Absolute Stability of Nonlinear Control Systems

Sufficient and necessary conditions for Lyapunov stability of genetic networks with SUM regulatory logic

“…If condition (I) holds, by LaSalle invariant theorem (Liao and Yu 2008), the zero solution of system (4) on the partial variablem is globally asymptotically stable. Similarly, if condition (II) holds, the zero solution of system (4) on the partial variablep is globally asymptotically stable.…”

“…Furthermore, there exists a positive constant vector 1 ¼ ½n 1 ; n 2 ; Á Á Á ; n n ; g 1 ; g 2 ; Á Á Á ; g n T [ 0, such that A T 1 [ 0 (Liao and Yu 2008). So, it holds that…”

“…Luo et al (2010) proposed some results of unconditional global exponential stability in sense of Lagrange for GRNs with time-variant delays and SUM logic. Some studies are about the stability in sense of Lyapunov (Yu et al 2009;Li et al 2010;Ren and Cao 2008;Hu et al 2013;Wang et al 2010;Wu 2011), while some other researches are about the stability in sense of Lagrange (Luo et al 2010;Liao and Yu 2008). Nevertheless, they are mostly sufficient condition, and the linear matrix inequalities (LMIs) are mainly used.…”

“…Since GRN is actually a Lurie system, which is a linear system plus nonlinear feedback term, and sufficient and necessary conditions for absolute stability of Lurie system are proposed [see references in Liao and Yu (2008)], people always raise the question whether the sufficient and necessary conditions for stability of GRNs exist, unlike the existing literature of sufficient conditions. By analyzing, GRNs with SUM logic are just with the excellent properties which Lurie systems have.…”

“…By analyzing, GRNs with SUM logic are just with the excellent properties which Lurie systems have. Therefore, we can take advantage of the core idea of reference (Liao and Yu 2008), and aim to study the GRNs with SUM logic.…”