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.

The pharmacokinetics of a diclofenac sodium was investigated in swine. A single intravenous (i.v.) or intramuscular (i.m.) injection of 5% diclofenac sodium (concentration = 2.5 mg · kg-1) was administered to 8 healthy pigs according to a two-period crossover design. The pharmacokinetic parameters were calculated by non-compartmental analysis with DAS2.1.1 software. After a single i.v. administration, the main pharmacokinetic parameters of diclofenac sodium injection in swine were as follows: the elimination half-time (T1/2β) was 1.32±0.34 h; the area under the curve (AUC) was (55.50±5.50 μg · mL-1 h; the mean residence time (MRT) was 1.60±0.28 h; the apparent volume of distribution (Vd) was 0.50±0.05 L · kg-1; and the body clearance (CLB) was 0.26±0.04 L · (h · kg)-1. After the single i.m. administration, the pharmacokinetic parameters were as follows: peak time (Tmax) was 1.19±0.26 h; and peak concentration (Cmax) was 11.61±5.99 μg mL-1. The diclofenac sodium has the following pharmacokinetic characteristics in swine: rapid absorption and elimination; high peak concentration; and bioavailability.