Controllability of Boolean control networks via the Perron–Frobenius theory

Abstract

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.

Observability of Boolean networks: A graph-theoretic approach

“…Algorithms for converting a BCN in the form (1) to its algebraic representation (2), and vice versa, may be found in Cheng andQi (2010a, 2009). As we will see below, one advantage of the ASSR is that it naturally suggests a graph-theoretic approach for studying BNs (see also Zhao et al (2010); Laschov and Margaliot (2012a);Yin (2011)). We first review some results on the observability of graphs from a recent paper by Jungers and Blondel (2011).…”

“…This representation has proved useful for studying BCNs in a control-theoretic framework. Examples include the analysis of disturbance decoupling (Cheng (2011)), controllability and observability (Cheng and Qi (2009) ;Zhao et al (2010); Laschov and Margaliot (2012a); Fornasini and Valcher (2012)), realization theory ), optimal control Margaliot (2011, 2012c,b)), and more (Cheng and Qi (2010a,b); ).…”

Minimum-Time Control of Boolean Networks

“…This representation proved useful for addressing control-theoretic problems for BCNs. Examples include the analysis of disturbance decoupling [16], controllability and observability [19], [48], realization theory [18], and more [20], [21], [15]. See the recent monograph [22] for a detailed presentation.…”

Controllability of Boolean control networks under asynchronous stochastic update with time delay

“…Recently, a novel mathematic method called the semi-tensor product (STP) was proposed by Cheng et al (2011), by which logical dynamics can be converted into linear representation and further analytical analysis can be implemented. Consequently, some impressive results have been achieved, especially for the in-depth research on control-related problems, such as controllability (Cheng and Qi, 2009;Li and Sun, 2011a;Laschov and Margaliot, 2012), observability (Cheng and Qi, 2009;Li et al, 2011), realization (Cheng et al, 2010), stability (Li and Sun, 2011b) and control for disturbance decoupling , etc. Wherein, controllability is a fundamental problem in control field.…”

“…Wherein, controllability is a fundamental problem in control field. Cheng and Qi (2009) introduced STP technique into BCNs and analyzed the controllability of logical systems, Li et al (2011) studied the controllability of BCNs with time delay in states and Laschov and Margaliot (2012) investigated the controllability of BCNs via Perron-Frobenius theory. However, all of the above studies focused on synchronous Boolean networks, and main results were based on a prerequisite that BCNs were deterministic systems and attractors were circular, obviously, which couldn't be held for asynchronous dynamics.…”