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.     

Functional method for the localization of invariant compact sets in discrete systems

We suggest a method for the localization of invariant compact sets in discrete autonomous systems. We describe the properties of the corresponding localizing sets. By using this method, we construct localizing sets for invariant compact sets of the discrete Henon system.     

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.     

Instability of closed invariant sets of semidynamical systems: Method of sign-constant Lyapunov functions

In the paper, a reduction principle for the instability property of a closed positively invariant set M for semidynamical systems is proved. The fact that the result is not traditional is stressed by the assumption on the existence of a closed positively invariant set with respect to which the set M has the attraction property. The corresponding instability theorem of the method of sign-constant Lyapunov functions is presented. The assertion thus obtained generalizes the well-known Chetaev and Krasovskii theorems for systems of ordinary differential equations, theorems on the instability with respect to some of the variables, and also the Shimanov and Hale theorems for systems with retarded argument. Illustrating examples are presented.     

Localization of invariant compact sets of nonautonomous systems

We generalize the localization method for invariant compact sets of an autonomous dynamical system to the case of a nonautonomous system of differential equations. By using this method, we solve the localization problem for the Vallis third-order dynamical system governing some processes in atmosphere dynamics over the Pacific Ocean. For this system, we construct a one-parameter family of localizing sets bounded by second-order surfaces and find the intersection of all sets of the family.     

On compact sets with a certain affine invariant

We give a complete characterization of finite-dimensional compact sets with the following property: all of their images under affine operators are symmetric (that is, have symmetry planes of certain dimensions). We also study the noncompact case; namely, we reveal a class of unbounded closed sets with this property and conjecture that this class is complete.     

Pointwise compact sets of measurable functions

If X is a compact Radon measure space, and A is a pointwise compact set of real-valued measurable functions on X, then A is compact for the topology of convergence in measure (Corollary 2H). Consequently, if X_o,..., X_n are Radon measure spaces, then a separately continuous real-valued function on X_o×X₁×...×X_n is jointly measurable (Theorem 3E). If we seek to generalize this work, we encounter some undecidable problems (§4).     

Localization of compact invariant sets of discrete-time control systems

Doklady Mathematics -     

Localization of invariant compact sets in uncertain discrete systems

A functional method for the localization of invariant compact sets in discrete autonomous systems is generalized to discrete systems with uncertainty. We describe the properties of the corresponding localizing sets. By using this method, we construct localizing sets for positively invariant compact sets of the discrete Henon system with uncertainty.     

Localization of invariant compact sets of families of discrete-time systems

The functional method of localization of invariant compact sets developed for continuous- and discrete-time dynamical systems is extended to families of discrete-time systems. Positively invariant compact sets are considered. As an example, the method is applied to the Hénon system with uncertain parameters.     

Localizing sets for invariant compact sets of continuous dynamical systems with a perturbation

A functional method of localization of invariant compact sets, which was earlier developed for autonomous continuous and discrete systems, is generalized to continuous dynamical systems with perturbations. We describe properties of the corresponding localizing sets. By using that method, we construct localizing sets for positively invariant compact sets of the Lorenz system with a perturbation.