Partial stability criteria for time-varying nonlinear systems

International Journal of Mathematics and Mathematical Sciences, 1995

In this paper, we investigate total stability, attractivity and uniform stability in terms of two measures of nonlinear differential systems under constant perturbations. Some sufficient conditions are obtained using Lyapunov's direct method. An example is also worked out.

Differential Equations and Dynamical Systems, 2014

This work is motivated by problem of practical stability of nonlinear time-varying system˙ () = (, ()) (0) = 0 where ∈ ℝ + , ∈ ℝ and : ℝ + ×ℝ −→ ℝ is continuous in and locally Lipschitz in. We give some sufficient conditions to guarantee practical uniform asymptotic stability in terms of convergence of solutions toward a neighborhood of the origin. An invariance principle is given when the origin is not an equilibrium point. The main result of this paper is illustrated by an example in three dimensional.

Proceedings of 1994 American Control Conference - ACC '94, 1994

This paper presents a Converse Lyapunov Function Theorem motivated by robust control analysis and design. Our result is based upon, but generalizes, various aspects of well-known classical theorems. In a unied and natural manner, it (1) includes arbitrary bounded disturbances acting on the system, (2) deals with global asymptotic stability, (3) results in smooth (innitely dierentiable) Lyapunov functions, and (4) applies to stability with respect to not necessarily compact invariant sets.

Automatica, 2006

We explicitly construct global strict Lyapunov functions for rapidly time-varying nonlinear control systems. The Lyapunov functions we construct are expressed in terms of oftentimes more readily available Lyapunov functions for the limiting dynamics which we assume are uniformly globally asymptotically stable. This leads to new sufficient conditions for uniform global exponential, uniform global asymptotic, and input-to-state stability of fast time-varying dynamics. We also construct strict Lyapunov functions for our systems using a strictification approach. We illustrate our results using several examples.

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Proceedings of the 36th IEEE Conference on Decision and Control, 2000

Within the Liapunov framework, sufficient conditions for exponential and uniform asymptotic stability of ordinary differential equations are proposed. Unlike with classical Liapunov theory, the time derivative of the Liapunov function, taken along solutions of the system, may have positive and negative values. Verification of the conditions of the main theorems may be harder than in the classical case. It is shown that the proposed conditions are useful for the investigation of the exponential stability of fast time-varying systems. This sets the stability study by means of averaging in a Liapunov context. In particular, it is established that exponential stability of the averaged system implies exponential stability of the original fast time-varying system.

Journal of Linear and Topological Algebra, 2016

In this paper, global uniform exponential stability of perturbed dynamical systems is studied by using Lyapunov techniques. The system presents a perturbation term which is bounded by an integrable function with the assumption that the nominal system is globally uniformly exponentially stable. Some examples in dimensional two are given to illustrate the applicability of the main results.

Proceedings of the IEEE Conference on Decision and Control, 2007

This paper develops Lyapunov and converse Lyapunov theorems for semistable nonlinear dynamical systems. Semistability is the property whereby the solutions of a dynamical system converge to (not necessarily isolated) Lyapunov stable equilibrium points determined by the system initial conditions. Specifically, we provide necessary and sufficient conditions for semistability and show that semistability implies the existence of a smooth Lyapunov function that decreases along the dynamical system trajectories such that the Lyapunov function satisfies inequalities involving the distance to the set of equilibria.

IEEE Transactions on Automatic Control, 1998

Abstract| A new su cient condition for asymptotic stability of ordinary di erential equations is proposed. Unlike classical Liapunov theory, the time derivative along solutions of the Liapunov function may have positive and negative values. The classical Liapunov approach may be regarded as an in nitesimal version of the present theorem. Veri cation in practical problems is harder than in the classical case; an example is included in order to indicate how the present theorem may be applied.

Discussiones Mathematicae. Differential Inclusions, Control and Optimization, 2015

We consider nonlinear non-autonomous multivariable systems governed by differential equations with differentiable linear parts. Explicit conditions for the exponential stability are established. These conditions are formulated in terms of the norms of the derivatives and eigenvalues of the variable matrices, and certain scalar functions characterizing the nonlinearity. Moreover, an estimate for the solutions is derived. It gives us a bound for the region of attraction of the steady state. As a particular case we obtain absolute stability conditions. Our approach is based on a combined usage of the properties of the "frozen" Lyapunov equation, and recent norm estimates for matrix functions. An illustrative example is given.