On convergence conditions for generalized Persidskii systems

Abstract

The convergence conditions for a class of generalized Persidskii systems and their discretized dynamics are introduced, which can be checked through linear (matrix) inequalities. The case of almost periodic convergence for this class of systems with almost periodic input is also studied. The proposed results are applied to a Lotka-Volterra model and opinion dynamics.

State observer design for bilinear Persidskii systems

“…The system (6) includes the generalized Persidskii systems studied in [6], [12], [11] (the first line in ( 6)), but also additional bilinear terms multiplied by the state components (the second line). The motivation for considering this class is that it encapsulates some models of interest, which have been studied both in the context of observer design and of stabilization [5], [3].…”

“…The model we consider in this paper is a generalization of the important class of Persidskii systems, first introduced in [1], [13] for stability analysis, also related with Lur'e systems studied in the absolute stability theory [17], [9]. Thus, the advantage of Persidskii systems consists in the existence of a well-investigated form of candidate Lyapunov functions, which was used in many cases [6], [12], [11], often resulting in stability conditions that can be formulated in the form of linear matrix inequalities. Moreover, this class of models has been found valuable in representing many physical and biological phenomena, and therefore, it has been studied from many viewpoints, including that of IOS-stability [12].…”

Annular short-time stability of generalized Persidskii systems

“…Following [28], [16], [22], [23], the stability analysis of (2) can be performed using a Lyapunov function…”

“…The considered class of nonlinear dynamics in this paper is called the generalized Persidskii systems, which have been extensively studied in the context of neural networks [15], biological models [16], and power systems [17], and whose original form was introduced in [18], [19], [20]. Recently the conditions of input-to-state stability, input-tooutput stability, and convergence, as well as the synthesis of a state observer have been established in [21], [16], [22], [23] for generalized Persidskii models. Note that most existing approaches to synthesizing Lyapunov functions for stability analysis in nonlinear dynamics involve various canonical forms of the studied differential equations, such as Lur'e systems [24], homogeneous models [25], Persidskii systems [18], and Lipschitz dynamics, and the presence of nonlinearities leads to that the established stability conditions can be rather complicated.…”

A class of nonlinear state observers for an SIS system counting primo-infections

“…Section II recalls the model under study (system (1) below), transformed by a change of coordinates into a Persidskii system form [16], [10], [9], for which a class of observers is proposed. The asymptotic convergence of the latter is established in Section III under adequate assumptions on the gain coefficients and using techniques inspired from [13], [14]. Numerical simulations are provided in Section IV before some concluding remarks in Section V. For the sake of space, all proofs are omitted and will be published in an upcoming research report.…”