Successive Overrelaxation Iteration for the Stability of Large Scale Systems

During the past two decades, the stability of large scaler systems has been studied by numerous authors; many results can be found in the books of Michel and Miller [12] and Siljak [14]. Recently, Liao [7-lo] used the lumped iterations to investigate stabilities of large scale systems and interval matrices. Iterations are the most important methods used to solve algebraic equations in number analysis (cf. Ortega and Rheinbadt [13], Vorga [lS], Young [16]). In this article, following Liao [9], we employ the successive overrelaxation iteration and the variation of constants formula (cf. Alekseev [l], Athanassov [2], Brauer [3,4,6], Brauer and Strauss [S]) to discuss the stability of large scale systems. Some simple algebraic criterion are obtained directly from coefficients of the system, and it is possible to circumvent the difficulty of finding Liapunov functions.