Practical partial stability of time-varying systems

Abstract

<p style='text-indent:20px;'>In this paper we investigate the practical asymptotic and exponential partial stability of time-varying nonlinear systems. We derive some sufficient conditions that guarantee practical partial stability of perturbed systems using Lyapunov's theory where a converse theorem is presented. Therefore, we generalize some works which are already made in the literature. Furthermore, we present some illustrative examples to verify the effectiveness of the proposed methods.</p>

Partial stability criteria for time-varying nonlinear systems

“…Consequently, M (δ) δ > θ, for all δ ≥ 0. The challenge is that we cannot apply the results given in [8] to show the practical y-stability of (2.9) although the system is globally uniformly practically exponentially y-stable (see Remark 4.2).…”

“…In this paper, we continue to generalize the existing results given in ( [4], [7], [8]) and we present a new Lyapunov function based practical partial stability analysis approach for a class of a timevarying systems. The practical partial stability of the considered system can be guaranteed if a scalar function is practical stable.…”

“…The question arises whether it is true that global uniform practical exponential stability implies the existence of Lyapunov functions such as described in the Theorem 2.1. The authors in ( [2], [4], [8]) have established a converse stability theorem for (2.1), when the origin is not an equilibrium point under some restrictions on the dynamic of the considered system. Unfortunately, converse theorems are often proven by assuming knowledge of the solutions and are therefore useless in practice.…”

“…But this technique is very restrictive when the term of perturbation is more complexity. Hence some researchers attempt to use the Lyapunov function associated to the nominal system to construct a new Lyapunov function that guarantees the stability of the considered system (4.1), see ( [1], [4], [5], [6], [7], [8]). Thus, we continue in this direction and we answer to the question mentioned in the Section 2. by building a Lyapunov function for a more general class of perturbed systems.…”

“…The question addressed in this paper is related to the study of the preservation of stability when considering a new system with a perturbation term ( [1], [2], [3], [4], [6]). Recently, for the time-varying perturbed systems ẋ = f (t, x) + g(t, x) (1.1) where f, g : R + × R n → R n are piecewise continuous in t, locally Lipschitz in x, the authors in ( [4], [7], [8]) studied the asymptotic and exponential stability of a class of system (1.1) with respect to a neighborhood of the origin approximated by a small ball of radius r > 0 centered at the origin in the sense that the trajectories approach a small compact set containing the origin based on the stability of the nominal system…”