Review on computational methods for Lyapunov functions

Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them. Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ di_erent methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function

Characterization of solid dispersions of carbamazepine and PEG6000/vinylcaprolactam/vinylacetate copolymer

Purpose. to investigate the solid-state properties and dissolution performances of solid dispersions of carbamazepine and a new marketed copolymer proposed for hot melt extrusion. Methods. Physical mixtures [PM] were prepared by mixing in mortar and pestle for 10 min carbamazepine [CBZ, polymorphic form III] and PEG6000/vinylcaprolactam/vinylacetate copolymer (Soluplus\uae BASF) in various drug/polymer ratios (1:9, 3:7, 1:1, 7:3 w/w). Co-melted formulations [CM] were prepared by heating the PM at 170\ub0C on a hot plate for 15 min, allowing to cool at room temperature and grinding the resulting glasses in a mortar. PXRD study, DSC analysis and dissolution test (paddle apparatus, 500mL of 0.0125% SLS solution, 37\ub10.5\ub0C, 100 rpm, UV detection at 287 nm) were performed on all samples. Results. drug-polymer solid-state interaction, resulting in complete loss of CBZ crystallinity was found to occur in CM with drug/polymer ratios 1:9 and 3:7 w/w. Accordingly, the glass transition recorded for these dispersions was lowered with respect to that of the pure polymer. Non-interacted crystalline CBZ was detected in all PM as well as in 1:1 and 7:3 CM formulations. Furthermore, non-interacted amorphous CBZ was thought to be present in the 7:3 CM formulation, resulting in drug crystallization up on heating in the DSC experiment. As far as dissolution properties are concerned, PM presented similar or only slightly higher dissolution rate with respect to the pure drug, whilst 1:9 and 3:7 CMF demonstrated a remarkable improvement of dissolution rate (the amount dissolved at t=10min doubled that of the pure CBZ). The 7:3 CM formulation was affected by a decreased dissolution rate probably caused by the formation of CBZ hydrate during the test. Conclusion. solid dispersion of CBZ with a new PEG6000/vinylcaprolactam/vinylacetate copolymer can be successfully prepared by comelting method, resulting in a high degree of drug-polymer interaction and consequently a remarkable improvement of drug dissolution rate