Uniform estimates of attracting sets of extended Lurie systems using LMIs

This research presents a systematic procedure to obtain estimates, via extended Lyapunov functions, of attracting sets of a class of nonlinear systems, as well as an estimate of their stability regions. The considered class of nonlinear systems, called in this note the extended Lurie system, consists of nonlinear systems like those of the Lurie problem where one of the nonlinear functions can violate the sector conditions of the Lurie problem around the origin. In case of nonautonomous systems the concept of absolute stability is extended and uniform estimates of the attracting set are obtained. Two classical nonlinear systems, the forced duffing equation and the Van der Pol system, are analyzed with the proposed procedure.

Revegetation on a Removed Topsoil: Effect on Aggregate Stability

The construction of a large reservoir on the Parana River (Selviria, MS, Brazil) disturbed the soil of an extensive agricultural area in which between 5 and 8m of topsoil were removed. In this area, a restoration process was carried out using revegetation with green manure without or with amendments (for 4 years), crops (2 years), and Brachiaria decumbens cultivation for 6 years. The following treatments were used: control plots, T0 (residual subsoil) and T1 (soil tillage without culture); plots with green manure and without amendments: T2 (velvet bean) and T3 (pigeon pea); plots with green manure and with amendments: T4 (limed + velvet bean), T5 (limed + pigeon pea); T6 (limed + gypsum + velvet bean) and T7 (limed + gypsum + pigeon-pea). They were arranged in randomized blocks. After 13 years of rehabilitation process, when the soil was cultivated with brachiaria, the structural stability in three depths was evaluated. Organic-matter content and others chemical properties did not show any relationship with the stability of aggregates of the experimental area as measured by mean weight diameter (MWD). Significant differences between depths were found for MWD and the other parameters measured. Nevertheless, there were no significant differences observed between treatments, independent of the adopted system of revegetation. By taking an absolute value of MWD, the stability of superficial layer was observed in the following sequence: T7 T5 T6 T1 T2 T3 T4 T0. The control plot (T0) gave the lowest value of MWD (1.76mm) in relation to the plots in restoration process. Treatment T7 was the most effective in recovering the stability of aggregates (2.63mm). However, treatments T5 and T6 displayed a similar value. After 13 years of revegetation practices, a slight recovery of the stability was observed, although this is still lower than stability in soils of similar edaphic conditions in the original topsoil of experimental area.