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Controllability of Boolean control networks via the Perron-Frobenius theory. (English) Zbl 1244.93026
Summary: Boolean Control Networks (BCNs) are recently attracting considerable interest as computational models for genetic and cellular networks. Addressing control-theoretic problems in BCNs may lead to a better understanding of the intrinsic control in biological systems, as well as to developing suitable protocols for manipulating biological systems using exogenous inputs. We introduce two definitions for controllability of a BCN, and show that a necessary and sufficient condition for each form of controllability is that a certain nonnegative matrix is irreducible or primitive, respectively. Our analysis is based on a result that may be of independent interest, namely, a simple algebraic formula for the number of different control sequences that steer a BCN between given initial and final states in a given number of time steps, while avoiding a set of forbidden states.

93B05 Controllability
90C59 Approximation methods and heuristics in mathematical programming
93B03 Attainable sets, reachability
92C42 Systems biology, networks
15B48 Positive matrices and their generalizations; cones of matrices
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