Explicit stability conditions for time-discrete vector Lur'e-type systems

^** Email: gilmi{at}cs.bgu.ac.il^*** Email: rmedina{at}ulagos.cl Non-linear discrete-time Lur'e-type systems are considered.Explicit stability conditions are suggested. They are formulatedin terms of the determinants of characteristic polynomials.Moreover, an estimate for the region of attraction of the zerosolution is derived.     

Finite attraction for nonlinear uncertain mechanical systems

This paper deals with the robust control problem for uncertainnonlinear mechanical systems with ideal and holonomic constraints.A robust control law having the form of state feedback is obtained.The control law consists of two parts: the first one is a nominalcontrol law which affects the behaviour of the nominal dynamicsfor the considered mechanical system; the second one is a learningcontrol law which reduces the influence of uncertainties onthe stability of the mechanical system. Under the action ofthe robust control law, the closed-loop system for the uncertainmechanical system reaches a region specified a short time inadvance.     

A system-theoretic framework for a wide class of systems I: Applications to numerical analysis

A system-theoretic framework is proposed, which allows the study of hybrid uncertain systems, which do not satisfy the so-called “semigroup property.” Characterizations of the notion of robust global asymptotic output stability (RGAOS) are given. Based on the provided characterizations, the qualitative behavior of hybrid systems obtained by time-discretization of systems of ordinary differential equations with a globally asymptotically stable equilibrium point, is studied.     

The domain of attraction and the stability region for stochastic partial differential equations with delays

In this paper, a stochastic partial differential equation with delays is considered. On the basis of the properties of nonnegative matrices, stochastic convolution and the inequality technique, sufficient conditions for determining the domain of p$p$th-moment attraction and the p$p$th-moment asymptotic stability region are obtained. An example is also discussed, to illustrate the efficiency of the results obtained.     

Lyapunov theorems for systems described by retarded functional differential equations

Lyapunov-like characterizations for non-uniform in time and uniform robust global asymptotic stability of uncertain systems described by retarded functional differential equations are provided.     

A quantitative study on the stability of delay discrete singular systems

For quadratic delay discrete singular systems, an algebraic criterion on the stability is established, and the size of the uniform stability region and asymptotic stability region around zero is estimated. Hence, the criterion is both qualitative and quantitative. With the computer techniques, the criterion dependent of delay is easy test and applies to the application in the practice. An illustrative simulation is given to illustrate the application of the obtained result.     

Prey-predator systems with intertwined basins of attraction

In this article, we show the existence of intertwined basins of attraction for a class of prey-predator systems, which improves the extant results by deleting some crucial and hard testing conditions.     

Intertwined basins of attraction of dynamical systems

In this paper, we investigate a kind of intertwining phenomenon and give a new definition of intertwined attractors of the dynamical systems and get some related results.     

Description of stability for linear time-invariant systems based on the first curvature

This paper focuses on using the first curvature κ(t) of trajectory to describe the stability of linear time-invariant system. We extend the results for two and three-dimensional systems (Wang, Sun, Song et al, arXiv:1808.00290) to n-dimensional systems. We prove that for a system $\text{◂=▸}\dot{r}(t)=\text{◂⋅▸}Ar(t)$, (a) if there exists a measurable set whose Lebesgue measure is greater than zero, such that $\text{◂≠▸}\underset{t\to +\infty }{\lim }\kappa (t)\ne 0$ or $\underset{t\to +\infty }{\lim }\kappa (t)$ does not exist for any initial value in this set, then the zero solution of the system is stable; (b) if the matrix A is invertible, and there exists a measurable set whose Lebesgue measure is greater than zero, such that $\text{◂=▸}\underset{t\to +\infty }{\lim }\kappa (t)=+\infty$ for any initial value in this set, then the zero solution of the system is asymptotically stable.     

Inner-approximating domains of attraction for discrete-time switched systems via Multi-step multiple Lyapunov-like functions

In this paper, we propose an iterative approach for estimating the domains of attraction for a class of discrete-time switched systems, where the state space is divided into several disjoint regions and each region is described by polynomial inequalities. At first, we introduce the basic concepts of Multi-step state subsequence, Multi-step state subspace, Multi-step basin of attraction and Multi-step multiple Lyapunov-like function. Secondly, beginning with an initial inner estimation, a theoretical framework is proposed for estimating the domain of attraction by iteratively calculating the Multi-step multiple Lyapunov-like functions. Thirdly, notice that the Multi-step state subspaces may be empty sets such that the corresponding constraints in the theoretical framework are redundant, we propose a numerical approach based on the homotopy continuation method to pre-check the non-emptiness of the Multi-step state subspaces, and then under-approximatively realize the framework by using S-procedure and sum of squares programming. At last, we implement our iterative approach and apply it to three discrete-time switched system examples with comparisons to existing methods in the literatures. These computation and comparison results show the advantages of our method.     

Guaranteed asymptotic stability for a class of uncertain linear dynamical systems

We consider a class of linear dynamical systems with bounded, Lebesgue-measurable uncertainties in the system and input matrices as well as in the input itself. A state feedback control is derived, which guarantees global, uniform asymptotic stability of the zero state; this control is continuous, except at the zero state.This paper is based in part on research supported by the National Science Foundation.     

On the asymptotic and practical stability of Persidskii-type systems with switching

In the paper, one class of differential systems with nonlinearities satisfying sector constraints is considered. We study the case where some of the sector constraints are given by linear inequalities, and some by nonlinear ones. It is assumed that the coefficients in the system can switch from one set of values to another. Sufficient conditions for the asymptotic and practical stability of the zero solution of the system are investigated using the direct Lyapunov method and the theory of differential inequalities. Restrictions on the switching law that provide a given region of attraction and ultimate bound for solutions of the system are obtained. An approach based on the construction of different differential inequalities for the Lyapunov function in different parts of the phase space is proposed, which makes it possible to improve the results obtained. The results are applied to the analysis of one automatic control system.     

On areas of attraction and repulsion in finite time dynamical systems and their numerical approximation

ABSTRACT

Stable and unstable fibre bundles with respect to a fixed point or a bounded trajectory are of great dynamical relevance in (non)autonomous dynamical systems. These sets are defined via an infinite limit process. However, the dynamics of several real world models are of interest on a short time interval only. This task requires finite time concepts of attraction and repulsion that have been recently developed in the literature. The main idea consists in replacing the infinite limit process by a monotonicity criterion and in demanding the end points to lie in a small neighbourhood of the reference trajectory. Finite time areas of attraction and repulsion defined in this way are fat sets and their dimension equals the dimension of the state space. We propose an algorithm for the numerical approximation of these sets and illustrate its application to several two- and three-dimensional dynamical systems in discrete and continuous time. Intersections of areas of attraction and repulsion are also calculated, resulting in finite time homoclinic orbits.     

Global well-posedness of coupled parabolic systems

The initial boundary value problem of a class of reaction-diffusion systems(coupled parabolic systems)with nonlinear coupled source terms is considered in order to classify the initial data for the global existence,finite time blowup and long time decay of the solution.The whole study is conducted by considering three cases according to initial energy:the low initial energy case,critical initial energy case and high initial energy case.For the low initial energy case and critical initial energy case the sufficient initial conditions of global existence,long time decay and finite time blowup are given to show a sharp-like condition.In addition,for the high initial energy case the possibility of both global existence and finite time blowup is proved first,and then some sufficient initial conditions of finite time blowup and global existence are obtained,respectively.     

Global existence and asymptotic behaviour for systems of nonlinear hyperbolic equations

We consider the initial-boundary value problem for a class of nonlinear hyperbolic equations system in a bounded domain. Using the potential well theory, the existence of global solutions is investigated. We also established the asymptotic behaviour of global solutions as t?→?+?∞ by applying the multiplier method.