output feedback stabilization control design for boolean control networks this paper investigates the output feedback stabilization of boolean control networks bcns by using the semi tensor product method and presents a number of new results first based on the algebraic expression of bcns a necessary and sufficient condition is presented for the existence of output feedback stabilizers second a constructive procedure is proposed to design output feedback stabilization controllers for bcns the study of an illustrative example shows that the new results obtained in this paper are very effective in designing output feedback stabilizers for bcns

invariant subset and set stability of boolean networks under arbitrary switching signals this study investigates the set stability of switched boolean networks bns with arbitrary switching signals based on invariant subsets set stability determines whether a bn converges to a given subset while arbitrary switching signals can be used to characterize uncertainties and disturbances an algorithm is proposed to calculate the largest invariant subset contained in any given set as this is the key step in establishing the set stability criterion based on this algorithm a necessary and sufficient condition for set stability is obtained the result is then used to solve the robust output synchronization problem and the robust node synchronization problem for bns in the presence of uncertainties and disturbances examples are presented to demonstrate the use of the proposed approach

reconstructing displacements from the solution to the periodic lippmann schwinger equation discretized on a uniform grid uniform grid solvers of the periodic lippmann schwinger equation have been introduced by moulinec and suquet for the numerical homogenization of heterogeneous materials based on the fast fourier transform these methods use the strain as main unknown and usually do not produce displacement fields while this is generally not perceived as a restriction for homogenization purposes some tasks might require kinematically admissible displacement fields in this paper we show how the numerical solution to the periodic lippmann schwinger equation can be post processed to reconstruct a displacement field our procedure is general it applies to any variant of the moulinec suquet solver the reconstruction is formulated as an auxiliary elastic equilibrium problem of a homogeneous material which is solved with displacement based finite elements taking advantage of the periodicity the uniformity of the grid and the homogeneity of the material the resulting linear system is formulated and solved efficiently in fourier space the cost of our procedure is lower than that of one iteration of the lippmann schwinger solver an application of this post processing procedure is proposed in which the reconstructed displacement field is used to compute a rigorous upper bound on the effective shear modulus of some model microstructure this is the pre peer reviewed version of the following article reconstructing displacements from the solution to the periodic lippmann schwinger equation discretized on a uniform grid which has been published in final form at 10 1002 nme 5263 this article may be used for non commercial purposes in accordance with wiley terms and conditions for self archiving sections 1 introduction and 5 numerical results have been significantly reworked in the published final version of this article the remainder of this article including sections 3 and 4 where the method itself is presented is essentially unchanged

number and length of attractors in a critical kauffman model with connectivity one the kauffman model describes a system of randomly connected nodes with dynamics based on boolean update functions though it is a simple model it exhibits very complex behavior for critical parameter values at the boundary between a frozen and a disordered phase and is therefore used for studies of real network problems we prove here that the mean number and mean length of attractors in critical random boolean networks with connectivity one both increase faster than any power law with network size we derive these results by generating the networks through a growth process and by calculating lower bounds

identification of boolean control networks in this paper the identification of boolean control networks is addressed first necessary and sufficient conditions are obtained for the identification of state equation from input state data then a necessary and sufficient condition for a controllable boolean network to be observable is presented based on these two results a necessary and sufficient condition for the identification from input output data is achieved to practically identify the model a numerical algorithm is proposed two particular cases i identification of systems with a known network graph ii identification of a higher order boolean network are also investigated finally the approximate identification for large size networks is explored

system decomposition with respect to inputs for boolean control networks abstract in this paper the problem of system decomposition with respect to inputs is investigated for boolean control networks bcns first based on the linear representation of bcns some algebraic equivalent conditions for the decomposability are obtained second the concept of perfect equal vertex partition pevp is proposed for bcns a necessary and sufficient graphical condition for the decomposability i e the existence of a pevp is proposed third for calculating a pevp an algorithm called vertex set uniting algorithm vsua is designed finally a biological example is analyzed to show the effectiveness of the proposed method

optimal infinite horizon control for probabilistic boolean networks external control of a genetic regulatory network is used for the purpose of avoiding undesirable states such as those associated with disease heretofore intervention has focused on finite horizon control i e control over a small number of stages this paper considers the design of optimal infinite horizon control for context sensitive probabilistic boolean networks pbns it can also be applied to instantaneously random pbns the stationary policy obtained is independent of time and dependent on the current state this paper concentrates on discounted problems with bounded cost per stage and on average cost per stage problems these formulations are used to generate stationary policies for a pbn constructed from melanoma gene expression data the results show that the stationary policies obtained by the two different formulations are capable of shifting the probability mass of the stationary distribution from undesirable states to desirable ones

observability of boolean networks a graph theoretic approach boolean networks bns are discrete time dynamical systems with boolean state variables and outputs bns are recently attracting considerable interest as computational models for genetic and cellular networks we consider the observability of bns that is the possibility of uniquely determining the initial state given a time sequence of outputs our main result is that determining whether a bn is observable is np hard this holds for both synchronous and asynchronous bns thus unless p np there does not exist an algorithm with polynomial time complexity that solves the observability problem we also give two simple algorithms with exponential complexity that solve this problem our results are based on combining the algebraic representation of bns derived by d cheng with a graph theoretic approach some of the theoretical results are applied to study the observability of a bn model of the mammalian cell cycle

controllability and observability of boolean control networks the controllability and observability of boolean control networks are investigated after a brief review on converting a logic dynamics to a discrete time linear dynamics with a transition matrix some formulas are obtained for retrieving network and its logical dynamic equations from this network transition matrix based on the discrete time dynamics the controllability via two kinds of inputs is revealed by providing the corresponding reachable sets precisely then the problem of observability is also solved by giving necessary and sufficient conditions

simultaneous stabilization for a set of boolean control networks abstract this paper investigates simultaneous stabilization of a collection of boolean control networks bcns by using the semi tensor product method and presents a number of new results first an algebraic expression of the bcns is obtained by the semi tensor product based on which some necessary and sufficient conditions are presented to solve the simultaneous stabilization problem by a free control sequence a state feedback control and an output feedback control respectively second using the column stacking form of matrices a new procedure is established to design both state feedback and output feedback controllers for the simultaneous stabilization problem the study of two illustrative examples shows that the new results obtained in this paper are very effective in solving simultaneous stabilization of a collection of bcns

realization of boolean control networks based on the linear expression of the dynamics of boolean networks the coordinate transformation of boolean variables is defined it follows that the state space coordinate transformation for the dynamics of boolean networks is revealed using it the invariant subspace for a boolean control network is defined then the structure of a boolean control network is analyzed and the controllable and observable normal forms and the kalman decomposition form are presented finally the realization problem including minimum realization of boolean control networks is investigated

further results on disturbance decoupling of mix valued logical networks this note investigates the disturbance decoupling problems ddps of mix valued logical control networks mlcns via semi tensor product method by using prime factor decomposition a unique y friendly subspace of mix valued logical networks mlns is derived a necessary and sufficient condition is obtained for the solvability of ddps an algorithm is proposed to find all disturbance decoupling controllers this approach is more operational than the existing results in which the y friendly subspace is not unique finally an illustrative numerical example is given to show the effectiveness of the proposed method

boolean dynamics of networks with scale free topology the dynamics of boolean networks with scale free topology are studied the existence of a phase transition from ordered to chaotic dynamics governed by the value of the scale free exponent of the network is shown analytically by analyzing the overlap between two distinct trajectories the phase diagram shows that the phase transition occurs for values of the scale free exponent in the open interval 2 2 5 since the boolean networks under study are directed graphs the scale free topology of the input connections and that of the output connections are studied separately ultimately these two topologies are shown to be equivalent a numerical study of the attractor structure of the configuration space reveals that this structure is similar in both networks with scale free topologies and networks with homogeneous random topologies however an important result of this work is that the fine tuning usually required to achieve stability in the dynamics of networks with homogeneous random topologies is no longer necessary when the network topology is scale free finally based on the results presented in this work it is hypothesized that the scale free topology favors the evolution and adaptation of network functioning from a biological perspective

input state incidence matrix of boolean control networks and its applications the input state incidence matrix of a control boolean network is proposed it is shown that this matrix contains complete information of the input state mapping using it an easily verifiable necessary and sufficient condition for the controllability of a boolean control network is obtained the corresponding control which drives a point to a given reachable point is designed moreover certain topological properties such as the fixed points and cycles of a boolean control network are investigated then as another application a sufficient condition for the observability is presented finally the results are extended to mix valued logical control systems

output tracking control of boolean control networks via state feedback this paper investigates the state feedback based output tracking control of boolean control networks bcns with a constant reference signal by using the semi tensor product method based on the algebraic expression of bcns and by constructing a series of reachable sets a general procedure is proposed for the design of the state feedback laws for bcns to track a constant reference signal the study of an illustrative example shows that the obtained new results are effective in designing state feedback based output tracking controllers for bcns to track a constant reference signal

control design for output tracking of delayed boolean control networks abstract this paper investigates the output tracking control of delayed boolean control networks dbcns by using the algebraic state space representation method firstly the output tracking control problem of dbcns is formulated and the dynamics of dbcns is converted into an equivalent algebraic form secondly based on the algebraic form and controllability of dbcns a necessary and sufficient condition is presented for the output tracking control of dbcns without state constraints thirdly a necessary and sufficient condition is presented for the output tracking control of dbcns with state constraints by constructing a constrained reachability matrix the study of an illustrative example shows the effectiveness of the obtained new results

redefined observability matrix for boolean networks and distinguishable partitions of state space this paper redefines the observability matrix and defines the observability index for boolean networks bns in the new definition a bn is observable if and only if the observability matrix is of full rank the observability matrix is calculated by a simple algorithm and is applied to the problem of distinguishability of partitions a partition of the state space is said to be distinguishable if any two distinct subsets are distinguishable and the finest distinguishable partition fdp is finer than any other distinguishable partition a necessary and sufficient condition is proposed for checking the distinguishability of any given partition and an algorithm is proposed for calculating the fdp the proposed results are illustrated by examples

stability and set stability in distribution of probabilistic boolean networks we propose a new concept stability in distribution sd of a probabilistic boolean network pbn which determines whether the probability distribution converges to the distribution of the target state namely a one point distributed random variable in a pbn stability with probability one stability in the stochastic sense and sd are equivalent the sd is easily generalized to subset stability i e to set stability in distribution ssd we prove that the transition probability from any state to an invariant subset or to a fixed point is nondecreasing in time this monotonicity is an important property in establishing stability criteria and in calculating or estimating the transient period we also obtain a verifiable necessary and sufficient condition for sd of pbns with independently and identically distributed switching we then show that sd problems of pbns with markovian switching and pbn synchronizations can be recast as ssd problems of markov chains after calculating the largest invariant subset of a markov chain in a given set by the newly proposed algorithm we propose a necessary and sufficient condition for ssds of markov chains the proposed method and results are supported by examples

invertibility and nonsingularity of boolean control networks invertibility is an interesting and classical control theoretic problem however there has been no result for the invertibility of boolean control networks bcns so far we first adopt the theory of symbolic dynamics to characterize it first it is shown that a bcn generates a continuous mapping from the space of input trajectories to the space of output trajectories based on it the concepts of nonsingularity and invertibility of bcns are first defined as the injectivity and bijectivity of the mapping respectively second combined symbolic dynamics with the semi tensor product stp of matrices an equivalent test criterion for invertibility is given easily computable algorithms to construct the inverse bcn for an invertible bcn are presented and it is proved that invertibility remains invariant under coordinate transformations third an equivalent test criterion for nonsingularity is given via defining a novel directed graph that is called weighted pair graph lastly as an application of invertibility to systems biology we prove that the bcn model proposed in faure et al 2006 is not invertible i e we prove that arbitrarily controlling mammalian cell cycles is unfeasible at the theoretical level

model construction of boolean network via observed data in this paper a set of data is assumed to be obtained from an experiment that satisfies a boolean dynamic process for instance the dataset can be obtained from the diagnosis of describing the diffusion process of cancer cells with the observed datasets several methods to construct the dynamic models for such boolean networks are proposed instead of building the logical dynamics of a boolean network directly its algebraic form is constructed first and then is converted back to the logical form firstly a general construction technique is proposed to reduce the size of required data the model with the known network graph is considered motivated by this the least in degree model is constructed that can reduce the size of required data set tremendously next the uniform network is investigated the number of required data points for identification of such networks is independent of the size of the network finally some principles are proposed for dealing with data with errors

on reachability and controllability of switched boolean control networks this paper investigates the reachability and controllability of switched boolean control networks bcns by using the semi tensor product method first a kind of switching input state transfer graph is defined based on which and the algebraic expression of switched bcns we thus propose a new kind of switching input state incidence matrix that contains the complete information of the switching input state transfer graph second by the switching input state incidence matrix several necessary and sufficient conditions are obtained for the reachability and controllability of switched bcns finally an algorithm is established to find a switching sequence and a control scheme to realize the reachability of switched bcns in the shortest time the study of an illustrative example shows that the new results obtained in this paper are very effective in investigating the reachability and controllability of switched bcns

controllability analysis and control design for switched boolean networks with state and input constraints this paper investigates the controllability analysis and the control design for switched boolean networks sbns with state and input constraints by using the semi tensor product method and presents a number of new results on their controllability optimal control and stabilization first the constrained incidence matrix is proposed for sbns based on which several necessary and sufficient conditions are obtained for the controllability second using the results on the controllability two algorithms are established to design proper switching sequence and controls that minimize the cost functional in a fixed designed shortest termination time third the constrained sbn is converted to an equivalent unconstrained one based on which some necessary and sufficient conditions are presented for the stabilization of the constrained sbn with open loop and closed loop controls respectively finally a practical example of the apoptosis network is studied by using the new results obtained in this paper the s

a linear representation of dynamics of boolean networks a new matrix product called semi tensor product of matrices is reviewed using it a matrix expression of logic is proposed where a logical variable is expressed as a vector a logical function is expressed as a multiple linear mapping under this framework a boolean network equation is converted into an equivalent algebraic form as a conventional discrete time linear system analyzing the transition matrix of the linear system formulas are obtained to show a the number of fixed points b the numbers of cycles of different lengths c transient period for all points to enter the set of attractors and d basin of each attractor the corresponding algorithms are developed and used to some examples

on boolean control networks with maximal topological entropy boolean control networks bcns are discrete time dynamical systems with boolean state variables and inputs that are interconnected via boolean functions bcns are recently attracting considerable interest as computational models for genetic and cellular networks with exogenous inputs the topological entropy of a bcn with m inputs is a nonnegative real number in the interval 0 m log 2 roughly speaking a larger topological entropy means that asymptotically the control is more powerful we derive a necessary and sufficient condition for a bcn to have the maximal possible topological entropy our condition is stated in the framework of cheng s algebraic state space representation of bcns this means that verifying this condition incurs an exponential time complexity we also show that the problem of determining whether a bcn with n state variables and m n inputs has a maximum topological entropy is np hard suggesting that this problem cannot be solved in general using a polynomial time algorithm

controllability decomposition of dynamic algebraic boolean control networks two types of controllability of dynamic algebraic boolean control networks are first presented the necessary and sufficient conditions of such controllability are derived a similar concept of coo

l1 gain analysis and model reduction problem for boolean control networks in this paper the weighted l1 gain analysis and l1 model reduction problem for boolean control networks are proposed and investigated via semi tensor product method first the input energy and output energy are described by pseudo boolean functions based on which the definition of the weighted l1 gain is introduced by constructing a co positive lyapunov function a sufficient condition is established to ensure that a boolean control network is not only internally asymptotically stable but also has an l1 gain no more than a given scalar along this line by virtue of the properties of semi tensor product the l1 model reduction problem of a boolean control network is defined and converted to the l1 gain problem of another boolean control network with more nodes a sufficient condition for the l1 model reduction problem is then derived immediately and an algorithm is presented to compute the matrices in the reduced order model finally two examples including the boolean model for biochemical oscillators in the cell cycle are displayed to show the feasibility of the theoretical results

controllability and observability of boolean control networks with time variant delays in states this brief investigates the controllability and observability of boolean control networks with not necessarily bounded time variant delays in states after a brief introduction to converting a boolean control network to an equivalent discrete time bilinear dynamical system via the semi tensor product of matrices the system is split into a finite number of subsystems constructed forest with no time delays by using the idea of splitting time that is proposed in this brief then the controllability and observability of the system are investigated by verifying any so called controllability constructed path and any so called observability constructed paths in the above forest respectively which generalize some recent relevant results matrix test criteria for the controllability and observability are given the corresponding control design algorithms based on the controllability theorems are given we also show that the computing complexity of our algorithm is much less than that of the existing algorithms

control of large scale boolean networks via network aggregation a major challenge to solve problems in control of boolean networks is that the computational cost increases exponentially when the number of nodes in the network increases we consider the problem of controllability and stabilizability of boolean control networks address the increasing cost problem by partitioning the network graph into several subnetworks and analyze the subnetworks separately easily verifiable necessary conditions for controllability and stabilizability are proposed for a general aggregation structure for acyclic aggregation we develop a sufficient condition for stabilizability it dramatically reduces the computational complexity if the number of nodes in each block of the acyclic aggregation is small enough compared with the number of nodes in the entire boolean network

observability of boolean control networks a unified approach based on finite automata the problem on how to determine the observability of boolean control networks bcns has been open for five years already in this technical note we propose a unified approach to determine all the four types of observability of bcns in the literature we define the concept of weighted pair graphs for bcns in the sense of each observability we use the so called weighted pair graph to transform a bcn to a finite automaton and then we use the automaton to determine observability in particular the two types of observability that rely on initial states and inputs in the literature are determined finally we show that no pairs of the four types of observability are equivalent which reveals the essence of nonlinearity of bcns

state feedback based output tracking control of probabilistic boolean networks abstract this paper studies the state feedback based output tracking control of probabilistic boolean control networks pbcns by using the semi tensor product of matrices a series of reachable sets with probability 1 is defined inductively for pbcns and some useful properties are obtained for the reachable sets based on the properties of the reachable sets a necessary and sufficient condition is presented for the state feedback based output tracking control of pbcns meanwhile a constructive procedure is proposed to design state feedback laws for pbcns to track a constant reference signal the study of two illustrative examples shows that the obtained new results are very effective