Stability Of Nonautonomous Systems Research Articles

Stability of nonautonomous systems on Fréchet spaces

Stability of nonautonomous systems on Fréchet spaces

Asymptotic Stability of Non-Autonomous Systems and a Generalization of Levinson’s Theorem

We study the asymptotic stability of non-autonomous linear systems with time dependent coefficient matrices { A ( t ) } t ∈ R . The classical theorem of Levinson has been an indispensable tool for the study of the asymptotic stability of non-autonomous linear systems. Contrary to constant coefficient system, having all eigenvalues in the left half complex plane does not imply asymptotic stability of the zero solution. Levinson’s theorem assumes that the coefficient matrix is a suitable perturbation of the diagonal matrix. Our objective is to prove a theorem similar to Levinson’s Theorem when the family of matrices merely admits an upper triangular factorization. In fact, in the presence of defective eigenvalues, Levinson’s Theorem does not apply. In our paper, we first investigate the asymptotic behavior of upper triangular systems and use the fixed point theory to draw a few conclusions. Unless stated otherwise, we aim to understand asymptotic behavior dimension by dimension, working with upper triangular with internal blocks adds flexibility to the analysis.

Boundary for Complete Set of Attractors for Forced–Damped Essentially Nonlinear Systems

Forced–damped essentially nonlinear oscillators can have a multitude of dynamic attractors. Generically, no analytic procedure is available to reveal all such attractors. For many practical and engineering applications, however, it might be not necessary to know all the attractors in detail. Knowledge of the zone in the state space (or the space of initial conditions), in which all the attractors are situated might be sufficient. We demonstrate that this goal can be achieved by relatively simple means—even for systems with multiple and unknown attractors. More specifically, this paper suggests an analytic procedure to determine the zone in the space of initial conditions, which contains all attractors of the essentially nonlinear forced–damped system for a given set of parameters. The suggested procedure is an extension of well-known Lyapunov functions approach; here we use it for analysis of stability of nonautonomous systems with external forcing. Consequently, instead of the complete state space of the problem, we consider a space of initial conditions and define a bounded trapping region in this space, so that for every initial condition outside this region, the dynamic flow will eventually enter it and will never leave it. This approach is used to find a special closed curve on the plane of initial conditions for a forced–damped strongly nonlinear oscillator with single-degree-of-freedom (single-DOF). Solving the equations of motion is not required. The approach is illustrated by the important benchmark example of x2n potential, including the celebrated Ueda oscillator for n = 2. Another example is the well-known model of forced–damped oscillator with double-well potential. We also demonstrate that the boundary curve, obtained by analytic tools, can be efficiently “tightened” numerically, yielding even stricter estimation for the zone of the existing attractors.

Stability and stabilization of positive switched systems with mode-dependent average dwell time

This paper investigates stability and stabilization of positive switched systems with mode-dependent average dwell time, which permits to each subsystem in the underlying systems to have its own average dwell time. First, by using the multiple linear copositive Lyapunov function, the stability analysis of continuous-time systems in the autonomous form is addressed based on the mode-dependent average dwell time switching strategy. Then, the stabilization of non-autonomous systems is considered. State-feedback controllers are constructed, and all the proposed conditions are solvable in terms of linear programming. The obtained results are also extended to discrete-time systems. Finally, the simulation examples are given to illustrate the correctness of the design. The switching strategy used in the paper seems to be more effective than the average dwell time switching by some comparisons.

Stability of [formula omitted]-fractional non-autonomous systems

In this manuscript, using Lyapunov’s direct method, the stability of non-autonomous systems within the frame of the q-Caputo fractional derivative is studied. The conditions for stability, uniform stability and asymptotic stability are discussed.

Feedback equivalence and global stabilization of nonstationary discrte systems

A problem is considered for the global stabilization of nonstationary nonlinear discrete systems without outputs. Here, the key instant is the establishment of equivalence of an arbitrary discrete system and a lossless system with the use of the appropriate feedback. The construction of control laws is performed on the basis of new methods for investigating the stability of nonautonomous systems of difference equations, using the theory of limit systems and Lyapunov limit functions.

Robust Stability and Stability Radius for Variational Control Systems

We consider an integral variational control system on a Banach space X and we study the connections between its uniform exponential stability and the (I(ℝ+, X), O(ℝ+, X)) stability, where I and O are Banach function spaces. We identify the viable classes of input spaces and output spaces related to the exponential stability of systems and provide optimization techniques with respect to the input space. We analyze the robustness of exponential stability in the presence of structured perturbations. We deduce general estimations for the lower bound of the stability radius of a variational control system in terms of input‐output operators acting on translation‐invariant spaces. We apply the main results at the study of the exponential stability of nonautonomous systems and analyze in the nonautonomous case the robustness of this asymptotic property.

Robustness with respect to small delays for exponential stability of non-autonomous systems

Robustness of stability with respect to small delays, e.g., motivated by feedback systems in control theory, is of great theoretical and practical important, but this property does not hold for many systems. In this paper, we introduce the conception of robustness with respect to small time-varying delays for exponential stability of the non-autonomous linear systems. Sufficient conditions are given for the non-autonomous systems to be robust, and examples are provided to illustrate that the conditions are satisfied for a large class of the non-autonomous parabolic systems.

On the stability of nonautonomous systems

In Kalitine (RAIRO Automatique/Systems Anal. Control 16(3) (1982) 275) the use of semi-definite Lyapunov functions for exploring the local stability of autonomous dynamical systems has been introduced. In this paper, we give an extension of the results of Kalitine (1982) that allows to study the local stability of nonautonomous differential systems. We give an application to the algebraic Riccati equation.

Stabilization of Non-Autonomous Systems with Respect to a Part of Variables by Means of Controlled Lyapunov Functions

Stabilization of Non-Autonomous Systems with Respect to a Part of Variables by Means of Controlled Lyapunov Functions

Stability of nonautonomous systems by Liapunov's direct method

This paper discusses asymptotic stability for nonautonomous systems by means of the direct method of Liapunov. The existence of a positive time-invariant Liapunov function with negative semi-definite derivative is assumed. The paper focuses on the extra c

Stability of nonautonomous systems

Stability of nonautonomous systems