A stochastic Boolean network (SBN) emerges as a more realistic model for gene regulatory networks than a deterministic Boolean network (BN). In order to reduce output sampling while ensuring a given estimation accuracy, this article proposes an event-triggered sampling strategy for the state estimation of SBNs. Under this strategy, the output is sampled when the one-step prediction mean error exceeds a prespecified threshold. An iterative algorithm for the state probability distribution is proposed based on the algebraic form of SBNs, which determines the optimal state estimation. A matrix inequality method is proposed to calculate the worst-case mean estimation error based on its monotonicity with time. Then, the range of sampling triggering thresholds that minimize the worst-case mean estimation error is obtained. This article demonstrates that the event-triggered sampling strategy can make a tradeoff between estimation error and sampling rate. It explains that the full sampling estimator is a special event-triggered sampling estimator. Finally, the proposed method is applied to BN models of the lac operon in Escherichia coli to analyze the relationship among the sampling triggering threshold, the sampling rate, and the estimation error.

Multiagent dynamic task assignment of forest fires is a complicated optimization problem because it requires the consideration of multiple factors, such as the spread speed of fires, firefighting speed of agents, the movement speed of agents, and the number of deployed agents. In this article, we investigate multiagent dynamic task assignment based on a forest fire point model, the objective of which is to minimize task completion time. First, we establish a model for the spread of fire and dynamic task assignments. Second, we prove that the optimal static task assignment always makes all task completion times the same under certain assumptions. Furthermore, we calculate the optimal solution to the static task assignment problem assuming no travel time for the agents, which provides the theoretical basis for the initial deployment and dynamic deployment. Third, we propose a dynamic task assignment scheme based on the global information, which ensures that every reassignment reduces the task completion time and makes all task completion times close to each other. Finally, the simulation is carried out on the MATLAB platform to verify the performance of the proposed dynamic task assignment scheme by comparing with a multistage global auction algorithm. We hope that this work provides insight for decision-makers designing reasonable assignment strategies based on the model and solving assignment optimization problem in different situations.Note to practitioners—The forest firefighting problem considered in this article is a typical multitask and multistage optimization problem. Many searching algorithms for multistage optimization problem are available in the existing literature. However, one of the main challenges is that the time of searching increases exponentially with the number of stages. This work first proves that the tasks are completed in the minimum amount of time, under the constraint of one-shot assignment. This finding helps us to evaluate the gap between the searching algorithm and the optimal solution. In addition, in practice, if the underlying dynamic process can be modeled or partially modeled, then we can predict the behavior of future stages and reduce the searching domain. If a model is available, then we can also adjust the assignment scheme dynamically based on the principle that each adjustment would reduce the total time of tasks completion. In this article, we establish a dynamical fire-spreading model and propose a model-based solution to the multistage optimization problems. The findings in this work can serve as a supplement to the existing optimization algorithms.

We investigate the stability of Boolean networks (BNs) with impulses triggered by both states and random factors. A hybrid index model is used to describe impulsive BNs. First, several necessary and sufficient conditions for forward completeness are obtained. Second, based on the stability criterion of probabilistic BNs and the forward completeness criterion, the necessary and sufficient conditions for the finite-time stability with probability one and the asymptotical stability in distribution are presented. The relationship between these two kinds of stability is discussed. Last, examples and time-domain simulations are provided to illustrate the obtained results.

This article studies the optimal control of probabilistic Boolean control networks (PBCNs) with the infinite horizon average cost criterion. By resorting to the semitensor product (STP) of matrices, a nested optimality equation for the optimal control problem of PBCNs is proposed. The Laurent series expression technique and the Jordan decomposition method derive a novel policy iteration-type algorithm, where finite iteration steps can provide the optimal state feedback law, which is presented. Finally, the intervention problem of the probabilistic Ara operon in E. coil, as a biological application, is solved to demonstrate the effectiveness and feasibility of the proposed theoretical approach and algorithms.

This paper presents an investigation of the optimal estimation of state for Boolean control networks subject to stochastic disturbances. The disturbances are modeled as independently and identically distributed processes that are assumed to be both mutually independent and independent of the current and the historical states. An iterative algorithm is proposed to calculate the conditional probability distribution of the state given the output measurements. This algorithm is applied to the problems of minimum mismatching estimation and maximum posterior estimation of the state. An example is provided to illustrate the proposed results.

This article investigates the partial stabilization problem of probabilistic Boolean control networks (PBCNs) under sample-data state-feedback control (SDSFC) with a control Lyapunov function (CLF) approach. First, the probability structure matrix of the considered PBCN is represented by a Boolean matrix, based on which, a new algebraic form of the system is obtained. Second, we convert the partial stabilization problem of PBCNs into the global set stabilization one. Third, we define CLF and its structural matrix under SDSFC. It is found that the existence of a CLF is equivalent to that of SDSFC. Then, a necessary and sufficient condition is obtained for the existence of CLF under SDSFC, based on which, all possible sample-data state-feedback controllers and corresponding structural matrices of CLF are designed by two different methods. Finally, examples are given to illustrate the efficiency of the obtained results.

In this article, we investigate the asymptotical feedback set stabilization in distribution of probabilistic Boolean control networks (PBCNs). We prove that a PBCN is asymptotically feedback stabilizable to a given subset if and only if (iff) it constitutes asymptotically feedback stabilizable to the largest control-invariant subset (LCIS) contained in this subset. We proposed an algorithm to calculate the LCIS contained in any given subset with the necessary and sufficient condition for asymptotical set stabilizability in terms of obtaining the reachability matrix. In addition, we propose a method to design stabilizing feedback based on a state-space partition. Finally, the results were applied to solve asymptotical feedback output tracking and asymptotical feedback synchronization of PBCNs. Examples were detailed to demonstrate the feasibility of the proposed method and results.

This brief reviews various definitions of observability for Boolean control networks (BCNs) and proposes a new one: output-feedback observability. This new definition applies to all BCNs whose initial states can be identified from the history of output measurements. A technique called parallel extension is then proposed to facilitate observability analysis. Furthermore, a technique called state transition graph reconstruction is proposed for analyzing the set reachability of BCNs, based on which new criteria for observability, single-input sequence observability, and arbitrary-input observability, are obtained. Using the proposed techniques, this brief proves that the problem of output-feedback observability can be recast as that of stabilizing a logic dynamical system with output feedback. Then, a necessary and sufficient condition for static output feedback observability is proposed. The relationships between the different definitions of observability are discussed, and the main results are illustrated with examples.