New Approach to the Stability of Chemical Reaction Networks: Piecewise Linear in Rates Lyapunov Functions

Abstract

Abstract-Piecewise-Linear in Rates (PWLR) Lyapunov functions are introduced for a class of Chemical Reaction Networks (CRNs). In addition to their simple structure, these functions are robust with respect to arbitrary monotone reaction rates, of which Mass-Action is a special case. The existence of such functions ensures the convergence of trajectories towards equilibria, and can be used to establish their asymptotic stability with respect to the corresponding stoichiometric compatibility class. We give the definition of these Lyapunov functions, prove their basic properties, and provide algorithms for constructing them. Examples are provided, relationship with consensus dynamics are discussed, and future directions are elaborated.

Construction of robust Lyapunov functions for reaction networks

“…This violates a the result that the Jacobians of GSNs can not have negative principle minors [8]. This implies that it can not admit a PWLR Lyapunov function.…”

“…Furthermore, the LaSalle's condition can be verified also by a graphical algorithm. For more details, refer to [8].…”

“…As the motivating example may indicate, it can be shown that the nonpositivity of every term in the expansion ofV is needed for it to be an RLF [8]. In order to verify this, it can be assumed, without loss of generality, that polyhedral partition (12) have the property that sgnẋ is constant in the interior of each region.…”

“…Within the class of networks that satisfy the necessary conditions outlined in [8], [9], our proposed algorithms are reasonably successful. In most examples, either a PWLR function is constructed, or a necessary condition is violated rendering the Lyapunov function inadmissable.…”

“…Recently, the monotonicity concept has been applied to provide graphical conditions for global convergence of some of these networks [7]. This paper reviews the recent work of the authors [8], [9], [10] that uncovered a new class of networks that can be analyzed robustly for any monotone kinetics. The next section reviews the required background on CRNs.…”