Box invariance for biologically-inspired dynamical systems

Abstract

In this paper, motivated by models drawn from biology, we introduce the notion of box invariant dynamical systems. We argue that box invariance, that is, the existence of a "box"-shaped positively invariant region, is a characteristic of many biologically-inspired dynamical models. Box invariance is also useful for the verification of stability and safety properties of such systems. This paper presents effective characterization of this notion for some classes of systems, computational results on checking box invariance, the study of the dynamical properties it subsumes, and a comparison with related concepts in the literature. The concept is illustrated using models derived from different case studies in biology.

Computation of polytopic invariants for polynomial dynamical systems using linear programming

“…Let us remark that rectangles form a subclass of polytopic invariants and in some sense, our work extends the work of [BH06,ATS09]. More precisely, we shall consider a dynamical system of the form:…”

“…However, when the computation of an invariant set is part of a bigger process such as controller synthesis or safety verification, it is sometimes preferable to have invariants given by polytopes that are easier to manipulate [ALBH07,SDI08]. For instance, for specific classes of polynomial systems such as multi-affine or quasi multi-affine systems, methods to obtain rectangular invariants have been developed in [BH06,ATS09].…”

Generating Box Invariants

“…The converse is not true; for example, the system dx 1 /dt = x 3 1 + x 1 is monotone but not multiaffine.…”

“…Computational results on box invariance of linear systems [3] and some preliminary results for nonlinear and hybrid systems [2] have been presented before. This paper develops these results further and identifies the classes of monotone, quasi-monotone, and uniformly quasi-monotone systems on which box invariants computation can be reduced to constraint solving.…”

“…We are interested in the case when Box (l, u) is a positively invariant set [2,3]. We say a hybrid system is box invariant if there exists a box that is also a positively invariant set.…”