Stability and synchronization of discrete-time neural networks with switching parameters and time-varying delays

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Dynamical principles in neuroscience

Dynamical modeling of neural systems and brain functions has a history of success over the last half century. This includes, for example, the explanation and prediction of some features of neural rhythmic behaviors. Many interesting dynamical models of learning and memory based on physiological experiments have been suggested over the last two decades. Dynamical models even of consciousness now exist. Usually these models and results are based on traditional approaches and paradigms of nonlinear dynamics including dynamical chaos. Neural systems are, however, an unusual subject for nonlinear dynamics for several reasons: (i) Even the simplest neural network, with only a few neurons and synaptic connections, has an enormous number of variables and control parameters. These make neural systems adaptive and flexible, and are critical to their biological function. (ii) In contrast to traditional physical systems described by well-known basic principles, first principles governing the dynamics of neural systems are unknown. (iii) Many different neural systems exhibit similar dynamics despite having different architectures and different levels of complexity. (iv) The network architecture and connection strengths are usually not known in detail and therefore the dynamical analysis must, in some sense, be probabilistic. (v) Since nervous systems are able to organize behavior based on sensory inputs, the dynamical modeling of these systems has to explain the transformation of temporal information into combinatorial or combinatorial-temporal codes, and vice versa, for memory and recognition. In this review these problems are discussed in the context of addressing the stimulating questions: What can neuroscience learn from nonlinear dynamics, and what can nonlinear dynamics learn from neuroscience?This work was supported by NSF Grant No. NSF/EIA-0130708, and Grant No. PHY 0414174; NIH Grant No. 1 R01 NS50945 and Grant No. NS40110; MEC BFI2003-07276, and Fundación BBVA

Qualitative Studies of Nonlinear Hybrid Systems

A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior. Hybrid systems arise in a wide variety of important applications in diverse areas, ranging from biology to computer science to air traffic dynamics. The interaction of continuous- and discrete-time dynamics in a hybrid system often leads to very rich dynamical behavior and phenomena that are not encountered in purely continuous- or discrete-time systems. Investigating the dynamical behavior of hybrid systems is of great theoretical and practical importance. The objectives of this thesis are to develop the qualitative theory of nonlinear hybrid systems with impulses, time-delay, switching modes, and stochastic disturbances, to develop algorithms and perform analysis for hybrid systems with an emphasis on stability and control, and to apply the theory and methods to real-world application problems. Switched nonlinear systems are formulated as a family of nonlinear differential equations, called subsystems, together with a switching signal that selects the continuous dynamics among the subsystems. Uniform stability is studied emphasizing the situation where both stable and unstable subsystems are present. Uniformity of stability refers to both the initial time and a family of switching signals. Stabilization of nonlinear systems via state-dependent switching signal is investigated. Based on assumptions on a convex linear combination of the nonlinear vector fields, a generalized minimal rule is proposed to generate stabilizing switching signals that are well-defined and do not exhibit chattering or Zeno behavior. Impulsive switched systems are hybrid systems exhibiting both impulse and switching effects, and are mathematically formulated as a switched nonlinear system coupled with a sequence of nonlinear difference equations that act on the switched system at discrete times. Impulsive switching signals integrate both impulsive and switching laws that specify when and how impulses and switching occur. Invariance principles can be used to investigate asymptotic stability in the absence of a strict Lyapunov function. An invariance principle is established for impulsive switched systems under weak dwell-time signals. Applications of this invariance principle provide several asymptotic stability criteria. Input-to-state stability notions are formulated in terms of two different measures, which not only unify various stability notions under the stability theory in two measures, but also bridge this theory with the existent input/output theories for nonlinear systems. Input-to-state stability results are obtained for impulsive switched systems under generalized dwell-time signals. Hybrid time-delay systems are hybrid systems with dependence on the past states of the systems. Switched delay systems and impulsive switched systems are special classes of hybrid time-delay systems. Both invariance property and input-to-state stability are extended to cover hybrid time-delay systems. Stochastic hybrid systems are hybrid systems subject to random disturbances, and are formulated using stochastic differential equations. Focused on stochastic hybrid systems with time-delay, a fundamental theory regarding existence and uniqueness of solutions is established. Stabilization schemes for stochastic delay systems using state-dependent switching and stabilizing impulses are proposed, both emphasizing the situation where all the subsystems are unstable. Concerning general stochastic hybrid systems with time-delay, the Razumikhin technique and multiple Lyapunov functions are combined to obtain several Razumikhin-type theorems on both moment and almost sure stability of stochastic hybrid systems with time-delay. Consensus problems in networked multi-agent systems and global convergence of artificial neural networks are related to qualitative studies of hybrid systems in the sense that dynamic switching, impulsive effects, communication time-delays, and random disturbances are ubiquitous in networked systems. Consensus protocols are proposed for reaching consensus among networked agents despite switching network topologies, communication time-delays, and measurement noises. Focused on neural networks with discontinuous neuron activation functions and mixed time-delays, sufficient conditions for existence and uniqueness of equilibrium and global convergence and stability are derived using both linear matrix inequalities and M-matrix type conditions. Numerical examples and simulations are presented throughout this thesis to illustrate the theoretical results

Impulsive Control of Dynamical Networks

Dynamical networks (DNs) consist of a large set of interconnected nodes with each node being a fundamental unit with detailed contents. A great number of natural and man-made networks such as social networks, food networks, neural networks, WorldWideWeb, electrical power grid, etc., can be effectively modeled by DNs. The main focus of the present thesis is on delay-dependent impulsive control of DNs. To study the impulsive control problem of DNs, we firstly construct stability results for general nonlinear time-delay systems with delayed impulses by using the method of Lyapunov functionals and Razumikhin technique. Secondly, we study 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. Then, a novel hybrid consensus protocol with dynamically changing interaction topologies is designed to take the time-delay into account in both the continuous-time communication among agents and the instant information exchange at discrete moments. We also study the consensus problem of networked multi-agent systems. Distributed delays are considered in both the agent dynamics and the proposed impulsive consensus protocols. Lastly, stabilization and synchronization problems of DNs under pinning impulsive control are studied. A pinning algorithm is incorporated with the impulsive control method. We propose a delay-dependent pinning impulsive controller to investigate the synchronization of linear delay-free DNs on time scales. Then, we apply the pinning impulsive controller proposed for the delay-free networks to stabilize time-delay DNs. Results show that the delay-dependent pinning impulsive controller can successfully stabilize and synchronize DNs with/without time-delay. Moreover, we design a type of pinning impulsive controllers that relies only on the network states at history moments (not on the states at each impulsive instant). Sufficient conditions on stabilization of time-delay networks are obtained, and results show that the proposed pinning impulsive controller can effectively stabilize the network even though only time-delay states are available to the pinning controller at each impulsive instant. We further consider the pinning impulsive controllers with both discrete and distributed time-delay effects to synchronize the drive and response systems modeled by globally Lipschitz time-delay systems. As an extension study of pinning impulsive control approach, we investigate the synchronization problem of systems and networks governed by PDEs

Nonlinear Systems

Open Mathematics is a challenging notion for theoretical modeling, technical analysis, and numerical simulation in physics and mathematics, as well as in many other fields, as highly correlated nonlinear phenomena, evolving over a large range of time scales and length scales, control the underlying systems and processes in their spatiotemporal evolution. Indeed, available data, be they physical, biological, or financial, and technologically complex systems and stochastic systems, such as mechanical or electronic devices, can be managed from the same conceptual approach, both analytically and through computer simulation, using effective nonlinear dynamics methods. The aim of this Special Issue is to highlight papers that show the dynamics, control, optimization and applications of nonlinear systems. This has recently become an increasingly popular subject, with impressive growth concerning applications in engineering, economics, biology, and medicine, and can be considered a veritable contribution to the literature. Original papers relating to the objective presented above are especially welcome subjects. Potential topics include, but are not limited to: Stability analysis of discrete and continuous dynamical systems; Nonlinear dynamics in biological complex systems; Stability and stabilization of stochastic systems; Mathematical models in statistics and probability; Synchronization of oscillators and chaotic systems; Optimization methods of complex systems; Reliability modeling and system optimization; Computation and control over networked systems

Discrete Time Systems

Discrete-Time Systems comprehend an important and broad research field. The consolidation of digital-based computational means in the present, pushes a technological tool into the field with a tremendous impact in areas like Control, Signal Processing, Communications, System Modelling and related Applications. This book attempts to give a scope in the wide area of Discrete-Time Systems. Their contents are grouped conveniently in sections according to significant areas, namely Filtering, Fixed and Adaptive Control Systems, Stability Problems and Miscellaneous Applications. We think that the contribution of the book enlarges the field of the Discrete-Time Systems with signification in the present state-of-the-art. Despite the vertiginous advance in the field, we also believe that the topics described here allow us also to look through some main tendencies in the next years in the research area

On Stability and Stabilization of Hybrid Systems

The thesis addresses the stability, input-to-state stability (ISS), and stabilization problems for deterministic and stochastic hybrid systems with and without time delay. The stabilization problem is achieved by reliable, state feedback controllers, i.e., controllers experience possible faulty in actuators and/or sensors. The contribution of this thesis is presented in three main parts. Firstly, a class of switched systems with time-varying norm-bounded parametric uncertainties in the system states and an external time-varying, bounded input is addressed. The problems of ISS and stabilization by a robust reliable $H_{\infty}$ control are established by using multiple Lyapunov function technique along with the average dwell-time approach. Then, these results are further extended to include time delay in the system states, and delay systems subject to impulsive effects. In the latter two results, Razumikhin technique in which Lyapunov function, but not functional, is used to investigate the qualitative properties. Secondly, the problem of designing a decentralized, robust reliable control for deterministic impulsive large-scale systems with admissible uncertainties in the system states to guarantee exponential stability is investigated. Then, reliable observers are also considered to estimate the states of the same system. Furthermore, a time-delayed large-scale impulsive system undergoing stochastic noise is addressed and the problems of stability and stabilization are investigated. The stabilization is achieved by two approaches, namely a set of decentralized reliable controllers, and impulses. Thirdly, a class of switched singularly perturbed systems (or systems with different time scales) is also considered. Due to the dominant behaviour of the slow subsystem, the stabilization of the full system is achieved through the slow subsystem. This approach results in reducing some unnecessary sufficient conditions on the fast subsystem. In fact, the singular system is viewed as a large-scale system that is decomposed into isolated, low order subsystems, slow and fast, and the rest is treated as interconnection. Multiple Lyapunov functions and average dwell-time switching signal approach are used to establish the stability and stabilization. Moreover, switched singularly perturbed systems with time-delay in the slow system are considered

18th IEEE Workshop on Nonlinear Dynamics of Electronic Systems: Proceedings

Proceedings of the 18th IEEE Workshop on Nonlinear Dynamics of Electronic Systems, which took place in Dresden, Germany, 26 – 28 May 2010.:Welcome Address ........................ Page I Table of Contents ........................ Page III Symposium Committees .............. Page IV Special Thanks ............................. Page V Conference program (incl. page numbers of papers) ................... Page VI Conference papers Invited talks ................................ Page 1 Regular Papers ........................... Page 14 Wednesday, May 26th, 2010 ......... Page 15 Thursday, May 27th, 2010 .......... Page 110 Friday, May 28th, 2010 ............... Page 210 Author index ............................... Page XII