New equivalent conditions of the asymptotical stability and stabilization of positive linear dynamical systems are investigated in this paper. The asymptotical stability of the positive linear systems means that there is a solution for linear inequalities systems. New necessary and sufficient conditions for the existence of solutions of the linear inequalities systems as well as the asymptotical stability of the linear dynamical systems are obtained. New conditions for the stabilization of the resultant closed-loop systems to be asymptotically stable and positive are also presented. Both the stability and the stabilization conditions can be easily checked by the so-called I-rank of a matrix and by solving linear programming (LP). The proposed LP has compact form and is ready to be implemented, which can be considered as an improvement of existing LP methods. Numerical examples are provided in the end to show the effectiveness of the proposed method.

Positively invariant sets play an important role in the theory and applications of dynamical systems. The stability in the sense of Lyapunov of the equilibrium x = 0 is equivalent to the existence of the ellipsoidal positively invariant sets. The constraints on the state and control vectors of dynamical systems can be formulated as polyhedral positively invariant sets in practical engineering problems. Numerical checking method of positive invariance of polyhedral sets is addressed in this paper. The validation of the positively invariant sets can be done by solving LPs which can be easily done numerically. It is illustrated by examples that our checking method is effective. Compared with the now existing algebraic methods, numerical checking method is an attractive method in that it’s easy to be implemented.

We investigated changes in concentrations of ADP (adiponectin), LEP (leptin), BHBA (beta-hydroxybutyric acid), NEFA (non-esterified fatty acid), Glucose (Glu) and INS (insulin) in serum of healthy perinatal dairy cows and cows with ketosis. Twenty-one healthy cows and seventeen cows with ketosis from a herd of a total 60 Holstein cows (near dry period i.e. 56 days antepartum) were selected. Blood was collected through the tail vein every 7 days, from 56 day antepartum to 56 day postpartum. Serum ADP, LEP, BHBA, NEFA, Glu, and INS concentrations were determined, and ketosis was diagnosed through serum BHBA (≥1.2 mmol/L). We showed the concentration of serum adipokines and energy balancing indices were stable during antepar- tum period. However, ADP concentration increased while LEP decreased, and there were a significant increase in cows with ketosis compared to that of in healthy cows. Serum BHBA and NEFA concentrations increased significantly at first, and then gradually decreased in both healthy cows and cows with ketosis. However, cows with ketosis showed higher concentrations of BHBA and NEFA which restored later. The serum concentration of Glu in both healthy dairy cows and cows with ketosis showed a decreasing trend. INS concentration in healthy cows was decreased while it was increased in cows with ketosis. The results reflect the extent of hypo- glycemia and lipid mobilization postpartum, suggest IR exists in cows with ketosis while serum ADP and LEP might play roles in the development of ketosis.