Invariance of asymptotic stability of perturbed linear systems on Hilbert space

The questions of stabilizability of structurally perturbed or uncertain linear systems in Hilbert space of the form are considered. The operatorA is assumed to be the infinitesimal generator of aC ₀-semigroup of contractionsT(t),t0, in a Hilbert spaceX;B is a bounded linear operator from another Hilbert spaceU toX; and {P(r),r } is a family of bounded or unbounded perturbations ofA inX, where is an arbitrary set, not necessarily carrying any topology. Sufficient conditions are presented that guarantee controllability and stabilizability of the perturbed system, given that the unperturbed system has similar properties. In particular, it is shown that, for certain classes of perturbations, weak and strong stabilizability properties are preserved for the same state feedback operator.This work was supported in part by the Natural Science and Engineering Research Council of Canada under Grant No. A7109.     

Liapunov asymptotic stability and Zhukovskij asymptotic stability

In this paper the Liapunov stability and Zhukovkij stability of motion in continuous dynamical systems are discussed. It is shown that each type of asymptotic stability is typically characterized by omega limit set of an orbit of this type of stability.     

About positive invariance and asymptotic stability

In this note we consider the differential equation , t0 with initial conditionx(0)G, whereG is a closed pointed convex cone. Our results pertain to positive invariance and asymptotic stability. A result by Stern, [3], will be improved. Subtangentiality will also be discussed as this notion is important in deriving results about positive invariance.     

Connections between stability and asymptotic stability of nonlinear switched systems

Here are considered nonlinear switched systems in which the switching occurs among a class of subsystems that are characterized by input–output properties stated in terms of L_p spaces of signals. The relationships between the L_p stability of each subsystem and the internal stability of the switched system are studied. In particular, conditions on the dwell time of the switching signals that guarantee the asymptotic stability of the overall system are provided. The connections among these conditions and the L_p input–output properties of the subsystems are investigated.     

An asymptotic stability involving collision and avoidance

Generally it is not easy task whether the stable systems governed by nonlinear ordinary differential equations are asymptotically stable or not. This problem often appears in studying a collision and avoidance control problem based on the stability theory. In this paper we devoted to finding conditions that the stable system obtained from the collision and avoidance control problem is asymptotically stable.     

Some refinements of asymptotic stability theory

Summary The concept of a Liapunov function with strongly negative definite derivative on an annulus is introduced. Sufficient conditions for stability and asymptotic stability are given without using any “definiteness” properties of Liapunov functions. Entrata in Redazione il 23 novembre 1970.     

Partial asymptotic stability via limiting equations

In this paper the limiting equations approach is applied to study the stability properties with respect to a part of the state variables for nonautonomous dynamical systems. Sufficient conditions are given for uniform asymptotic eventual strongly partial stability and for uniform asymptotic partial stability. An application of the results is given.     

The asymptotic stability of difference equations

In this paper, the linear delay difference equation x_n+1 −x_n = −a_nx_n−k, where a_n ≥ 0 (n ≥ 0) any nonnegative coefficient sequence, is considered and its stability properties are investigated. The obtained result is extended to the nonlinear difference equation x_n+1 − x_n = f_n(x_n−k).     

On the asymptotic stability of equilibrium

The classical criterion of asymptotic stability of the zero solution of equations x^′=f(t,x) is that there exists a positive definite function V which has infinitesimal upper bound such that is negative definite. In this paper we prove that if is bounded then the condition that is negative definite can be weakened and replaced by that and is negative definite.     

Conley index condition for asymptotic stability

In this paper, we use Conley index theory to develop necessary conditions for stability of equilibrium and periodic solutions of nonlinear continuous-time systems. The Conley index is a topological generalization of the Morse theory which has been developed to analyze dynamical systems using topological methods. In particular, the Conley index of an invariant set with respect to a dynamical system is defined as the relative homology of an index pair for the invariant set. The Conley index can then be used to examine the structure of the system invariant set as well as the system dynamics within the invariant set, including system stability properties. Efficient numerical algorithms using homology theory have been developed in the literature to compute the Conley index and can be used to deduce the stability properties of nonlinear dynamical systems.     

Evolutionary and asymptotic stability in symmetric multi-player games

We provide a classification of symmetric three-player games with two strategies and investigate evolutionary and asymptotic stability (in the replicator dynamics) of their Nash equilibria. We discuss similarities and differences between two-player and multi-player games. In particular, we construct examples which exhibit a novel behavior not found in two-player games.Received October 2001/Revised May 2003     

Parameter estimation and asymptotic stability in stochastic filtering

In this paper, we study the problem of estimating a Markov chain X$X$ (signal) from its noisy partial information Y$Y$, when the transition probability kernel depends on some unknown parameters. Our goal is to compute the conditional distribution process P{_Xn∣_Yn,…,_Y1}$\mathbb{P}\{X_n\mid Y_n,\dots ,Y_1\}$, referred to hereafter as the optimal filter. Following a standard Bayesian technique, we treat the parameters as a non-dynamic component of the Markov chain. As a result, the new Markov chain is not going to be mixing, even if the original one is. We show that, under certain conditions, the optimal filters are still going to be asymptotically stable with respect to the initial conditions. Thus, by computing the optimal filter of the new system, we can estimate the signal adaptively.     

Cohomology and asymptotic stability of 1-dimensional continua

We prove that a 1-dimensional continuum carrying a flow without singular points is homeomorphic to the unit circle if its first Čech cohomology group with integer coefficients is isomorphic to ℤ. As an application of this we obtain that an asymptotically stable invariant 1-dimensional continuum of a flow on a locally compact ANR, which does not contain singular points, must be a periodic orbit.     

Third-order nilpotency, nice reachability and asymptotic stability

We consider an affine control system whose vector fields span a third-order nilpotent Lie algebra. We show that the reachable set at time T using measurable controls is equivalent to the reachable set at time T using piecewise-constant controls with no more than four switches. The bound on the number of switches is uniform over any final time T. As a corollary, we derive a new sufficient condition for stability of nonlinear switched systems under arbitrary switching. This provides a partial solution to an open problem posed in [D. Liberzon, Lie algebras and stability of switched nonlinear systems, in: V. Blondel, A. Megretski (Eds.), Unsolved Problems in Mathematical Systems and Control Theory, Princeton Univ. Press, 2004, pp. 203-207].