On a Class of Lur’e Dynamical Systems with State-Dependent Set-Valued Feedback

Set-Valued and Variational Analysis - Using a new implicit discretization scheme, we study in this paper the existence and uniqueness of strong solutions for a class of Lur’e dynamical...     

Martelli’s Chaos in Inverse Limit Dynamical Systems and Hyperspace Dynamical Systems

In this paper we investigate Martelli’s chaos of inverse limit dynamical systems and hyperspace dynamical systems which are both induced from dynamical systems on a compact metric space. We give the implication of Martelli’s chaos among those systems. More precisely, we show that inverse limit dynamical system is Martelli’s chaos if and only if so is original system, and we prove that hyperspace dynamical system is Martelli’s chaos implies original system is Martelli’s chaos if the orbit of every single point set of original system is unstable in hyperspace dynamical system.     

Synchronization Criterion for Lur’e Systems via Delayed PD Controller

In this paper, the effects of a time varying delay on a chaotic drive-response synchronization are considered. Using a delayed feedback proportional-derivative (PD) controller scheme, a delay-dependent synchronization criterion is derived for chaotic systems represented by the Lur’e system with sector and slope restricted nonlinearities. The derived criterion is a sufficient condition for the absolute stability of the error dynamics between the drive and the response systems. By the use of a convex representation of the nonlinearity and the discretized Lyapunov-Krasovskii functional, stability condition is obtained via the LMI formulation. The condition represented in the terms of linear matrix inequalities (LMIs) can be solved by the application of convex optimization algorithms. The effectiveness of the work is verified through numerical examples.     

Penalty Algorithms in Hilbert Spaces

We analyze the classical penalty algorithm for nonlinear programming in Hilbert spaces and obtain global convergence results, as well as asymptotic superlinear convergence order. These convergence results generalize similar results obtained for finite-dimensional problems. Moreover, the nature of the algorithms allows us to solve the unconstrained subproblems in finite-dimensional spaces.     

Sampled-data control of Lur’e systems

The main purpose of this paper is to fill a gap in the literature concerning the problem of designing a sampled-data control law for continuous-time Lur’e systems. The goal is to design a state feedback sampled-data control for this class of nonlinear systems preserving global asymptotic stability and minimizing a guaranteed quadratic cost. The main challenge towards the solution of the proposed problem is to handle this class of nonlinear system in order to propose less conservative design conditions expressed through differential linear matrix inequalities - (DLMIs). Bellman’s Principle of Optimality applied together with the Popov–Lyapunov function that emerges from the celebrated Popov Stability Criterion is the key issue to obtain the reported results. Two examples are solved for illustration.     

Some Dynamical Properties in Set-valued Discrete Systems

1 Introduction A discrete dynamical system can be expressed as xn+1 ?f(xn), n = 0,1,2,... where X is a metric space and f : X →X is a continuous map. The study of it tells us how the points in the base space X moved. Nevertheless, this is not enough for the researches of biological species, demography, numerical simulation and attractors (see [1], [2]). It is necessary to know how the subsets of X moved. In this direction, we consider the set-valued discrete system associated to f, An+1 = (f|-)(An), n = 0,1,2,... where (f|-) is the natural extension of f to K(X) (the class of all compact subsets of X).     

Lorentz–Minkowski Distances in Hilbert Spaces

For arbitrary pre-Hilbert spaces X of dimension at least 2, we define the notion of Lorentz–Minkowski distance. We determine all Lorentz transformations, i.e. all distance preserving mappings of X into itself, introduce relativistic addition, and characterize the 2-point-invariants of the group of bijective Lorentz tranformations and, especially, the distance in question.     

Hidden Oscillations in Dynamical Systems. 16 Hilbert’s Problem,Aizerman’s and Kalman’s Conjectures,Hidden Attractors in Chua’s Circuits

The present survey is devoted to efficient methods for localization of hidden oscillations in dynamical systems. Their application to Hilbert’s sixteenth problem for quadratic systems, Aizerman’s problem, and Kalman’s problem on absolute stability of control systems, and to the localization of chaotic hidden attractors (the basin of attraction of which does not contain neighborhoods of equilibria) is considered. The synthesis of the describing function method with the applied bifurcation theory and numerical methods for computing hidden oscillations is described.     

Cyclic Hilbert Spaces and Connes’ Embedding Problem

Let M be a II ₁-factor with trace τ, the finite dimensional subspaces of L ²(M, τ) are not just common Hilbert spaces, but they have an additional structure. We introduce the notion of a cyclic linear space by taking these additional properties as axioms. In Sect. 3 we formulate the following problem: “does every cyclic Hilbert space embed into L ²(M, τ), for some M?”. An affirmative answer would imply the existence of an algorithm to check Connes’ embedding Conjecture. In Sect. 4 we make a first step towards the answer of the previous question.     

Strong Convergence in Hilbert Spaces via Γ-Duality

We analyze a primal-dual pair of problems generated via a duality theory introduced by Svaiter. We propose a general algorithm and study its convergence properties. The focus is a general primal-dual principle for strong convergence of some classes of algorithms. In particular, we give a different viewpoint for the weak-to-strong principle of Bauschke and Combettes and unify many results concerning weak and strong convergence of subgradient type methods.     

Painlev′e Prop erty and Integrability of Polynomial Dynamical Systems

The main purpose of this paper is to investigate the connection between the Painlev′e property and the integrability of polynomial dynamical systems. We show that if a polynomial dynamical system has P...     

Small-gain stability theorems for positive Lur’e inclusions

Positivity - Stability results are presented for a class of differential and difference inclusions, so-called positive Lur’e inclusions which arise, for example, as the feedback...     

Novel Delay-Dependent Stabilization Criterion for Lur’e Systems with Sector-Restricted Nonlinearities and External Disturbances via PD Feedback Approach

In this paper, the effects of time delay on control of Lur’e systems are considered. Using a convex representation of the nonlinearity, a novel delay-dependent stability criterion is derived for the Lur’e systems via proportional-derivative (PD) feedback law. The Lyapunov–Krasovskii functional based on the delay discretization approach is used for the purpose of obtaining the stability condition by utilizing Projection Lemma. The criterion is utilized to not only guarantee stability of systems but also reduce the effect of external disturbance to an norm constraint. Finally, two numerical examples show the effectiveness of the proposed method.     

Periodic Nonuniform Dynamical Sampling in ℓ2(ℤ) and Shift-Invariant Spaces

In this article, we mainly study the periodic nonuniform dynamical sampling in ?²(?) and shift-invariant spaces. We first provide a su?cient and necessary condition for c∈?²(?) which can be reconstructed by its spatial and temporal samples. Then we give a concrete example to show that the su?cient and necessary condition is feasible. Finally, we discuss the periodic nonuniform dynamic sampling problem in shift-invariant spaces.     

Asymptotical synchronization for chaotic Lur’e systems using sampled-data control

This paper presents a new method for the asymptotical synchronization of two identical chaotic Lur’e systems using sampling control. The method is based on a new Lyapunov–Krasovskii functional (LKF) in the framework of an input delay method. Compared with existing works, the new LKF makes full use of the information on the nonlinear part of the system and introduces a novel term, which waives the common positive requirement of each LKF term to guarantee the positive of the whole LKF. A typical Chua’s circuit is given to verify the effectiveness of the proposed method.     

Synchronization of chaotic Lur’e systems with quantized sampled-data controller

In this paper, we consider the master–slave synchronization problem of chaotic Lur’e systems. It is assumed that only quantized sampled measurements are available for the controller. By modeling the synchronization error system as an input-delay system and constructing a new Lyapunov functional, a new sufficient condition and feedback controller design method for global exponential asymptotical synchronization of master and slave system are obtained. The proposed approach has taken the feature of sample-induced delay into consideration and simulation results show the less conservativeness.     

Almost sure convergence of stochastic integrals in Hilbert Spaces ∗

It is proved that, under the usual measurability and the square-integrability conditions, the stochastic integral of an operator-valued integrand with respect to a continuous local martingale in Hilbert spaces can be defined for almost every sample path via a sequence of simple progressively measurable approximations. The result generalizes McKeanls construction of the It6 integral with respect to a one-dimensional Brownian motion