system analysis via integral quadratic constraints this paper introduces a unified approach to robustness analysis with respect to nonlinearities time variations and uncertain parameters from an original idea by yakubovich 1967 the approach has been developed under a combination of influences from the western and russian traditions of control theory it is shown how a complex system can be described using integral quadratic constraints iqc for its elementary components a stability theorem for systems described by iqcs is presented that covers classical passivity dissipativity arguments but simplifies the use of multipliers and the treatment of causality a systematic computational approach is described and relations to other methods of stability analysis are discussed last but not least the paper contains a summarizing list of iqcs for important types of system components

on boundedness of solutions of state periodic systems a multivariable cell structure approach many dynamical systems are periodic with respect to several state variables this periodicity typically leads to the coexistence of multiple invariant solutions equilibria or limit cycles as a consequence while there are many classical techniques for analysis of boundedness and stability of such systems most of these only permit to establish local properties motivated by this a new sufficient criterion for global boundedness of solutions of such a class of nonlinear systems is presented the proposed method is inspired by the cell structure approach developed by leonov and noldus and characterized by two main advances first the conventional cell structure framework is extended to the case of dynamics which are periodic with respect to multiple states second by introducing the notion of a leonov function the usual definiteness requirements of standard lyapunov functions are relaxed to sign indefinite functions furthermore it is shown that under mild additional conditions the existence of a leonov function also ensures input to state stability i e robustness with respect to exogenous perturbations the performance of the proposed approach is demonstrated via the global analysis of boundedness of trajectories for a nonlinear system

a new hybrid iterative algorithm for variational inequalities let h be a real hilbert space let f h h be a strongly monotone and lipschitzian mapping let t n n 1 h h be an infinite family of non expansive mappings with common fixed points set n 1 fix t n a we devise an iterative algorithmy n x email protected nf x n x n 1 email protected n y email protected nw ny n n 0 where l n is a sequence in 0 a n is a sequence in 0 1 and w n is the w mapping we prove that the sequence x n converges in norm to x email protected n 1 fix t n which is the unique solution of the following variational inequality 0 email protected email protected n 1 fix t n

an eulerian approach to the analysis of krause s consensus models in this paper we analyze a class of multiagent consensus dynamical systems inspired by krause s original model as in krause s model the basic assumption is the so called bounded confidence two agents can influence each other only when their state values are below a given distance threshold r we study the system under an eulerian point of view considering possibly continuous probability distributions of agents and we present original convergence results the limit distribution is always necessarily a convex combination of delta functions at least r far apart from each other in other terms these models are locally aggregating the eulerian perspective provides the natural framework for designing a numerical algorithm by which we obtain several simulations in 1 and 2 dimensions

a relaxed extragradient like method for a generalized mixed equilibrium problem a general system of generalized equilibria and a fixed point problem abstract very recently takahashi and takahashi s takahashi w takahashi strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a hilbert space nonlinear anal 69 2008 1025 1033 suggested and analyzed an iterative method for finding a common solution of a generalized equilibrium problem and a fixed point problem of a nonexpansive mapping in a hilbert space in this paper based on takahashi takahashi s iterative method and well known extragradient method we introduce a relaxed extragradient like method for finding a common solution of a generalized mixed equilibrium problem a general system of generalized equilibria and a fixed point problem of a strictly pseudocontractive mapping in a hilbert space and then obtain a strong convergence theorem utilizing this theorem we establish some new strong convergence results in fixed point problems variational inequalities mixed equilibrium problems and systems of generalized equilibria

distributed multi agent optimization with state dependent communication we study distributed algorithms for solving global optimization problems in which the objective function is the sum of local objective functions of agents and the constraint set is given by the intersection of local constraint sets of agents we assume that each agent knows only his own local objective function and constraint set and exchanges information with the other agents over a randomly varying network topology to update his information state we assume a state dependent communication model over this topology communication is markovian with respect to the states of the agents and the probability with which the links are available depends on the states of the agents we study a projected multi agent subgradient algorithm under state dependent communication the state dependence of the communication introduces significant challenges and couples the study of information exchange with the analysis of subgradient steps and projection errors we first show that the multi agent subgradient algorithm when used with a constant stepsize may result in the agent estimates to diverge with probability one under some assumptions on the stepsize sequence we provide convergence rate bounds on a disagreement metric between the agent estimates our bounds are time nonhomogeneous in the sense that they depend on the initial starting time despite this we show that agent estimates reach an almost sure consensus and converge to the same optimal solution of the global optimization problem with probability one under different assumptions on the local constraint sets and the stepsize sequence

finite time semistability and consensus for nonlinear dynamical networks this paper focuses on semistability and finite time stability analysis and synthesis of systems having a continuum of equilibria semistability is the property whereby the solutions of a dynamical system converge to lyapunov stable equilibrium points determined by the system initial conditions in this paper we merge the theories of semistability and finite time stability to develop a rigorous framework for finite time semistability in particular finite time semistability for a continuum of equilibria of continuous autonomous systems is established continuity of the settling time function as well as lyapunov and converse lyapunov theorems for semistability are also developed in addition necessary and sufficient conditions for finite time semistability of homogeneous systems are addressed by exploiting the fact that a homogeneous system is finite time semistable if and only if it is semistable and has a negative degree of homogeneity unlike previous work on homogeneous systems our results involve homogeneity with respect to semistable dynamics and require us to adopt a geometric description of homogeneity finally we use these results to develop a general framework for designing semistable protocols in dynamical networks for achieving coordination tasks in finite time

on the solution of the minimal rational interpolation problem it is shown that the reachability and the observability indices of appropriately defined pairs of matrices provide the key for solving the minimal rational bitangen tial interpolation problem

parabolic pde based multi agent formation control on a cylindrical surface abstractthis paper considers the modelling and control design of the multi agent systems in the 3 d space the communication graph of the agents is a mesh grid 2 d cylindrical surface different from most existing literatures where the agents are modelled by ordinary differential equations odes we treat the agents as a continuum in this paper more specifically we model the collective dynamics of the agents by two reaction advection diffusion 2 d partial differential equations pdes the pde states represent the agent positions and the equilibria correspond to possible formation manifolds these pdes can be open loop unstable and the boundary stabilisation problem of the pdes on the cylindrical surface is solved using the backstepping method an all explicit observer based output control scheme is constructed which is distributed in the sense that each agent only needs local information closed loop exponential stability in the l2 h1 and h2 spaces is proved for the controller designs numerical

nonlinear proximal point algorithms using bregman functions with applications to convex programming a bregman function is a strictly convex differentiable function that induces a well behaved distance measure or d function on euclidean space this paper shows that for every bregman function there exists a nonlinear version of the proximal point algorithm and presents an accompanying convergence theory applying this generalization of the proximal point algorithm to convex programming one obtains the d function proximal minimization algorithm of censor and zenios and a wide variety of new multiplier methods these multiplier methods are different from those studied by kort and bertsekas and include nonquadratic variations on the proximal method of multipliers

differential inequalities in multi agent coordination and opinion dynamics modeling abstract many distributed algorithms for multi agent coordination employ the simple averaging dynamics referred to as the laplacian flow besides the standard consensus protocols examples include but are not limited to algorithms for aggregation and containment control target surrounding distributed optimization and models of opinion formation in social groups in spite of their similarities each of these algorithms has been studied using separate mathematical techniques in this paper we show that stability and convergence of many coordination algorithms involving the laplacian flow dynamics follow from the general consensus dichotomy property of a special differential inequality the consensus dichotomy implies that any solution to the differential inequality is either unbounded or converges to a consensus equilibrium in this paper we establish the dichotomy criteria for differential inequalities and illustrate their applications to multi agent coordination and opinion dynamics modeling

an interior point method with bregman functions for the variational inequality problem with paramonotone operators we present an algorithm for the variational inequality problem on convex sets with nonempty interior the use of bregman functions whose zone is the convex set allows for the generation of a sequence contained in the interior without taking explicitly into account the constraints which define the convex set we establish full convergence to a solution with minimal conditions upon the monotone operatorf weaker than strong monotonicity or lipschitz continuity for instance and including cases where the solution needs not be unique we apply our algorithm to several relevant classes of convex sets including orthants boxes polyhedra and balls for which bregman functions are presented which give rise to explicit iteration formulae up to the determination of two scalar stepsizes which can be found through finite search procedures 1998 the mathematical programming society inc published by elsevier science b v

sparse jurdjevic quinn stabilization of dissipative systems abstract for control affine systems with a proper lyapunov function the classical jurdjevic quinn procedure see jurdjevic and quinn 1978 gives a well known and widely used method for the design of feedback controls that asymptotically stabilize the system to some invariant set in this procedure all controls are in general required to be activated i e nonzero at the same time in this paper we give sufficient conditions under which this stabilization can be achieved by means of sparse feedback controls i e feedback controls having the smallest possible number of nonzero components we thus obtain a sparse version of the classical jurdjevic quinn theorem we propose three different explicit stabilizing control strategies depending on the method used to handle possible discontinuities arising from the definition of the feedback a time varying periodic feedback a sampled feedback and a hybrid hysteresis we illustrate our results by applying them to opinion formation models thus recovering and generalizing former results for such models

approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method in this paper we introduce an iterative method based on the extragradient method for finding a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping in a real hilbert space furthermore we prove that the studied iterative method strongly converges to a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping under some mild conditions imposed on algorithm parameters

finite time consensus using stochastic matrices with positive diagonals we discuss the possibility of reaching consensus in finite time using only linear iterations with the additional restrictions that the update matrices must be stochastic with positive diagonals and consistent with a given graph structure we show that finite time average consensus can always be achieved for connected undirected graphs for directed graphs we show some necessary conditions for finite time consensus including strong connectivity and the presence of a simple cycle of even length

stability of multiagent systems with time dependent communication links we study a simple but compelling model of network of agents interacting via time dependent communication links the model finds application in a variety of fields including synchronization swarming and distributed decision making in the model each agent updates his current state based upon the current information received from neighboring agents necessary and or sufficient conditions for the convergence of the individual agents states to a common value are presented thereby extending recent results reported in the literature the stability analysis is based upon a blend of graph theoretic and system theoretic tools with the notion of convexity playing a central role the analysis is integrated within a formal framework of set valued lyapunov theory which may be of independent interest among others it is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence even for the simple models discussed in the present paper

rational interpolation and state variable realizations abstract the problem is considered of passing from interpolation data for a real rational transfer function matrix to a minimal state variable realization of the transfer function matrix the tool is a loewner matrix which is a generalization of the standard hankel matrix of linear system realization theory and which possesses a decomposition into a product of generalized observability and controllability matrices

signed bounded confidence models for opinion dynamics abstract the aim of this paper is to modify continuous time bounded confidence opinion dynamics models so that changes of opinion intended as changes of the sign of the initial states are never induced during the evolution such sign invariance can be achieved by letting opinions of different sign localized near the origin interact negatively or neglect each other or even repel each other in all cases it is possible to obtain sign preserving bounded confidence models with state dependent connectivity and with a clustering behavior similar to that of a standard bounded confidence model

synchronization of diffusively coupled limit cycle oscillators we develop analytical and numerical conditions to determine whether limit cycle oscillations synchronize in diffusively coupled systems we examine two classes of systems reaction diffusion pdes with neumann boundary conditions and compartmental odes where compartments are interconnected through diffusion terms with adjacent compartments in both cases the uncoupled dynamics are governed by a nonlinear system that admits an asymptotically stable limit cycle we provide two time scale averaging methods for certifying stability of spatially homogeneous time periodic trajectories in the presence of sufficiently small or large diffusion and develop methods using the structured singular value for the case of intermediate diffusion we highlight cases where diffusion stabilizes or destabilizes such trajectories

lagrangian duality and related multiplier methods for variational inequality problems we consider a new class of multiplier interior point methods for solving variational inequality problems with maximal monotone operators and explicit convex constraint inequalities developing a simple lagrangian duality scheme which is combined with the recent logarithmic quadratic proximal lqp theory introduced by the authors we derive three algorithms for solving the variational inequality vi problem this provides a natural extension of the methods of multipliers used in convex optimization and leads to smooth interior point multiplier algorithms we prove primal dual and primal dual convergence under very mild assumptions eliminating all the usual assumptions used until now in the literature for related algorithms applications to complementarity problems are also discussed

on eigenvalues of laplacian matrix for a class of directed signed graphs abstract the eigenvalues of the laplacian matrix for a class of directed graphs with both positive and negative weights are studied the laplacian matrix naturally arises in a wide range of applications involving networks first a class of directed signed graphs is studied in which one pair of nodes either connected or not is perturbed with negative weights a necessary and sufficient condition is proposed to attain the following objective for the perturbed graph the real parts of the non zero eigenvalues of its laplacian matrix are positive under certain assumption on the unperturbed graph it is established that the objective is achieved if and only if the magnitudes of the added negative weights are smaller than an easily computable upper bound this upper bound is shown to depend on the topology of the unperturbed graph it is also pointed out that the obtained condition can be applied in a recursive manner to deal with multiple edges with negative weights secondly for directed graphs a subset of pairs of nodes are identified where if any of the pairs is connected by an edge with infinitesimal negative weight the resulting laplacian matrix will have at least one eigenvalue with negative real part illustrative examples are presented to show the applicability of our results

brief paper synchronization in networks of identical linear systems the paper investigates the synchronization of a network of identical linear state space models under a possibly time varying and directed interconnection structure the main result is the construction of a dynamic output feedback coupling that achieves synchronization if the decoupled systems have no exponentially unstable mode and if the communication graph is uniformly connected the result can be interpreted as a generalization of classical consensus algorithms stronger conditions are shown to be sufficient but to some extent also necessary to ensure synchronization with the diffusive static output coupling often considered in the literature

on borwein wiersma decompositions of monotone linear relations monotone operators are of basic importance in optimization as they generalize simultaneously subdifferential operators of convex functions and positive semidefinite not necessarily symmetric matrices in 1970 asplund a monotone convergence theorem for sequences of nonlinear mapping in nonlinear functional analysis american math society providence ri 1970 pp 1 9 studied the additive decomposition of a maximal monotone operator as the sum of a subdifferential operator and an irreducible monotone operator in 2007 borwein and wiersma siam j optim 18 2007 pp 946 960 introduced another additive decomposition where the maximal monotone operator is written as the sum of a subdifferential operator and a skew monotone operator both decompositions are variants of the well known additive decomposition of a matrix via its symmetric and skew part this paper presents a detailed study of the borwein wiersma decomposition of a maximal monotone linear relation we give sufficient conditions and characterizations for a maximal monotone linear relation to be borwein wiersma decomposable and show that borwein wiersma decomposability implies asplund decomposability we exhibit irreducible linear maximal monotone operators without full domain thus answering one of the questions raised by borwein and wiersma the borwein wiersma decomposition of any maximal monotone linear relation is made quite explicit in hilbert space

multi agent deployment in 3 d via pde control this paper introduces a methodology for modelling analysis and control design of a large scale system of agents deployed in 3 d space the agents communication graph is a mesh grid disk 2 d topology in polar coordinates treating the agents as a continuum we model the agents collective dynamics by complex valued reaction diffusion 2 d partial differential equations pdes in polar coordinates whose states represent the position coordinates of the agents due to the reaction term in the pdes the agents can achieve a rich family of 2 d deployment manifolds in 3 d space which correspond to the pdes equilibrium as determined by the boundary conditions unfortunately many of these deployment surfaces are open loop unstable to stabilize them a heretofore open and challenging problem of pde stabilization by boundary control on a disk has been solved in this paper using a new class of explicit backstepping kernels that involve the poisson kernel a dual observer which is also explicit allows to estimate the positions of all the agents as needed in the leaders feedback by only measuring the position of their closest neighbors hence an all explicit control scheme is found which is distributed in the sense that each agent only needs local information closed loop exponential stability in the l 2 h 1 and h 2 spaces is proved for both full state and output feedback designs numerical simulations illustrate the proposed approach for 3 d deployment of discrete agents

synchronization of networks of nonidentical euler lagrange systems with uncertain parameters and communication delays this paper addresses the problem of synchronizing networks of nonidentical nonlinear dynamical systems described by euler lagrange equations which are assumed fully actuated with their states available for measurement but with unknown parameters the only assumption made on the communication graph is that it is connected moreover the communication is subject to constant time delays which are also unknown the main result of the paper is the construction of an adaptive controller that achieves global full state synchronization i e the difference between the agents positions and velocities asymptotically converges to zero if a desired trajectory for all systems is given a slight modification to the proposed scheme achieves also full state synchronization simulations using a ten robot manipulator network are used to illustrate the performance of the proposed schemes

an interior point iterative maximum likelihood reconstruction algorithm incorporating upper and lower bounds with application to spect transmission imaging the algorithm we consider here is a block iterative or ordered subset version of the inferior point algorithm for transmission reconstruction our algorithm is an interior point method because each vector of the iterative sequence x sup k k 0 1 2 satisfies the constraints a sub j x sub j sup k b sub j j 1 j because it is a block iterative algorithm that reconstructs the transmission attenuation map and places constraints above and below the pixel values of the reconstructed image we call it the bitab method computer simulations using the three dimensional mathematical cardiac and torso phantom reveal that the bitab algorithm in conjunction with reasonably selected prior upper and lower bounds has the potential to improve the accuracy of the reconstructed attenuation coefficients from truncated fan beam transmission projections by suitably selecting the bounds it is possible to restrict the over estimation of coefficients outside the fully sampled region that results from reconstructing truncated fan beam projections with iterative transmission algorithms such as the maximum likelihood gradient type algorithm

smooth methods of multipliers for complementarity problems this paper describes several methods for solving nonlinear complementarity problems a general duality framework for pairs of monotone operators is developed and then applied to the monotone complementarity problem obtaining primal dual and primal dual formulations we derive bregman function based generalized proximal algorithms for each of these formulations generating three classes of complementarity algorithms the primal class is well known the dual class is new and constitutes a general collection of methods of multipliers or augmented lagrangian methods for complementarity problems in a special case it corresponds to a class of variational inequality algorithms proposed by gabay by appropriate choice of bregman function the augmented lagrangian subproblem in these methods can be made continuously differentiable the primal dual class of methods is entirely new and combines the best theoretical features of the primal and dual methods some preliminary computation shows that this class of algorithms is effective at solving many of the standard complementarity test problems

convergence properties for discrete time nonlinear systems three similar convergence notions are considered two of them are the long established notions of convergent dynamics and incremental stability the other is the more recent notion of contraction analysis all three convergence notions require that all solutions of a system converge to each other in this note we investigate the differences between these convergence properties for discrete time and time varying nonlinear systems by comparing the properties in pairs and using examples we also demonstrate a time varying smooth lyapunov function characterization for each of these convergence notions and with appropriate assumptions we provide several sufficient conditions to establish relationships between these properties in terms of lyapunov functions

decentralised minimum time consensus we consider the discrete time dynamics of a network of agents that exchange information according to a nearest neighbour protocol under which all agents are guaranteed to reach consensus asymptotically we present a fully decentralised algorithm that allows any agent to compute the final consensus value of the whole network in finite time using the minimum number of successive values of its own state history we show that the minimum number of steps is related to a jordan block decomposition of the network dynamics and present an algorithm to compute the final consensus value in the minimum number of steps by checking a rank condition of a hankel matrix of local observations furthermore we prove that the minimum number of steps is related to graph theoretical notions that can be directly computed from the laplacian matrix of the graph and from the minimum external equitable partition

practical stability and stabilization presents a practical stability result for dynamical systems depending on a small parameter this result is applied to a practical stability analysis of fast time varying systems studied in averaging theory and of highly oscillatory systems studied by sussmann and liu 1999 furthermore the problem of practically stabilizing control affine systems with drift is discussed

graph diameter eigenvalues and minimum time consensus we consider the problem of achieving average consensus in the minimum number of linear iterations on a fixed undirected graph we are motivated by the task of deriving lower bounds for consensus protocols and by the so called definitive consensus conjecture which states that for an undirected connected graph g with diameter d there exist d matrices whose nonzero pattern complies with the edges in g and whose product equals the all ones matrix our first result is a counterexample to the definitive consensus conjecture which is the first improvement of the diameter lower bound for linear consensus protocols we then provide some algebraic conditions under which this conjecture holds which we use to establish that all distance regular graphs satisfy the definitive consensus conjecture

convergence analysis of a proximal like minimization algorithm using bregman functions an alternative convergence proof of a proximal like minimization algorithm using bregman functions recently proposed by censor and zenios is presented the analysis allows the establishment of a global convergence rate of the algorithm expressed in terms of function values

iterative algorithms for a general system of generalized nonlinear mixed composite type equilibria in this paper we consider and study a general system of generalized nonlinear mixed composite type equilibria in hilbert spaces first we prove the existence and uniqueness of the solution for this system of generalized nonlinear mixed composite type equilibria second the mann iterative method with errors is extended to develop some new iterative algorithms for finding approximate solutions for this system of generalized nonlinear mixed composite type equilibria we also derive the strong convergence of the sequences generated by these iterative algorithms in hilbert spaces

the primal dual algorithm as a constraint set manipulation device a general primal dual algorithm for linearly constrained optimization problems is formulated in which the dual variables are updated by a dual algorithmic operator convergence is proved under the assumption that the dual algorithmic operator implies asymptotic feasibility of the primal iterates with respect to the linear constraints a general result relating the minimal values of an infinite sequence of constrained problems to the minimal value of a limiting problem constrained by the limit of the sequence of constraints sets is established and invoked the applicability of the general theory is demonstrated by analyzing a specific dual algorithmic operator this leads to the mart algorithm for constrained entropy maximization used in image reconstruction from projections

convergence of type symmetric and cut balanced consensus seeking systems we consider continuous time consensus seeking systems whose time dependent interactions are cut balanced in the following sense if a group of agents influences the remaining ones the former group is also influenced by the remaining ones by at least a proportional amount models involving symmetric interconnections and models in which a weighted average of the agent values is conserved are special cases we prove that such systems always converge we give a sufficient condition on the evolving interaction topology for the limit values of two agents to be the same conversely we show that if our condition is not satisfied then these limits are generically different these results allow treating systems where the agent interactions are a priori unknown being for example random or determined endogenously by the agent values

distributed nonlinear control of mobile autonomous multi agents abstract this paper studies the distributed nonlinear control of mobile autonomous agents with variable and directed topology a new distributed nonlinear design scheme is presented for multi agent systems modeled by double integrators with the new design the outputs of the controlled agents asymptotically converge to each other as long as a mild connectivity condition is satisfied moreover the velocity derivative of the output of each agent can be restricted to be within any specified neighborhood of the origin which is of practical interest for systems under such physical constraint the new design is still valid if one of the agents is a leader and the control objective is to achieve leader following as an illustration of the generality and effectiveness of the presented methodology the formation control of a group of unicycle mobile robots with nonholonomic constraints is revisited instead of assuming the point robot model the unicycle model is transformed into two double integrators by dynamic feedback linearization and the proposed distributed nonlinear design method is used to overcome the singularity problem caused by the nonholonomic constraint by properly restricting the velocities simulation results are included to illustrate the theoretical results

a proximal point method for the variational inequality problem in banach spaces in this paper we prove well definedness and weak convergence of the generalized proximal point method when applied to the variational inequality problem in reflexive banach spaces the proximal version we consider makes use of bregman functions whose original definition for finite dimensional spaces has here been properly extended to our more general framework

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

convergent dynamics a tribute to boris pavlovich demidovich we review and pay tribute to a result on convergent systems by the russian mathematician boris pavlovich demidovich in a sense demidovich s approach forms a prelude to a field which is now called incremental stability of dynamical systems developments on incremental stability are reviewed from a historical perspective

distributed model predictive control of linear discrete time systems with local and global constraints abstract this paper proposes a distributed model predictive control dmpc approach for a family of discrete time linear systems with local uncoupled and global coupled constraints the proposed approach is based on the dual problem of a mpc optimization problem involving all systems this dual problem is then distributedly solved based on the alternating direction multiplier method admm with several known simplifications when the network of systems is large or sparsely connected the computation of the optimal control using admm can be expensive the proposed approach mitigates this problem by allowing early termination of the admm process this is made possible via a finite time consensus algorithm that determines the satisfaction of the termination condition and by appropriate tightening of the coupled constraints under reasonable assumptions the approach is guaranteed to converge to a small neighborhood of the optimal so long as the network is connected recursive feasibility and exponential stability of the closed loop system are shown the performance of the proposed approach is demonstrated by a numerical example

robust synchronization for multistable systems in this note we study a robust synchronization problem for multistable systems evolving on manifolds within an input to state stability iss framework based on a recent generalization of the classical iss theory to multistable systems a robust synchronization protocol is designed with respect to a compact invariant set of the unperturbed system the invariant set is assumed to admit a decomposition without cycles that is with neither homoclinic nor heteroclinic orbits numerical simulation examples illustrate our theoretical results

optimal control of discrete time fractional multi agent systems abstract this paper deals with control strategies for discrete time fractional multi agent systems by using the discrete fractional order operator we introduce memory effects to the considered problem necessary optimality conditions for discrete time fractional optimal control problems with single and double summator dynamics are proved and simulation results are presented

central paths generalized proximal point methods and cauchy trajectories in riemannian manifolds we study the relationships between three concepts which arise in connection with variational inequality problems central paths defined by arbitrary barriers generalized proximal point methods where a bregman distance substitutes for the euclidean one and cauchy trajectory in riemannian manifolds first we prove that under rather general hypotheses the central path defined by a general barrier for a monotone variational inequality problem is well defined bounded and continuous and converges to the analytic center of the solution set with respect to the given barrier thus generalizing results which deal only with complementarity problems and with the logarithmic barrier next we prove that a sequence generated by the proximal point method with the bregman distance naturally induced by the barrier function converges precisely to the same point furthermore for a certain class of problems including linear programming such a sequence is contained in the central path making the concepts of central path and generalized proximal point sequence virtually equivalent finally we prove that for this class of problems the central path also coincides with the cauchy trajectory in the riemannian manifold defined on the positive orthant by a metric given by the hessian of the barrier i e a curve whose direction at each point is the negative gradient of the objective function at that point in the riemannian metric

continuous time average preserving opinion dynamics with opinion dependent communications we study a simple continuous time multiagent system related to krause s model of opinion dynamics each agent holds a real value and this value is continuously attracted by every other value differing from it by less than 1 with an intensity proportional to the difference we prove convergence to a set of clusters with the agents in each cluster sharing a common value and provide a lower bound on the distance between clusters at a stable equilibrium under a suitable notion of multiagent system stability to better understand the behavior of the system for a large number of agents we introduce a variant involving a continuum of agents we prove under some conditions the existence of a solution to the system dynamics convergence to clusters and a nontrivial lower bound on the distance between clusters finally we establish that the continuum model accurately represents the asymptotic behavior of a system with a finite but large number of agents

feedback mechanisms for global oscillations in lure systems the paper presents two mechanisms for global oscillations in feedback systems based on bifurcations in absolutely stable systems the external characterization of the oscillators provides the basis for a energy based dissipativity theory for oscillators thereby opening new possibilities for rigorous stability analysis of high dimensional systems and interconnected oscillators

inexact proximal point methods for variational inequality problems we present a new family of proximal point methods for solving monotone variational inequalities our algorithm has a relative error tolerance criterion in solving the proximal subproblems our convergence analysis covers a wide family of regularization functions including double regularizations recently introduced by silva eckstein and humes jr siam j optim 12 2001 pp 238 261 and the bregman distance induced by h x sum i 1 n x i log x i we do not use in our analysis the assumption of paramonotonicity which is standard in proving convergence of bregman based proximal methods

matrix fraction description from frequency samples abstract it is shown how the controllability indices in a minimal state space realization of a real rational transfer matrix may be calculated from evaluations of this transfer matrix at a sufficient number of discrete points in the frequency domain subsequently a procedure for obtaining coprime matrix fraction descriptions of the transfer matrix is given an example is used to illustrate the results

global analysis of piecewise linear systems using impact maps and surface lyapunov functions this paper presents an entirely new constructive global analysis methodology for a class of hybrid systems known as piecewise linear systems pls this methodology infers global properties of pls solely by studying the behavior at switching surfaces associated with pls the main idea is to analyze impact maps i e maps from one switching surface to the next switching surface such maps are known to be unfriendly maps in the sense that they are highly nonlinear multivalued and not continuous we found however that an impact map induced by an linear time invariant flow between two switching surfaces can be represented as a linear transformation analytically parametrized by a scalar function of the state this representation of impact maps allows the search for surface lyapunov functions sulf to be done by simply solving a semidefinite program allowing global asymptotic stability robustness and performance of limit cycles and equilibrium points of pls to be efficiently checked this new analysis methodology has been applied to relay feedback on off and saturation systems where it has shown to be very successful in globally analyzing a large number of examples in fact it is still an open problem whether there exists an example with a globally stable limit cycle or equilibrium point that cannot be successfully analyzed with this new methodology examples analyzed include systems of relative degree larger than one and of high dimension for which no other analysis methodology could be applied this success in globally analyzing certain classes of pls has shown the power of this new methodology and suggests its potential toward the analysis of larger and more complex pls

on dual convergence and the rate of primal convergence of bregman s convex programming method bregman s method is an iterative algorithm for solving optimization problems with convex objective and linear inequality constraints it generates two sequences one the primal one is known to converge to the solution of the problem under the assumption of smoothness of the objective function at the solution it is proved that the other sequence the dual one converges to a solution of the dual problem and that the rate of convergence of the primal sequence is at least linear

an outer approximation method for the variational inequality problem we study two outer approximation schemes applied to the variational inequality problem in reflexive banach spaces first we propose a generic outer approximation scheme and its convergence analysis unifies a wide class of outer approximation methods applied to the constrained optimization problem as is standard in this setting boundedness and optimality of weak limit points are proved to hold under two alternative conditions i boundedness of the feasible set or ii coerciveness of the operator to develop a convergence analysis where i and ii do not hold we consider a second scheme in which the approximated subproblems use a coercive approximation of the original operator under conditions alternative to both i and ii we obtain standard convergence results furthermore when the space is uniformly convex we establish full strong convergence of the second scheme to a solution

selective pattern formation control autonomous pattern formation phenomena are ubiquitous throughout nature the goal of this paper is to show the possibility to effectively generate various desired spatial patterns by guiding such phenomena suitably to this end we employ a reaction diffusion system as a mathematical model and formulate and solve a novel pattern formation control problem first we describe the control objective in terms of spatial spectrum consensus which enables utilize recent advances on networked control system theory next the effectiveness of the proposed control law is evaluated theoretically by exploiting the center manifold theorem and also numerically by simulation the turing instabilities play a crucial role throughout the paper

a new generalized consensus problem and its crt based solution this technical note introduces a new generalized consensus problem where nodes in a network are guaranteed to reach the final consensus on a common vector whose elements are exactly the initial values arbitrarily chosen by nodes we propose a fully decentralized algorithm able to solve the above problem and derive conditions to guarantee that consensus is reached in a finite number of steps in particular we prove that in a network composed by n nodes the proposed consensus problem is solvable in at most 2n steps moreover we introduce a finite field solution based on the chinese remainder theorem crt able to reduce the complexity of the proposed approach in the case of capacity constraints and we discuss an illustrative case study

a generalized proximal point algorithm for the variational inequality problem in a hilbert space we consider a generalized proximal point method for solving variational inequality problems with monotone operators in a hilbert space it differs from the classical proximal point method as discussed by rockafellar for the problem of finding zeroes of monotone operators in the use of generalized distances called bregman distances instead of the euclidean one these distances play not only a regularization role but also a penalization one forcing the sequence generated by the method to remain in the interior of the feasible set so that the method becomes an interior point one under appropriate assumptions on the bregman distance and the monotone operator we prove that the sequence converges weakly if and only if the problem has solutions in which case the weak limit is a solution if the problem does not have solutions then the sequence is unbounded we extend similar previous results for the proximal point method with bregman distances which dealt only with the finite dimensional case and which applied only to convex optimization problems or to finding zeroes of monotone operators which are particular cases of variational inequality problems

on variable step relaxed projection algorithm for variational inequalities projection algorithms are practically useful for solving variational inequalities vi however some among them require the knowledge related to vi in advance such as lipschitz constant usually it is impossible in practice this paper studies the variable step basic projection algorithm and its relaxed version under weakly co coercive condition the algorithms discussed need not know constant function associated with the co coercivity or weak co coercivity and the step size is varied from one iteration to the next under certain conditions the convergence of the variable step basic projection algorithm is established for the practical consideration we also give the relaxed version of this algorithm in which the projection onto a closed convex set is replaced by another projection at each iteration and latter is easy to calculate the convergence of relaxed scheme is also obtained under certain assumptions finally we apply these two algorithms to the split feasibility problem sfp

rescaling and stepsize selection in proximal methods using separable generalized distances this paper presents a convergence proof technique for a broad class of proximal algorithms in which the perturbation term is separable and may contain barriers enforcing interval constraints there are two key ingredients in the analysis a mild regularity condition on the differential behavior of the barrier as one approaches an interval boundary and a lower stepsize limit that takes into account the curvature of the proximal term we give two applications of our approach first we prove subsequential convergence of a very broad class of proximal minimization algorithms for convex optimization where different stepsizes can be used for each coordinate applying these methods to the dual of a convex program we obtain a wide class of multiplier methods with subsequential convergence of both primal and dual iterates and independent adjustment of the penalty parameter for each constraint the adjustment rules for the penalty parameters generalize a well established scheme for the exponential method of multipliers the results may also be viewed as a generalization of recent work by ben tal and zibulevsky siam j optim 7 1997 pp 347 366 and auslender teboulle and ben tiba comput optim appl 12 1999 pp 31 40 math oper res 24 1999 pp 645 668 on methods derived from varphi divergences the second application established full convergence under a novel stepsize condition of bregman function based proximal methods for general monotone operator problems over a box prior results in this area required strong restrictive assumptions on the monotone operator

distributed least squares solver for network linear equations in this paper we study the problem of finding the least square solutions of over determined linear algebraic equations over networks in a distributed manner each node has access to one of the lin

distributed estimation of laplacian eigenvalues via constrained consensus optimization problems from the recent literature we know that some consecutive measurements of the consensus protocol can be used to compute the exact average of the initial condition in this paper we show that these measurements can also be used for estimating the laplacian eigenvalues of the graph representing the network as recently shown in the literature by solving the factorization of the averaging matrix the laplacian eigenvalues can be inferred herein the problem is posed as a constrained consensus problem formulated two fold the first formulation direct approach yields a non convex optimization problem solved in a distributed way by means of the method of lagrange multipliers the second formulation indirect approach is obtained after an adequate re parameterization the problem is then convex and is solved by using the distributed subgradient algorithm and the alternating direction method of multipliers admm the proposed algorithms allow estimating the actual laplacian eigenvalues with high accuracy

observability and coarse graining of consensus dynamics through the external equitable partition using the intrinsic relationship between the external equitable partition eep and the spectral properties of the graph laplacian we characterize convergence and observability properties of consensus dynamics on networks in particular we establish the relationship between the original consensus dynamics and the associated consensus of the quotient graph under varied initial conditions and characterize the asymptotic convergence to the synchronization manifold under nonuniform input signals we also show that the eep with respect to a node can reveal nodes in the graph with an increased rate of asymptotic convergence to the consensus value as characterized by the second smallest eigenvalue of the quotient laplacian finally we show that the quotient graph preserves the observability properties of the full graph and how the inheritance by the quotient graph of particular aspects of the eigenstructure of the full laplacian underpins the observability and convergence properties of the system

a bode plot characterization of all stabilizing controllers in this technical note we consider continuous time control systems and present a new characterization of the nyquist criterion in terms of bode plots of the plant and the controller this gives a nonparametric and model independent characterization of arbitrary order stabilizing controllers the result shows that the frequency response of any stabilizing controller must satisfy constraints on its magnitude phase and rate of change of phase at certain frequencies that are imposed by the frequency response of the inverse plant

local lyapunov functions for consensus in switching nonlinear systems this note presents two theorems on asymptotic state consensus of continuous time nonlinear multi agent systems the agents reside in mathbb r m and have switching interconnection topologies both the first theorem formulated in terms of the states of individual agents and the second theorem formulated in terms of the pairwise states for pairs of agents can be interpreted as variants of lyapunov s second method the two theorems complement each other the second provides stronger convergence results under weaker graph topology assumptions whereas the first often can be applied in a wider context in terms of the structure of the right hand sides of the systems the second theorem also sheds some new light on well known results for consensus of nonlinear systems where the right hand sides of the agents dynamics are convex combinations of directions to neighboring agents for such systems instead of proving consensus by using the theory of contracting convex sets a local quadratic lyapunov function can be used

well posedness of the limiting equation of a noisy consensus model in opinion dynamics abstract this paper establishes the global well posedness of the nonlinear fokker planck equation for a noisy version of the hegselmann krause model the equation captures the mean field behavior of a classic multiagent system for opinion dynamics we prove the global existence uniqueness nonnegativity and regularity of the weak solution we also exhibit a global stability condition which delineates a forbidden region for consensus formation this is the first nonlinear stability result derived for the hegselmann krause model

equilibrium independent passivity we extend the traditional notion of passivity to a forced system whose equilibrium is dependent on the control input by defining equilibrium independent passivity a system property characterized by a dissipation inequality centered at an arbitrary equilibrium point we provide a necessary input output condition which can be tested for systems of arbitrary dimension and sufficient conditions to certify this property for scalar systems an example from network stability analysis is presented which demonstrates the utility of this new definition we then proceed to numerical certification of equilibrium independent passivity using sum of squares programming finally through numerical examples we show that equilibrium independent passivity is less restrictive than incremental passivity

emergent behavior in flocks we provide a model for both continuous and discrete time describing the evolution of a flock our model is parameterized by a constant beta capturing the rate of decay which in our model is polynomial of the influence between birds in the flock as they separate in space our main result shows that when beta 1 2 convergence of the flock to a common velocity is guaranteed while for betages1 2 convergence is guaranteed under some condition on the initial positions and velocities of the birds only

analysis of interconnected oscillators by dissipativity theory this paper employs dissipativity theory for the global analysis of limit cycles in particular dynamical systems of possibly high dimension oscillators are regarded as open systems that satisfy a particular dissipation inequality it is shown that this characterization has implications for the global stability analysis of limit cycle oscillations i in isolated oscillators ii in interconnections of oscillators and iii for the global synchrony analysis in interconnections of identical oscillators

step size sequence design for finite time average consensus in secure wireless sensor networks this paper concerns the study of average consensus in wireless sensor networks with aim of providing a way to reach consensus in a finite number of steps in particular we investigate the design of consensus protocols when for security reasons for instance the underlying graph is constrained to be strongly regular or distance regular the proposed design method is based on parameters of the intersection array characterizing the underlying graph with this protocol at execution time average consensus is achieved in a number of steps equal to the diameter of the graph i e the smallest possible number of steps to achieve consensus

the evolution of beliefs over signed social networks we study the evolution of opinions or beliefs over a social network modeled as a signed graph the sign attached to an edge in this graph characterizes whether the corresponding individuals or end nodes are friends positive links or enemies negative links pairs of nodes are randomly selected to interact over time and when two nodes interact each of them updates its opinion based on the opinion of the other node and the sign of the corresponding link this model generalizes the degroot model to account for negative links when two adversaries interact their opinions go in opposite directions we provide conditions for convergence and divergence in expectation in mean square and in almost sure sense and exhibit phase transition phenomena for these notions of convergence depending on the parameters of the opinion update model and on the structure of the underlying graph we establish a no survivor theorem stating that the difference in opinions of any two nodes diverges whenever opinions in the network diverge as a whole we also prove a live or die lemma indicating that almost surely the opinions either converge to an agreement or diverge finally we extend our analysis to cases where opinions have hard lower and upper limits in these cases we study when and how opinions may become asymptotically clustered to the belief boundaries and highlight the crucial influence of strong or weak structural balance of the underlying network on this clustering phenomenon

brief input to state stability for a class of lurie systems we analyze input to state stability iss for the feedback interconnection of a linear block and a nonlinear element this study is of importance for establishing robustness against actuator nonlinearities and disturbances in the absolute stability framework we prove iss from a positive real property of the linear block by restricting the sector nonlinearity to grow unbounded as its argument tends to infinity when this growth condition is violated examples show that the iss property is lost the result is used to give a simple proof of boundedness for negative resistance oscillators such as the van der pol oscillator in a separate application we relax the minimum phase assumption of an earlier boundedness result for systems with nonlinearities that grow faster than linear