Consensus of second-order discrete-time multi-agent systems with fixed topology

This paper studies the consensus of second-order discrete-time multi-agent systems with fixed topology. First, we formulate the problem and give some preliminaries. Then, by algebraic graph theory and matrix theory, the convergence of system matrix is analyzed. Our main results indicate that the consensus of second-order system can be achieved if and only if the topology graph has a directed spanning tree and the values of the scaling parameters satisfy a range. The eigenvalues of the corresponding Laplacian matrix play a key role in reaching consensus. Finally, numerical simulations are given to illustrate the results.     

Consensus in Byzantine asynchronous systems

This paper presents a consensus protocol resilient to Byzantine failures. It uses signed and certified messages and is based on two underlying failure detection modules. The first is a muteness failure detection module of the class . The second is a reliable Byzantine behaviour detection module. More precisely, the first module detects processes that stop sending messages, while processes experiencing other non-correct behaviours (i.e., Byzantine) are detected by the second module. The protocol is resilient to F faulty processes, Fmin((n−1)/2,C) (where C is the maximum number of faulty processes that can be tolerated by the underlying certification service).The approach used to design the protocol is new. While usual Byzantine consensus protocols are based on failure detectors to detect processes that stop communicating, none of them use a module to detect their Byzantine behaviour (this detection is not isolated from the protocol and makes it difficult to understand and prove correct). In addition to this modular approach and to a consensus protocol for Byzantine systems, the paper presents a finite state automaton-based implementation of the Byzantine behaviour detection module. Finally, the modular approach followed in this paper can be used to solve other problems in asynchronous systems experiencing Byzantine failures.     

Consensus seeking in multi-agent systems via hybrid protocols with impulse delays

This paper studies the consensus problem of multi-agent systems with both fixed and switching topologies. A hybrid consensus protocol is proposed to take into consideration of continuous-time communications among agents and delayed instant information exchanges on a sequence of discrete times. Based on the proposed algorithms, the multi-agent systems with the hybrid consensus protocols are described in the form of impulsive systems or impulsive switching systems. By employing results from matrix theory and algebraic graph theory, some sufficient conditions for the consensus of multi-agent systems with fixed and switching topologies are established, respectively. Our results show that, for small impulse delays, the hybrid consensus protocols can solve the consensus problem if the union of continuous-time and impulsive-time interaction digraphs contains a spanning tree frequently enough. Simulations are provided to demonstrate the effectiveness of the proposed consensus protocols.     

Consensus of the second-order multi-agent systems with an active leader and coupling time delay

This article investigates the consensus problem of the second-order multi-agent systems with an active leader and coupling time delay in direct graph. One decentralized state control rule is constructed for each agent to track the active leader and it is proved that the proposed control scheme enables the consensus to be obtained when the adjacency topology is fixed/switched. Simulation results show effectiveness of the proposed theoretical analysis.     

Consensus of heterogeneous multi-agent systems with linear and nonlinear dynamics

In this paper, we perform an in-depth study about the consensus problem of heterogeneous multi-agent systems with linear and nonlinear dynamics.Specifically, this system is composed of two classes of agents respectively described by linear and nonlinear dynamics. By the aid of the adaptive method and Lyapunov stability theory, the mean consensus problem is realized in the framework of first-order case and second-order case under undirected and connected networks.Still, an meaningful example is provided to verify the effectiveness of the gained theoretical results. Our study is expected to establish a more realistic model and provide a better understanding of consensus problem in the multi-agent system.     

Metrizability of spaces of holomorphic functions with the Nachbin topology

In this paper we prove, among other things, that the space of all holomorphic functions on an open subset U of a complex metrizable space E, endowed with the Nachbin ported topology, is metrizable only if E has finite dimension. This answers a question by Mujica in [J. Mujica, Gérmenes holomorfos y funciones holomorfas en espacios de Fréchet, Publicaciones del Departamento de Teoría de Funciones, Universidad de Santiago, Spain, 1978].     

Consensus protocol design for discrete‐time networks of multiagent with time‐varying delay via logarithmic quantizer

This article considers the problem of consensus for discrete‐time networks of multiagent with time‐varying delays and quantization. It is assumed that the logarithmic quantizer is utilized between the information flow through the sensor of each agent, and its quantization error is included in the proposed method. By constructing a suitable Lyapunov‐Krasovskii functional and utilizing matrix theory, a new consensus criterion for the concerned systems is established in terms of linear matrix inequalities (LMIs) which can be easily solved by various effective optimization algorithms. Based on the consensus criterion, a designing method of consensus protocol is introduced. One numerical example is given to illustrate the effectiveness of the proposed method. © 2014 Wiley Periodicals, Inc. Complexity 21: 163–176, 2015     

On split common fixed point problems

Based on the convergence theorem recently proved by the second author, we modify the iterative scheme studied by Moudafi for quasi-nonexpansive operators to obtain strong convergence to a solution of the split common fixed point problem. It is noted that Moudafi's original scheme can conclude only weak convergence. As a consequence, we obtain strong convergence theorems for split variational inequality problems for Lipschitz continuous and monotone operators, split common null point problems for maximal monotone operators, and Moudafi's split feasibility problem.     

Consensus reaching in committees

In this paper, we apply a consensus model to decision-making in committees that have to choose one or more alternatives from a set of alternatives. The model does not use a voting rule nor a set of winning coalitions. Every decision maker evaluates each alternative with respect to given criteria. The criteria may be of unequal importance to a decision maker. Decision makers may be advised by a chairman to adjust their preferences, i.e., to change their evaluation of some alternative(s) or/and the importance of the criteria, in order to obtain a better consensus. The consensus result should satisfy constraints concerning the consensus degree and the majority degree. A simple example is presented.     

An existence theorem for systems of boundary value problems

We prove an existence theorem for , , in , using the shooting method. The function is supposed to be asymptotically linear.

     

Well-posedness of mixed variational inequalities, inclusion problems and fixed point problems

We generalize the concept of well-posedness to a mixed variational inequality and give some characterizations of its well-posedness. Under suitable conditions, we prove that the well-posedness of a mixed variational inequality is equivalent to the well-posedness of a corresponding inclusion problem. We also discuss the relations between the well- posedness of a mixed variational inequality and the well-posedness of a fixed point problem. Finally, we derive some conditions under which a mixed variational inequality is well-posed. This work was supported by the National Natural Science Foundation of China (10671135) and Specialized Research Fund for the Doctoral Program of Higher Education (20060610005). The research of the third author was partially support by NSC 95-2221-E-110-078.     

Formation control for second-order multi-agent systems with time-varying delays under directed topology

In this paper, we investigate a formation control problem for second-order multi-agent systems with directed graph interconnection topologies that contain time-varying coupling delays. By using a special multiple leaders’ framework, sufficient conditions are obtained for both time-invariant and time-varying formations as well as for time-varying formations for trajectory tracking, which guarantees the attainment of the formations is at exponentially converging speeds. Some numerical simulations are also conducted to validate the theoretical results.     

AgentGeom: a multiagent system for pedagogical support in geometric proof problems

This paper aims, first, to describe the fundamental characteristics and workings of the AgentGeom artificial tutorial system, which is designed to help students develop knowledge and skills related to problem solving, mathematical proof in geometry, and the use of mathematical language. Following this, we indicate the manner in which a secondary school student can appropriate these abilities through interactions with the system. Our system uses strategic messages of the agent tutor in an argumentative process that collaborates with a student in the construction of a proof.     

A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces

In this paper, we introduce two iterative schemes by the general iterative method for finding a common element of the set of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. Then, we prove two strong convergence theorems for nonexpansive mappings to solve a unique solution of the variational inequality which is the optimality condition for the minimization problem. These results extended and improved the corresponding results of Marino and Xu [G. Marino, H.K. Xu, A general iterative method for nonexpansive mapping in Hilbert spaces, J. Math. Anal. Appl. 318 (2006) 43-52], S. Takahashi and W. Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (1) (2007) 506-515], and many others.     

On convergence of functions of normal operators in strong operator topology

The theorem in [E. B. Davies, J. London Math. Soc. (2), 4, 435–436 (1972)] is extended. As a corollary, Kadison’s theorem on strong continuity of operator functions is refined.     

赋予hit—or-miss拓扑超空间动力系统的拓扑熵

设（X,d,f）为拓扑动力系统,其中X为局部紧第二可数Hausdorff空间,d为紧型度量,f为完备映射,用2^x和f分别表示由X的所有非空闭子集和所有闭子集构成的集族,（2^x,ρ,2^f）和（f,ρ,2^f）为由（X,d,f）诱导的赋予hit—or—miss拓扑的超空间动力系统．本文研究了h（X,d,f）和h（2^...     

Disturbance decoupling and model matching problems for discrete-time systems with time-varying delays

In this paper, the disturbance decoupling problem and the model matching problem for discrete-time linear systems with time-varying delays are considered. Solvability of the above problems is characterized by means of structural necessary and sufficient conditions that can be checked by algorithmic procedures. The basic method used to analyze the considered problems consists in representing the discrete-time linear systems with time-varying delays as switching linear systems, whose properties can be studied by a powerful structural approach. In this way, the considered control problems can be reduced to the corresponding problems for switched linear systems, whose solvability has been recently characterized.     

Extending partial automorphisms and the profinite topology on free groups

A class of structures is said to have the extension property for partial automorphisms (EPPA) if, whenever and are structures in , finite, , and are partial automorphisms of extending to automorphisms of , then there exist a finite structure in and automorphisms of extending the . We will prove that some classes of structures have the EPPA and show the equivalence of these kinds of results with problems related with the profinite topology on free groups. In particular, we will give a generalisation of the theorem, due to Ribes and Zalesskii stating that a finite product of finitely generated subgroups is closed for this topology.

     

Global optimization of truss topology with discrete bar areas—Part I: theory of relaxed problems

This two part paper considers the classical problem of finding a truss design with minimal compliance subject to a given external force and a volume bound. The design variables describe the cross-section areas of the bars. While this problem is well-studied for continuous bar areas, we treat here the case of discrete areas. This problem is of major practical relevance if the truss must be built from pre-produced bars with given areas. As a special case, we consider the design problem for a single bar area, i.e., a 0/1-problem. In contrast to heuristic methods considered in other approaches, Part I of the paper together with Part II present an algorithmic framework for the calculation of a global optimizer of the underlying large-scaled mixed integer design problem. This framework is given by a convergent branch-and-bound algorithm which is based on solving a sequence of nonconvex continuous relaxations. The main issue of the paper and of the approach lies in the fact that the relaxed nonlinear optimization problem can be formulated as a quadratic program (QP). Here the paper generalizes and extends the available theory from the literature. Although the Hessian of this QP is indefinite, it is possible to circumvent the non-convexity and to calculate global optimizers. Moreover, the QPs to be treated in the branch-and-bound search tree differ from each other just in the objective function. In Part I we give an introduction to the problem and collect all theory and related proofs for the treatment of the original problem formulation and the continuous relaxed problems. The implementation details and convergence proof of the branch-and-bound methodology and the large-scale numerical examples are presented in Part II.     

Plurisubharmonic and holomorphic functions relative to the plurifine topology

A weak and a strong concept of plurifinely plurisubharmonic and plurifinely holomorphic functions are introduced. Strong will imply weak. The weak concept is studied further. A function f is weakly plurifinely plurisubharmonic if and only if it is locally bounded from above in the plurifine topology and f°h is finely subharmonic for all complex affine-linear maps h. As a consequence, the regularization in the plurifine topology of a pointwise supremum of such functions is weakly plurifinely plurisubharmonic, and it differs from the pointwise supremum at most on a pluripolar set. Weak plurifine plurisubharmonicity and weak plurifine holomorphy are preserved under composition with weakly plurifinely holomorphic maps.