# 2008 Research Summary

## A Reachability Algorithm for Biologically Inspired Piecewise-Affine Hybrid Systems

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Anil Jayanti Aswani and Claire Tomlin

We describe a reachability algorithm for a class of piecewise-affine (PWA) hybrid systems relevant to biological systems. The algorithm is proven to be computable and to terminate provided that a reasonable set of constraints are met, including: the vector field of the PWA hybrid system has a diagonal structure, and the hybrid system does not have any trajectory cycles. These abstract conditions are shown to be easily verifiable for a certain class of qualitative biological models which describe species (e.g., protein) interactions in terms of promotion or inhibition between species. We show that if these qualitative interactions are structured such that there is no negative feedback, then the hybrid system generated from the biological model contains no trajectory cycles. Equivalently, negative feedback in the biological model is a necessary condition for the presence of limit cycles, centers, and foci. The given reachability algorithm can be guaranteed to converge for these biological models. A simple example of applying the algorithm to a biological model is given.

Figure 1: Subsystem of *Drosophila melanogaster* segment polarity network

Figure 2: Reach set convergent to ([CID], [iWG], [SLP]) = (0,0,0)