The Consensus of Different Fractional-Order Chaotic Multiagent Systems Using Adaptive Protocols

This paper is concerned with the adaptive consensus problem of incommensurate chaotic fractional order multiagent systems. Firstly, we introduce fractional-order derivative in the sense of Caputo and the classical stability theorem of linear fractional order systems; also, algebraic graph theory and sufficient conditions are presented to ensure the consensus for fractional multiagent systems. Furthermore, adaptive protocols of each agent using local information are designed and a detailed analysis of the leader-following consensus is presented. Finally, some numerical simulation examples are also given to show the effectiveness of the proposed results.

Exact stability for Turán’s Theorem

Turán's Theorem says that an extremal Kr+1-free graph is r-partite. The Stability Theorem of Erdős and Simonovits shows that if a Kr+1-free graph with n vertices has close to the maximal tr(n) edges, then it is close to being r-partite. In this paper we determine exactly the Kr+1-free graphs with at least m edges that are farthest from being r-partite, for any m≥tr(n)−δrn2. This extends work by Erdős, Győri and Simonovits, and proves a conjecture of Balogh, Clemen, Lavrov, Lidický and Pfender.

Distributed Consensus Algorithm for Nonholonomic Wheeled Mobile Robot

This paper focuses on consensus of the nonholonomic wheeled mobile robotic systems whose geometric center and centroid do not coincide. A consensus control algorithm for mobile robots based on the nonstandard chain systems is proposed. Firstly, coordinate transformation is used to transform the nonholonomic robotic systems into the nonstandard chain model. Then, a distributed cooperative control algorithm is designed, and the Lyapunov stability theorem and LaSalle invariance principle are used to prove that each state of the mobile robot is consensus. Finally, the effectiveness of the algorithm is proved through numerical simulation.

Robust Stabilization and Observer-Based Stabilization for a Class of Singularly Perturbed Bilinear Systems

An infinite-bound stabilization of a system modeled as singularly perturbed bilinear systems is examined. First, we present a Lyapunov equation approach for the stabilization of singularly perturbed bilinear systems for all ε∈(0, ∞). The method is based on the Lyapunov stability theorem. The state feedback constant gain can be determined from the admissible region of the convex polygon. Secondly, we extend this technique to study the observer and observer-based controller of singularly perturbed bilinear systems for all ε∈(0, ∞). Concerning this problem, there are two different methods to design the observer and observer-based controller: one is that the estimator gain can be calculated with known bounded input, the other is that the input gain can be calculated with known observer gain. The main advantage of this approach is that we can preserve the characteristic of the composite controller, i.e., the whole dimensional process can be separated into two subsystems. Moreover, the presented stabilization design ensures the stability for all ε∈(0, ∞). A numeral example is given to compare the new ε-bound with that of previous literature.

Distributed Observer Design for Linear Systems under Time-Varying Communication Delay

The paper investigates the state estimation problem of general continuous-time linear systems with the consideration of time-varying communication delay. A solution is proposed in terms of the networked distributed observer, which consists of multiple local observers. Each local observer relies on only part of the system output and exchanges information with neighbors through undirected links modeled by a prespecified communication graph. A simple approach for computing observer parameters is presented by solving a parametric algebraic Riccati equation. Furthermore, by the Lyapunov–Krasovskii stability theorem, an upper bound of the delay could be calculated explicitly and together with the conditions of joint observability and connectivity of the communication graph; the resulting distributed observers work coordinately to achieve an asymptotic estimate of the full plant state. An illustrative example is provided to confirm the analytical results.

Multistability Analysis of State-Dependent Switched Hopfield Neural Networks With The Gaussian-Wavelet-Type Activation Function*

This paper studies the multistability of state-dependent switched Hopfield neural networks (SSHNNs) with the Gaussian-wavelet-type activation function. The coexistence and stability of multiple equilibria of SSHNNs are proved. By using Brouwer's fixed point theorem, it is obtained that the SSHNNs can have at least 7n or 6n equilibria under a specified set of conditions. By using the strictly diagonally dominance matrix (SDDM) theorem and Lyapunov stability theorem, 4n or 5n locally stable (LS) equilibria are obtained, respectively. Compared with the conventional Hopfield neural networks (HNNs) without state-dependent switching or SSHNNs with other kinds of activation functions, SSHNNs with this type of activation functions can have more LS equilibria, which implies that SSHNNs with Gaussian-wavelet-type activation functions can have even larger storage capacity and would be more dominant in associative memory application. Last, some simulation results are given to verify the correctness of the theoretical results.

Exponential Stability of Impulsive Stochastic Delay System Based on Razumikhin Method and Its Application to Chaos Control

In this paper, the exponential stability of a stochastic delay system with impulsive signal is considered, and stability theorem of this system is proposed based on the Lyapunov–Razumikhin method; the convergence rate is also given, which gives theoretical foundation to chaos control and synchronization using the impulsive method. Finally, the classic delay chaos system with white noise and impulsive signal is employed to verify the feasibility and effectiveness of our theorem.

Absence of bubbling phenomena for non-convex anisotropic nearly umbilical and quasi-Einstein hypersurfaces

We prove that, for every closed (not necessarily convex) hypersurface Σ in ℝ n + 1 {\mathbb{R}^{n+1}} and every p > n {p>n} , the L p {L^{p}} -norm of the trace-free part of the anisotropic second fundamental form controls from above the W 2 , p {W^{2,p}} -closeness of Σ to the Wulff shape. In the isotropic setting, we provide a simpler proof. This result is sharp since in the subcritical regime p ≤ n {p\leq n} , the lack of convexity assumptions may lead in general to bubbling phenomena. Moreover, we obtain a stability theorem for quasi-Einstein (not necessarily convex) hypersurfaces and we improve the quantitative estimates in the convex setting.

On the rotational symmetry of 3-dimensional κ-solutions

In a recent paper, Brendle showed the uniqueness of the Bryant soliton among 3-dimensional κ-solutions. In this paper, we present an alternative proof for this fact and show that compact κ-solutions are rotationally symmetric. Our proof arose from independent work relating to our Strong Stability Theorem for singular Ricci flows.

Fixed time synchronization of a class of chaotic systems based via the saturation control

In this paper, we discuss the fixed time synchronization of a class of chaotic systems based on the backstepping control with disturbances. A new and important fixed time stability theorem is presented. The upper bound estimate formulas of the settling time are also given which are different from the existing results in the literature. Based on the new fixed time stability theorem, a novel saturation controller for the fixed time synchronization a class of chaotic systems is proposed via the backstepping method. Finally, the new chaotic system is taken as an example to illustrate the applicability of the obtained theory.