Review Special Issues

Impulsive strategies in nonlinear dynamical systems: A brief overview

  • The studies of impulsive dynamical systems have been thoroughly explored, and extensive publications have been made available. This study is mainly in the framework of continuous-time systems and aims to give an exhaustive review of several main kinds of impulsive strategies with different structures. Particularly, (i) two kinds of impulse-delay structures are discussed respectively according to the different parts where the time delay exists, and some potential effects of time delay in stability analysis are emphasized. (ii) The event-based impulsive control strategies are systematically introduced in the light of several novel event-triggered mechanisms determining the impulsive time sequences. (iii) The hybrid effects of impulses are emphatically stressed for nonlinear dynamical systems, and the constraint relationships between different impulses are revealed. (iv) The recent applications of impulses in the synchronization problem of dynamical networks are investigated. Based on the above several points, we make a detailed introduction for impulsive dynamical systems, and some significant stability results have been presented. Finally, several challenges are suggested for future works.

    Citation: Haitao Zhu, Xinrui Ji, Jianquan Lu. Impulsive strategies in nonlinear dynamical systems: A brief overview[J]. Mathematical Biosciences and Engineering, 2023, 20(2): 4274-4321. doi: 10.3934/mbe.2023200

    Related Papers:

    [1] Yasong Sun, Jiazi Zhao, Xinyu Li, Sida Li, Jing Ma, Xin Jing . Prediction of coupled radiative and conductive heat transfer in concentric cylinders with nonlinear anisotropic scattering medium by spectral collocation method. AIMS Energy, 2021, 9(3): 581-602. doi: 10.3934/energy.2021028
    [2] Hong-Wei Chen, Fu-Qiang Wang, Yang Li, Chang-Hua Lin, Xin-Lin Xia, He-Ping Tan . Numerical design of dual-scale foams to enhance radiation absorption. AIMS Energy, 2021, 9(4): 842-853. doi: 10.3934/energy.2021039
    [3] Chao-Jen Li, Peiwen Li, Kai Wang, Edgar Emir Molina . Survey of Properties of Key Single and Mixture Halide Salts for Potential Application as High Temperature Heat Transfer Fluids for Concentrated Solar Thermal Power Systems. AIMS Energy, 2014, 2(2): 133-157. doi: 10.3934/energy.2014.2.133
    [4] Hong-Yu Pan, Chuang Sun, Xue Chen . Transient thermal characteristics of infrared window coupled radiative transfer subjected to high heat flux. AIMS Energy, 2021, 9(5): 882-898. doi: 10.3934/energy.2021041
    [5] Caliot Cyril, Flamant Gilles . Pressurized Carbon Dioxide as Heat Transfer Fluid: In uence of Radiation on Turbulent Flow Characteristics in Pipe. AIMS Energy, 2014, 1(2): 172-182. doi: 10.3934/energy.2014.2.172
    [6] Matteo Moncecchi, Davide Falabretti, Marco Merlo . Regional energy planning based on distribution grid hosting capacity. AIMS Energy, 2019, 7(3): 264-284. doi: 10.3934/energy.2019.3.264
    [7] Khadim Ndiaye, Stéphane Ginestet, Martin Cyr . Thermal energy storage based on cementitious materials: A review. AIMS Energy, 2018, 6(1): 97-120. doi: 10.3934/energy.2018.1.97
    [8] Kokou Aménuvéla Toka, Yawovi Nougbléga, Komi Apélété Amou . Optimization of hybrid photovoltaic-thermal systems integrated into buildings: Impact of bi-fluid exchangers and filling gases on the thermal and electrical performances of solar cells. AIMS Energy, 2024, 12(5): 1075-1095. doi: 10.3934/energy.2024051
    [9] Muluken Biadgelegn Wollele, Abdulkadir Aman Hassen . Design and experimental investigation of solar cooker with thermal energy storage. AIMS Energy, 2019, 7(6): 957-970. doi: 10.3934/energy.2019.6.957
    [10] Gemma Graugés Graell, George Xydis . Solar Thermal in the Nordics. A Belated Boom for All or Not?. AIMS Energy, 2022, 10(1): 69-86. doi: 10.3934/energy.2022005
  • The studies of impulsive dynamical systems have been thoroughly explored, and extensive publications have been made available. This study is mainly in the framework of continuous-time systems and aims to give an exhaustive review of several main kinds of impulsive strategies with different structures. Particularly, (i) two kinds of impulse-delay structures are discussed respectively according to the different parts where the time delay exists, and some potential effects of time delay in stability analysis are emphasized. (ii) The event-based impulsive control strategies are systematically introduced in the light of several novel event-triggered mechanisms determining the impulsive time sequences. (iii) The hybrid effects of impulses are emphatically stressed for nonlinear dynamical systems, and the constraint relationships between different impulses are revealed. (iv) The recent applications of impulses in the synchronization problem of dynamical networks are investigated. Based on the above several points, we make a detailed introduction for impulsive dynamical systems, and some significant stability results have been presented. Finally, several challenges are suggested for future works.



    Abbreviations: Aij: area of wall cell (i,j), m2; e: energy carried by sample Monte Carlo bundle, W; Eb(T): spectral blackbody emissive power at temperature T, W/(m2·m); H: total exercise times; L: length of the domain, m; M, N: lattice indices in square mesh corresponding to x, y position, respectively; n: refractive index of medium; Nij: number of sample bundles emitted by cell (i,j); Nijkl: number of sample bundles emitted by cell (i,j) and absorbed by cell (k,l); Nij: number of sample bundles emitted by cell (i,j) in the BDMC method; Nijkl: number of sample bundles emitted by cell (i,j) and absorbed by cell (k,l)in the BDMC method; N0: the number density of sampling bundles for a reference case; Nr: the number of sample bundles chosen for a reference volume cell Vr; P: performance parameter for Monte Carlo simulation; RDλ,ijkl: radiative exchange factor of cell (i,j) to cell (k,l) for non-gray medium in the TMC method; ¯RDλ,ijkl: modified form of RDλ,ijkl; RDijkl: radiative exchange factor of cell (i,j) to cell (k,l) for gray medium in the TMC method; ¯RDijkl: modified form of RDijkl; RDijkl: radiative exchange factor of cell (i,j) to cell (k,l) in the BDMC method; ¯RDijkl: modified form of RDijkl; sn, ˆsn: bundles trajectories and coupled bundles trajectories; t: time, s; T: absolute temperature, K; Vij: volume of medium cell (i,j), m3; Wij: bundle weight of cell (i,j); δijkl: reciprocity error of a couple of radiative exchange factor by the TMC method; δijkl: reciprocity error of a couple of radiative exchange factor by the BDMC method; ΔP: variation of performance parameter; Δr: relative error of reciprocity; ελ: spectral emissivity; Φλ,ij: spectral radiative energy emitted from cell Vij (or Aij); Φλ,ijkl: spectral radiative energy emitted from cell Vij (or Aij), that is absorbed by cell Vkl (or Akl); Φijkl: net radiative energy exchange, W; γ: standard deviation; κaλ: spectral absorption coefficient of medium; κe: extinction coefficient, m-1; σ: Stefan-Boltzmann constant, (σ=5.67×108W/m2K4); τ: optical thickness (τ=κeL); ω: scattering albedo of medium. Subscripts, a: medium absorption; h: hth sampling; i,j: order number of cell; r: reference cell; w: wall

    Radiative heat transfer in participating medium is described by radiative transfer equation (RTE), which is an integro-differential equation. Many methods have been developed to solve the RTE, the common used numerical methods such as the discrete ordinate method (DOM) and the finite volume method (FVM), rely on both spatial discretization and angular discretization, which are difficult to solve the RTE with high accuracy. Monte Carlo (MC) method is a stochastic statistical method based on the physical processes, it has been applied to solve radiative heat transfer in various participating media [1,2,3,4,5,6]. Moreover, the results predicted by MC method can often be treated as benchmark solutions due to its high accuracy of solution [7,8].

    However, for the statistical nature of the MC method, the high computational cost is still a considerable disadvantage of MC simulation. Many attempts have been made in order to improve the computational efficiency of the method [9]. For example, several sampling approaches were developed to improve the speed of convergence, such as the importance sampling method [10], the rejection sampling method [11], the differential sampling method [12] and the weight-equivalent sampling method [13]. In addition, parallel computing technique was introduced into the MC simulation to improve computation efficiency evidently [1,9]. Howell [9] analyzed the advantages of various programming strategies of the MC method for radiative heat transfer in absorbing and scattering medium. All of these improvements were mainly mathematical efforts and hardware improvement of computer, which are universal to the MC simulations.

    It has been noticed that the physical feature of thermal radiation transfer can also provide possibility to improve the MC simulation of radiative heat transfer in participating medium. For example, Walters [14] developed a reverse method based on the reciprocity principle for radiative heat transfer in a generalized enclosure containing an absorbing, emitting and scattering medium, the reverse method was proved to be efficient. Cherkaoui et al. [15,16] developed a net exchange Monte Carlo method based on a net-exchange formulation, provided an efficient way of systematically fulfilling the reciprocity principle, the computing time was proved much smaller than the conventional Monte Carlo approach. Lataillade [17] and his cooperators applied the net exchange Monte Carlo approach for radiative heat transfer in optically thick medium with spectral dependent radiative properties. Eymet et al. [18] extended this method to absorbing, emitting, and scattering media. Tessé et al. [19] improved the forward Monte Carlo (FMC) method based on the reciprocity principle, the method was used for radiative transfer in real gases, and proved to be a better choice for optically thick or nearly isothermal media compared with the forward Monte Carlo method. In addition, the reverse Monte Carlo (RMC) methods, based on the reversibility of radiative transfer trajectory, has been developed to solve the radiative transfer in absorbing and scattering media [20,21,22,23,24]. Kovtanyuk et al. [25] presented a recursive algorithm based on modification of Monte Carlo method, the modified method was used to solve the coupled radiative and conductive heat transfer in an absorbing and scattering medium, and was proved to be more accurate. Soucasse et al. [26] proposed a Monte Carlo formulation for radiative transfer in quasi-isothermal media which consists in directly computing the difference between the actual radiative field and the equilibrium radiative field at the minimum temperature in the medium.

    In the present study, a bidirectional Monte Carlo (BDMC) method based on reversibility of bundle trajectory and reciprocity of radiative energy exchange was developed to solve thermal radiation transfer in absorbing and scattering medium. Two types of sampling models for MC simulation were presented, namely the equivalent sampling and the weight sampling. The equivalent sampling was chosen for the uniform mesh while the weight sampling was more suitable to the non-uniform mesh. The bidirectional information of tracing a sampling bundle was utilized by the BDMC method, the solution precision or efficiency can be evidently improved. Radiative heat transfer in a two-dimensional rectangular domain with absorbing and scattering media was solved by the BDMC method and the TMC method, respectively. The radiative exchange factors and the temperature profiles were investigated, in addition, the performance parameter defined by Howell [9] was also calculated to evaluate the two MC methods.

    Radiative heat transfer in a two-dimensional (2-D) rectangular domain with absorbing, emitting and/or scattering medium was investigated. Figure 1 shows the 2-D rectangular geometry as well as the coordinate system. The four walls were assumed to be diffuse and gray. Radiative properties, such as the absorption coefficient, the scattering coefficient were assumed to be constant. The rectangular domain was divided into M × N = MN cells, any of which was depicted by Vij (medium cell) or Aij (wall cell), wherein the subscripts I∈[1, M] and j∈[1, N]. Monte Carlo method has robust adaptability and can be extended to more complex cases.

    Figure 1.  Schematic diagram of radiative transfer in a rectangular domain with participating medium in the BDMC method.

    In the present study, the radiative exchange factor RDλ,ijkl was introduced to decouple the solution of radiative transfer from that of the temperature profile. It was defined as a fraction of the spectral radiative energy emitted from cell Vij (or Aij), that is absorbed by cell Vkl (or Akl) [27,28].

    RDλ,ijkl=Φλ,ijkl/Φλ,ij (1)

    As Eq 1, where Φλ,ij is the spectral radiative energy emitted from cell Vij (or Aij), Φλ,ijkl is the spectral radiative energy emitted from cell Vij (or Aij), and absorbed by cell Vkl (or Akl), taking into account possible wall reflections. It is obvious that the conservation relation M,Nk=1,l=1RDλ,ijkl=1 is tenable according to the definition of radiative exchange factors. Note that the radiative exchange factor depends only on the system geometry and the radiative properties distribution of the medium [29]. The reciprocity relation between a couple of radiative exchange factors was given by [30].

    (κaλ)ijVijRDλ,ijkl=(κaλ)klVklRDλ,klij (2)

    or

    4(κaλ)ijVijRDλ,ijkl=(ελ)klAklRDλ,klij (3)

    As Eqs 2 and 3, where, κaλ is the spectral absorption coefficient of the medium, while ελ is the spectral emissivity of the boundary wall. For convenience, the radiative exchange factors were usually modified as

    ¯RDλ,ijkl=4(κaλ)ijVijRDλ,ijkl (4)

    or

    ¯RDλ,ijkl=(ελ)ijAijRDλ,ijkl (5)

    here, Eqs 4 and 5 are used for volume and wall cells, respectively. The reciprocity relation can be written as

    ¯RDλ,ijkl=¯RDλ,klij (6)

    As Eq 6, after solving the radiative exchange factors, the net radiative energy exchange Φijkl from Vij to Vkl can be calculated from

    Φijkl=0¯RDλ,ijkl[Ebλ(Tij)Ebλ(Tkl)]dλ (7)

    As Eq 7, where, is the Stefan-Boltzmann constant ( = 5.67×10-8 W/m2K4), E(T) is the spectral blackbody emissive power at temperature T.

    In the MC simulation, Φλ,ij is represented by a large number of independent sampling bundles, the propagating process of each sampling bundle can be tracked and counted. For example, the number of total sampling bundles for Φλ,ij is Nij, each bundle with the same energy of eij=Φλ,ij/Nij. If Nijkl bundles among them are finally absorbed by Vkl, then, Φλ,ijkl=eijNijkl, the radiative exchange factor can be calculated from [31].

    RDλ,ijkl=Nijkl/Nij (8)

    As Eq 8, the number Nijkl can be counted from the MC simulation, however, only an approximate value of Nijkl can be obtained because of the pseudo randomicity [32] in the sampling process. In fact, the reciprocity correlation shown in Eq 6 cannot be strictly satisfied, and random errors always exist. In order to obtain more accurate results, one need to increase the sampling bundles Nij, which results in the increasing computing cost.

    In the TMC method, the propagation trajectories of Nij sampling bundles for cell Vij are tracked and counted to get the value of Nijkl for any cell Vkl. Similarly, the number Nklij is obtained after tracing another Nkl propagation trajectories of the sampling bundles for cell Vkl. Wherein, only the forward propagation information of a trajectory is used, the number of sampling bundles for cell Vij is Nij.

    According to the reversibility of light propagation, for a tracing trajectory from Vkl to Vij, a bundle from Vij can also propagate to Vkl along the reverse direction of the tracing trajectory. The information of the reverse direction can be used in the MC simulation, the bidirectional information can be obtained from the forward tracing, which results in the development of the present BDMC method.

    In order to simplify the description of the BDMC method, the formula in the following sections were given for gray medium with gray walls. The basic idea of BDMC is to calculate the radiative exchange factor by the reversibility of light beams. According to the reversibility of light beams, the transmission path of the bundles received by cells can also be used as the transmission path of the bundles transmitted by cells. In this way, the effective information of sampling bundles if fully utilized without increasing the number of sampling bundles and calculation amount. Theoretically, the statistical sampling bundles in calculation can be doubled. And it can better satisfy the relationship of the reciprocal of radiative exchange factor. Figure 1 shows the schematic diagram of radiative transfer in a rectangular domain with participating medium in the BDMC method, it can be noticed that there are Nij tracing trajectories departing from cell Vij, and M,Nk=1,l=1Nklij tracing trajectories getting into cell Vij at the same time. According to the reversibility of bundle trajectories, there must be M,Nk=1,l=1Nklij tracing trajectories depart from cell Vij in the reverse directions. It is believed that the M,Nk=1,l=1Nklij trajectories are independent of the emitted Nij trajectories because of the randomness of the sampling process. Thus, the number of total sampling bundles for Vij can be counted anew as

    Nij=Nij+M,Nk=1,l=1Nklij (9)

    while the number of bundles emitted by Vij and absorbed by Vkl is counted anew as

    Nijkl=Nijkl+Nklij (10)

    then, the radiative exchange factor RDijkl in the BDMC method can be calculated by

    RDijkl=NijklNij=Nijkl+NklijNij+M,Nk=1,l=1Nklij (11)

    where, RDijkl is used to denote the radiative exchange factor in the BDMC method. The information of the bidirectional trajectories has been used, as is shown in Eq 11, it indicates that the effective sampling bundles are increased, while the computing cost remains unchanged. Therefore, more accurate results can be predicted from Eq 11 than that from Eq 8 under the same computing cost.

    It should be pointed out that Eqs 9 and 11 are valid only if the sampling bundles for cell Vij and any other cells are equivalent. According to the reversibility of light beams, the transmission path of the bundles received by cells can also be used as the transmission path of the bundles transmitted by cells. In this way, the effective information of sampling bundles if fully utilized without increasing the number of sampling bundles and calculation amount. In other words, the forward and reverse bundles in the BDMC method should have the same contribution for radiative exchange factors calculation. To use Eq 11, the equivalent sampling was introduced, in which the number of sampling bundles for cell (i, j) was determined by

    Nij=4(κa)ijVijN0orNij=εijAijN0 (12)

    where N0 is the sampling density for a reference case defined as

    N0=Nr4(κa)rVr (13)

    where, Nr is the number of sample bundles of the reference cell Vr, the subscript r refers to the reference cell.

    The bidirectional counting of sampling bundles employed in the BDMC method is equivalent to double the sampling number if the random error does not exist. In fact, any of the Monte Carlo methods always accompanied by random error. For the BDMC method, that leads to Nij2Nij and the energy of a bundle is not strictly equal to other cells. For any cell, it can be easily demonstrated by Eq 11, and the radiative exchange factors calculated by the BDMC method satisfied the conservation relation strictly. But the reciprocity relation between a couple of radiative exchange factors cannot be satisfied strictly. According to Eq 11, the following equation can be obtained

    RDijkl(Nij+M,Nk=1,l=1Nklij)=RDklij(Nkl+M,Ni=1,j=1Nijkl) (14)

    According to Eq 8, for gray medium, Nijkl=NijRDijkl, Nklij=NklRDklij. Therefore,

    RDijkl(Nij+M,Nk=1,l=1NklRDklij)=RDklij(Nkl+M,Ni=1,j=1NijRDijkl) (15)

    replace Nij and Nkl according to Eq 12, in addition, considering Eqs 4 and 5, then Eq 15 can be transformed into

    ¯RDijkl+RDijklM,Nk=1,l=1¯RDklij=¯RDklij+RDklijM,Ni=1,j=1¯RDijkl (16)

    where, ¯RDijkl and ¯RDklij are the modified forms for RDijkl and RDklij, respectively.

    Supposing the reciprocity errors for a couple of radiative exchange factors ¯RDijkl and ¯RDklij predicted in the TMC method are δijkl and δklij, respectively, one can write

    ¯RDklij=¯RDijkl+δijkland¯RDijkl=¯RDklij+δklij (17)

    where, it is obvious that δklij=δijkl.

    For radiative exchange factors predicted in the BDMC method, similar equations can be written as follows

    ¯RDklij=¯RDijkl+δijkland¯RDijkl=¯RDklij+δklij (18)

    where, δijkl and δklij are the reciprocity errors for a couple of radiative exchange factors ¯RDijkl and ¯RDklij in the BDMC method, and δklij=δijkl.Combine Eqs 16–18, then

    δklij=RDklijM,Ni=1,j=1(¯RDklij+δklij)RDijklM,Nk=1,l=1(¯RDijkl+δijkl) (19)

    Considering the conservation relation M,Ni=1,j=1RDλ,klij=1, M,Nk=1,l=1RDλ,ijkl=1, δklij=δijkl, δklij=δijkl and Eq 4, then, M,Ni=1,j=1¯RDklij=4(κa)klVkl, and M,Nk=1,l=1¯RDijkl=4(κa)ijVij, moreover, considering Eq 18, the following error relation can be derived from Eq 19

    δijkl=12(RDijklM,Nk=1,l=1δijkl+RDklijM,Ni=1,j=1δijkl) (20)

    therefore,

    |δijkl|12(RDijkl|M,Nk=1,l=1δijkl|+RDklij|M,Ni=1,j=1δijkl|) (21)

    Because of the randomicity of the reciprocity error δijkl, the value of |M,Nk=1,l=1δijkl| would be very close to 0 if M and N tend to infinity. The formulae |M,Nk=1,l=1δijkl|=O(|δijkl|) and |M,Nk=1,l=1δijkl|=O(|δijkl|) would be valid for most of the discrete elements as M×N was large in the present study, where O(|δijkl|) is less than |δijkl|. Note that the values of RDijkl and RDklij do not greater than unity, therefore

    |δijkl|O(|δijkl|) (22)

    The reciprocity error for a couple of radiative exchange factors predicted by the BDMC method is always smaller than those predicted by the TMC method.

    For some cases, the equivalent sampling may be inconvenient or inaccurate. For example, if the differences of apparent radiation characteristics and/or cell volume are very large among different cells, the equivalent sampling will result in a very small sampling number for a cell, and may be a very large sampling number for another. The former is not expected for the statistical analysis, while the latter increase computational cost.

    The weight sampling was introduced into the BDMC method to avoid the above shortages of the equivalent sampling. For weight sampling, the number of sampling bundles for a cell was determined only based on the statistical request, but the weight of a sampling bundle was taken into account in the final statistical calculation. If the number of sampling bundles for cell Vij was Nij, then, its bundle weight Wij was defined as

    Wij=(κa)ijVijNr(κa)rVrNij,orWij=εijAijNr4(κa)rVrNij (23)

    where Nr is the number of sample bundles for reference cell Vr. Then, the radiative exchange factor RDijkl in the BDMC method should be calculated by

    RDijkl=Nijkl+NklijWkl/WijNij+M,Nk=1,l=1(NklijWkl/Wij) (24)

    Eq 11 should be substituted by Eq 24 if the weight sampling is employed.

    Radiative transfer in a rectangular domain with absorbing, emitting and isotropic scattering gray medium was solved separately by the BDMC method and the TMC method. The radiative exchange factors, the radiative equilibrium temperature profiles, and the performance parameter defined by Farmer and Howell [1,9], predicted by the BDMC method were compared with those predicted by the TMC method. In addition, the performance of the equivalent sampling and the weight sampling in the MC simulation were also examined.

    Table 1.  Discrete parameters of the uniform and non-uniform mesh systems.
    (i[2,M1], j[2,N1])
    mesh number mesh size
    uniform mesh M×N=21×21 Δxi,j=Δyi,j=const
    non-uniform mesh M×N=21×21 Δxi,j=ΔxcK|iM+12|, Δyi,j=ΔycK|jN+12|

     | Show Table
    DownLoad: CSV

    The radiative properties of the medium considered were supposed to be uniform and depicted by constant extinction coefficient κe, scattering albedo ω, and refractive index n = 1.0. The length and width of the rectangular domain were Lx=Ly=L. Optical thickness was defined as τ=κeL. The walls were assumed to be diffuse and gray, with constant emissivity of ε. The surfaces were imposed to constant temperature of TWE, TWW, TWN, and TWS, respectively, see Figure 1. The dimensionless coordinates were defined as X=x/Lx, and Y=y/Ly. The simulations were conducted using both uniform and non-uniform grids. The discrete parameters of the two types of mesh systems were listed in Table 1, where, xc = yc were the width and length of the largest volume cell in the non-uniform mesh system, and xc = yc, the parameter K = 1.04, M = N = 21, the mesh sizes in the non-uniform mesh system were expressed as Δxij=ΔxcK|i11|, and Δyij=ΔycK|j11|. For medium cells, i.e., i[2,M1], and j[2,N1], the maximum width of the cells was Δxi,j=Δxc for i = 11, and the minimum width of the cells was Δxi,j=ΔxcK9 for i = 2 or 20. Similarly, the maximum length of the cells was Δyi,j=Δyc for j = 11, while the minimum length of the cells was Δyi,j=ΔycK9 for j = 2 or 20. The maximum/minimum ratio of the mesh size is (ΔxcΔyc)/(ΔxcK9ΔycK9) = 2.026.

    The sampling bundles needed in the MC simulation should be estimated as the results were calculated based on statistical computation. Figures 2 and 3 show the radiative equilibrium temperature profiles at dimensionless locations of Y = 0.3 and Y = 0.5 predicted by the TMC method with different sampling bundles of Nb = 104, 105, and 106, respectively. Figure 2 shows the results for absorbing medium, while the results shown in Figure 3 were for absorbing and isotropic scattering medium with scattering albedo of = 0.5. For both the two cases, the wall temperatures were imposed separately as TWE = TWW = 1000 K and TWN = TWS = 1500 K, the emissivity of the walls were assumed to be constant and equal to 0.5, the optical thickness = keL = 15. For both the absorbing and absorbing-scattering medium, the predicted temperature profiles were found to be close to each other for different sampling bundles of Nb = 104, 105, and 106, moreover, the temperature difference for Nb = 105 and Nb = 106 were less than 0.5%, therefore, the results were considered to reach convergence for sampling bundles Nb = 105 in the present study, and the sampling bundles of Nb = 105 was also chosen as the reference sampling bundles Nr in the BDMC method.

    Figure 2.  Radiative equilibrium temperature profiles predicted by the TMC method in absorbing, emitting, non-scattering medium at locations Y = 0.3 and 0.5.
    Figure 3.  Radiative equilibrium temperature profiles predicted by the TMC method in absorbing, emitting and isotropic scattering medium (scattering albedo = 0.5) at Y = 0.3 and 0.5.

    In this section, the radiative exchange factors predicted by the BDMC method were compared with those predicted by the TMC method, the numerical simulations were implemented using both uniform and non-uniform grids, meanwhile, the equivalent sampling and the weight sampling were also employed separately. The predicted radiative exchange factors always satisfy the conservation relation strictly, therefore, only the reciprocity for radiative exchange factors was examined in the present study. The relative error of the reciprocity for a couple of radiative exchange factors was defined as

    Δr=2|¯RDijkl¯RDklij|¯RDijkl+¯RDklij×100% (25)

    where, ¯RDijkl refers not only to the modified form of the radiative exchange factors given by Eq 8 for the TMC method, but also to those given by Eq 11 (BDMC method, equivalent sampling) and by Eq 24 (BDMC method, weight sampling).

    Table 2 shows the modified radiative exchange factors predicted by the BDMC method and the TMC method employing the equivalent sampling and the uniform grids as well as the corresponding relative error Dr defined by Eq 25. The emissivity of the walls were taken as = 0.5, the optical thicknesses of the rectangle medium along the length and width direction were τ=1.0, the reference sampling bundles was taken as Nr=105. It indicates that the reciprocity error for a couple of radiative exchange factors predicted by the BDMC method was at least an order of magnitude less than those predicted by the TMC method with the same sampling bundles. The maximum relative error for all the elements considered did not exceed 0.58% in the BDMC method, while it reached 30.1% in the TMC method.

    Table 2.  Comparison of radiative exchange factors predicted by the BDMC method and the TMC method using the equivalent sampling and the uniform grids.
    (ε=0.5, τ=1.0, ω=0.0, Nr=105, Ni,j=7.22×107)
    Cell 1 Cell 2 TMC method BDMC method
    (i, j) (k, l ) ¯RDi,jk,l ¯RDk,li,j Δr(%) ¯RDi,jk,l ¯RDk,li,j Δr(%)
    (11, 11) (1, 11) 1.115 × 10-4 1.065 × 10-4 4.59 1.091 × 10-4 1.091 × 10-4 0.00
    (11, 11) (11, 1) 1.044 × 10-4 1.098 × 10-4 5.04 1.071 × 10-4 1.066 × 10-4 0.47
    (11, 11) (11, 21) 1.014 × 10-4 1.065 × 10-4 4.90 1.040 × 10-4 1.041 × 10-4 0.10
    (11, 11) (21, 11) 1.021 × 10-4 1.098 × 10-4 7.27 1.060 × 10-4 1.061 × 10-4 0.09
    (11, 11) (2, 2) 1.410 × 10-5 1.747 × 10-5 21.3 1.579 × 10-5 1.586 × 10-5 0.44
    (11, 11) (2, 20) 1.936 × 10-5 1.684 × 10-5 13.9 1.811 × 10-5 1.813 × 10-5 0.11
    (11, 11) (20, 2) 1.305 × 10-5 1.768 × 10-5 30.1 1.537 × 10-5 1.546 × 10-5 0.58
    (11, 11) (20, 20) 1.452 × 10-5 1.873 × 10-5 25.3 1.663 × 10-5 1.670 × 10-5 0.42

     | Show Table
    DownLoad: CSV

    Table 3 shows the radiative exchange factors predicted by the BDMC method using the equivalent sampling and the weight sampling in a non-uniform mesh system. The computing parameters were ε=0.5, τ=1.0, ω=0.0, and Nr=105. The reciprocity error for a couple of radiative exchange factors predicted by the BDMC method with the equivalent sampling was unacceptable for the non-uniform mesh, it indicates that the equivalent sampling was not recommended to solve radiative transfer in the BDMC method with non-uniform grids. If there is obvious difference of heat capacity between cells, the equivalent sampling will lead to big difference of sampling number for the same cells. The calculation accuracy will come under influence if the equivalent sampling is applied. However, the BDMC method with the weight sampling can predict quite satisfactory results for all the cells considered.

    Table 3.  Comparison of radiative exchange factors predicted by the BDMC method with the equivalent sampling and the weight sampling using the non-uniform grids.
    (ε=0.5, τ=1.0, ω=0.0, Nr=105, Ni,j=8.12×107)
    Cell 1 Cell 2 BDMC (equivalent sampling) BDMC (weight sampling)
    (i, j) (k, l ) ¯RDi,jk,l ¯RDk,li,j Δr(%) ¯RDi,jk,l ¯RDk,li,j Δr(%)
    (11, 11) (1, 11) 6.730 × 10-4 5.000 × 10-4 29.5 9.413 × 10-4 9.414 × 10-4 0.01
    (11, 11) (11, 1) 6.697 × 10-4 4.950 × 10-4 30.0 9.364 × 10-4 9.364 × 10-4 0.00
    (11, 11) (11, 21) 6.541 × 10-4 4.855 × 10-4 29.6 9.257 × 10-4 9.239 × 10-4 0.19
    (11, 11) (21, 11) 6.654 × 10-4 4.940 × 10-4 29.6 9.335 × 10-4 9.332 × 10-4 0.03
    (11, 11) (2, 2) 6.993 × 10-5 2.837 × 10-5 84.6 4.374 × 10-5 4.384 × 10-5 0.23
    (11, 11) (2, 20) 7.047 × 10-5 2.855 × 10-5 84.7 5.052 × 10-5 5.049 × 10-5 0.06
    (11, 11) (20, 2) 6.849 × 10-5 2.783 × 10-5 84.4 4.442 × 10-5 4.452 × 10-5 0.22
    (11, 11) (20, 20) 7.317 × 10-5 2.971 × 10-5 84.5 4.378 × 10-5 4.386 × 10-5 0.18

     | Show Table
    DownLoad: CSV

    Table 4 shows the radiative exchange factors predicted by the BDMC method and the TMC method employing the weight sampling and the non-uniform grids. It can be seen that the reciprocity error in the TMC method was larger than those in the BDMC method for all the cells considered. The BDMC method showed greatly superior to the TMC method when the weight sampling and the non-uniform grids was used.

    Table 4.  Comparison of radiative exchange factors by the BDMC method and the TMC method using the weight sampling and the non-uniform grids.
    (ε=0.5, τ=1.0, ω=0.0, Nr=105, Ni,j=8.12×107)
    Cell 1 Cell 2 TMC method BDMC method
    (i, j) (k, l) ¯RDi,jk,l ¯RDk,li,j (i, j) (k, l) ¯RDi,jk,l ¯RDk,li,j
    (11, 11) (1, 11) 9.492 × 10-4 9.344 × 10-4 (11, 11) (1, 11) 9.492 × 10-4 9.344 × 10-4
    (11, 11) (11, 1) 9.485 × 10-4 9.252 × 10-4 (11, 11) (11, 1) 9.485 × 10-4 9.252 × 10-4
    (11, 11) (11, 21) 9.180 × 10-4 9.344 × 10-4 (11, 11) (11, 21) 9.180 × 10-4 9.344 × 10-4
    (11, 11) (21, 11) 9.428 × 10-4 9.252 × 10-4 (11, 11) (21, 11) 9.428 × 10-4 9.252 × 10-4
    (11, 11) (2, 2) 4.156 × 10-5 4.596 × 10-5 (11, 11) (2, 2) 4.156 × 10-5 4.596 × 10-5
    (11, 11) (2, 20) 5.664 × 10-5 4.446 × 10-5 (11, 11) (2, 20) 5.664 × 10-5 4.446 × 10-5
    (11, 11) (20, 2) 4.223 × 10-5 4.667 × 10-5 (11, 11) (20, 2) 4.223 × 10-5 4.667 × 10-5
    (11, 11) (20, 20) 3.988 × 10-5 4.772 × 10-5 (11, 11) (20, 20) 3.988 × 10-5 4.772 × 10-5

     | Show Table
    DownLoad: CSV

    The advantage of the BDMC method was further verified by comparing the predicted temperature profiles. For gray medium, the radiative equilibrium temperature Tij of cell (i, j) satisfies the energy equation

    4(κa)ijVijT4ij=M,Nk=1,l=1¯RDklijT4kl (26)

    or

    εijAijT4ij=M,Nk=1,l=1¯RDklijT4kl (27)

    Utilizing the predictions of radiative exchange factors, the radiative equilibrium temperature profile can be solved iteratively from Eq 26 or Eq 27. In the iteration, the iteration is stopped by setting the residuals. The number of iterations is not certain, which is related to the residuals.

    Figure 4 shows the radiative equilibrium temperature profiles for an absorbing, emitting, and non-scattering medium predicted by the TMC method and the BDMC method employing the equivalent sampling and the uniform grids. The wall temperatures were imposed separately as TWE = TWW =1000 K and TWN = TWS =1500 K, the emissivity of the walls were assumed to be constant and equal to 0.5, the optical thickness = keLx = keLy = 15. First, the BDMC predictions were very close to the TMC predictions, it indicates that the development of the present BDMC for predicting temperature profiles is correct, in addition, the BDMC predictions were shown smoother than those predicted by the TMC method, the BDMC method converged faster than the TMC method. Similar superior of the BDMC method to the TMC method can be seen from Figure 5, where the weight sampling and the non-uniform grids were employed and the medium was absorbing and isotropic scattering with scattering albedo of = 0.5.

    Figure 4.  Radiative equilibrium temperature profiles predicted by the TMC method and the BDMC method with equivalent sampling using the uniform grid. (ε=0.5, τ=1.0, ω=0, Nr=105, Ni,j=7.22×107).
    Figure 5.  Radiative equilibrium temperature profiles predicted by the TMC method and the BDMC method with weight sampling using the non-uniform grid. (ε=0.5, τ=1.0, ω=0, Nr=105, Ni,j=8.12×107).

    Figure 6 shows the comparison of temperature profiles predicted by the BDMC method with non-uniform grids employing different sampling models of the equivalent sampling and the weight sampling. The temperature profiles predicted by using the weight sampling was smooth, while the predictions using equivalent sampling model contains some small rectangles, this may due to the fact that the number of the sampling bundles for different cells employing equivalent sampling were significantly different, which lead to large random error and affected the temperature profiles.

    Figure 6.  Radiative equilibrium temperature profiles predicted by the BDMC method with two different sampling models using the non-uniform grid. (ε=0.5, τ=1.0, ω=0.5, Nr=105, Ni,j=8.12×107).

    Figure 7 shows the radiative equilibrium temperature profiles predicted by the TMC method and the BDMC method with a relative small number of weight sampling bundles using the non-uniform grids. The temperature profiles predicted by the TMC method became unacceptable as the reference sampling bundles decreased to Nr = 103, while the BDMC method can still predict better results, the BDMC method converged faster than the TMC method. Therefore, the BDMC method would be a better choice if the sampling bundles for radiation computation is limited or efficient computation is required.

    Figure 7.  Radiative equilibrium temperature profiles predicted by the TMC method and the BDMC method with a smaller amount of weight sampling using the non-uniform grid. (ε=0.5, τ=1.0, ω=0, Nr=103, Ni,j=5.86×105).

    Farmer and Howell introduced the performance parameter P to evaluate various MC methods or strategies [1,9], the performance parameter was defined as

    P=γ2t (28)

    where, t is the CPU time spent by the concerned MC simulation and γ2 is the variance of the results. A good method or strategy for the MC simulation tends to minimize the performance parameter P. The variance γ2 is given by

    γ2=1HHh=1γ2h (29)

    where, γ2h is the variance of the hth exercise, h[1,H], and H is the total exercise times, γ2h can be calculated from

    γ2h=1M2N2M,Ni=1,j=1M,Nk=1,l=1[(¯RDijkl)h(¯RDijkl)a]2 (30)

    where, (¯RDijkl)h is the predicted value of ¯RDijkl in the hth exercise, and (¯RDijkl)a is the "exact value" of ¯RDijkl.

    It has been discussed that the acceptable numerical results can be obtained if the sampling bundles Nb in the MC simulation reached 105. In order to compute the performance parameters for the TMC method and the BDMC method, the "exact solution" of the radiative exchange factors should be firstly obtained. The approximate "exact solution" can be obtained by increasing the sampling bundles of the MC simulation due to the high precision of the MC method. To do so, the reference sampling bundles were taken as Nr = 106 and Nr = 108, respectively, the predicted results for the two cases were found to be nearly unchanged, and the maximum relative error for any of the radiative exchange factors correspond to different cells was less than 0.1% as the reference sampling bundles Nr increased from 106 to 108. Therefore, the results employing Nr = 108 were treated as the "exact solution" in the present study.

    For each exercise, only the initial random number and sampling order were changed, therefore, the CPU time spent by each exercise should be nearly unchanged. The total exercise times was set as H = 10. The performance parameter of the BDMC method and the TMC method were written as PBDM and PTM, respectively, while the performance increment ΔP was defined as

    ΔP=PTMPBDMPTM×100% (31)

    Table 5 reports the performance parameter for the TMC method and the BDMC method using the equivalent sampling and the uniform grids. It indicates that the performance parameter for the TMC method was always larger than that for the BDMC method, in addition, the performance increment decrease with the increasing sampling bundles. Table 6 shows the performance parameter for the TMC method and the BDMC method employing the weight sampling and the non-uniform grids. Similar conclusions can be drawn as those employing the equivalent sampling and the uniform grids.

    Table 5.  Performance comparison of the TMC method and the BDMC method using the equivalent sampling and the uniform grids.
    (ε=0.5, ω=0.5, Nr=103)
    PBDM PTM ΔP
    τ=1.0
    Ni,j=7.22×105
    8.24 × 10-10 1.38 × 10-9 40.2 %
    τ=0.01
    Ni,j=3.64×107
    6.13 × 10-10 7.91 × 10-10 22.4%

     | Show Table
    DownLoad: CSV
    Table 6.  Performance comparison of the TMC method and the BDMC method using the weight sampling and the non-uniform grids.
    (ε=0.5, ω=0.5, Nr=103)
    PBDM PTM ΔP
    τ=1.0
    Ni,j=8.12×105
    9.98 × 10-10 1.57 × 10-9 36.56 %
    τ=0.01
    Ni,j=4.54×107
    1.78 × 10-10 2.11 × 10-10 15.68%

     | Show Table
    DownLoad: CSV

    Table 7 shows the performance parameter for the BDMC method with equivalent sampling and the weight sampling using the non-uniform grids. The performance parameter for BDMC with the equivalent sampling was found to be larger than that with the weight sampling, this indicate that the weight sampling was more suitable for non-uniform grids in the BDMC simulations.

    Table 7.  Performance comparison of the equivalent sampling and the weight sampling in the BDMC method using the non-uniform grids.
    (ε=0.5, ω=0.5, Nr=103)
    weight sampling equivalent sampling ΔP
    τ=1.0
    Ni,j=8.12×105
    9.98 × 10-10 2.55 × 10-9 60.84%
    τ=0.01
    Ni,j=4.54×107
    1.78 × 10-10 6.91 × 10-10 74.24%

     | Show Table
    DownLoad: CSV

    The BDMC method along with the equivalent sampling and the weight sampling were developed to solve radiative transfer in absorbing and scattering medium. The BDMC method approximately doubles the information from the bundle tracing to possess a high efficiency by making the best use of the reversibility of the bundle trajectory. The formula for the BDMC method and the corresponding error analysis were derived and presented. Radiative heat transfer in a two-dimensional rectangular domain of absorbing and/or scattering medium were solved by the TMC method and the BDMC method, respectively. The reciprocity of the radiative exchange factors, the radiative equilibrium temperature profiles, and the performance parameter predicted by the two MC methods were examined and compared. The results showed that (1) the BDMC method can greatly improve the reciprocity satisfaction of radiative exchange factors, which is helpful for temperature profile solution, (2) the BDMC method was more accurate or efficient than the TMC method, (3) in the BDMC simulations, the weight sampling was found to be more flexible than the equivalent sampling.

    The authors acknowledge the financial support by National Science Foundation of China (No. 51776052) and Aeronautical Science Foundation of China (No.201927077002).

    The authors declare no conflict of interest.

    Xiaofeng Zhang: Software, Formal analysis Data Curation, Writing - Original Draft, Visualization. Qing Ai: Conceptualization, Methodology, Validation, Resources, Writing - Review & Editing, Supervision, Funding acquisition. Kuilong Song: Validation, Formal analysis, Writing - Review & Editing, Heping Tan: Writing - Review & Editing, Supervision, Project administration.



    [1] W. Haddad, V. Chellaboina, S. Nersesov, Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control, Princeton University Press, 2004. https://doi.org/10.1515/9781400865246
    [2] A. M. Samoilenko, N. A. Perestyuk, Impulsive Differential Equations, World Scientific, 1995. https://doi.org/10.1142/2892
    [3] I. Stamova, Stability Analysis of Impulsive Functional Differential Equations, De Gruyter, 2009. https://doi.org/10.1515/9783110221824
    [4] T. Yang, Impulsive Control Theory, Springer Science & Business Media, 2001.
    [5] V. Lakshmikantham, D. Bainov, P. Simeonov, Theory of Impulsive Differential Equations, World Scientific, 1989. https://doi.org/10.1142/0906
    [6] S. Dashkovskiy, P. Feketa, Input-to-state stability of impulsive systems and their networks, Nonlinear Anal. Hybrid Syst., 26 (2017), 190–200. https://doi.org/10.1016/j.nahs.2017.06.004 doi: 10.1016/j.nahs.2017.06.004
    [7] A. Ignatyev, A. Soliman, Asymptotic stability and instability of the solutions of systems with impulse action, Math. Notes, 80 (2006), 491–499. https://doi.org/10.1007/s11006-006-0167-7 doi: 10.1007/s11006-006-0167-7
    [8] A. N. Michel, L. Hou, D. Liu, Stability of Dynamical Systems, Birkhäuser Cham, 2008. https://doi.org/10.1007/978-3-319-15275-2
    [9] R. Goebel, R. G. Sanfelice, A. R. Teel, Hybrid Dynamical Systems: Modeling Stability, and Robustness, Princeton University Press, 2012. https://doi.org/10.23943/princeton/9780691153896.001.0001
    [10] H. Zhu, X. Li, S. Song, Input-to-state stability of nonlinear impulsive systems subjects to actuator saturation and external disturbance, IEEE Trans. Cyber., 2021 (2021). https://doi.org/10.1109/TCYB.2021.3090803
    [11] J. Lu, D. W. C. Ho, J. Cao, J. Kurths, Exponential synchronization of linearly coupled neural networks with impulsive disturbances, IEEE Trans. Neural Networks, 22 (2011), 329–336. https://doi.org/10.1109/TNN.2010.2101081 doi: 10.1109/TNN.2010.2101081
    [12] Y. Guo, Y. Shen, W. Gui, Asymptotical stability of logic dynamical systems with random impulsive disturbances, IEEE Trans. Automa. Control, 66 (2021), 513–525. https://doi.org/10.1109/TAC.2020.2985302 doi: 10.1109/TAC.2020.2985302
    [13] Z. He, C. Li, Z. Cao, H. Li, Periodicity and global exponential periodic synchronization of delayed neural networks with discontinuous activations and impulsive perturbations, Neurocomputing, 431 (2021), 111–127. https://doi.org/10.1016/j.neucom.2020.09.080 doi: 10.1016/j.neucom.2020.09.080
    [14] J. Yang, J. Lu, J. Lou, Y. Liu, Synchronization of drive-response boolean control networks with impulsive disturbances, Appl. Math. Comput., 364 (2020), 124679. https://doi.org/10.1016/j.amc.2019.124679 doi: 10.1016/j.amc.2019.124679
    [15] B. Jiang, J. Lu, X. Li, K. Shi, Impulsive control for attitude stabilization in the presence of unknown bounded external disturbances, Int. J. Robust Nonlinear Control, 32 (2022), 1316–1330. https://doi.org/10.1002/rnc.5889 doi: 10.1002/rnc.5889
    [16] X. Li, C. Zhu, Saturated impulsive control of nonlinear systems with applications, Automatica, 142 (2022), 110375. https://doi.org/10.1016/j.automatica.2022.110375 doi: 10.1016/j.automatica.2022.110375
    [17] B. Liu, B. Xu, Z. Sun, Incremental stability and contraction via impulsive control for continuous-time dynamical systems, Nonlinear Anal. Hybrid Syst., 39 (2021), 100981. https://doi.org/10.1016/j.nahs.2020.100981 doi: 10.1016/j.nahs.2020.100981
    [18] H. Li, A. Liu, Asymptotic stability analysis via indefinite lyapunov functions and design of nonlinear impulsive control systems, Nonlinear Anal. Hybrid Syst., 38 (2020), 100936. https://doi.org/10.1016/j.nahs.2020.100936 doi: 10.1016/j.nahs.2020.100936
    [19] H. Ren, P. Shi, F. Deng, Y. Peng, Fixed-time synchronization of delayed complex dynamical systems with stochastic perturbation via impulsive pinning control, J. Franklin Inst., 357 (2020), 12308–12325. https://doi.org/10.1016/j.jfranklin.2020.09.016 doi: 10.1016/j.jfranklin.2020.09.016
    [20] Y. Wang, X. Li, S. Song, Input-to-state stabilization of nonlinear impulsive delayed systems: An observer-based control approach, IEEE/CAA J. Autom. Sin., 9 (2022), 1273–1283. https://doi.org/10.1109/JAS.2022.105422 doi: 10.1109/JAS.2022.105422
    [21] H. Zhu, P. Li, X. Li, Input-to-state stability of impulsive systems with hybrid delayed impulse effects, J. Appl. Anal. Comput., 9 (2019), 777–795. https://doi.org/10.11948/2156-907X.20180182 doi: 10.11948/2156-907X.20180182
    [22] H. Zhu, P. Li, X. Li, H. Akca, Input-to-state stability for impulsive switched systems with incommensurate impulsive switching signals, Commun. Nonlinear Sci. Numer. Simul., 80 (2020), 104969. https://doi.org/10.1016/j.cnsns.2019.104969 doi: 10.1016/j.cnsns.2019.104969
    [23] Y. Wang, J. Lu, J. Liang, J. Cao, M. Perc, Pinning synchronization of nonlinear coupled Lur'e networks under hybrid impulses, IEEE Trans. Circuits Syst. II Express Briefs, 66 (2018), 432–436. https://doi.org/10.1109/TCSII.2018.2844883 doi: 10.1109/TCSII.2018.2844883
    [24] N. Wang, X. Li, J. Lu, F. Alsaadi, Unified synchronization criteria in an array of coupled neural networks with hybrid impulses, Neural Networks, 101 (2018), 25–32. https://doi.org/10.1016/j.neunet.2018.01.017 doi: 10.1016/j.neunet.2018.01.017
    [25] W. Wang, C. Huang, C. Huang, J. Cao, J. Lu, L. Wang, Bipartite formation problem of second-order nonlinear multi-agent systems with hybrid impulses, Appl. Math. Comput., 370 (2020), 124926. https://doi.org/10.1016/j.amc.2019.124926 doi: 10.1016/j.amc.2019.124926
    [26] B. Jiang, B. Li, J. Lu, Complex systems with impulsive effects and logical dynamics: A brief overview, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 1273–1299. https://doi.org/10.3934/dcdss.2020369 doi: 10.3934/dcdss.2020369
    [27] P. Li, X. Li, J. Lu, Input-to-state stability of impulsive delay systems with multiple impulses, IEEE Trans. Autom. Control, 66 (2021), 362–368. https://doi.org/10.1109/TAC.2020.2982156 doi: 10.1109/TAC.2020.2982156
    [28] T. Ensari, S. Arik, Global stability analysis of neural networks with multiple time varying delays, IEEE Trans. Autom. Control, 50 (2005), 1781–1785. https://doi.org/10.1109/TAC.2005.858634 doi: 10.1109/TAC.2005.858634
    [29] C. Murguia, R. Fey, H. Nijmeijer, Network synchronization using invariant-manifold-based diffusive dynamic couplings with time-delay, Automatica, 57 (2015), 34–44. https://doi.org/10.1016/j.automatica.2015.03.031 doi: 10.1016/j.automatica.2015.03.031
    [30] H. Freedman, J. Wu, Periodic solutions of single-species models with periodic delay, SIAM J. Math. Anal., 23 (1992), 689–701. https://doi.org/10.1137/0523035 doi: 10.1137/0523035
    [31] A. Martin, S. Ruan, Predator-prey models with delay and prey harvesting, J. Math. Biol., 43 (2001), 247–267. https://doi.org/10.1007/s002850100095 doi: 10.1007/s002850100095
    [32] H. Zhu, R. Rakkiyappan, X. Li, Delayed state-feedback control for stabilization of neural networks with leakage delay, Neural Networks, 105 (2018), 249–255. https://doi.org/10.1016/j.neunet.2018.05.013 doi: 10.1016/j.neunet.2018.05.013
    [33] X. Li, X. Yang, Lyapunov stability analysis for nonlinear systems with state-dependent state delay, Automatica, 112 (2020), 108674. https://doi.org/10.1016/j.automatica.2019.108674 doi: 10.1016/j.automatica.2019.108674
    [34] X. Wu, W. Zhang, Y. Tang, pth moment stability of impulsive stochastic delay differential systems with markovian switching, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 1870–1879. https://doi.org/10.1016/j.cnsns.2012.12.001 doi: 10.1016/j.cnsns.2012.12.001
    [35] S. Dashkovskiy, M. Kosmykov, A. Mironchenko, L. Naujok, Stability of interconnected impulsive systems with and without time delays, using lyapunov methods, Nonlinear Anal. Hybrid Syst., 6 (2012), 899–915. https://doi.org/10.1016/j.nahs.2012.02.001 doi: 10.1016/j.nahs.2012.02.001
    [36] X. Liu, K. Zhang, Input-to-state stability of time-delay systems with delay-dependent impulses, IEEE Trans. Autom. Control, 65 (2020), 1676–1682. https://doi.org/10.1109/TAC.2019.2930239 doi: 10.1109/TAC.2019.2930239
    [37] H. Yang, X. Wang, S. Zhong, L. Shu, Synchronization of nonlinear complex dynamical systems via delayed impulsive distributed control, Appl. Math. Comput., 320 (2018), 75–85. https://doi.org/10.1016/j.amc.2017.09.019 doi: 10.1016/j.amc.2017.09.019
    [38] G. Wang, Y. Liu, J. Lu, Z. Wan, Stability analysis of totally positive switched linear systems with average dwell time switching, Nonlinear Anal. Hybrid Syst., 36 (2020), 100877. https://doi.org/10.1016/j.nahs.2020.100877 doi: 10.1016/j.nahs.2020.100877
    [39] Z. H. Guan, Z. W. Liu, G. Feng, Y. W. Wang, Synchronization of complex dynamical networks with time-varying delays via impulsive distributed control, IEEE Trans. Circuits Syst. I Regular Papers, 57 (2010), 2182–2195. https://doi.org/10.1109/TCSI.2009.2037848 doi: 10.1109/TCSI.2009.2037848
    [40] X. Li, J. Cao, D. W. C. Ho, Impulsive control of nonlinear systems with time-varying delay and applications, IEEE Trans. Cybern., 50 (2020), 2661–2673. https://doi.org/10.1109/TCYB.2019.2896340 doi: 10.1109/TCYB.2019.2896340
    [41] B. Jiang, J. Lu, X. Li, J. Qiu, Input/output-to-state stability of nonlinear impulsive delay systems based on a new impulsive inequality, Int. J. Robust Nonlinear Control, 29 (2020), 6164–6178. https://doi.org/10.1002/rnc.4712 doi: 10.1002/rnc.4712
    [42] X. Li, X. Yang, S. Song, Lyapunov conditions for finite-time stability of time-varying time-delay systems, Automatica, 103 (2019), 135–140. https://doi.org/10.1016/j.automatica.2019.01.031 doi: 10.1016/j.automatica.2019.01.031
    [43] X. Li, X. Zhang, S. Song, Effect of delayed impulses on input-to-state stability of nonlinear systems, Automatica, 76 (2017), 378–382. https://doi.org/10.1016/j.automatica.2016.08.009 doi: 10.1016/j.automatica.2016.08.009
    [44] X. Li, S. Song, Stabilization of delay systems: Delay-dependent impulsive control, IEEE Trans. Autom. Control, 62 (2017), 406–411. https://doi.org/10.1109/TAC.2016.2530041 doi: 10.1109/TAC.2016.2530041
    [45] H. Wang, S. Duan, C. Li, L. Wang, T. Huang, Stability of impulsive delayed linear differential systems with delayed impulses, J. Franklin Inst., 352 (2015), 3044–3068. https://doi.org/10.1016/j.jfranklin.2014.12.009 doi: 10.1016/j.jfranklin.2014.12.009
    [46] X. Li, S. Song, J. Wu, Exponential stability of nonlinear systems with delayed impulses and applications, IEEE Trans. Autom. Control, 64 (2019), 4024–4034. https://doi.org/10.1109/TAC.2019.2905271 doi: 10.1109/TAC.2019.2905271
    [47] X. Li, P. Li, Stability of time-delay systems with impulsive control involving stabilizing delays, Automatica, 124 (2021), 109336. https://doi.org/10.1016/j.automatica.2020.109336 doi: 10.1016/j.automatica.2020.109336
    [48] J. Lu, B. Jiang, W. X. Zheng, Potential impact of delay on stability of impulsive control systems, IEEE Trans. Autom. Control, 67 (2021), 5179–5190. https://doi.org/10.1109/TAC.2021.3120672 doi: 10.1109/TAC.2021.3120672
    [49] J. P. Hespanha, D. Liberzon, A. R. Teel, Lyapunov conditions for input-to-state stability of impulsive systems, Automatica, 44 (2008), 2735–2744. https://doi.org/10.1016/j.automatica.2008.03.021 doi: 10.1016/j.automatica.2008.03.021
    [50] Y. Wang, J. Lu, Y. Lou, Halanay-type inequality with delayed impulses and its applications, Sci. China Inf. Sci., 62 (2019), 192–206. https://doi.org/10.1007/s11432-018-9809-y doi: 10.1007/s11432-018-9809-y
    [51] S. Luo, F. Deng, W. H. Chen, Stability and stabilization of linear impulsive systems with large impulse-delays: A stabilizing delay perspective, Automatica, 127 (2021), 109533. https://doi.org/10.1016/j.automatica.2021.109533 doi: 10.1016/j.automatica.2021.109533
    [52] T. Jiao, W. Zheng, S. Xu, Unified stability criteria of random nonlinear time-varying impulsive switched systems, IEEE Trans. Circuits Syst. I Regular Papers, 67 (2020), 3099–3112. https://doi.org/10.1109/TCSI.2020.2983324 doi: 10.1109/TCSI.2020.2983324
    [53] W. Zhu, D. Wang, L. Liu, G. Feng, Event-based impulsive control of continuous-time dynamic systems and its application to synchronization of memristive neural networks, IEEE Trans. Neural Networks Learn. Syst., 29 (2017), 3599–3609. https://doi.org/10.1109/TNNLS.2017.2731865 doi: 10.1109/TNNLS.2017.2731865
    [54] B. Li, Z. Wang, Q. L. Han, Input-to-state stabilization of delayed differential systems with exogenous disturbances: The event-triggered case, IEEE Trans. Syst. Man Cybern. Syst., 49 (2019), 1099–1109. https://doi.org/10.1109/TSMC.2017.2719960 doi: 10.1109/TSMC.2017.2719960
    [55] K. Zhang, B. Gharesifard, E. Braverman, Event-triggered control for nonlinear time-delay systems, IEEE Trans. Autom. Control, 67 (2022), 1031–1037. https://doi.org/10.1109/TAC.2021.3062577 doi: 10.1109/TAC.2021.3062577
    [56] L. Meng, H. Bao, Synchronization of delayed complex dynamical networks with actuator failure by event-triggered pinning control, Phys. A Stat. Mech. Appl., 606 (2022), 128138. https://doi.org/10.1016/j.physa.2022.128138 doi: 10.1016/j.physa.2022.128138
    [57] Y. Bao, Y. Zhang, B. Zhang, Fixed-time synchronization of coupled memristive neural networks via event-triggered control, Appl. Math. Comput., 411 (2021),: 126542. https://doi.org/10.1016/j.amc.2021.126542
    [58] P. Li, W. Zhao, J. Cheng, Event-triggered control for exponential stabilization of impulsive dynamical systems, Appl. Math. Comput., 413 (2022), 126608. https://doi.org/10.1016/j.amc.2021.126608 doi: 10.1016/j.amc.2021.126608
    [59] K. Zhang, B. Gharesifard, Hybrid event-triggered and impulsive control for time-delay systems, Nonlinear Anal. Hybrid Syst., 43 (2021), 101109. https://doi.org/10.1016/j.nahs.2021.101109 doi: 10.1016/j.nahs.2021.101109
    [60] Y. Zou, Z. Zeng, Event-triggered impulsive control on quasi-synchronization of memristive neural networks with time-varying delays, Neural Networks, 110 (2019), 55–65. https://doi.org/10.1016/j.neunet.2018.09.014 doi: 10.1016/j.neunet.2018.09.014
    [61] X. Tan, J. Cao, X. Li, Consensus of leader-following multiagent systems: A distributed event-triggered impulsive control strategy, IEEE Trans. Cybern., 49 (2019), 792–801. https://doi.org/10.1109/TCYB.2017.2786474 doi: 10.1109/TCYB.2017.2786474
    [62] B. Jiang, J. Lu, X. Li, J. Qiu, Event-triggered impulsive stabilization of systems with external disturbances, IEEE Trans. Autom. Control, 67 (2022), 2116–2122. https://doi.org/10.1109/TAC.2021.3108123 doi: 10.1109/TAC.2021.3108123
    [63] S. Shanmugasundaram, K. Udhayakumar, D. Gunasekaran, R. Rakkiyappan, Event-triggered impulsive control design for synchronization of inertial neural networks with time delays, Neurocomputing, 483 (2022), 322–332. https://doi.org/10.1016/j.neucom.2022.02.023 doi: 10.1016/j.neucom.2022.02.023
    [64] X. Li, X. Yang, J. Cao, Event-triggered impulsive control for nonlinear delay systems, Automatica, 117 (2020), 108981. https://doi.org/10.1016/j.automatica.2020.108981 doi: 10.1016/j.automatica.2020.108981
    [65] B. Liu, D. J. Hill, Z. Sun, Stabilisation to input-to-state stability for continuous-time dynamical systems via event-triggered impulsive control with three levels of events, IET Control Theory Appl., 12 (2018), 1167–1179. https://doi.org/10.1049/iet-cta.2017.0820 doi: 10.1049/iet-cta.2017.0820
    [66] X. Li, H. Zhu, S. Song, Input-to-state stability of nonlinear systems using observer-based event-triggered impulsive control, IEEE Trans. Syst. Man Cybern. Syst., 51 (2021), 6892–6900. https://doi.org/10.1109/TSMC.2020.2964172 doi: 10.1109/TSMC.2020.2964172
    [67] H. Zhu, J. Lu, J. Lou, Event-triggered impulsive control for nonlinear systems: The control packet loss case, IEEE Trans. Circuits Syst. II Express Briefs, 69 (2022), 3204–3208. https://doi.org/10.1109/TCSII.2022.3140346 doi: 10.1109/TCSII.2022.3140346
    [68] E. I. Verriest, F. Delmotte, M. Egerstedt, Control of epidemics by vaccination, in Proceedings of the 2005 American Control Conference, (2005), 985–990. https://doi.org/10.1109/ACC.2005.1470088
    [69] C. Briata, E. I. Verriest, A new delay-sir model for pulse vaccination, Biomed. Signal Process. Control, 4 (2009), 272–277. https://doi.org/10.3182/20080706-5-KR-1001.01742 https://doi.org/10.3182/20080706-5-KR-1001.01742 doi: 10.3182/20080706-5-KR-1001.01742
    [70] Y. V. Orlov, Discontinuous Systems: Lyapunov Analysis and Robust Synthesis under Uncertainty Conditions, Springer Science & Business Media, 2008.
    [71] X. Yang, X. Cao, A new approach to autonomous rendezvous for spacecraft with limited impulsive thrust: Based on switching control strategy, Aerosp. Sci. Technol., 43 (2015), 454–462. https://doi.org/10.1016/j.ast.2015.04.007 doi: 10.1016/j.ast.2015.04.007
    [72] L. A. Sobiesiak, C. J. Damaren, Optimal continuous/impulsive control for lorentz-augmented spacecraft formations, J. Guid. Control Dyn., 38 (2015), 151–157. https://doi.org/10.2514/1.G000334 doi: 10.2514/1.G000334
    [73] D. Auckly, L. Kapitanski, W. White, Control of nonlinear underactuated systems, Commun. Pure Appl. Math., 53 (2000), 354–369. https://doi.org/10.1002/(SICI)1097-0312(200003)53:3354::AID-CPA33.0.CO;2-U doi: 10.1002/(SICI)1097-0312(200003)53:3354::AID-CPA33.0.CO;2-U
    [74] N. Kant, R. Mukherjee, D. Chowdhury, H. K. Khalil, Estimation of the region of attraction of underactuated systems and its enlargement using impulsive inputs, IEEE Trans. Rob., 35 (2019), 618–632. https://doi.org/10.1109/TRO.2019.2893599 doi: 10.1109/TRO.2019.2893599
    [75] A. Churilov, A. Medvedev, A. Shepeljavyi, Mathematical model of non-basal testosterone regulation in the male by pulse modulated feedback, Automatica, 45 (2009), 78–85. https://doi.org/10.1016/j.automatica.2008.06.016 doi: 10.1016/j.automatica.2008.06.016
    [76] V. Badri, M. J. Yazdanpanah, M. S. Tavazoei, Global stabilization of lotka-volterra systems with interval uncertainty, IEEE Trans. Autom. Control, 64 (2018), 1209–1213. https://doi.org/10.1109/TAC.2018.2845659 doi: 10.1109/TAC.2018.2845659
    [77] E. I. Verriest, P. Pepe, Time optimal and optimal impulsive control for coupled differential difference point delay systems with an application in forestry, in Topics in Time Delay Systems, (2009), 255–265. https://doi.org/10.1007/978-3-642-02897-7_22
    [78] J. Zhang, J. Lu, M. Xing, J. Liang, Synchronization of finite field networks with switching multiple communication channels, IEEE Trans. Network Sci. Eng., 8 (2021), 2160–2169. https://doi.org/10.1109/TNSE.2021.3079631 doi: 10.1109/TNSE.2021.3079631
    [79] A. Arenas, A. Diaz-Guilera, C. J. Pˊerez-Vicente, Synchronization processes in complex networks, Phys. D Nonlinear Phenom., 224 (2006), 27–34. https://doi.org/10.1016/j.physd.2006.09.029 doi: 10.1016/j.physd.2006.09.029
    [80] A. Arenas, A. Diaz-Guilera, C. J. Pˊerez-Vicente, Synchronization reveals topological scales in complex networks, Phys. Rev. Lett., 96 (2006), 114102. https://doi.org/10.1103/PhysRevLett.96.114102 doi: 10.1103/PhysRevLett.96.114102
    [81] V. Belykh, I. Belykh, M. Hasler, K. Nevidin, Cluster synchronization in three-dimensional lattices of diffusively coupled oscillators, Int. J. Bifurcation Chaos, 13 (2003), 755–779. https://doi.org/10.1142/S0218127403006923 doi: 10.1142/S0218127403006923
    [82] W. Lu, T. Chen, Quad-condition, synchronization, consensus of multiagents, and antisynchronization of complex networks, IEEE Trans. Cybern., 51 (2021), 3384–3388. https://doi.org/10.1109/TCYB.2019.2939273 doi: 10.1109/TCYB.2019.2939273
    [83] L. Zhang, J. Zhong, J. Lu, Intermittent control for finite-time synchronization of fractional-order complex networks, Neural Networks, 144 (2021), 11–20. https://doi.org/10.1016/j.neunet.2021.08.004 doi: 10.1016/j.neunet.2021.08.004
    [84] L. Zhang, Y. Li, J. Lou, J. Qiu, Bipartite asynchronous impulsive tracking consensus for multi-agent systems, Front. Inf. Technol. Electron. Eng., 2022 (2022), 1–11. https://doi.org/10.1631/FITEE.2100122 doi: 10.1631/FITEE.2100122
    [85] H. Fan, K. Shi, Y. Zhao, Global μ-synchronization for nonlinear complex networks with unbounded multiple time delays and uncertainties via impulsive control, Phys. A Stat. Mech. Appl., 599 (2022), 127484. https://doi.org/10.1016/j.physa.2022.127484 doi: 10.1016/j.physa.2022.127484
    [86] Q. Cui, L. Li, J. Lu, A. Alofi, Finite-time synchronization of complex dynamical networks under delayed impulsive effects, Appl. Math. Comput., 430 (2022), 127290. https://doi.org/10.1016/j.amc.2022.127290 doi: 10.1016/j.amc.2022.127290
    [87] T. Chen, X. Liu, W. Lu, Pinning complex networks by a single controller, IEEE Trans. Syst. Man Cybern. Syst., 54 (2007), 1317–1326. https://doi.org/10.1109/TCSI.2007.895383 doi: 10.1109/TCSI.2007.895383
    [88] J. Lu, D. W. C. Ho, J. Cao, A unified synchronization criterion for impulsive dynamical networks, Automatica, 46 (2010), 1215–1221. https://doi.org/10.1016/j.automatica.2010.04.005 doi: 10.1016/j.automatica.2010.04.005
    [89] X. Ji, J. Lu, J. Lou, J. Qiu, K. Shi, A unified criterion for global exponential stability of quaternion-valued neural networks with hybrid impulses, Int. J. Robust Nonlinear Control, 30 (2020), 8098–8116. https://doi.org/10.1002/rnc.5210 doi: 10.1002/rnc.5210
    [90] R. Li, H. Wu, J. Cao, Exponential synchronization for variable-order fractional discontinuous complex dynamical networks with short memory via impulsive control, Neural Networks, 148 (2022), 13–22. https://doi.org/10.1016/j.neunet.2021.12.021 doi: 10.1016/j.neunet.2021.12.021
    [91] Q. Fu, S. Zhong, K. Shi, Exponential synchronization of memristive neural networks with inertial and nonlinear coupling terms: Pinning impulsive control approaches, Appl. Math. Comput., 402 (2021), 126169. https://doi.org/10.1016/j.amc.2021.126169 doi: 10.1016/j.amc.2021.126169
    [92] X. Ji, J. Lu, B. Jiang, J. Zhong, Network synchronization under distributed delayed impulsive control: Average delayed impulsive weight approach, Nonlinear Anal. Hybrid Syst., 44 (2022), 101148. https://doi.org/10.1016/j.nahs.2021.101148 doi: 10.1016/j.nahs.2021.101148
    [93] A. D'Jorgea, A. Andersona, A. Ferramoscab, A. H. Gonzˊaleza, M. Actis, On stability of nonzero set-point for nonlinear impulsive control systems, Syst. Control Lett., 165 (2022), 105244. https://doi.org/10.1016/j.sysconle.2022.105244 doi: 10.1016/j.sysconle.2022.105244
    [94] J. Liu, L. Guo, M. Hu, Z. Xu, Y. Yang, Leader-following consensus of multi-agent systems with delayed impulsive control, IMA J. Math. Control Inf., 33 (2016), 137–146. https://doi.org/10.1093/imamci/dnu037 doi: 10.1093/imamci/dnu037
    [95] S. Dashkovskiy, A. Mironchenko, Input-to-state stability of nonlinear impulsive systems, SIAM J. Math. Anal., 51 (2013), 1962–1987. https://doi.org/10.1137/120881993 doi: 10.1137/120881993
    [96] P. Feketa, N. Bajcinca, On robustness of impulsive stabilization, Automatica, 104 (2019), 48–56. https://doi.org/10.1016/j.automatica.2019.02.056 doi: 10.1016/j.automatica.2019.02.056
    [97] P. Feketa, S. Bogomolov, T. Meurer, Safety verification for impulsive systems, IFAC-Papers OnLine, 53 (2020), 1949–1954. https://doi.org/10.1016/j.ifacol.2020.12.2589 doi: 10.1016/j.ifacol.2020.12.2589
    [98] P. Feketa, V. Klinshov, L. L¨ucken, A survey on the modeling of hybrid behaviors: How to account for impulsive jumps properly, Commun. Nonlinear Sci. Numer. Simul., 103 (2021), 105955. https://doi.org/10.1016/j.cnsns.2021.105955 doi: 10.1016/j.cnsns.2021.105955
    [99] P. Feketa, N. Bajcinca, Average dwell-time for impulsive control systems possessing iss-lyapunov function with nonlinear rates, in Proceedings of the 18th European Control Conference, (2019), 3686–3691. https://doi.org/10.23919/ECC.2019.8796238
    [100] A. S. Morse, Supervisory control of families of linear set-point controllers-part i. exact matching, IEEE Trans. Autom. Control, 41 (1996), 1413–1431. https://doi.org/10.1109/9.539424 doi: 10.1109/9.539424
    [101] C. Briat, A. Seuret, Convex dwell-time characterizations for uncertain linear impulsive systems, IEEE Trans. Autom. Control, 57 (2012), 3241–3246. https://doi.org/10.1109/TAC.2012.2200379 doi: 10.1109/TAC.2012.2200379
    [102] J. C. Geromel, P. Colaneri, Stability and stabilization of continuous-time switched linear systems, SIAM J. Control Optim., 45 (2006), 1915–1930. https://doi.org/10.1137/050646366 doi: 10.1137/050646366
    [103] C. Briat, A. Seuret, A looped-functional approach for robust stability analysis of linear impulsive systems, Syst. Control Lett., 61 (2012), 980–988. https://doi.org/10.1016/j.sysconle.2012.07.008 doi: 10.1016/j.sysconle.2012.07.008
    [104] S. Dashkovskiy, V. Slynko, Stability conditions for impulsive dynamical systems, Math. Control Signals Syst., 34 (2022), 95–128, 2022. https://doi.org/10.1007/s00498-021-00305-y doi: 10.1007/s00498-021-00305-y
    [105] S. Dashkovskiy, V. Slynko, Dwell-time stability conditions for infinite dimensional impulsive systems, Automatica, 147 (2023), 110695. https://doi.org/10.1016/j.automatica.2022.110695 doi: 10.1016/j.automatica.2022.110695
    [106] J.L. Mancilla-Aguilar, H. Haimovich, P. Feketa, Uniform stability of nonlinear time-varying impulsive systems with eventually uniformly bounded impulse frequency, Nonlinear Anal. Hybrid Syst., 38 (2020), 100933. https://doi.org/10.1016/j.nahs.2020.100933 doi: 10.1016/j.nahs.2020.100933
    [107] C. Briat, Dwell-time stability and stabilization conditions for linear positive impulsive and switched systems, Nonlinear Anal. Hybrid Syst., 24 (2017), 198–226. https://doi.org/10.1016/j.nahs.2017.01.004 doi: 10.1016/j.nahs.2017.01.004
    [108] J. Tan, C. Li, T. Huang, Stability of impulsive systems with time window via comparison method, Int. J. Control Autom. Syst., 13 (2015), 1346–1350. https://doi.org/10.1007/s12555-014-0197-y doi: 10.1007/s12555-014-0197-y
    [109] E. Fridman, Introduction to Time-Delay Systems: Analysis and Control, Springer, 2014. https://doi.org/10.1007/978-3-319-09393-2
    [110] J. K. Hale, Theory of Functional Differential Equations, Springer, 1971. https://doi.org/10.1007/BFb0060406
    [111] S. Dashkovskiy, P. Feketa, Asymptotic properties of zeno solutions, Nonlinear Anal. Hybrid Syst., 30 (2018), 256–265. https://doi.org/10.1016/j.nahs.2018.06.005 doi: 10.1016/j.nahs.2018.06.005
    [112] J. P. Hespanha, A. S. Morse, Stability of switched systems with average dwell-time, in Proceedings of the 38th IEEE Conference on Decision and Control, (1999), 2655–2660. https://doi.org/10.1109/CDC.1999.831330
    [113] T. Yang, L. Chua, Impulsive stabilization for control and synchronization of chaotic systems: Theory and application to secure communication, IEEE Trans. Circuits Syst. I Fundam. Theory Appl., 44 (1997), 976–988. https://doi.org/10.1109/81.633887 doi: 10.1109/81.633887
    [114] E. Kaslik, S. Sivasundaram, Impulsive hybrid discrete-time hopfield neural networks with delays and multistability analysis, Neural Networks, 24 (2011), 370–377. https://doi.org/10.1016/j.neunet.2010.12.008 doi: 10.1016/j.neunet.2010.12.008
    [115] S. Duan, H. Wang, L. Wang, T. Huang, C. Li, Impulsive effects and stability analysis on memristive neural networks with variable delays, IEEE Trans. Neural Networks Learn. Syst., 28 (2017), 476–481. https://doi.org/10.1109/TNNLS.2015.2497319 doi: 10.1109/TNNLS.2015.2497319
    [116] R. Rakkiyappan, A. Chandrasekar, S. Lakshmanan, J. H. Park, H. Y. Jung, Effects of leakage time-varying delays in markovian jump neural networks with impulse control, Neurocomputing, 121 (2013), 365–378. https://doi.org/10.1016/j.neucom.2013.05.018 doi: 10.1016/j.neucom.2013.05.018
    [117] Z. Tang, J. Park, Impulsive effects on quasi-synchronization of neural networks with parameter mismatches and time-varying delay, IEEE Trans. Neural Networks Learn. Syst., 29 (2018), 908–919. https://doi.org/10.1109/TNNLS.2017.2651024 doi: 10.1109/TNNLS.2017.2651024
    [118] K. Gu, J. Chen, V. L. Kharitonov, Stability of Time-Delay Systems, Springer Science & Business Media, 2003. https://doi.org/10.1007/978-1-4612-0039-0
    [119] N. N. Krasovskii, Stability of Motion: Applications of Lyapunov's Second Method to Differential Systems and Equations with Delay, Stanford University Press, 1963.
    [120] B. Zhou, Construction of strict lyapunov-krasovskii functionals for time-varying time delay systems, Automatica, 107 (2019), 382–397. https://doi.org/10.1016/j.automatica.2019.05.058 https://doi.org/10.1016/j.jfranklin.2020.05.051
    [121] Q. L. Han, On stability of linear neutral systems with mixed time delays: A discretized lyapunov functional approach, Automatica, 41 (2005), 1209–1218. https://doi.org/10.1016/j.automatica.2005.01.014 doi: 10.1016/j.automatica.2005.01.014
    [122] I. Haidar, P. Pepe, Lyapunov-krasovskii characterizations of stability notions for switching retarded systems, IEEE Trans. Autom. Control, 66 (2021), 437–443. https://doi.org/10.1109/TAC.2020.2979754 doi: 10.1109/TAC.2020.2979754
    [123] T. H. Lee, H. M. Trinh, J. H. Park, Stability analysis of neural networks with time-varying delay by constructing novel lyapunov functionals, IEEE Trans. Neural Networks Learn. Syst., 29 (2018), 4238–4247. https://doi.org/10.1109/TNNLS.2017.2760979 doi: 10.1109/TNNLS.2017.2760979
    [124] X. Liu, Q. Wang, The method of lyapunov functionals and exponential stability of impulsive systems with time delay, Nonlinear Anal. Theory Methods Appl., 66 (2007), 1465–1484. https://doi.org/10.1016/j.na.2006.02.004 doi: 10.1016/j.na.2006.02.004
    [125] Z. Luo, J. Shen, Stability of impulsive functional differential equations via the liapunov functional, Appl. Math. Lett., 22 (2009), 163–169. https://doi.org/10.1016/j.aml.2008.03.004 doi: 10.1016/j.aml.2008.03.004
    [126] M. A. Davo, A. Banos, F. Gouaisbaut, S. Tarbouriech, A. Seuret, Stability analysis of linear impulsive delay dynamical systems via looped-functionals, Automatica, 81 (2017), 107–114. https://doi.org/10.1016/j.automatica.2017.03.029 doi: 10.1016/j.automatica.2017.03.029
    [127] J. Liu, X. Liu, W. Xie, Input-to-state stability of impulsive and switching hybrid systems with time-delay, Automatica, 47 (2011), 899–908. https://doi.org/10.1016/j.automatica.2011.01.061 doi: 10.1016/j.automatica.2011.01.061
    [128] X. Sun, W. Wang, Integral input-to-state stability for hybrid delayed systems with unstable continuous dynamics, Automatica, 48 (2012), 2359–2364. https://doi.org/10.1016/j.automatica.2012.06.056 doi: 10.1016/j.automatica.2012.06.056
    [129] C. Briat, Theoretical and numerical comparisons of looped functionals and clock-dependent Lyapunov functions???The case of periodic and pseudo-periodic systems with impulses, Int. J. Robust Nonlinear Control, 26 (2016), 2232–2255. https://doi.org/10.1002/rnc.3405 doi: 10.1002/rnc.3405
    [130] J. J Nieto, R. R. Lopez, New comparison results for impulsive integro-differential equations and applications, J. Math. Anal. Appl., 328 (2007), 1343–1368. https://doi.org/10.1016/j.jmaa.2006.06.029 doi: 10.1016/j.jmaa.2006.06.029
    [131] Q. Wu, H. Zhang, L. Xiang, J. Zhou, A generalized halanay inequality on impulsive delayed dynamical systems and its applications, Chaos Solitons Fractals, 45 (2012), 56–62. https://doi.org/10.1016/j.chaos.2011.09.010 doi: 10.1016/j.chaos.2011.09.010
    [132] R. Kumar, U. Kumar, S. Das, J. Qiu, J. Lu, Effects of heterogeneous impulses on synchronization of complex-valued neural networks with mixed time-varying delays, Inf. Sci., 551 (2021), 228–244. https://doi.org/10.1016/j.ins.2020.10.064 doi: 10.1016/j.ins.2020.10.064
    [133] A. R. Teel, A. Subbaraman, A. Sferlazza, Stability analysis for stochastic hybrid systems: a survey, Automatica, 50 (2014), 2435–2456. https://doi.org/10.1016/j.automatica.2014.08.006 doi: 10.1016/j.automatica.2014.08.006
    [134] W. Hu, Q. Zhu, Stability criteria for impulsive stochastic functional differential systems with distributed-delay dependent impulsive effects, IEEE Trans. Syst. Man Cybern. Syst., 51 (2021), 2027–2032. https://doi.org/10.1109/TSMC.2019.2905007 doi: 10.1109/TSMC.2019.2905007
    [135] K. Rengamannar, G. P. Balakrishnan, M. Palanisamy, M. Niezabitowski, Exponential stability of non-linear stochastic delay differential system with generalized delay-dependent impulsive points, Appl. Math. Comput., 382 (2020), 125344. https://doi.org/10.1016/j.amc.2020.125344 doi: 10.1016/j.amc.2020.125344
    [136] H. Chen, P. Shi, C. C. Lim, Synchronization control for neutral stochastic delay markov networks via single pinning impulsive strategy, IEEE Trans. Syst. Man Cybern. Syst., 50 (2020), 5406–5419. https://doi.org/10.1109/TSMC.2018.2882836 doi: 10.1109/TSMC.2018.2882836
    [137] X. Wu, Y. Tang, W. Zhang, Input-to-state stability of impulsive stochastic delayed systems under linear assumptions, Automatica, 66 (2016), 195–204. https://doi.org/10.1016/j.automatica.2016.01.002 doi: 10.1016/j.automatica.2016.01.002
    [138] W. Hu, Q. Zhu, H. R. Karimi, Some improved razumikhin stability criteria for impulsive stochastic delay differential systems, IEEE Trans. Autom. Control, 64 (2019), 5207–5213. https://doi.org/10.1109/TAC.2019.2911182 doi: 10.1109/TAC.2019.2911182
    [139] H. Xu, Q. Zhu, New criteria on exponential stability of impulsive stochastic delayed differential systems with infinite delays, Commun. Nonlinear Sci. Numer. Simul., 111 (2022), 106460. https://doi.org/10.1016/j.cnsns.2022.106460 doi: 10.1016/j.cnsns.2022.106460
    [140] X. Liu, G. Ballinger, Uniform asymptotic stability of impulsive delay differential equations, Comput. Math. Appl., 41 (2001), 903–915. https://doi.org/10.1016/S0898-1221(00)00328-X doi: 10.1016/S0898-1221(00)00328-X
    [141] S. Zhang, A new technique in stability of infinite delay differential equations, Comput. Math. Appl., 44 (2002), 1275–1287. https://doi.org/10.1016/S0898-1221(02)00255-9 doi: 10.1016/S0898-1221(02)00255-9
    [142] X. Li, J. Cao, An impulsive delay inequality involving unbounded time-varying delay and applications, IEEE Trans. Autom. Control, 62 (2017), 3618–3625. https://doi.org/10.1109/TAC.2017.2669580 doi: 10.1109/TAC.2017.2669580
    [143] X. Li, D. Peng, Uniform stability of nonlinear systems with state-dependent delay, Automatica, 137 (2011), 110098. https://doi.org/10.1016/j.automatica.2021.110098 doi: 10.1016/j.automatica.2021.110098
    [144] Y. Tang, X. Wu, P. Shi, F. Qian, Input-to-state stability for nonlinear systems with stochastic impulses, Automatica, 113 (2020), 108766. https://doi.org/10.1016/j.automatica.2019.108766 doi: 10.1016/j.automatica.2019.108766
    [145] N. Zhang, S. Huang, W. Li, Stability of stochastic delayed semi-markov jump systems with stochastic mixed impulses: A novel stochastic impulsive differential inequality, J. Franklin Inst., 2022 (2022). https://doi.org/10.1016/j.jfranklin.2022.06.033
    [146] M. Yao, G. Wei, D. Ding, W. Li, Output-feedback control for stochastic impulsive systems under round-robin protocol, Automatica, 143 (2022), 110394. https://doi.org/10.1016/j.automatica.2022.110394 doi: 10.1016/j.automatica.2022.110394
    [147] W. Zhang, Y. Tang, Q. Miao, W. Du, Exponential synchronization of coupled switched neural networks with mode-dependent impulsive effects, IEEE Trans. Neural Networks Learn. Syst., 24 (2013), 1316–1326. https://doi.org/10.1109/TNNLS.2013.2257842 doi: 10.1109/TNNLS.2013.2257842
    [148] X. Yang, X. Li, Q. Xi, P. Duan, Review of stability and stabilization for impulsive delayed systems, Math. Biosci. Eng., 15 (2018), 1495–1515. https://doi.org/10.3934/mbe.2018069 doi: 10.3934/mbe.2018069
    [149] A. Vinodkumar, T. Senthilkumar, S. Hariharan, J. Alzabut, Exponential stabilization of fixed and random time impulsive delay differential system with applications, Math. Biosci. Eng., 18 (2021), 2384–2400. https://doi.org/10.3934/mbe.2021121 doi: 10.3934/mbe.2021121
    [150] B. Hu, Z. Wang, M. Xu, D. Wang, Quasilinearization method for an impulsive integro-differential system with delay, Math. Biosci. Eng., 19 (2022), 612–623. https://doi.org/10.3934/mbe.2022027 doi: 10.3934/mbe.2022027
    [151] Z. Xiong, X. Li, M. Ye, Q. Zhang, Finite-time stability and optimal control of an impulsive stochastic reaction-diffusion vegetation-water system driven by lˊevy process with time-varying delay, Math. Biosci. Eng., 18 (2021), 8462–8498. https://doi.org/10.3934/mbe.2021419 doi: 10.3934/mbe.2021419
    [152] C. Lu, B. Li, L. Zhou, L. Zhang, Survival analysis of an impulsive stochastic delay logistic model with lˊevy jumps, Math. Biosci. Eng., 16 (2019), 3251–3271. https://doi.org/10.3934/mbe.2019162 doi: 10.3934/mbe.2019162
    [153] L. Gao, D. Wang, G. Wang, Further results on exponential stability for impulsive switched nonlinear time-delay systems with delayed impulse effects, Appl. Math. Comput., 268 (2015), 186–200. https://doi.org/10.1016/j.amc.2015.06.023 doi: 10.1016/j.amc.2015.06.023
    [154] J. Sun, Q. L. Han, X. Jiang, Impulsive control of time-delay systems using delayed impulse and its application to impulsive master–slave synchronization, Phys. Lett. A, 372 (2008), 6375–6380. https://doi.org/10.1016/j.physleta.2008.08.067 doi: 10.1016/j.physleta.2008.08.067
    [155] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Springer Science & Business Media, 2013.
    [156] N. Wouw, P. Naghshtabrizi, M. Cloosterman, J. P. Hespanha, Tracking control for sampled-data systems with uncertain timevarying sampling intervals and delays, Int. J. Robust Nonlinear Control, 20 (2010), 387–411. https://doi.org/10.1002/rnc.1433 doi: 10.1002/rnc.1433
    [157] W. H. Chen, D. Wei, X. Lu, Exponential stability of a class of nonlinear singularly perturbed systems with delayed impulses, J. Franklin Inst., 350 (2013), 2678–2709. https://doi.org/10.1016/j.jfranklin.2013.06.012 doi: 10.1016/j.jfranklin.2013.06.012
    [158] A. Khadra, X. Liu, X. Shen, Impulsively synchronizing chaotic systems with delay and applications to secure communication, Automatica, 41 (2005), 1491–1502. https://doi.org/10.1016/j.automatica.2005.04.012 doi: 10.1016/j.automatica.2005.04.012
    [159] W. H. Chen, W. X. Zheng, Input-to-state stability and integral input-to-state stability of nonlinear impulsive systems with delays, Automatica, 45 (2009), 1481–1488. https://doi.org/10.1016/j.automatica.2009.02.005 doi: 10.1016/j.automatica.2009.02.005
    [160] W. H. Chen, W. X. Zheng, Exponential stability of nonlinear time-delay systems with delayed impulse effects, Automatica, 47 (2011), 1075–1083. https://doi.org/10.1016/j.automatica.2011.02.031 doi: 10.1016/j.automatica.2011.02.031
    [161] X. Liu, K. Zhang, Synchronization of linear dynamical networks on time scales: Pinning control via delayed impulses, Automatica, 72 (2016), 147–152. https://doi.org/10.1016/j.automatica.2016.06.001 doi: 10.1016/j.automatica.2016.06.001
    [162] B. Jiang, J. Lu, Y. Liu, Exponential stability of delayed systems with average-delay impulses, SIAM J. Control Optim., 58 (2020), 3763–3784. https://doi.org/10.1137/20M1317037 doi: 10.1137/20M1317037
    [163] K. Zhang, E. Braverman, Event-triggered impulsive control for nonlinear systems with actuation delays, IEEE Trans. Autom. Control, 2022 (2022). https://doi.org/10.1109/TAC.2022.3142127
    [164] X. Li, J. Wu, Stability of nonlinear differential systems with state-dependent delayed impulses, Automatica, 64 (2016), 63–69. https://doi.org/10.1016/j.automatica.2015.10.002 doi: 10.1016/j.automatica.2015.10.002
    [165] H. Akca, R. Alassar, V. Covachev, Z. Covacheva, E. Al-Zahrani, Continuous-time additive hopfield-type neural networks with impulses, J. Math. Anal. Appl., 290 (2004), 436–451. https://doi.org/10.1016/j.jmaa.2003.10.005 doi: 10.1016/j.jmaa.2003.10.005
    [166] W. H. Chen, W. X. Zheng, The effect of delayed impulses on stability of impulsive time-delay systems, IFAC Proc. Volumes, 44 (2011), 6307–6312. https://doi.org/10.3182/20110828-6-IT-1002.02984 doi: 10.3182/20110828-6-IT-1002.02984
    [167] X. Li, J. Wu, Sufficient stability conditions of nonlinear differential systems under impulsive control with state-dependent delay, IEEE Trans. Autom. Control, 63 (2018), 306–311. https://doi.org/10.1109/TAC.2016.2639819 doi: 10.1109/TAC.2016.2639819
    [168] X. Zhang, X. Li, Input-to-state stability of non-linear systems with distributed delayed impulses, IET Control Theory Appl., 11 (2017), 81–89. https://doi.org/10.1049/iet-cta.2016.0469 doi: 10.1049/iet-cta.2016.0469
    [169] Q. Cui, L. Li, J. Cao, Stability of inertial delayed neural networks with stochastic delayed impulses via matrix measure method, Neurocomputing, 471 (2022), 70–78. https://doi.org/10.1016/j.neucom.2021.10.113 doi: 10.1016/j.neucom.2021.10.113
    [170] X. Yang, Z. Yang, Synchronization of ts fuzzy complex dynamical networks with time-varying impulsive delays and stochastic effects, Fuzzy Sets Syst., 235 (2014), 25–43. https://doi.org/10.1016/j.fss.2013.06.008 doi: 10.1016/j.fss.2013.06.008
    [171] X. Yang, X. Li, J. Lu, Z. Cheng, Synchronization of time-delayed complex networks with switching topology via hybrid actuator fault and impulsive effects control, IEEE Trans. Cybern., 50 (2020), 4043–4052. https://doi.org/10.1109/TCYB.2019.2938217 doi: 10.1109/TCYB.2019.2938217
    [172] P. Rubbioni, Asymptotic stability of solutions for some classes of impulsive differential equations with distributed delay, Nonlinear Anal. Real World Appl., 61 (2021), 103324. https://doi.org/10.1016/j.nonrwa.2021.103324 doi: 10.1016/j.nonrwa.2021.103324
    [173] X. Liu, K. Zhang, Stabilization of nonlinear time-delay systems: Distributed-delay dependent impulsive control, Syst. Control Lett., 120 (2018), 17–22. https://doi.org/10.1016/j.sysconle.2018.07.012 doi: 10.1016/j.sysconle.2018.07.012
    [174] Y. Zhao, X. Li, J. Cao, Global exponential stability for impulsive systems with infinite distributed delay based on flexible impulse frequency, Appl. Math. Comput., 386 (2020), 125467. https://doi.org/10.1016/j.amc.2020.125467 doi: 10.1016/j.amc.2020.125467
    [175] X. Li, D. Peng, J. Cao, Lyapunov stability for impulsive systems via event-triggered impulsive control, IEEE Trans. Autom. Control, 65 (2020), 4908–4913. https://doi.org/10.1109/TAC.2020.2964558 doi: 10.1109/TAC.2020.2964558
    [176] W. Du, S. Leung, Y. Tang, A. Vasilakos, Differential evolution with event-triggered impulsive control, IEEE Trans. Cybern., 47 (2017), 244–257. https://doi.org/10.1109/TCYB.2015.2512942 doi: 10.1109/TCYB.2015.2512942
    [177] X. Li, P. Li, Input-to-state stability of nonlinear systems: Event-triggered impulsive control, IEEE Trans. Autom. Control, 67 (2022), 1460–1465. https://doi.org/10.1109/TAC.2021.3063227 doi: 10.1109/TAC.2021.3063227
    [178] X. Li, Y. Wang, S. Song, Stability of nonlinear impulsive systems: Self-triggered comparison system approach, IEEE Trans. Autom. Control, 2022 (2011). https://doi.org/10.1109/TAC.2022.3209441
    [179] E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Autom. Control, 34 (1989), 435–443. https://doi.org/10.1109/9.28018 doi: 10.1109/9.28018
    [180] W. Liu, P. Li, X. Li, Impulsive systems with hybrid delayed impulses: Input-to-state stability, Nonlinear Anal. Hybrid Syst., 46 (2022), 101248. https://doi.org/10.1016/j.nahs.2022.101248 doi: 10.1016/j.nahs.2022.101248
    [181] J. Mancilla-Aguilar, H. Haimovich, Uniform input-to-state stability for switched and time-varying impulsive systems, IEEE Trans. Autom. Control, 65 (2020), 5028–5042. https://doi.org/10.1109/TAC.2020.2968580 doi: 10.1109/TAC.2020.2968580
    [182] C. Ning, Y. He, M. Wu, S. Zhou, Indefinite lyapunov functions for input-to-state stability of impulsive systems, Inf. Sci., 436 (2018), 343–351. https://doi.org/10.23919/ChiCC.2018.8483927 https://doi.org/10.1016/j.ins.2018.01.016
    [183] S. Dashkovskiy, P. Feketa, Input-to-state stability of impulsive systems with different jump maps, IFAC-Papers OnLine, 49 (2016), 1073–1078.
    [184] P. Feketa, N. Bajcinca, Stability of nonlinear impulsive differential equations with non-fixed moments of jumps, in Proceedings of the 17th European Control Conference, (2018), 900–905. https://doi.org/10.23919/ECC.2018.8550434
    [185] N. Zhang, X. Wang, W. Li, Stability for multi-linked stochastic delayed complex networks with stochastic hybrid impulses by dupire itˆo's formula, Nonlinear Anal. Hybrid Syst., 45 (2022), 101200. https://doi.org/10.1016/j.nahs.2022.101200 doi: 10.1016/j.nahs.2022.101200
    [186] C. W. Wu, L. Chua, Synchronization in an array of linearly coupled dynamical systems, IEEE Trans. Circuits Syst. I Fundam. Theory Appl., 42 (1995), 430–447.
    [187] B. Jiang, J. Lu, J. Lou, J. Qiu, Synchronization in an array of coupled neural networks with delayed impulses: Average impulsive delay method, Neural Networks, 121 (2020), 452–460. https://doi.org/10.1016/j.neunet.2019.09.019 doi: 10.1016/j.neunet.2019.09.019
    [188] Y. Fiagbedzi, A. Pearson, A multistage reduction technique for feedback stabilizing distributed time-lag systems, Automatica, 23 (1987), 311–326. https://doi.org/10.1016/0005-1098(87)90005-7 doi: 10.1016/0005-1098(87)90005-7
    [189] X. Ji, J. Lu, B. Jiang, K. Shi, Distributed synchronization of delayed neural networks: Delay-dependent hybrid impulsive control, IEEE Trans. Network Sci. Eng., 9 (2021), 634–647. https://doi.org/10.1109/TNSE.2021.3128244 doi: 10.1109/TNSE.2021.3128244
    [190] W. He, F. Qian, J. Cao, Pinning-controlled synchronization of delayed neural networks with distributed-delay coupling via impulsive control, Neural Networks, 85 (2017), 1–9. https://doi.org/10.1016/j.neunet.2016.09.002 doi: 10.1016/j.neunet.2016.09.002
    [191] Z. Huang, J. Cao, J. Li, H. Bin, Quasi-synchronization of neural networks with parameter mismatches and delayed impulsive controller on time scales, Nonlinear Anal. Hybrid Syst., 33 (2019), 104–115. https://doi.org/10.1016/j.nahs.2019.02.005 doi: 10.1016/j.nahs.2019.02.005
    [192] D. Ding, Z. Tang, J. H. Park, Y. Wang, Z. Ji, Dynamic self-triggered impulsive synchronization of complex networks with mismatched parameters and distributed delay, IEEE Trans. Cybern., 2022 (2022). https://doi.org/10.1109/TCYB.2022.3168854
    [193] D. Antunes, J. P. Hespanha, C. Silvestre, Stability of networked control systems with asynchronous renewal links: An impulsive systems approach, Automatica, 49 (2013), 402–413. https://doi.org/10.1016/j.automatica.2012.11.033 doi: 10.1016/j.automatica.2012.11.033
    [194] C. Yuan, F. Wu, Delay scheduled impulsive control for networked control systems, IEEE Trans. Control Network Syst., 4 (2017), 587–597. https://doi.org/10.1109/TCNS.2016.2541341 doi: 10.1109/TCNS.2016.2541341
    [195] X. Yang, J. Lu, D. W. C. Ho, Q. Song, Synchronization of uncertain hybrid switching and impulsive complex networks, Appl. Math. Modell., 59 (2018), 379–392. https://doi.org/10.1016/j.apm.2018.01.046 doi: 10.1016/j.apm.2018.01.046
    [196] J. Hu, G. Sui, X. Lv, X. Li, Fixed-time control of delayed neural networks with impulsive perturbations, Nonlinear Anal. Modell. Control, 23 (2018), 904–920. https://doi.org/10.15388/NA.2018.6.6 doi: 10.15388/NA.2018.6.6
    [197] J. Lu, L. Li, D. W. C. Ho, J. Cao, Collective Behavior in Complex Networked Systems under Imperfect Communication, Springer, 2021. https://doi.org/10.1007/978-981-16-1506-1
    [198] X. Li, S. Song, Impulsive Systems with Delays, Springer, 2022. https://doi.org/10.1007/978-981-16-4687-4
    [199] C. Louembet, D. Arzelier, G. Deaconu, Robust rendezvous planning under maneuver execution errors, J. Guid. Control Dyn., 38 (2015), 76–93. https://doi.org/10.2514/1.G000391 doi: 10.2514/1.G000391
    [200] M. Brentari, S. Urbina, D. Arzelier, C. Louembet, L. Zaccarian, A hybrid control framework for impulsive control of satellite rendezvous, IEEE Trans. Control Syst. Technol., 27 (2019), 1537–1551. https://doi.org/10.1109/ACC.2016.7526843 https://doi.org/10.1109/TCST.2018.2812197
    [201] G. Deaconu, C. Louembet, A. Thˊeron, A two-impulse method for stabilizing the spacecraft relative motion with respect to a periodic trajectory, in Proceedings of the 51st IEEE Conference on Decision and Control (CDC), (2012), 6541–6546. https://doi.org/10.1109/CDC.2012.6426542
    [202] W. Fehse, Automated Rendezvous and Docking of Spacecraft, Cambridge University Press, 2003.
    [203] P. S. Rivadeneira, C. H. Moog, Impulsive control of single-input nonlinear systems with application to hiv dynamics, Appl. Math. Comput., 218 (2012), 8462–8474. https://doi.org/10.1016/j.amc.2012.01.071 doi: 10.1016/j.amc.2012.01.071
    [204] M. Legrand, E. Comets, G. Aymard, R. Tubiana, C. Katlama, B. Diquet, An in vivo pharmacokinetic/pharmacodynamic model for antiretroviral combination, HIV Clin. Trials, 4 (2003), 170–183. https://doi.org/10.1310/77YN-GDMU-95W3-RWT7 doi: 10.1310/77YN-GDMU-95W3-RWT7
    [205] P. S. Rivadeneira, C. H. Moog, Observability criteria for impulsive control systems with applications to biomedical engineering processes, Automatica, 55 (2015), 125–131. https://doi.org/10.1016/j.automatica.2015.02.042 doi: 10.1016/j.automatica.2015.02.042
  • This article has been cited by:

    1. V. A. Kuznetsov, Modern Methods for Numerical Simulation of Radiation Heat Transfer in Selective Gases (Review), 2022, 69, 0040-6015, 702, 10.1134/S0040601522080043
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3033) PDF downloads(279) Cited by(0)

Figures and Tables

Figures(2)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog