controllability on principal fibre bundles with compact structure group abstract in this paper we consider the controllability problem for systems defined on principal fibre bundles in which the set of associated vector fields is invariant under the action of the structure group and projects down on to a set of vector fields on the base space it is shown that if the system is accessible and the structure group is compact then the system is controllable if and only if the system is controllable when projected onto the base space the connection between these systems and systems admitting symmetries generated by free and proper group actions is described and various examples are given relating to systems on homogeneous spaces

a new stability analysis of delayed cellular neural networks in this paper the global asymptotic stability of delayed cellular neural networks dcnns is investigated a novel criterion for the stability using the lyapunov stability theory and linear matrix inequality lmi framework is presented the criterion expressed by lmis is delay dependent the result is less conservative than those established in the earlier references a numerical example is given to show the effectiveness of proposed method

on the linearization via a restricted class of dynamic feedback we deal with the problem of linearization of nonlinear systems using a restricted class of dynamic compensator which is composed of chains of integrators followed by static feedback in the vector field formulation we present an upper bound on the number of integrators to linearize nonlinear systems and show that the bound is sharp when n 2 spl ges m spl ges 2 this results in the checkable necessary and sufficient conditions

optimal controllers and output feedback stabilization abstract in this paper the output feedback stabilizability problem is explored in terms of control lyapunov functions sufficient conditions for stabilization are provided for a certain class of systems by means of output feedback stabilizers that can be obtained from an optimization problem our main results extends those developed in 31 and generalize a theorem due to sontag 23

optimal control of systems possessing symmetries it is shown that a symmetry in an optimization problem induces a decomposition of the optimal feedback control law into factors one factor can be calculated algebraically and depends only on the symmetry the other factor corresponds to a lower dimensional optimization problem this gives a priori information about the structure of the optimal feedback control law and indicates a possibly more efficient method for optimizing such systems

analytical existence of solutions to a system of nonlinear equations with application we study the existence of analytical solutions to a system of nonlinear equations under constraints linked to the analysis of a road safety measure without computing second derivatives we formally demonstrate this existence of solutions by applying a matrix inversion principle through schur complement to a subsystem of equations derived from the main system the analytical results thus obtained are used to construct a simple iterative procedure to look for optimal solutions as well as an initial solution adapted to data of each case study we illustrate our results with simulated accident data obtained from the setting up of a road safety measure the numerical solutions thus obtained are then compared to those given through a classic newton raphson type approach directly applied to the main system

quotients of fully nonlinear control systems in this paper we introduce and study quotients of fully nonlinear control systems our definition is inspired by categorical definitions of quotients as well as recent work on abstractions of affine control systems we show that quotients exist under mild regularity assumptions and characterize the structure of the quotient state input space this allows one to understand how states and inputs of the quotient system are related to states and inputs of the original system we also introduce a notion of projectability which turns out to be equivalent to controlled invariance this allows one to regard previous work on symmetries partial symmetries and controlled invariance as leading to special types of quotients we also show the existence of quotients that are not induced by symmetries or controlled invariance such decompositions have a potential use in a theory of hierarchical control based on quotients

stabilization of hamiltonian systems with nonholonomic constraints based on time varying generalized canonical transformations this paper is concerned with the stabilization of nonholonomic systems in port controlled hamiltonian formulae based on time varying generalized canonical transformations a special class of time varying generalized canonical transformations are introduced which modify the kinetic energy of the original system without changing the generalized hamiltonian structure with passivity utilizing these transformations time varying asymptotically stabilizing controllers for the nonholonomic hamiltonian systems are derived since the proposed method is a natural generalization of passivity based control for conventional holonomic systems it is expected that the tools developed for conventional systems will be applicable to nonholonomic systems based on the proposed method

approximate reduction of dynamic systems abstract the reduction of dynamic systems has a rich history with many important applications related to stability control and verification reduction of nonlinear systems is typically performed in an exact manner as is the case with mechanical systems with symmetry which unfortunately limits the type of systems to which it can be applied the goal of this paper is to consider a more general form of reduction termed approximate reduction in order to extend the class of systems that can be reduced using notions related to incremental stability we give conditions on when a dynamic system can be projected to a lower dimensional space while providing hard bounds on the induced errors i e when it is behaviourally similar to a dynamic system on a lower dimensional space these concepts are illustrated on a series of examples

controllability and observability for affine nonlinear hamiltonian systems it is shown that an affine nonlinear hamiltonian system is controllable if and only if it is observable in the sense that strong accessibility implies local weak observability and vice versa furthermore it is shown that a nonminimal hamiltonian system can be reduced to a locally weakly observable and strongly accessible system in such a way that the reduced system is again hamiltonian

controlled lagrangians and the stabilization of mechanical systems ii potential shaping for pt i see ibid vol 45 p 2253 70 2000 we extend the method of controlled lagrangians cl to include potential shaping which achieves complete state space asymptotic stabilization of mechanical systems the cl method deals with mechanical systems with symmetry and provides symmetry preserving kinetic shaping and feedback controlled dissipation for state space stabilization in all but the symmetry variables potential shaping complements the kinetic shaping by breaking symmetry and stabilizing the remaining state variables the approach also extends the method of controlled lagrangians to include a class of mechanical systems without symmetry such as the inverted pendulum on a cart that travels along an incline

the structure of nonlinear control systems possessing symmetries a concept of symmetry is defined for general nonlinear control systems it is shown under various technical conditions that nonlinear control systems with symmetries admit local and or global decompositions in terms of lower dimensional subsystems and feedback loops the structure of the individual subsystems is dependent on the structure of the symmetry group for example if the symmetry group is abelian one of the subsystems is a quadrature an additional feature of the decomposition is that the state space dimensions of the subsystems sum to the state space dimension of the original system

bifurcations stability and monotonicity properties of a delayed neural network model abstract a delay differential equation modelling an artificial neural network with two neurons is investigated a linear stability analysis provides parameter values yielding asymptotic stability of the stationary solutions these can lose stability through either a pitchfork or a hopf bifurcation which is shown to be supercritical at appropriate parameter values an interaction takes place between the pitchfork and hopf bifurcations conditions are also given for the set of initial conditions that converge to a stable stationary solution to be open and dense in the functional phase space analytic results are illustrated with numerical simulations

controllability distributions for nonlinear control systems the purpose of this paper is to relate the notion of controlled invariance to a notion of controllability for nonlinear control systems by means of the linear ideas of a subsystem and a quotient system we derive a very structured description of nonlinear systems

asymptotic feedback stabilization a sufficient condition for the existence of control lyapunov functions abstract in this paper we study the feedback stabilization problem for a wide class of nonlinear systems that are affine in the control we offer sufficient conditions for the existence of control lyapunov functions that according to 3 23 and 28 30 guarantee stabilization at a specified equilibrium by means of a feedback law which is smooth except possibly at the equilibrium we note that the results of the paper present a local nature

energy and entropy based stabilization for lossless dynamical systems via hybrid controllers a novel class of dynamic energy based hybrid controllers is proposed as a means for achieving enhanced energy dissipation in lossless dynamical systems these dynamic controllers combine a logical switching architecture with continuous dynamics to guarantee that the system plant energy is strictly decreasing across switchings the general framework leads to closed loop systems described by impulsive differential equations in addition we construct hybrid dynamic controllers that guarantee that the closed loop system is consistent with basic thermodynamic principles in particular the existence of an entropy function for the closed loop system is established that satisfies a hybrid clausius type inequality special cases of energy based and entropy based hybrid controllers involving state dependent switching are described

robust stability analysis for interval cohen grossberg neural networks with unknown time varying delays in this paper robust stability problems for interval cohen grossberg neural networks with unknown time varying delays are investigated using linear matrix inequality m matrix theory and halanay inequality techniques new sufficient conditions independent of time varying delays are derived to guarantee the uniqueness and the global robust stability of the equilibrium point of interval cohen grossberg neural networks with time varying delays all these results have no restriction on the rate of change of the time varying delays compared to some existing results these new criteria are less conservative and are more convenient to check two numerical examples are used to show the effectiveness of the present results

on some necessary and sufficient conditions for a recurrent neural network model with time delays to generate oscillations in this paper the existence of oscillations for a class of recurrent neural networks with time delays between neural interconnections is investigated by using the fixed point theory and liapunov functional we prove that a recurrent neural network might have a unique equilibrium point which is unstable this particular type of instability combined with the boundedness of the solutions of the system will force the network to generate a permanent oscillation some necessary and sufficient conditions for these oscillations are obtained simple and practical criteria for fixing the range of parameters in this network are also derived typical simulation examples are presented

zeros at infinity for affine nonlinear control systems a definition of zeros at infinity for affine nonlinear control systems is proposed the definition is local which means that we exclude certain singularities we argue the reasonableness of our definition by showing its relevance to the problem of nonlinear decoupling in particular we give a necessary and sufficient condition for the solvability of the general regular decoupling problem for affine systems in terms of the zeros at infinity

symmetric failures in symmetric control systems abstract this paper discusses the fault tolerance of symmetric systems with respect to controllability which is a fundamental characteristic of control systems in particular we reveal the underlying mathematical mechanism of the loss of controllability for symmetric systems induced by failures based on the decomposition of the symmetric systems into subsystems under the symmetry the controllability of the entire system can be discussed by checking that of each subsystem the analysis of the fault tolerance in this paper is an extension of this idea with the aid of the chain adapted transformation matrix for the decomposition the result is shown as a necessary condition for symmetric systems to retain the controllability despite some symmetric failures we also discuss sufficient conditions

on the almost periodic solution of cellular neural networks with distributed delays by exponential dichotomy about differential equations a formal almost periodic solution aps of a class of cellular neural networks cnns with distributed delays is obtained then within different normed spaces several sufficient conditions guaranteeing the existence and uniqueness of an aps are proposed using two fixed point theorems based on the continuity property and some inequality techniques two theorems insuring the global stability of the unique aps are given comparing with known literatures all conclusions are drawn with slacker restrictions e g do not require the integral of the kernel function determining the distributed delays from zero to positive infinity to be one and the activation functions to be bounded etc besides all criteria are obtained by different ways finally two illustrative examples show the validity and that all criteria are easy to check and apply

symmetry and bifurcations of momentum mappings the zero set of a momentum mapping is shown to have a singularity at each point with symmetry the zero set is diffeomorphic to the product of a manifold and the zero set of a homogeneous quadratic function the proof uses the kuranishi theory of deformations among the applications it is shown that the set of all solutions of the yang mills equations on a lorentz manifold has a singularity at any solution with symmetry in the sense of a pure gauge symmetry similarly the set of solutions of einstein s equations has a singularity at any solution that has spacelike killing fields provided the spacetime has a compact cauchy surface

a new approach for instantaneous pole placement with recurrent neural networks and its application in control of nonlinear time varying systems in this paper we first show that online computation of feedback gain used for pole placement of nonlinear systems in recent years is not reliable and then we present a new approach for instantaneous pole placement and apply it with dynamical recurrent neural networks for online computation of feedback gain because of high speed convergence of neural network to feedback gain we can apply this method for pole placement of nonlinear time varying systems one strategy for realization of this method is instantaneous linearization as we do here by simulation the advantage of the proposed method is a global uniform asymptotical exponential stability guaes of closed loop system around the equilibrium point

disturbance decoupling by observation feedback for hamiltonian systems in this note we show that if disturbance decoupling by dynamic observation feedback for a linear hamiltonian system is possible it is also possible using a hamiltonian compensator this gives an example of a theory in which we look for solutions of control problems which remain in the same class of systems

energy dissipating hybrid control for impulsive dynamical systems abstract a novel class of fixed order energy based hybrid controllers is proposed as a means for achieving enhanced energy dissipation in nonsmooth euler lagrange hybrid port controlled hamiltonian and lossless impulsive dynamical systems these dynamic controllers combine a logical switching architecture with hybrid dynamics to guarantee that the system plant energy is strictly decreasing across switchings the general framework leads to hybrid closed loop systems described by impulsive differential equations special cases of energy based hybrid controllers involving state dependent switching are described and an illustrative numerical example is given to demonstrate the efficacy of the proposed approach

reduction and non linear controllability of symmetric distributed systems this paper considers the reduction problem for distributed control systems in particular we consider controllability of systems containing multiple instances of diffeomorphic components where the overall system dynamics is invariant with respect to a discrete group action a subclass of such systems are systems with a set of identical components where the overall system dynamics are invariant with respect to physically interchanging these components the main result is a proposition which shows that for an equivalence class of symmetric systems of this type controllability of the entire class of systems can be determined by analysing the member of the equivalence class with the smallest state space the reduction methods developed are illustrated by considering the controllability of a team of mobile robots and a platoon of underwater vehicles

stabilization of relative equilibria for underactuated systems on riemannian manifolds steady motions of mechanical systems with symmetry are stabilized via a potential shaping and damping control a general methodology based on riemannian geometry tools provides exponential stabilization over the full phase space

schur complements and statistics abstract in this paper we discuss various properties of matrices of the type s h ge 1 f which we call the schur complement of e in a e f g h the matrix e is assumed to be nonsingular when e is singular or rectangular we consider the generalized schur complements s h ge f where e is a generalized inverse of e a comprehensive account of results pertaining to the determinant the rank the inverse and generalized inverses of partitioned matrices and the inertia of a matrix is given both for schur complements and for generalized schur complements we survey the known results in a historical perspective and obtain several extensions numerous applications in numerical analysis and statistics are included the paper ends with an exhaustive bibliography of books and articles related to schur complements

absolute exponential stability of recurrent neural networks with generalized activation function in this paper the recurrent neural networks rnns with a generalized activation function class is proposed in this proposed model every component of the neuron s activation function belongs to a convex hull which is bounded by two odd symmetric piecewise linear functions that are convex or concave over the real space all of the convex hulls are composed of generalized activation function classes the novel activation function class is not only with a more flexible and more specific description of the activation functions than other function classes but it also generalizes some traditional activation function classes the absolute exponential stability aest of the rnn with a generalized activation function class is studied through three steps the first step is to demonstrate the global exponential stability ges of the equilibrium point of original rnn with a generalized activation function being equivalent to that of rnn under all vertex functions of convex hull the second step transforms the rnn under every vertex activation function into neural networks under an array of saturated linear activation functions because the ges of the equilibrium point of three systems are equivalent the next stability analysis focuses on the ges of the equilibrium point of rnn system under an array of saturated linear activation functions the last step is to study both the existence of equilibrium point and the ges of the rnn under saturated linear activation functions using the theory of m matrix in the end a two neuron rnn with a generalized activation function is constructed to show the effectiveness of our results

a remark on nonlinear accessibility conditions and infinite prolongations the lie rank condition for strong nonlinear accessibility is interpreted via the differential geometry of jets and prolongations of infinite order it yields to a new lie algebraic criterion and to the consideration of first integrals which apply to nonlinear systems in quite general form the latter characterization in particular is valid without distinguishing between the system variables and is independent of time scalings weak accessibility is also considered

necessary and sufficient conditions for state equivalence to a nonlinear discrete time observer canonical form in this technical note we obtain necessary and sufficient conditions for a multi input multi output discrete time nonlinear system to be state equivalent to a nonlinear observer form and for an uncontrolled multi output system to be state equivalent to a linear observer form we adopt a geometric approach and the proofs are constructive with respect to the required coordinate change

observability and controllability for smooth nonlinear systems the definition of a smooth nonlinear system as proposed recently by willems is elaborated as a natural generalization of the more common definitions of a smooth nonlinear input output system mini

global bilinearization of non linear systems and the existence of volterra series a global volterra series is developed for an analytic system defined on an analytic manifold m local bilinearizations are pieced together using the theory of fibre bundles giving rise to twisted bilinear systems denned globally on m

controlled invariance for nonlinear systems necessary and sufficient conditions are derived for a b invariance called here controlled invariance for nonlinear systems dot x f x u the obtained results generalize and elucidate already known results about systems dot x a x sum min i 1 max m u i b i x a new and direct differential geometric interpretation of the concept of controlled invariance and the derived conditions is given

stabilizability of nonholonomic control systems abstract we consider here the problem of stabilization of nonlinear hamiltonian control systems with nonholonomic constraints such systems are of interest in various applications including the control of robotic systems we discuss the formulation of hamiltonian systems with nonholonomic constraints in particular considering the differences between the dirac theory of constraints the classical theory of nonholonomic constraints and weber s theory of generalized nonholonomic constraints stabilization of systems with these constraints is discussed using the theory of hamiltonian control systems and a reduced stabilization criterion is given

controlled invariance by static output feedback for nonlinear systems the paper deals with the notion of static output feedback for nonlinear systems necessary and sufficient conditions are derived for c a b invariance here called measured controlled invariance for nonlinear control systems

stabilization and feedback equivalence to linear coupled oscillators sufficient conditions for the system xdot f x u t g x x r2n to be stabilized via a smooth state feedback are given it is also shown that certain lie algebraic conditions are necessary and sufficient for the system to be feedback equivalent to a system of controlled linear coupled oscillators which is feedback stabilizable and which is proposed as a reference model for stabilization problems two overlapping sets of sufficient conditions which guarantee the existence and the explicit construction of stabilizing smooth state feedback controls are therefore given and their connections are discussed

abstractions of hamiltonian control systems given a control system and a desired property an abstracted system is a reduced system that preserves the property of interest while ignoring modeling detail in previous work abstractions of linear and nonlinear control systems were considered while preserving reachability properties in this paper we consider the abstraction problem for hamiltonian control systems where in addition to the property of interest we also preserve the hamiltonian structure of the control system we show how the hamiltonian structure of control systems can be exploited to simplify the abstraction process we then focus on local accessibility preserving abstractions and provide conditions under which local accessibility properties of the abstracted hamiltonian system are equivalent to the local accessibility properties of the original hamiltonian control system

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

the analysis of zero dynamics of nonlinear control systems with symmetries zero dynamics of nonlinear control systems with symmetries are studied in this paper it is shown that the zero dynamics of a symmetric system is also symmetric and admits a special form of cascade decomposition a local zero dynamics is always possible to be extended to the semiglobal case the relation between the zero dynamics and the quotient system is also discussed

rotational motion control of a spacecraft this note describes a systematic procedure for the control synthesis of a rigid spacecraft using the energy shaping method the geometric concept of a mechanical system in a coordinate independent form is used to derive a control algorithm for the euler poincare equations the main result of this note is a specialization of the method on the unit quaternion group this note is concluded with the examples of the potential functions and implementation for the three axis attitude control problem

when is a symplectic quotient an orbifold let k be a compact lie group of positive dimension we show that for most unitary k modules the corresponding symplectic quotient is not regularly symplectomorphic to a linear symplectic orbifold the quotient of a unitary module of a finite group when k is connected we show that even a symplectomorphism to a linear symplectic orbifold does not exist our results yield conditions that preclude the symplectic quotient of a hamiltonian k manifold from being locally isomorphic to an orbifold as an application we determine which unitary su2 modules yield symplectic quotients that are z graded regularly symplectomorphic to a linear symplectic orbifold we similarly determine which unitary circle representations yield symplectic quotients that admit a regular diffeomorphism to a linear symplectic orbifold

equivalence of nonlinear systems to input output prime forms the problem of transforming nonlinear control systems into input output prime forms is dealt with using state space static state feedback and also output space transformations necessary and sufficient geometric conditions for the solvability of this problem are obtained the results obtained generalize well known results both on feedback linearization as well as input output decoupling of nonlinear systems it turns out that from a computational point of view the output space transformation is the crucial step that is performed by constructing rectifying coordinates for a nested sequence of distributions on the output manifold

decomposable representations and lagrangian submanifolds of moduli spaces associated to surface groups the importance of explicit examples of lagrangian submanifolds of moduli spaces is revealed by papers such as dostoglou and salamon ann of math 2 139 3 581 640 1994 and salamon proceedings of the international congress of mathematicians vol 1 2 zurich 1994 pp 526 536 birkhauser basel 1995 given a 3 manifold m with boundary m dostoglou and salamon use such examples to obtain a proof of the atiyah floer conjecture relating the symplectic floer homology of the representation space hom 1 m u u associated to an explicit pair of lagrangian submanifolds of this representation space and the instanton homology of the 3 manifold m in the present paper we construct a lagrangian submanifold of the space of representations mathcal m g l hom mathcal c pi g l u u of the fundamental group g lof a punctured riemann surface g l into an arbitrary compact connected lie group u this lagrangian submanifold is obtained as the fixed point set of an anti symplectic involution hat beta defined on mathcal m g l we show that the involution hat beta is induced by a form reversing involution defined on the quasi hamiltonian space u times u g times mathcal c 1 times cdots times mathcal c l the fact that hat beta has a non empty fixed point set is a consequence of the real convexity theorem for group valued momentum maps proved in schaffhauser a real convexity theorem for quasi hamiltonian actions submitted 25 p 2007 http arxiv org abs math 0705 0858 the notion of decomposable representation provides a geometric interpretation of the lagrangian submanifold thus obtained

schur complements and its applications to symmetric nonnegative and z matrices in linear algebra appl 177 1992 137 smith proved that if h is a hermitian semidefinite matrix and a is a nonsingular principal submatrix then the eigenvalues of the schur complement h a interlace those of h in this paper we refine the latter result and use it to derive eigenvalues interlacing results on an irreducible symmetric nonnegative matrix that involve perron complements for an irreducible symmetric nonnegative matrix we give lower and upper bounds for its spectral radius and also a lower bound for the maximal spectral radius of its principal submatrices of a fixed order we apply our results to an irreducible symmetric z matrix and to the adjacency matrix or the general laplacian matrix of a connected weighted graph the equality cases for the bounds for spectral radii or least eigenvalues are also examined

controllability canonical phase variable form for non linear time variable systems a controllability canonical form for non linear time variable systems with a control variable appearing linearly is introduced the global controllability of this nonlinear phase variable form is given independently of the non linearities a transformation to the controllability canonical form requires the regularity of a matrix which can be considered the controllability matrix of a non linear time variable system a problem which still remains unsolved is the actual computation of the non linear controllability canonical form for a given system

a schur complement approach for computing subcovariance matrices arising in a road safety measure modelling this paper deals with the determination of the analytical expression of a real partitioned matrix inverse a formal inversion method is suggested through schur complements and the elements of any inverse matrix block are explained irrespective of the size of the considered matrix the suggested approach does not need any matrix inversion numerical program this methodology is applied to a fisher information matrix relating to a multidimensional modelling of road accident data when a road safety measure is applied on different experiment sites an example of formal calculation and interpretation is given to support our approach

symmetries conservation laws and time reversibility for hamiltonian systems with external forces a system theoretic framework is given for the description of hamiltonian systems with external forces and partial observations of the state it is shown how symmetries and conservation laws can be defined within this framework a generalization of noether s theorem is obtained finally a precise definition of time reversibility is given and its consequences are explored

symmetry reduction of optimal control systems and principal connections this paper explores the role of symmetries and reduction in nonlinear control and optimal control systems the focus of the paper is to give a geometric framework of symmetry reduction of optimal control systems as well as to show how to obtain explicit expressions of the reduced system by exploiting the geometry in particular we show how to obtain a principal connection to be used in the reduction for various choices of symmetry groups as opposed to assuming such a principal connection is given or choosing a particular symmetry group to simplify the setting our result synthesizes some previous works on symmetry reduction of nonlinear control and optimal control systems affine and kinematic optimal control systems are of particular interest we explicitly work out the details for such systems and also show a few examples of symmetry reduction of kinematic optimal control problems

nonlinear observability and duality developing the method introduced by hermann and krener 3 we construct quasi minimal realizations for complete systems solving the eventual non regularity in the c case the results obtained are dual of those known for the accessibility problem

time reversible hamiltonian systems it is shown that transfer matrices satisfying g s g s g t t s have a minimal hamiltonian realization with an energy which is the sum of potential and kinetic energy yielding the time reversibility of the equations furthermore connections are made with an associated gradient system the theory is applied to electrical networks resulting in a characterization of lct networks

linearization and input output decoupling for general nonlinear systems necessary and sufficient conditions are obtained for the linearization and input decoupling by state feedback of general nonlinear systems it is shown how these conditions can be derived from the already known conditions for affine nonlinear systems thereby also elucidating the existing theory for affine systems

reduction of hamiltonian systems with symmetry abstract this paper presents an alternative phase space reduction process for hamiltonian systems with symmetry using a lie algebra of infinitesimal symmetries instead of a lie group of symmetries this approach avoids the use of the various conditions on the action of the group or of the isotropy subgroup necessary for the usual construction of a reduced phase space quotient with a globally defined symplectic manifold structure the existence of such a reduced phase space is proved without recourse to the existence of an equivariant moment map instead it is assumed that the orbits of the action associated to the lie algebra can be described as the level sets of some analytic map an assumption that holds in many of the known examples the relation between the two reduction processes is discussed in particular some cases where the marsden weinstein process leads to a quotient space which is not a smooth manifold are shown to be amenable to the method now proposed the natural control theoretic interpretation of the techniques used is also discussed

equilibria and their bifurcations in a recurrent neural network involving iterates of a transcendental function some practical models contain so complicated mathematical expressions that it is hard to determine the number and distribution of all equilibria not mentioning the qualitative properties and bifurcations of those equilibria the three node recurrent neural network system with two free weight parameters originally introduced by ruiz owens and townley in 1997 is such a system for which the equation of equilibria involves transcendental function and its iterates not computing coordinates of its equilibria in this paper we display an effective technique to determine the number and distribution of its equilibria without full information about equilibria our method enables to further study qualitative properties of those equilibria and discuss their saddle node pitchfork and hopf bifurcations by approximating center manifolds

controlled lagrangians and the stabilization of mechanical systems i the first matching theorem we develop a method for the stabilization of mechanical systems with symmetry based on the technique of controlled lagrangians the procedure involves making structured modifications to the lagrangian for the uncontrolled system thereby constructing the controlled lagrangian the euler lagrange equations derived from the controlled lagrangian describe the closed loop system where new terms in these equations are identified with control forces since the controlled system is lagrangian by construction energy methods can be used to find control gains that yield closed loop stability we use kinetic shaping to preserve symmetry and only stabilize systems module the symmetry group the procedure is demonstrated for several underactuated balance problems including the stabilization of an inverted planar pendulum on a cart moving on a line and an inverted spherical pendulum on a cart moving in the plane

equations of motion for hamiltonian systems with constraints in this paper the problem of obtaining the equations of motion for hamiltonian systems with constraints is considered conditions are given which ensure that the phase space points satisfying the primary and secondary constraints form a symplectic manifold on which the resulting equations of motion are hamiltonian and uniquely determined

the fisher information matrix for linear systems estimation of parameters of linear systems is a problem often encountered in applications the cramer rao lower bound gives a lower bound on the variance of any unbiased parameter estimation method and therefore provides an important tool in the assessment of a parameter estimation method and for experimental design here we study the calculation of the fisher information matrix the inverse of the cramer rao lower bound from a system theoretic point of view a number of results appear in the literature that deal with the case where the stationary data is given as the output of a linear system driven by gaussian noise the non stationary situation where the data is the output of a linear system with gaussian measurement noise is rarely considered despite its importance in applications a general description will be given for fisher information for such data in terms of a derivative system for a uniformly sampled data set of impulse response type a closed form expression can be given for the fisher information using the solution of a lyapunov equation

improved delay dependent asymptotic stability criteria for delayed neural networks this brief is concerned with asymptotic stability of neural networks with uncertain delays two types of uncertain delays are considered one is constant while the other is time varying the discretized lyapunov krasovskii functional lkf method is integrated with the technique of introducing the free weighting matrix between the terms of the leibniz newton formula the integrated method leads to the establishment of new delay dependent sufficient conditions in form of linear matrix inequalities for asymptotic stability of delayed neural networks dnns a numerical simulation study is conducted to demonstrate the obtained theoretical results which shows their less conservatism than the existing stability criteria

local observability of nonlinear differential algebraic equations daes from the linearization along a trajectory the main results of this note establish sufficient rank conditions for local observability of nonlinear differential algebraic equations dae systems near a known trajectory associated with a given control verification of the observability rank condition is addressed and the relationship with previous work on observability of dae is indicated

symmetries and conservation laws for hamiltonian systems with inputs and outputs a generalization of noether s theorem the definition of symmetries and conservation laws for autonomous i e without external cases hamiltonian systems are generalized to hamiltonian systems with inputs and outputs it is shown that a symmetry implies the existence of a conservation law and vice versa thereby generalizing noether s theorem for autonomous hamiltonian systems

stabilization of hamiltonian systems 1 hamiltonian systems in this paper we will be concerned with the stabilization by feedback of hamiltonian systems in order to facilitate our discussions especially when applying lyapunov s second method we will restrict ourselves to a particular although natural subclass of hamiltonian systems given in the following way l let q be an n dimensional smooth manifold denoting the configuration space and let t q be the cotangent bundle denoting the phase space or srufe space furthermore there is a smooth m dimensional output manifold y m n and a smooth output map c q y smooth will mean c or ck with k sufficiently big although we shall restrict ourselves in the second part of section 2 to analytic data for simplicity we take c to be submersive so rank dc q m we assume that the system on t q has an internal energy which is the sum of a kinetic energy k and a porentiul energy v this means that there exists a riemannian metric on q in local coordinates q q j for q given by

hamiltonian systems with a discrete symmetry abstract if m m is an antisymplectic involution of a symplectic manifold m then the fixed set of is a lagrangian submanifold l m moreover there exist cotangent bundle coordinates in a neighborhood of l in m such that in these coordinates maps a covector into its negative thus classical examples which have a discrete symmetry such as the restricted three body problems are locally like a reversible system

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

stratified symplectic spaces and reduction let m w be a hamiltonian g space with proper momentum map j m g it is well known that if zero is a regular value of j and g acts freely on the level set j 0 then the reduced space mo j 0 g is a symplectic manifold we show that if the regularity assumptions are dropped the space mo is a union of symplectic manifolds i e it is a stratified symplectic space arms et al 2 proved that mo possesses a natural poisson bracket using their result we study hamiltonian dynamics on the reduced space in particular we show that hamiltonian flows are strata preserving and give a recipe for lifting a reduced hamiltonian flow to the level set j 0 finally we give a detailed description of the stratification of mo and prove the existence of a connected

maximum likelihood methods for log linear models when expected frequencies are subject to linear constraints abstract hypotheses relating to log linear models may be defined in terms of constraints on the frequencies or on the logarithms of the frequencies a two step algorithm is developed to allow fitting models containing constraints of both types this algorithm is shown to yield maximum likelihood estimates of the log linear parameters three examples are presented to illustrate cases in which these models are of interest

global stability analysis of neural networks with multiple time varying delays in this note we study the equilibrium and stability properties of neural networks with time varying delays our main results give sufficient conditions for the existence uniqueness and global asymptotic stability of the equilibrium point the proposed conditions establish the relationships between network parameters of the neural systems and the delay parameters the obtained results are applicable to all continuous nonmonotonic neuron activation functions and do not require the interconnection matrices to be symmetric some examples are also presented to compare our results with the previous results derived in the literature

singular angular momentum mappings we algebraically reduce the system consisting of a nonrelativistic particle moving in r with vanishing angular momentum after analyzing the conical structure of the constraint set 1 0 we use algebraic geometric techniques to explicitly construct the reduced poisson algebra of rotationally invariant observables this procedure enables us to completely identify the effects of the singularity in j 0 on the system we then group theoretically reduce the system and compare our results with those obtained algebraically

frustration stability and delay induced oscillations in a neural network model the effect of time delays on the linear stability of equilibria in an artificial neural network of hopfield type is analyzed the possibility of delay induced oscillations occurring is characterized in terms of properties of the not necessarily symmetric connection matrix of the network such oscillations are possible exactly when the network is frustrated equivalently when the signed digraph of the matrix does not require the perron property nonlinear analysis centre manifold computation of a three unit frustrated network is presented giving the nature of the bifurcations taking place a supercritical hopf bifurcation is shown to occur and a codimension two bifurcation is unfolded

proportional integral controllers for minimum phase nonaffine in control systems we show that stabilizing tracking proportional integral pi controllers can be constructed for minimum phase nonaffine in control systems the constructed pi controller is an equivalent realization of an approximate dynamic inversion controller this equivalence holds only for the time response when applied to the unperturbed system even when restricted to unperturbed minimum phase linear time invariant systems their closed loop robustness properties differ this shows that in general properties that do not define the equivalence relation for systems controllers are not preserved under such equivalence transformations

stabilization of relative equilibria discusses the problem of obtaining feedback laws to asymptotically stabilize relative equilibria of mechanical systems with symmetry we show how to stabilize an internally unstable relative equilibrium using internal actuators the methodology is that of potential shaping but the system is allowed to be underactuated i e have fewer actuators than the dimension of the shape space the theory is illustrated with the problem of stabilization of the cowboy relative equilibrium of the double spherical pendulum

realization theory for hamiltonian systems we prove that control systems of the hamiltonian type have a quasi minimal realization on a state space of minimal dimension of one of two types a hamiltonian system or the suspension of a time dependent hamiltonian system these realizations are unique up to a symplectomorphism in the first case or a canonical transformation in the second one when the lie algebra of the initial system sigma assumed to be analytic is finite dimensional we obtain a characterization of the state space and the dynamics of the quasi minimal realization in terms of the coadjoint actions of lie groups associated with sigma an example is given applying these results using control theoretic methods we prove the kostant kirillov souriau theorem the technique used for systems with finite dimensional lie algebra is closely related to the theory developed by those authors

existence learning and replication of periodic motions in recurrent neural networks a class of recurrent neural networks is shown to possess a stable limit cycle a gradient type algorithm is used to modify the parameters of the network so that it learns and replicates autonomously a time varying periodic signal the results are applied to controlling the repetitive motion of a two link robot manipulator

robust synchronization of an array of coupled stochastic discrete time delayed neural networks this paper is concerned with the robust synchronization problem for an array of coupled stochastic discrete time neural networks with time varying delay the individual neural network is subject to parameter uncertainty stochastic disturbance and time varying delay where the norm bounded parameter uncertainties exist in both the state and weight matrices the stochastic disturbance is in the form of a scalar wiener process and the time delay enters into the activation function for the array of coupled neural networks the constant coupling and delayed coupling are simultaneously considered we aim to establish easy to verify conditions under which the addressed neural networks are synchronized by using the kronecker product as an effective tool a linear matrix inequality lmi approach is developed to derive several sufficient criteria ensuring the coupled delayed neural networks to be globally robustly exponentially synchronized in the mean square the lmi based conditions obtained are dependent not only on the lower bound but also on the upper bound of the time varying delay and can be solved efficiently via the matlab lmi toolbox two numerical examples are given to demonstrate the usefulness of the proposed synchronization scheme

an exact cholesky decomposition and the generalized inverse of the variance covariance matrix of the multinomial distribution with applications a symbolic formula is given for the square root free cholesky decomposition of the variance covariance matrix of the multinomial distribution the evaluation of the symbolic cholesky factors requires much fewer arithmetic operations than does the general cholesky algorithm since the symbolic formula is not affected by an ill conditioned matrix it is particularly useful when the elements of a probability vector are of quite different orders of magnitude a simpler formula is obtained for pederson s procedure of sampling from a multinomial population an explicit formula of the moore penrose inverse of the variance covariance matrix is given as well as a symmetric representation of a multinormal density approximation to the multinomial distribution these formulae facilitate symmetric manipulation of the matrix and are useful in statistical modelling and computation involving the logistic density transformation of the multinomial distribution and in computer simulations of dynamic models in population genetics each element of the cholesky factors is given interesting probabilistic interpretations