The incidence energy $\text{I}\text{E}(G)$, defined as the sum of the singular values of the incidence matrix of $G$, is a much studied quantity with well known applications in chemical physics. In this paper, we derived the closed-form formulae expressing the incidence energy of the 3.12.12 lattice, triangular kagomé lattice, and $S(m,n)$ lattice, respectively. Simultaneously, the explicit asymptotic values of the incidence energy in these lattices are obtained by utilizing the applications of analysis method with the help of software calculation.

In this paper, we consider the robust $H_{\infty }$ control problem for a class of discrete time-delay stochastic systems with randomly occurring nonlinearities. The parameter uncertainties enter all the system matrices; the stochastic disturbances are both state and control dependent, and the randomly occurring nonlinearities obey the sector boundedness conditions. The purpose of the problem addressed is to design a state feedback controller such that, for all admissible uncertainties, nonlinearities, and time delays, the closed-loop system is robustly asymptotically stable in the mean square, and a prescribed $H_{\infty }$ disturbance rejection attenuation level is also guaranteed. By using the Lyapunov stability theory and stochastic analysis tools, a linear matrix inequality (LMI) approach is developed to derive sufficient conditions ensuring the existence of the desired controllers, where the conditions are dependent on the lower and upper bounds of the time-varying delays. The explicit parameterization of the desired controller gains is also given. Finally, a numerical example is exploited to show the usefulness of the results obtained.

This paper investigates the finite-time $H_{\infty }$ fault estimation problem for linear time-delay systems, where the delay appears in both state and measurement equations. Firstly, the design of finite horizon $H_{\infty }$ fault estimation is converted into a minimum problem of certain quadratic form. Then we introduce a stochastic system in Krein space, and a sufficient and necessary condition for the minimum is derived by applying innovation analysis approach and projection theory. Finally, a solution to the $H_{\infty }$ fault estimation is obtained by recursively computing a partial difference Riccati equation, which has the same dimension as the original system. Compared with the conventional augmented approach, the solving of a high dimension Riccati equation is avoided.

The packing problem of unit equilateral triangles not only has the theoretical significance but also offers broad prospects in material processing and network resource optimization. Because this problem is nondeterministic polynomial (NP) hard and has the feature of continuity, it is necessary to limit the placements of unit equilateral triangles before optimizing and obtaining approximate solution (e.g., the unit equilateral triangles are not allowed to be rotated). This paper adopts a new quasi-human strategy to study the packing problem of unit equilateral triangles. Some new concepts are put forward such as side-clinging action, and an approximation algorithm for solving the addressed problem is designed. Time complexity analysis and the calculation results indicate that the proposed method is a polynomial time algorithm, which provides the possibility to solve the packing problem of arbitrary triangles.

A partial synchronization problem in an oscillator network is considered. The concept on a principal quasi-submatrix corresponding to the topology of a cluster is proposed for the first time to study partial synchronization. It is shown that partial synchronization can be realized under the condition depending on the principal quasi-submatrix, but not distinctly depending on the intercluster couplings. Obviously, the dimension of any principal quasi-submatrix is usually far less than the one of the network topology matrix. Therefore, our criterion provides us a novel index of partial synchronizability, which reduces the network size when the network is composed of a great mount of nodes. Numerical simulations are carried out to confirm the validity of the method.

We propose a kind of evolving network which shows tree structure. The model is a combination of preferential attachment model and uniform model. We show that the proportional degree sequence ${\left\{p_k\right\}}_{k>1}$ obeys power law, exponential distribution, and other forms according to the relation of $k$ and parameter $m$.

A class of fractional-order BAM neural networks with delays in the leakage terms is considered. By using inequality technique and analysis method, several delay-dependent sufficient conditions are established to ensure the uniform stability of such networks. Moreover, the sufficient conditions guaranteeing the existence, uniqueness, and stability of the equilibrium point are also obtained. In addition, three simulation examples are given to demonstrate the effectiveness of the obtained results.

Real-time pricing DSM (demand side management) is widely used to dynamically change or shift the electricity consumption in the smart grid. In this paper, a game decision making scheme is proposed in the smart grid with DSM. The interaction between two retailers and their wholesaler is modeled as a two-stage game model. Considering the competition between two retailers, two different game models are developed in terms of the different action order between retailers and their wholesaler. Through analyzing the equilibrium revenues of the retailers for different situations we find that although the wholesaler expects to decentralize certain management powers to the retailers, it has retained the right to change the rules of the game and frequently reneged on the promises. More specifically, the law should ensure that any change of the revenue-sharing formula must go through certain legal procedures. Imposing legal restrictions on the wholesaler’s discretionary policy suggests that the time-inconsistency problem is mitigated. Numerical simulation shows the effectiveness of proposed scheme.

This paper proposes a switched approach to robust stabilization of a collection of coupled networked controlled systems (NCSs) with node devices acting over a limited communication channel. We suppose that the state information of every subsystem is split into different packets and only one packet of the subsystem can be transmitted at a time. Multiple NCSs with norm-bounded parameter uncertainties and multiple transmissions are modeled as a periodic switched system in this paper. State feedback controllers can be constructed in terms of linear matrix inequalities. A numerical example is given to show that a collection of uncertain NCSs with the problem of limited communication can be effectively stabilized via the designed controller.

The impulsive complex-valued neural networks with three kinds of time delays including leakage delay, discrete delay, and distributed delay are considered. Based on the homeomorphism mapping principle of complex domain, a sufficient condition for the existence and uniqueness of the equilibrium point of the addressed complex-valued neural networks is proposed in terms of linear matrix inequality (LMI). By constructing appropriate Lyapunov-Krasovskii functionals, and employing the free weighting matrix method, several delay-dependent criteria for checking the global $\mu$-stability of the complex-valued neural networks are established in LMIs. As direct applications of these results, several criteria on the exponential stability, power-stability, and log-stability are obtained. Two examples with simulations are provided to demonstrate the effectiveness of the proposed criteria.

The problem of $H_{\infty }$ control for network-based 2D systems with missing measurements is considered. A stochastic variable satisfying the Bernoulli random binary distribution is utilized to characterize the missing measurements. Our attention is focused on the design of a state feedback controller such that the closed-loop 2D stochastic system is mean-square asymptotic stability and has an $H_{\infty }$ disturbance attenuation performance. A sufficient condition is established by means of linear matrix inequalities (LMIs) technique, and formulas can be given for the control law design. The result is also extended to more general cases where the system matrices contain uncertain parameters. Numerical examples are also given to illustrate the effectiveness of proposed approach.