On stability and controllability of conjunctive Boolean networks

Boolean networks;Discrete time dynamics;Networked control systems

Gao, Zuguang ; Başar ; Tamer

关键词: Boolean networks;    Discrete time dynamics;    Networked control systems;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/98333/GAO-THESIS-2017.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

A Boolean network (BN) is a finite state discrete time dynamical system. At each step, each variable takes a value from a binary set. The value update rule for each variable is a local function which depends only on a selected subset of variables. BNs have been used in modeling gene regulatory networks. We focus in this thesis on a special class of BNs, termed as conjunctive Boolean networks (CBNs). A BN is conjunctive if the associated value update rule is comprised of only AND operations.It is known that any trajectory of a finite dynamical system will enter a periodic orbit. Periodic orbits of a CBN are now completely understood. We first characterize in this thesis all periodic orbits of a CBN. In particular, we establish a bijection between the set of periodic orbits and the set of binary necklaces of a certain length. We further investigate the stability of a periodic orbit. Specifically, we perturb a state in the periodic orbit by changing the value of a single entry of the state. The trajectory, with the perturbed state being the initial condition, will enter another (possibly the same) periodic orbit in finite time steps. We then provide a complete characterization of all such transitions from one periodic orbit to another. In particular, we construct a digraph, with the vertices being the periodic orbits, and the (directed) edges representing the transitions among the orbits. We call such a digraph the stability structure of the CBN.We then investigate the orbit-controllability and state-controllability of a CBN. We ask the question of how one can steer a CBN to enter any periodic orbit or to reach any final state, from any initial state. Suppose that there is a selected subset of variables whose values can be controlled for some finite time steps, while other variables still follow the value update rule during all time. We establish in the thesis a necessary and sufficient condition for this subset such that the trajectory, with any initial condition, will enter any desired periodic orbit or reach any final state. We also provide algorithms specifying the methods of manipulating the values of these variables to realize these control goals.