Positive diagonal solutions to the Lyapunov equations

On common quadratic lyapunov functions for pairs of stable lti systems whose system matrices are in companion form

“…Lemma 4.1 [10], [11]: Consider the LTI systems 1). Thus, the nonsingularity of these two pencils is a necessary condition for the existence of a CQLF for the systems 6 A , 6 A .…”

An algebraic characterization ofB-convergent Runge-Kutta methods

“…Sufficient conditions were presented already by Frank et al [13,14]; they showed that if the method is algebraically stable and also satisfies a slightly different algebraic condition -known as "diagonal stability" in matrix theory [1,2] -then it is B-convergent on ~u for arbitrary /~]R. In this paper it will be proved that for # > 0 these two conditions are necessary as well, under some mild apriori assumptions on the method. This result shows that there * This paper was written while J. Schneid was visiting the Centre for Mathematics and Computer Science with an Erwm-Schrfdinger shpend from the Fonds zur F6rderung der wissenschafthchen Forschung can be a different behaviour of methods when applied to dissipative problems (#=0) or to strictly dissipative problems (# < 0), as it is known from the work of Spijker, Dekker, Kraaijevanger and Schneid [21,10,20] that for/z < 0 algebraic stability on its own is already sufficient, and also necessary, for B-convergence on flu.…”

“…For any given Runge-Kutta method it is straightforward to check whether (2.5) is satisfied. Some necessary and sufficient conditions for (2.7) to hold can be found in [1] and [2]. For example, it is known that (2.7) implies that all principles minors of A are positive, and this is also sufficient in case s = 2.…”

Some results on the stability of positive switched linear systems

“…Before proceeding we note that the existence of diagonal Lyapunov functions is an important research area in its own right, and that several papers have appeared on this topic [12], [13], [15], [16], [17]. These functions arise in the study of decentralised and interconnected systems, [18], [19], as well as in the study of neural networks, and asynchronous computation [20].…”