Finite-time stability with respect to a closed invariant set for a class of discontinuous systems

This paper discusses the problem of finite-time stability with respect to a closed, but not necessarily compact, invariant set for a class of nonlinear systems with discontinuous right-hand sides in the sense of the Filippov solutions. When the Lyapunov function is Lipschitz continuous and regular, the Lyapunov theorem on finite-time stability with respect to a closed invariant set is presented.

A Note on Invariant Sets

The lower bound of Hausdorff dimension of the invariant set is given.

On the new results of global attractive set and positive invariant set of the Lorenz chaotic system and the applications to chaos control and synchronization

Constructing a family of generalized Lyapunov functions, a new method is proposed to obtain new global attractive set and positive invariant set of the Lorenz chaotic system. The method we proposed greatly simplifies the complex proofs of the two famous estimations presented by the Russian scholar Leonov. Our uniform formula can derive a series of the new estimations. Employing the idea of intersection in set theory, we extract a new Leonov formula-like estimation from the family of the estimations. With our method and the new estimation, one can confirm that there are no equilibrium, periodic solutions, almost periodic motions, wandering motions or other chaotic attractors outside the global attractive set. The Lorenz butterfly-like singular attractors are located in the global attractive set only. This result is applied to the chaos control and chaos synchronization. Some feedback control laws are obtained to guarantee that all the trajectories of the Lorenz systems track a periodic solution, or globally stabilize an unstable (or locally stable but not globally asymptotically stable) equilibrium. Further, some new global exponential chaos synchronization results are presented. Our new method and the new results are expected to be applied in real secure communication systems.

Regularity of Invariant Sets in Variable Internal Damped Wave Equations

In this paper we prove that every compact invariant subset■associated with the semigroup{Sn,k(t)}t≥0 generated by wave equations with variable damping,either in the interior or on the boundary of the domainΩwhereΩ■R^3 is a smooth bounded domain,in H^10(Ω)×L^2(Ω)is in fact bounded in D(B0)×H^10(Ω)As an application of our results,we obtain the upper-semicontinuity for global attractor of the weakly damped semilinear wave equation in the norm of H^1(Ω)×L^2(Ω)when the interior variable damping converges to theboundary damping in the sense of distributions.

Stabilizing model predictive control scheme for piecewise affine systems with maximal positively invariant terminal set

An efficient algorithm is proposed for computing the solution to the constrained finite time optimal control （CFTOC） problem for discrete-time piecewise affine （PWA） systems with a quadratic performance index. The maximal positively invariant terminal set, which is feasible and invariant with respect to a feedback control law, is computed as terminal target set and an associated Lyapunov function is chosen as terminal cost. The combination of these two components guarantees constraint satisfaction and closed-loop stability for all time. The proposed algorithm combines a dynamic programming strategy with a multi-parametric quadratic programming solver and basic polyhedral manipulation. A numerical example shows that a larger stabilizable set of states can be obtained by the proposed algorithm than precious work.

La Shalle＇s invariant-set-theory based asymptotic synchronization of duffing system with unknown parameters

A novel La Shalle＇s invariant set theory （LSIST） based adaptive asymptotic synchronization （LSISAAS） method is proposed to asymptotically synchronize Duffing system with unknown parameters which also are considered as system states. The LSISASS strategy depends on the only information, i.e. one state of the master system. According to the LSIST, the LSISASS method can asymptotically synchronize fully the states of the master system and the unknown system parameters as well. Simulation results also validate that the LSISAAS approach can obtain asymptotic synchronization.

Heteroclinic Cycles in a Class of 3-Dimensional Piecewise Affine Systems

This paper explores the existence of heteroclinic cycles and corresponding chaotic dynamics in a class of 3-dimensional two-zone piecewise affine systems. Moreover, the heteroclinic cycles connect two saddle foci and intersect the switching manifold at two points and the switching manifold is composed of two perpendicular planes.

Hybrid Dynamic Variables-Dependent Event-Triggered Fuzzy Model Predictive Control

This article focuses on dynamic event-triggered mechanism(DETM)-based model predictive control(MPC) for T-S fuzzy systems.A hybrid dynamic variables-dependent DETM is carefully devised,which includes a multiplicative dynamic variable and an additive dynamic variable.The addressed DETM-based fuzzy MPC issue is described as a “min-max” optimization problem(OP).To facilitate the co-design of the MPC controller and the weighting matrix of the DETM,an auxiliary OP is proposed based on a new Lyapunov function and a new robust positive invariant(RPI) set that contain the membership functions and the hybrid dynamic variables.A dynamic event-triggered fuzzy MPC algorithm is developed accordingly,whose recursive feasibility is analysed by employing the RPI set.With the designed controller,the involved fuzzy system is ensured to be asymptotically stable.Two examples show that the new DETM and DETM-based MPC algorithm have the advantages of reducing resource consumption while yielding the anticipated performance.

Better Hausdorff Dimension Estimations of Quadratic and Cubic Functions' Julia Sets

More accurate Hausdorff dimension estimations of Julia sets for two simple functions are given by the methods of composition mapping and invariant set of contraction mapping. For quadratic function fc （ z ） = z^2 ＋ c（c ∈^C）, the range of parameter c is expanded largely and a result on the Hausdorff dimension of its Julia set is gained. Similarly, a better result is obtained for cubic function fc（z） = z^3 ＋ c（c ∈ ^C）.

Existence and Concentration of Sign-Changing Solutions of Quasilinear Choquard Equation

In this paper, we study the following quasilinear equation of choquard type: where A(x,t) is given real functions on RN × R and with N ≥ 3, 1 p N, max{N-2p,1} α N, , and ε > 0 is a small parameter, Iα is the Riesz potential. We establish for small ε the existence of a sequence of sign-changing solutions concentrating near a given local minimum point of the bounded potential function V by using the method of invariant sets of descending flow, perturbation method and truncation technique. .

Discrete Conley index and bifurcation points

In this paper, a sufficient condition for the existence of bifurcation points for discrete dynamical systems is presented. The relation between two families of systems is further discussed, and a sufficient condition for determining whether they may have the similar bifurcation points is given.

Event-triggered MPC Design for Distributed Systems With Network Communications

This paper deals with the communication problem in the distributed system, considering the limited battery power in the wireless network and redundant transmission among nodes. We design an event-triggered model predictive control(ET-MPC) strategy to reduce the unnecessary communication while promising the system performance. On one hand, for a linear discrete time-invariant system, a triggering condition is derived based on the Lyapunov stability. Here, in order to further reduce the communication rate, we enforce a triggering condition only when the Lyapunov function will exceed its value at the last triggered time, but an average decrease is guaranteed. On the other hand, the feasibility is ensured by minimizing and optimizing the terminal constrained set between the maximal control invariant set and the target terminal set. Finally, we provide a simulation to verify the theoretical results. It's shown that the proposed strategy achieves a good trade-off between the closed-loop system performance and communication rate.

Stability analysis for continuous-time systems with actuator saturation

The aim of this paper is to study the determination of the stability regions for continuous-time systems subject to actuator saturation. Using an affine saturation-dependent Lyapunov function, a new method is proposed to obtain the estimation of the domain of attraction of the closed-loop system. A family of linear matrix inequalities （LMIs） that provides sufficient conditions for the existence of this type of Lyapunov function are presented. The results obtained in this paper can reduce the conservativeness compared with the existing ones. Numerical examples are given to illustrate the effectiveness of the proposed results.

THE POSITIVE INVARIANT SET OF THE DIFFERENTIAL EQUATION ON BANACH SPACE

A sufficient condition of the existence of a positive invariant set in the differential equation on Banach space is given, which improves a theorem of Mitio Nagumo.

Existence of solutions for fourth-order boundary value problem with parameter

This paper is a sequel to a previous paper （Yang, Y. and Zhang, J. H. Existence of solutions for some fourth-order boundary value problems with parameters. Nonlinear Anal. 69（2）, 1364-1375 （2008）） in which the nontrivial solutions to the fourthorder boundary value problems were studied. In the current work with the same conditions near infinity but different near zero, the positive, negative, and sign-changing solutions are obtained by the critical point theory, retracting property, and invariant sets.

ASYMPTOTIC PROPERTIES OF FUNCTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES

The paper is devoted to the asymptotic properties of functional differential equations in Banach spaces.The criteria of the invariant and attracting sets are obtained.Particularly, the sufficient condition of asymptotic stability of the equilibrium point is given as the system has an equilibrium point.Several examples are also worked out to demonstrate the validity of the results.

Constrained predictive control of nonlinear stochastic systems

The receding horizon control（RHC） problem is considered for nonlinear Markov jump systems which can be represented by Takagi-Sugeno fuzzy models subject to constraints both on control inputs and on observe outputs.In the given receding horizon,for each mode sequence of the T-S modeled nonlinear system with Markov jump parameter,the cost function is optimized by constraints on state trajectories,so that the optimization control input sequences are obtained in order to make the state into a terminal invariant set.Out of the receding horizon,the stability is guaranteed by searching a state feedback control law.Based on such stability analysis,a linear matrix inequality approach for designing receding horizon predictive controller for nonlinear systems subject to constraints both on the inputs and on the outputs is developed.The simulation shows the validity of this method.

Reconfigurability evaluation method for input-constrained control systems

This paper proposes a quantitative reconfigurability evaluation method for control systems with actuator saturation and additive faults from the perspective of system stability.Placing the saturated feedback law in the convex hull of a group of auxiliary linear controls,the sufficient reconfigurability conditions for the system under additive faults are derived using invariant sets.These conditions are then expressed as linear matrix inequalities(LMIs)and applied to quantify the degree of reconfigurability for the fault system.The largest fault magnitude for which the system can be stabilized,the largest initial state domain from which all the trajectories are convergent,and the minimum final state domain to which the trajectories will converge are investigated.The effectiveness of the proposed method is illustrated through an application example.

Deterministic Multicultural Dynamic Networks: Seeking a Balance between Attractive and Repulsive Forces

An important issue in society is the attempt to balance communities working co-operatively and cohesively with one another while allowing members the ability to retain individuality and fostering an environment of cultural diversity. We seek to study the cohesive properties of a culturally diverse dynamic social network. By considering a multi-agent dynamic network, we seek to model a social structure and find conditions under which cohesion and coexistence are maintained. We present a specific illustration that serves to establish the framework in which explicit sufficient conditions in terms of system parameters are found for which the network is cohesive. By utilizing Lyapunov’s Second Method and comparison equations, we are able to find such conditions for the given illustration. Further, for the illustration, we decompose the cultural state domain into invariant sets and consider the behavior of members within each set. Moreover, we analyze the relative cultural affinity between individual members relative to the center of the social network. We also demonstrate how conservative the estimates are using Euler type numerical approximation schemes based on the given illustration. We are then able to consider how changes in the various parameters affect the dynamics of the illustrated network. By gaining such insight into the behavior of the illustrated network, we are able to better understand the impact of both attractive and repulsive influences on the network. This leads to establishing a schema for helping when creating policies and practices catered to promoting both diversity and cohesion within a cultural network.

Design of switched linear systems in the presence of actuator saturation and L-infinity disturbances

This paper considers the problem of disturbance tolerance/rejection of a switched system resulting from a family of linear systems subject to actuator saturation and E-infinity disturbances. For a given set of linear feedback gains, a given switching scheme and a given bound on the E-infinity norm of the disturbances, conditions are established, in terms of linear or bilinear matrix inequalities, under which a set of a certain form is invariant for a given switched linear system in the presence of actuator saturation and E-infinity disturbances, and the closed-loop system possesses a certain level of disturbance rejection capability. With these conditions, the design of feedback gains and switching scheme can be formulated and solved as constrained optimization problems. Disturbance tolerance is measured by the largest bound on the disturbances for which the trajectories starting from a given set remain bounded. Disturbance rejection is measured either by the E-infinity norm of the system output or by the system＇s ability to steer its state into and/or keep it within a small neighborhood of the origin. In the event that all systems in the family are identical, the switched system reduces to a single system under a switching feedback law. Simulation results show that such a single system under a switching feedback law could have stronger disturbance tolerance/rejection capability than a single linear feedback law can.