On Observer Design for a Class of Nonlinear Systems Including Unknown Time-Delay

The observer design for nonlinear systems with unknown, bounded, time-varying delays, on both input and state, is still an open problem for researchers. In this paper, a new observer design for a class of nonlinear system with unknown, bounded, time-varying delay was presented. For the proof of the observer stability, a Lyapunov–Krasovskii function was chosen. Sufficient assumptions are provided to prove the practical stability of the proposed observer. Furthermore, the exponential convergence of the observer was proved in the case of a constant time delay. Simulation results were shown to illustrate the feasibility of the proposed strategy.

     

The Hilbert number of a class of differential equations

The notion of Hilbert number from polynomial differential systems in the plane of degree $n$ can be extended to the differential equations of the form \[\dfrac{dr}{d\theta}=\dfrac{a(\theta)}{\displaystyle \sum_{j=0}^{n}a_{j}(\theta)r^{j}} \eqno(*)\] defined in the region of the cylinder $(\tt,r)\in \Ss^1\times \R$ where the denominator of $(*)$ does not vanish. Here $a, a_0, a_1, \ldots, a_n$ are analytic $2\pi$--periodic functions, and the Hilbert number $\HHH(n)$ is the supremum of the number of limit cycles that any differential equation $(*)$ on the cylinder of degree $n$ in the variable $r$ can have. We prove that $\HHH(n)= \infty$ for all $n\ge 1$.     

Sur le principe de superposition et l'équation de Riccati

In this Note we study the conditions under which a system of ordinary differential equations admits a nonlinear superposition of the solutions, and also the linearization of such systems. A particular study is established for the Riccati equation. To cite this article: S. Rezzag et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Graft copolymerization of acrylic acid onto polyamide fibers

The grafting of acrylic acid (AA) monomer (CH₂CHCOOH) on polyamide 6.6 monofilaments (PA 6.6) using benzoyl peroxide (BPO) as initiator was carried out in order to enhance the hydrophilic nature of fibers. The grafting rate depends on the AA concentration, the BPO concentration, the time and the temperature of reaction.The best conditions for optimum rate of grafting were obtained with a AA concentration of 0.5 M, a BPO concentration of 0.03 M, a reaction temperature of T = 85 °C and a reaction time of 120 mn.The fiber surface has been investigated by many experimental techniques of characterization such as Fourier transform infrared spectroscopy (FTIR), calorimetric analysis (DSC), scanning electron microscopy (SEM), and contact angle measurements.The effect of grafting of acrylic acid onto PA 6.6 fibers on their moisture and mechanical resistances was analyzed from water sorption and elongation at break measurements.The analysis of the experimental data shows clearly the efficiency of the grafting reaction used, leading to a significant increase of the hydrophilic character of the PA 6.6 surface.     

LIMIT CYCLES OF THIRD-ORDER DIFFERENTIAL EQUATION

In this paper, we investigate a third-order differential equation. Based on the averaging theory, we obtain sufficient conditions for the existence of periodic solutions to the equation.     

LIMIT CYCLES OF A GENERALIZED LINARD DIFFERENTIAL EQUATION VIA AVERAGING THEORY

We apply the averaging theory of first and second orders to a generalized Liénard differential equation to study the maximum number of limit cycles of the system.