Asymptotic stability analysis of autonomous systems by applying the method of localization of compact invariant sets

The asymptotic stability and global asymptotic stability of equilibria in autonomous systems of differential equations are analyzed. Conditions for asymptotic stability and global asymptotic stability in terms of compact invariant sets and positively invariant sets are proved. The functional method of localization of compact invariant sets is proposed for verifying the fulfillment of these conditions. Illustrative examples are given.     

Global asymptotic stability analysis by the localization method of invariant compact sets

We study the asymptotic stability and the global asymptotic stability of equilibria of autonomous systems of differential equations. We prove necessary and sufficient conditions for the global asymptotic stability of an equilibrium in terms of invariant compact sets and positively invariant sets. To verify these conditions, we use some results of the localization method for invariant compact sets of autonomous systems. These results are related to finding sets that contain all invariant compact sets of the system (localizing sets) and to the behavior of trajectories of the system with respect to localizing sets. We consider an example of a system whose equilibrium belongs to the critical case.     

Lyapunov stability for impulsive control affine systems

This article studies several notions of Lyapunov stability for impulsive control affine systems in the setting of nonautonomous dynamical systems. It presents some relations between the stability of an impulsive control affine system and the stability of its adjacent control system. Stability of compact sets and their components are specially investigated. Lyapunov functionals are employed to characterize each type of stability of closed sets.     

Boundedness of orbits and stability of closed sets

This paper explores fundamental connections between boundedness of orbits, and stability and attractivity of closed sets. For this purpose, the paper considers topological notions of stability and attractivity which do not depend on a metric. The notions considered are characterized in terms of restricted prolongations and positive limit sets, and connections with boundedness are studied. Finally, the notion of nontangency is used to give a Lyapunov result for Lyapunov stability, attractivity and boundedness of orbits. Unlike previous sufficient conditions for boundedness, our result does not require the Lyapunov function to be proper or weakly proper. Examples are provided to illustrate the results.     

Application of the direct method of Lyapunov to the analysis of integral sets

The concepts are defined of the stability, attraction, and asymptotic stability of the integral sets of systems of ordinary differential equations. With the use of the second method of Lyapunov, a number of theorems are proved concerning the stability of integral sets.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 10, pp. 1342–1348, October, 1992.     

Differential games with fixed terminal time and estimation of the instability degree of sets in these games

This paper contributes to the theory of differential games. A game problem of bringing a conflict-controlled system to a compact target set is analyzed. Sets in the position space that terminate on the target set and are not stable bridges are considered. The notion of stability defect of these sets is examined. It is demonstrated how the notion of stability defect can be used to construct sets with relatively good geometry that are at the same time convenient for the first player to play the game successfully.     

Robust stability of Switched Boolean Networks with function perturbation

In genetic regulatory networks, gene mutations are one of natural phenomena, which attract much attention by biological researchers. In modeling gene networks using switched Boolean networks (SBNs), gene mutations can be described by function perturbations, which is a meaningful issue in analyzing function perturbation of SBNs. This paper studies robust stability of SBNs with function perturbation. With the help of semi-tensor product (STP) of matrices, one equivalent algebraic form of SBNs is established. By constructing two state sets, a criterion for global stability of SBNs under arbitrary switching signals is proposed. In order to relax the conditions of global stability, pointwise stabilizability and consistent stabilizability of SBNs are further considered. Based on state reachable sets, several criteria are established for the proposed kinds of stability. Finally, the obtained results are verified by two examples and lac operon in the Escherichia coli, respectively.     

A foundation for competence set analysis

For each decision problem, there is a competence set consisting of knowledge, information and skills for its effective solution. How the decision-maker acquires and expands his/her competence set plays a key role in the process and quality of decision-making. This paper provides a mathematical foundation for studying competence sets, their expansion processes and stability. Time and cost functions for expansion, reachable domains, effective expansion using minimal spanning trees, random set decomposition of competence sets, marginal analysis, connectivity, metrization and stability of competence sets are some key concepts introduced.     

The stability of the solution sets for set optimization problems via improvement sets

ABSTRACT

The aim of this paper is to investigate the stability of the solution sets for set optimization problems via improvement sets. Firstly, we consider the relations among the solution sets for optimization problem with set optimization criterion. Then, the closeness and the convexity of solution sets are discussed. Furthermore, the upper semi-continuity, Hausdorff upper semi-continuity and lower semi-continuity of solution mappings to parametric set optimization problems via improvement sets are established under some suitable conditions. These results extend and develop some recent works in this field.     

On contraction of nonlinear difference equations with time‐varying delays

General nonlinear difference equations with time‐varying delays are considered. Explicit criteria for contraction of such equations are presented. Then some simple sufficient conditions for global exponential stability of equilibria and for stability of invariant sets are derived. Furthermore, explicit criteria for existence, uniqueness and global exponential stability of periodic solutions are derived. Finally, the obtained results are applied to time‐varying discrete‐time neural networks with delay.     

稳定性理论中几个基本定理的推广

本文推广了经典的稳定性,有界性定理及关于渐近稳定的充分条件。去掉了原始定理中,函数的导数在其定义域内常负或定负的要求,并应用于集合稳定性判定.     

On the stable containment of two sets

This paper studies the stability of the set containment problem. Given two non-empty sets in the Euclidean space which are the solution sets of two systems of (possibly infinite) inequalities, the Farkas type results allow to decide whether one of the two sets is contained or not in the other one (which constitutes the so-called containment problem). In those situations where the data (i.e., the constraints) can be affected by some kind of perturbations, the problem consists of determining whether the relative position of the two sets is preserved by sufficiently small perturbations or not. This paper deals with this stability problem as a particular case of the maintaining of the relative position of the images of two set-valued mappings; first for general set-valued mappings and second for solution sets mappings of convex and linear systems. Thus the results in this paper could be useful in the postoptimal analysis of optimization problems with inclusion constraints.      

Global Behaviour of Solutions of Nonlinear Delay Difference Equations*

In this paper global asymptotic stability and asymptotic behaviour of solutions of nonlinear delay difference equations has been studied and a few sets of sufficient conditions for global asymptotic stability are derived.     

Stability of Equilibria of Discrete-Time Systems and Localization of Invariant Compact Sets

The stability (or asymptotic stability) of equilibria of an time-invariant discrete-time system can be verified with the use of stability and asymptotic stability criteria stated in terms of invariant sets. An earlier proposed method reduces the verification of these criteria to some set operations. However, the method is analytical and hard to implement. We propose another approach to the verification of these criteria based on the functional method for localizing invariant compact sets.     

Global attracting set,exponential stability and stability in distribution of SPDEs with jumps

A novel approach to the global attracting sets of mild solutions for stochastic functional partial differential equations driven by Lévy noise is presented. Consequently, some new sufficient conditions ensuring the existence of the global attracting sets of mild solutions for the considered equations are established. As applications, some new criteria for the exponential stability in mean square of the considered equations is obtained. Subsequently, by employing a weak convergence approach, we try to establish some stability conditions in distribution of the segment processes of mild solutions to stochastic delay partial differential equations with jumps under some weak conditions. Some known results are improved. Lastly, some examples are investigated to illustrate the theory.     

Lyapunov direct method for semidynamical systems

The stability of closed invariant sets of semidynamical systems defined on an arbitrary metric space is analyzed. The main theorems of Lyapunov’s second method for the uniform stability and uniform asymptotic stability (local and global) are stated. Illustrative examples are given.     

Stability of Parametric Quasivariational Inequality of the Minty Type

In this paper, stability of a parametric quasivariational inequality of the Minty type is studied via various sufficient conditions characterizing upper and lower semicontinuity of the solution sets as well as the approximate solution sets. Sufficient conditions ensuring upper semicontinuity of the approximate solution sets of an optimization problem with quasivariational inequality constraints are also presented.     

On Essential Stable Sets of Solutions in Set Optimization Problems

This paper introduces the essential stability for set optimization problems. Some kinds of essential stable sets of weakly minimal and minimal solutions are shown. The graph of minimal solution mappings is not necessarily closed, which is different from weakly minimal solution mappings. The existence of minimum essential sets of minimal solutions is proved.     

Über lokale stabilitätsmengen in der parametrischen quadratischen optimierung

In the present paper the concept of so-called local stability sets introduced by F. No?i?ka in parametric linear programming is extend to the case of convex quadratic programs parametrized in the objective function and in the right-hand side of the linear constraints. The continuity of the optimal-set-map and of the extreme value function holds on these sets. It is shown that the local stability sets are connected. Examples which illustrate the necessity of certain assumptions are given.