynomials let d g k be a complex bounded symmetric domain of tube type in a complex jordan algebra v and let dr j d d be its real form in a formally real euclidean jordan algebra j v dr h l is a bounded realization of the symmetric cone in j we consider representations of h that are gotten by the generalized segal bargmann transform from a unitary g space of holomorphic functions on d to an l2 space on dr we prove that in the unbounded realization the inverse of the unitary part of the restriction map is actually the laplace transform we find the extension to d of the spherical functions on dr and find their expansion in terms of the l spherical polynomials on d which are jack symmetric polynomials we prove that the coefficients are orthogonal polynomials in an l2 space the measure being the harish chandra plancherel measure multiplied by the symbol of the berezin transform we prove the difference equation and recurrence relation for those polynomials by considering the action of the lie algebra and the cayley transform on the polynomials on d finally we use the laplace transform to study generalized laguerre functions on symmetric cones

yang mills theory and the segal bargmann transform we use a variant of the segal bargmann transform to study canonically quantized yang mills theory on a space time cylinder with a compact structure group k the non existent lebesgue measure on the space of connections is approximated by a gaussian measure with large variance the segal bargmann transform is then a unitary map from the l 2 space over the space of connections to a holomorphicl 2 space over the space of complexified connections with a certain gaussian measure this transform is given roughly bye t1a 2 followed by analytic continuation here 1a is the laplacian on the space of connections and is the hamiltonian for the quantized theory on the gauge trivial subspace consisting of functions of the holonomy around the spatial circle the segal bargmann transform becomes e t1 k 2 followed by analytic continuation where 1k is the laplacian for the structure group k this result gives a rigorous meaning to the idea that 1a reduces to 1k on functions of the holonomy by letting the variance of the gaussian measure tend to infinity we recover the standard realization of the quantized yang mills theory on a space time cylinder namely 1 1k is the hamiltonian and l 2 k is the hilbert space as a byproduct of these considera tions we find a new one parameter family of unitary transforms from l 2 k to certain holomorphicl 2 spaces over the complexification of k this family of transformations interpolates between the two previously known unitary transformations our work is motivated by results of landsman and wren and uses probabilistic techniques similar to those of gross and malliavin

rigid formation control of double integrator systems in this chapter we move our focus from formation control systems modelled by single integrators e g those considered in previous chapters to formation systems modelled by double integrators two kinds of double integrator formation systems are considered namely formation stabilization systems and flocking control systems novel observations on the measurement requirement the null space and eigenvalues of the system jacobian matrix will be provided which reveal important properties of system dynamics and the associated convergence results we also establish some new links between single integrator formation systems and double integrator formation systems via a parameterized hamiltonian system which in addition provide novel stability criteria for different equilibria in double integrator formation systems by using available results from single integrator formation systems

similarity solutions for the two dimensional nonstationary ideal mhd equations a similarity analysis of the non linear two dimensional non stationary ideal mhd equations is presented in the case of a magnetic field perpendicular to the isentropic motion of the plasma the authors establish the complete lie algebra of infinitesimal symmetries the laws of conservation are mentioned the similarity method for partial differential equations as a procedure for reducing the number of independent variables is applied repeatedly finally they obtain systems of ordinary differential equations for similarity solutions of the mhd equations considered

the segal bargmann transform for path groups letkbe a connected lie group of compact type and letw k denote the set of continuous paths ink starting at the identity and with time interval 0 1 thenw k forms an infinite dimensional group under the operation of pointwise multiplication let denote the wiener measure onw k we construct an analog of the segal bargmann transform forw k letkcbe the complexification ofk w kc the set of continuous paths inkcstarting at the identity and the wiener measure onw kc our transform is a unitary map ofl2 w k onto the holomorphic subspace ofl2 w kc by analogy with the classical transform our transform is given by convolution with the wiener measure followed by analytic continuation we prove that the transform forw k is nicely related by means of the ito map to the classical segal bargmann transform for the path space in the lie algebra ofk

a subelliptic taylor isomorphism on infinite dimensional heisenberg groups let g denote an infinite dimensional heisenberg like group which is a class of infinite dimensional step 2 stratified lie groups we consider holomorphic functions on g that are square integrable with respect to a heat kernel measure which is formally subelliptic in the sense that all appropriate finite dimensional projections are smooth measures we prove a unitary equivalence between a subclass of these square integrable holomorphic functions and a certain completion of the universal enveloping algebra of the cameron martin lie subalgebra the isomorphism defining the equivalence is given as a composition of restriction and taylor maps

on the kakutani ito segal gross and segal bargmann hall isomorphisms recently gross has shown that the kakutani ito segal isomorphism theorem has an extension from the setting of gaussian measure on a vector space to heat kernel measure pt on a simply connected lie group g of compact type the isomorphism relates l2 pt to a certain completion of the universal enveloping algebra of g lie g gross proves this result using the kakutani ito segal theorem and an infinite dimensional calculus associated to g valued brownian motion hijab has greatly simplified and clarified gross proof hiiab s proof avoids most but not all of the infinite dimensional analysis in the original proof in this paper we will build on hijab s proof to give a completely finite dimensional non probabilistic proof of gross isomorphism theorem the proof given here relies heavily on hall s beautiful extension of the segal bargmann transformation to the setting of compact lie groups this theorem relating l2 pt to a certain l2 space of holomorphic functions l2 gc h on the complexified lie group gc will also be generalized to lie groups of compact type in the process it is shown how to characterize in terms of summability conditions on all the derivatives at the identity in gc those holomorphic functions which are in l2 gc

global stabilization of rigid formations in the plane this paper considers the problem of distributed control of rigid formation shapes in the plane for multi agent systems a constructive perturbation method is proposed and combined with the conventional gradient control law the proposed control law stabilizes the desired rigid formation in a global sense for all initial conditions except the case when a pair of communicating agents happen to have the same initial location it also avoids collisions between any two communicating agents during the motion simulation results are provided to illustrate the effectiveness of the control algorithm

symmetry groups and similarity solutions of mhd equations a similarity analysis of the nonlinear nonstationary real mhd equations is presented for several cases of real mhd the complete lie algebra of infinitesimal symmetries is calculated using subalgebras of this lie algebra reduced systems for similarity solutions can be obtained some similarity solutions of the ideal mhd equations are discussed

the pulsrodon in 2 1 dimensional magneto gasdynamics hamiltonian structure and integrability an elliptic vortex type ansatz introduced into a 2 1 dimensional system governing rotating homentropic magneto gasdynamics with a parabolic gas law is shown to lead to a finite dimensional nonlinear dynamical system which admits exact analytical solution in terms of an elliptic function and integral representation the dynamical system is demonstrated to be hamiltonian and equivalent to the stationary reduction of the integrable nonlinear schrodinger equation coupled with a steen ermakov pinney equation a novel magneto gasdynamic analogue of the pulsrodon of shallow water f plane theory is isolated thereby confined and time periodic magneto gasdynamic flows are constructed explicitly

a class of accelerated solutions of the two dimensional ideal magnetohydrodynamic equations a particular class of exact solutions of the two dimensional time dependent magnetohydrodynamic equations for an ideal isothermal plasma is presented the associated flows have non vanishing acceleration a special form of the mapping between eulerian and lagrangian coordinates is assumed and the acceleration term has to have the form of a potential force the class of potentials compatible with these assumptions is derived and the constraints imposed by the vanishing resistivity then lead to a restriction of the admissible flow fields some explicit solutions are constructed and their properties are investigated

square integrable holomorphic functions on infinite dimensional heisenberg type groups we introduce a class of non commutative complex infinite dimensional heisenberg like lie groups based on an abstract wiener space the holomorphic functions which are also square integrable with respect to a heat kernel measure on these groups are studied in particular we establish a unitary equivalence between the square integrable holomorphic functions and a certain completion of the universal enveloping algebra of the lie algebra of this class of groups using quasi invariance of the heat kernel measure we also construct a skeleton map which characterizes globally defined functions from the l 2 closure of holomorphic polynomials by their values on the cameron martin subgroup

geometric quantization and the generalized segal bargmann transform for lie groups of compact type let k be a connected lie group of compact type and let t k be its cotangent bundle this paper considers geometric quantization of t k first using the vertical polarization and then using a natural kahler polarization obtained by identifying t k with the complexified group k the first main result is that the hilbert space obtained by using the kahler polarization is naturally identifiable with the generalized segal bargmann space introduced by the author from a different point of view namely that of heat kernels the second main result is that the pairing map of geometric quantization coincides with the generalized segal bargmann transform introduced by the author this means that the pairing map in this case is a constant multiple of a unitary map for both results it is essential that the half form correction be included when using the kahler polarization

distributed formation stabilization using relative position measurements in local coordinates in this paper we present a novel distributed method to stabilize a set of agents moving in a two dimensional environment to a desired rigid formation in our approach each agent computes its control input using the relative positions of a set of formation neighbors but contrary to most existing works this information is expressed in the agent s own independent local coordinate frame without requiring any common reference the controller is based on the minimization of a lyapunov function that includes locally computed rotation matrices which are required due to the absence of a common orientation our contribution is that the proposed distributed coordinate free method achieves global stabilization to a rigid formation with the agents using only partial information of the team does not require any leader units and is applicable to both single integrator or unicycle agents to guarantee global stability we require that the network induced by the agent interactions belongs to a certain class of undirected rigid graphs in two dimensions which we explicitly characterize the performance of the proposed method is illustrated with numerical simulations

holomorphic sobolev spaces and the generalized segal bargmann transform we consider the generalized segal bargmann transform ct for a compact group k introduced in b c hall j funct anal 122 1994 103 151 let kc denote the complexification of k we give a necessary and sufficient pointwise growth condition for a holomorphic function on kc to be in the image under ct of c 1 k we also characterize the image under ct of sobolev spaces on k the proofs make use of a holomorphic version of the sobolev embedding theorem

three dimensional formation merging control under directed and switching topologies this paper concentrates on the formation merging control problem for a leader follower network the objective is to control a team of agents called followers such that they are merged with another team of agents called leaders to form a single globally rigid formation a method based on graph laplacian is introduced to address this problem each follower selects its interaction neighbors and interaction weights according to the given target configuration the graph modeling the interaction topology of all the agents is directed and time varying first by assuming that the synchronized velocity of the leaders is known to all the followers a necessary and sufficient condition is derived to ensure uniform asymptotic formation merging second we relax this assumption and consider that the velocity of the leaders is known to only a subset of followers for which the same necessary and sufficient condition is obtained with the help of an internal model for velocity synchronization

heat kernel analysis and cameron martin subgroup for infinite dimensional groups abstract the heat kernel measure t is constructed on gl h the group of invertible operators on a complex hilbert space h this measure is determined by an infinite dimensional lie algebra g and a hermitian inner product on it the cameron martin subgroup g cm is defined and its properties are discussed in particular there is an isometry from the l 2 t closure of holomorphic polynomials into a space h t g cm of functions holomorphic on g cm this means that any element from this l 2 t closure of holomorphic polynomials has a version holomorphic on g cm in addition there is an isometry from h t g cm into a hilbert space associated with the tensor algebra over g the latter isometry is an infinite dimensional analog of the taylor expansion as examples we discuss a complex orthogonal group and a complex symplectic group

the large n limit of the segal bargmann transform on un we study the two parameter segal bargmann transform b n on the unitary group un for large n acting on matrix valued functions that are equivariant under the adjoint action of the group the transform has a meaningful limit gs t as n which can be identified as an operator on the space of complex laurent polynomials we introduce the space of trace polynomials and use it to give effective computational methods to determine the action of the heat operator and thus the segal bargmann transform we prove several concentration of measure and limit theorems giving a direct connection from the finite dimensional transform b n to its limit gs t we characterize the operator gs t through its inverse action on the standard polynomial basis finally we show that in the case s t the limit transform gt t is the free hall transform g t introduced by biane

stabilisation of infinitesimally rigid formations of multi robot networks this article considers the design of a formation control for multivehicle systems that uses only local information the control is derived from a potential function based on an undirected infinitesimally rigid graph that specifies the target formation a potential function is obtained from the graph from which a gradient control is derived under this controller the target formation becomes a manifold of equilibria for the multivehicle system it is shown that infinitesimal rigidity is a sufficient condition for local asymptotical stability of the equilibrium manifold a complete study of the stability of the regular polygon formation is presented and results for directed graphs are presented as well finally the controller is validated experimentally

free convolution operators and free hall transform abstract we define an extension of the polynomial calculus on a w probability space by introducing an algebra c x i i i which contains polynomials this extension allows us to define transition operators for additive and multiplicative free convolution it also permits us to characterize the free segal bargmann transform and the free hall transform introduced by biane in a manner which is closer to classical definitions finally we use this extension of polynomial calculus to prove two asymptotic results on random matrices the convergence for each fixed time as n tends to of the distribution of the brownian motion on the linear group gl n c to the distribution of a free multiplicative circular brownian motion and the convergence of the classical hall transform on u n to the free hall transform

inverse eigenvalue problems a collection of inverse eigenvalue problems are identified and classified according to their characteristics current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper this exposition also reveals many open questions that deserve further study an extensive bibliography of pertinent literature is attached

time varying formation control for general linear multi agent systems with switching directed topologies time varying formation analysis and design problems for multi agent systems with general linear dynamics and switching directed interaction topologies are investigated different from the previous results the formation in this paper can be defined by specified piecewise continuously differentiable vectors and the switching topologies are directed firstly necessary and sufficient conditions for general linear multi agent systems with switching directed topologies to achieve time varying formations are proposed where a description of the feasible time varying formation set and approaches to expand the feasible formation set are given then an explicit expression of the time varying formation reference function is derived to describe the macroscopic movement of the whole formation an approach to assign the motion modes of the formation reference is provided moreover an algorithm consisting of four steps to design the formation protocol is presented in the case where the given time varying formation belongs to the feasible formation set it is proven that by designing the formation protocol using the proposed algorithm time varying formation can be achieved by multi agent systems with general linear dynamics and switching directed topologies if the dwell time is larger than a positive threshold finally numerical simulations are presented to demonstrate the effectiveness of the theoretical results

lyapunov stability theory of nonsmooth systems this paper develops nonsmooth lyapunov stability theory and lasalle s invariance principle for a class of nonsmooth lipschitz continuous lyapunov functions and absolutely continuous state trajectories computable tests based on filipov s differential inclusion and clarke s generalized gradient are derived the primary use of these results is in analyzing the stability of equilibria of differential equations with discontinuous right hand side such as in nonsmooth dynamic systems or variable structure control

minimal time for the continuity equation controlled by a localized perturbation of the velocity vector field in this work we study the minimal time to steer a given crowd to a desired configuration the control is a vector field representing a perturbation of the crowd velocity localized on a fixed control set we will assume that there is no interaction between the agents we give a characterization of the minimal time both for microscopic and macroscopic descriptions of a crowd we show that the minimal time to steer one initial configuration to another is related to the condition of having enough mass in the control region to feed the desired final configuration the construction of the control is explicit providing a numerical algorithm for computing it we finally give some numerical simulations