This note considers globally finite-time synchronization of coupled networks with Markovian topology and distributed impulsive effects. The impulses can be synchronizing or desynchronizing with ...certain average impulsive interval. By using M-matrix technique and designing new Lyapunov functions and controllers, sufficient conditions are derived to ensure the synchronization within a setting time, and the conditions do not contain any uncertain parameter. It is demonstrated theoretically and numerically that the number of consecutive impulses with minimum impulsive interval of the desynchronizing impulsive sequence should not be too large. It is interesting to discover that the setting time is related to initial values of both the network and the Markov chain. Numerical simulations are provided to illustrate the effectiveness of the theoretical analysis.
This technical note presents analytical investigations of reachability and controllability of Boolean control networks (BCNs) with pinning controllers. Based on semi-tensor product (STP) of matrices, ...BCNs with pinning controllers are converted into a discrete-time algebraic system. A formula is derived to calculate the number of different control sequences steering BCNs between two states in a given step, and then several necessary and sufficient criteria are derived for reachability and controllability of BCNs with pinning controllers. Moreover, we make a comparison among three forms of BCNs, which have similar algebraic representations. Finally, we obtain some efficient conditions to judge the dynamic structure of BCNs.
This paper investigates the pinning control for the disturbance decoupling problem (DDP) of Boolean networks (BNs) with disturbances. First, the solvability of DDP in BCNs is defined. Then, ...rank-conditions-based pinning control is proposed. Moreover, rank-conditions-based pinning state feedback controllers are designed for the DDP of BNs and the range of controllers' number is obtained. In addition, rank-conditions-based pinning output feedback controllers for the DDP of BNs are also discussed. An example is given to show the effectiveness of the obtained results.
In this paper, we propose a new approach to investigate the controllability and reachability of probabilistic Boolean control networks (PBCNs) with forbidden states. We first give a simple algebraic ...formula for the transition probability between two states in a given number of time-step, while avoiding a set of forbidden states. Then we construct the controllability matrix based on a new operator, and some necessary and sufficient conditions are obtained for the controllability and reachability of PBCNs. A numerical example is given to illustrate the efficiency of the obtained results.
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In this paper, we investigate the global stability of quaternion-valued neural networks (QVNNs) with time-varying delays. On one hand, in order to avoid the noncommutativity of quaternion ...multiplication, the QVNN is decomposed into four real-valued systems based on Hamilton rules: <inline-formula> <tex-math notation="LaTeX">ij=-ji=k,~jk=-kj=i </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">ki=-ik=j </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">i^{2}=j^{2}=k^{2}=ijk=-1 </tex-math></inline-formula>. With the Lyapunov function method, some criteria are, respectively, presented to ensure the global <inline-formula> <tex-math notation="LaTeX">\mu </tex-math></inline-formula>-stability and power stability of the delayed QVNN. On the other hand, by considering the noncommutativity of quaternion multiplication and time-varying delays, the QVNN is investigated directly by the techniques of the Lyapunov-Krasovskii functional and the linear matrix inequality (LMI) where quaternion self-conjugate matrices and quaternion positive definite matrices are used. Some new sufficient conditions in the form of quaternion-valued LMI are, respectively, established for the global <inline-formula> <tex-math notation="LaTeX">\mu </tex-math></inline-formula>-stability and exponential stability of the considered QVNN. Besides, some assumptions are presented for the two different methods, which can help to choose quaternion-valued activation functions. Finally, two numerical examples are given to show the feasibility and the effectiveness of the main results.
This note presents further results based on the recent paper J. Liang, H. Chen, and J. Lam, "An improved criterion for controllability of Boolean control networks," IEEE Trans. Autom. Control , ...vol. 62, no. 11, pp. 6012-6018, Nov. 2017. After some optimizations, the conventional method can be more efficient than the method used in the above paper. We also propose an improved method via combining the well known Tarjan's algorithm and depth-first search technique for the controllability analysis of Boolean control networks (BCNs). As a result, the computational complexity will not exceed <inline-formula><tex-math notation="LaTeX">O(N^2)</tex-math></inline-formula> with <inline-formula><tex-math notation="LaTeX">N=2^n</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">n</tex-math></inline-formula> is the number of state-variables in a BCN.
This article considers asymptotic stability and stabilization of Markovian jump Boolean networks (MJBNs) with stochastic state-dependent perturbation. By defining an augmented random variable as the ...product of the canonical form of switching signal and state variable, asymptotic stability of an MJBN with perturbation is converted into the set stability of a Markov chain (MC). Then, the concept of induced equations is proposed for an MC, and the corresponding criterion is subsequently derived for asymptotic set stability of an MC by utilizing the solutions of induced equations. This criterion can be, respectively, examined by either a linear programming algorithm or a graphical algorithm. With regards to the stabilization of MJBNs, the time complexity is reduced to a certain extent. Furthermore, all time-optimal signal-based state feedback controllers are designed to stabilize an MJBN towards a given target state. Finally, the feasibility of the obtained results is demonstrated by two illustrative biological examples.
This brief presents an analytical study of synchronization in an array of coupled deterministic Boolean networks (BNs) with time delay. Two kinds of models are considered. In one model, the outputs ...contain time delay, while in another one, the outputs do not. One restriction in this brief is that the state delay and output delay are restricted to be equal. By referring to the algebraic representations of logical dynamics and using the techniques of semitensor product of matrices, some necessary and sufficient conditions are derived for the synchronization of delay-coupled BNs. Examples including a practical epigenetic example are given for illustration.
This brief investigates the partial and complete synchronization of two Boolean control networks (BCNs). Necessary and sufficient conditions for partial and complete synchronization are established ...by the algebraic representations of logical dynamics. An algorithm is obtained to construct the feedback controller that guarantees the synchronization of master and slave BCNs. Two biological examples are provided to illustrate the effectiveness of the obtained results.
This paper proposes a quaternion-valued one-layer recurrent neural network approach to resolve constrained convex function optimization problems with quaternion variables. Leveraging the novel ...generalized Hamilton-real (GHR) calculus, the quaternion gradient-based optimization techniques are proposed to derive the optimization algorithms in the quaternion field directly rather than the methods of decomposing the optimization problems into the complex domain or the real domain. Via chain rules and Lyapunov theorem, the rigorous analysis shows that the deliberately designed quaternion-valued one-layer recurrent neural network stabilizes the system dynamics while the states reach the feasible region in finite time and converges to the optimal solution of the considered constrained convex optimization problems finally. Numerical simulations verify the theoretical results.
