Processing math: 79%
Research article

Synchronization for discrete coupled fuzzy neural networks with uncertain information via observer-based impulsive control

  • This paper discussed the synchronization of impulsive fuzzy neural networks (FNNs) with uncertainty of information exchange. Since the data of neural networks (NNs) cannot be completely measured in reality, we designed an observer-based impulsive controller on the basis of the partial measurement results and achieved the purpose of reducing the communication load and the controller load of FNNs. In terms of the Lyapunov stability theory, an impulsive augmented error system (IAES) was established and two sufficient criteria to guarantee the synchronization of our FNNs system were obtained. Finally, we demonstrated the validity of the results by a numerical example.

    Citation: Weisong Zhou, Kaihe Wang, Wei Zhu. Synchronization for discrete coupled fuzzy neural networks with uncertain information via observer-based impulsive control[J]. Mathematical Modelling and Control, 2024, 4(1): 17-31. doi: 10.3934/mmc.2024003

    Related Papers:

    [1] Mashael S. Maashi, Yasser Ali Reyad Ali, Abdelwahed Motwakel, Amira Sayed A. Aziz, Manar Ahmed Hamza, Amgad Atta Abdelmageed . Anas platyrhynchos optimizer with deep transfer learning-based gastric cancer classification on endoscopic images. Electronic Research Archive, 2023, 31(6): 3200-3217. doi: 10.3934/era.2023162
    [2] Chetan Swarup, Kamred Udham Singh, Ankit Kumar, Saroj Kumar Pandey, Neeraj varshney, Teekam Singh . Brain tumor detection using CNN, AlexNet & GoogLeNet ensembling learning approaches. Electronic Research Archive, 2023, 31(5): 2900-2924. doi: 10.3934/era.2023146
    [3] Jinjiang Liu, Yuqin Li, Wentao Li, Zhenshuang Li, Yihua Lan . Multiscale lung nodule segmentation based on 3D coordinate attention and edge enhancement. Electronic Research Archive, 2024, 32(5): 3016-3037. doi: 10.3934/era.2024138
    [4] Yixin Sun, Lei Wu, Peng Chen, Feng Zhang, Lifeng Xu . Using deep learning in pathology image analysis: A novel active learning strategy based on latent representation. Electronic Research Archive, 2023, 31(9): 5340-5361. doi: 10.3934/era.2023271
    [5] Hui-Ching Wu, Yu-Chen Tu, Po-Han Chen, Ming-Hseng Tseng . An interpretable hierarchical semantic convolutional neural network to diagnose melanoma in skin lesions. Electronic Research Archive, 2023, 31(4): 1822-1839. doi: 10.3934/era.2023094
    [6] Yi Dong, Jinjiang Liu, Yihua Lan . A classification method for breast images based on an improved VGG16 network model. Electronic Research Archive, 2023, 31(4): 2358-2373. doi: 10.3934/era.2023120
    [7] Chengyong Yang, Jie Wang, Shiwei Wei, Xiukang Yu . A feature fusion-based attention graph convolutional network for 3D classification and segmentation. Electronic Research Archive, 2023, 31(12): 7365-7384. doi: 10.3934/era.2023373
    [8] Shixiong Zhang, Jiao Li, Lu Yang . Survey on low-level controllable image synthesis with deep learning. Electronic Research Archive, 2023, 31(12): 7385-7426. doi: 10.3934/era.2023374
    [9] Haijun Wang, Wenli Zheng, Yaowei Wang, Tengfei Yang, Kaibing Zhang, Youlin Shang . Single hyperspectral image super-resolution using a progressive upsampling deep prior network. Electronic Research Archive, 2024, 32(7): 4517-4542. doi: 10.3934/era.2024205
    [10] Ahmed Abul Hasanaath, Abdul Sami Mohammed, Ghazanfar Latif, Sherif E. Abdelhamid, Jaafar Alghazo, Ahmed Abul Hussain . Acute lymphoblastic leukemia detection using ensemble features from multiple deep CNN models. Electronic Research Archive, 2024, 32(4): 2407-2423. doi: 10.3934/era.2024110
  • This paper discussed the synchronization of impulsive fuzzy neural networks (FNNs) with uncertainty of information exchange. Since the data of neural networks (NNs) cannot be completely measured in reality, we designed an observer-based impulsive controller on the basis of the partial measurement results and achieved the purpose of reducing the communication load and the controller load of FNNs. In terms of the Lyapunov stability theory, an impulsive augmented error system (IAES) was established and two sufficient criteria to guarantee the synchronization of our FNNs system were obtained. Finally, we demonstrated the validity of the results by a numerical example.



    In this paper we consider a semilinear hyperbolic-parabolic system evolving on a finite planar network composed from nodes connected by m oriented arcs Ii,

    {tui+λixvi=0,tvi+λixui=uixϕiβivi,t0,xIi,i=1,...,m;tϕi=Dixxϕi+aiuibiϕi, (1.1)

    the system is complemented by initial, boundary and transmission conditions at the nodes (see Section 2).

    We are interested in the study of stationary solutions and asymptotic behaviour of global solutions of the problem.

    The above system has been proposed as a model for chemosensitive movements of bacteria or cells on an artificial scaffold [12]. The unknown u stands for the cells concentration, λv is the average flux and ϕ is the chemo-attractant concentration. In particular, the model turns out to be useful to describe the process of dermal wound healing, when the stem cells in charge of the reparation of dermal tissue (fibroblasts) create an extracellular matrix and move along it to fill the wound, driven by chemotaxis; tissue engineers use artificial scaffolds, constituted by a network of crossed polymeric threads, inserting them within the wound to accelerate the process (see [13,20,25]). In the above mathematical model, the arcs of the graph mimic the fibers of the scaffold; each of them is characterized by a tipical velocity λi, a friction coefficient βi, a diffusion coefficient Di, and a production rate ai and a degradation one bi; the functions ui,ϕi are the densities of fibroblasts and chemoattractant on each arc.

    Starting from the Keller-Segel paper [18] in 1970 until now, a lot of articles have been devoted to PDE models in domains of Rn for chemotaxis phenomena. The parabolic (or parabolic-elliptic) Patlak-Keller-Segel system is the most studied model [17,23,22]; in recent years, hyperbolic models have been introduced too, in order to avoid the unrealistic infinite speed of propagation of cells, occurring in parabolic models [8,9,14,23,24,1,15,16,11].

    In [11] the Cauchy and the Neumann problems for the system in (1.1), respectively in R and in bounded intervals of R, are studied, providing existence of global solutions and stability of constant states results.

    Recently an interest in these mathematical models evolving on networks is arising, due to their applications in the study of biological phenomena and traffic flows, both in parabolic cases [2,6,21] and in hyperbolic ones [10,7,26,12,3].

    We notice that the transmission conditions for the unknowns, at each inner node, which complement the equations on networks, are crucial characteristics of the model, since they are the coupling among the solution's components on each arc.

    Most of the studies carried out until now, consider continuity conditions at each inner node for the density functions [7,6,21]; nevertheless, the eventuality of discontinuities at the nodes seems a more appropriate framework to decribe movements of individuals or traffic flows phenomena [5].

    For these reason in [12], transmission conditions which link the values of the density functions at the nodes with the fluxes, without imposing any continuity, are introduced; these conditions guarantee the fluxes conservation at each inner node, and, at the same time, the m-dissipativity of the linear spatial differential operators, a crucial property in the proofs of existence of local and global solutions contained in that paper.

    In this paper we focus our attention on stationary solutions to problem (1.1) complemented by null fluxes boundary conditions and by the same transmission conditions of [12] (see next section and Section 3 in [12] for details). We consider acyclic networks and we prove the existence and uniqueness of the stationary solution with fixed mass of cells mi=1Iiui(x)dx, under the assumption that the mass is suitably small. If the quantity aibi does not vary with the index i, we easily show that such solution is a constant state on the whole network. We notice that, in the case of acyclic networks, although the transmission conditions do not set the continuity of the density u at the inner nodes, the fluxes conservation at those nodes and the boundary null fluxes conditions imply the absence of jumps discontinuities at the inner vertexes, for the component u of a stationary solution.

    For general networks and the parameters ai and bi in the same range as above, it is easy to show that, for any fixed mass, a stationary solution constant on the whole network exists and the constant values of the densities are determined by the mass. In this case we also obtain a uniqueness result: in the set of stationary solutions with small density u in H1-norm and fixed mass of cells, the constant state on the network is the unique element.

    Finally we study the large time behaviour of global solutions on general networks, when the ratio between ai and bi is constant. We consider initial data with fixed small mass, which are small perturbations of the constant state on the network with the same mass, then we prove that such state is the asymptotic profile of the solutions corresponding to the data. So, we point out that, for small global solutions to our problem, the discontinuities at the inner nodes vanish when t goes to infinity, since their asymptotic profiles are continuous functions on the whole network.

    The study of the asymptotic behaviour provide informations about the evolution of a small mass of individuals moving on a network driven by chemotaxis: suitable initial distributions of individuals and chemoattractant, for large time evolve towards constant distributions on the network, preserving the mass of individuals.

    We recall that the stability of the constant solutions to this system, considered on bounded interval in R, is studied in [11] and stationary solutions and asymptotic behaviour for a linear system of uncoupled conservation laws on network are studied in [19].

    Finally, in [3] the authors introduce a numerical scheme to approximate the solutions to the problem (2.1); in that paper transmission conditions are set for the Riemann invariants of the hyperbolic part of the system, w±i=12(ui±vi), and are equivalent to our ones for some choices of the transmission coefficients. The tests presented there, in the case of acyclic graph and dissipative transmission coefficients, show an asymptotic behaviour of the solutions which agrees with our theoretical results.

    The paper is organized as follows. In Section 2 we give the statement of the problem and, in particular, we introduce the transmission conditions, while in Section 3 we prove the results about existence and uniqueness of stationary solutions. The last section is devoted to study the asymptotic behaviour of solutions; the results obtained in this section constitute the sequel and the development of the result of existence of global solutions in [12] and the proofs are based on the same techniques and use simple modifications of the a priori estimates obtained in [12].

    We consider a finite connected graph G=(V,A) composed by a set V of n nodes (or vertexes) and a set A of m oriented arcs, A={Ii:iM={1,2,...,m}}.

    Each node is a point of the plane and each oriented arc Ii is an oriented segment joining two nodes.

    We use ej, jJ, to indicate the external vertexes (or boundary vertexes) of the graph, i.e. the vertexes belonging to only one arc, and by Ii(j) the external arc outgoing or incoming in the external vertex ej.

    Moreover, we use Nν, νP, to denote the inner nodes; for each of them we consider the set of incoming arcs Aνin={Ii:iIν} and the set of the outgoing ones Aνout={Ii:iOν}; let Mν=IνOν.

    In this paper, a path in the graph is a sequence of arcs, two by two adjacent, without taking into account orientations. Moreover, we call acyclic a graph which does not contain cycles: for each couple of nodes, there exists a unique path with no repeated arcs connecting them (an example of acyclic graph is in Fig. 1).

    Figure 1. 

    Example of acyclic network; the highlighted arcs form the path linking the nodes N4 and N5.

    .

    Each arc Ii is considered as a one dimensional interval (0,Li). A function f defined on A is a m-tuple of functions fi, iM, each one defined on Ii; fi(Nν) denotes fi(0) if Nν is the initial point of the arc Ii and fi(Li) if Nν is the end point, and similarly for f(ej).

    We set Lp(A):={f:fiLp(Ii),iM}, Hs(A):={f:fiHs(Ii),iM} and

    f2:=iMfi2, fHs:=iMfiHs.

    We consider the evolution of the following one-dimensional problem on the graph G

    {tui+λixvi=0,tvi+λixui=uixϕiβivi,t0, xIi, iM,tϕi=Dixxϕi+aiuibiϕi, (2.1)

    where ai0,λi bi,Di,βi>0.

    We complement the system with the initial conditions

    ui0,vi0H1(Ii), ϕi0H2(Ii)  for iM ; (2.2)

    the boundary conditions at each outer point ej are the null flux conditions

    vi(j)(ej,t)=0,t>0, jJ, (2.3)
    ϕi(j)x(ej,t)=0t>0, jJ. (2.4)

    In addition, at each inner node Nν we impose the following transmission conditions for the unknown ϕ

    {Diϕix(Nν,t)=jMνανij(ϕj(Nν,t)ϕi(Nν,t)),    iIν, t>0,Diϕix(Nν,t)=jMνανij(ϕj(Nν,t)ϕi(Nν,t)),  iOν, t>0,ανij0, ανij=ανji  for all i,jMν, (2.5)

    which imply the continuity of the flux at each node, for all t>0,

    iIνDiϕix(Nν,t)=iOνDiϕix(Nν,t).

    For the unknonws v and u we impose the transmission conditions

    {λivi(Nν,t)=jMνKνij(uj(Nν,t)ui(Nν,t)), iIν, t>0,λivi(Nν,t)=jMνKνij(uj(Nν,t)ui(Nν,t)),   iOν, t>0,Kνij0, Kνij=Kνji  for all i,jMν .  (2.6)

    These conditions ensure the conservation of the flux of the density of cells at each node Nν, for t>0,

    iIνλivi(Nν,t)=iOνλivi(Nν,t),

    which corresponds to the conservation of the total mass

    iMIiui(x,t)dx=iMIiu0i(x)dx,

    i.e. no death nor birth of individuals occours during the observation.

    Motivations for the constraints on the coefficients in the transmission conditions can be found in [12].

    Finally, we impose the following compatibility conditions

    ui0,vi0,ϕi0 satisfy conditions (2.3)-(2.6) for all iM . (2.7)

    Existence and uniqueness of local solutions to problem (2.1)-(2.7),

    u,vC([0,T];H1(A))C1([0,T];L2(A)),ϕC([0,T];H2(A))C1([0,T];L2(A)) 

    are achieved in [12] by means of the linear contraction semigroup theory coupled with the abstract theory of nonhomogeneous and semilinear evolution problems; in fact, the transmission conditions (2.5) and (2.6) allows us to prove that the linear differential operators in (2.1) are m-dissipative and then, to apply the Hille-Yosida-Phillips Theorem (see [4]). The existence of global solutions when the initial data are small in (H1(A))2×H2(A) norm is proved [12] too; this result holds under the further assumption

     for all νP, for some kMν,Kνik0 for all iMν,ik . (2.8)

    In this section we approach the question of existence and uniqueness of stationary solutions of problem (2.1)-(2.8), with fixed mass

    μ:=iMIiui(x)dx0,

    in the case of an acyclic network (see Section 2). We look for stationary solutions (u,v,ϕ)(H1(A))2×H2(A).

    Obviously, the flux v of a stationary solution has to be constant on each arc and has to be null on the external arcs; in the case of acyclic networks, the boundary and transmission conditions (2.3), (2.6) force it to be null on each arc. In order to prove this fact we consider an internal arc Ij and its initial node Nμ; we consider the set

    Q={νP:Nν is linked to Nμ by a path not covering Ij} 

    (see Fig. 1: if, for example, j=9 then μ=4, Q={1,2,3,5} and the arcs in bold type form the path which links the nodes N5 and N4).

    At each node the conservation of the flux of the density of cells, stated in Section 2, holds; then

    νQ{μ}(iIνλivi(Nν)iOνλivi(Nν))=0 .

    Since, for all iM, vi(x) is constant on Ii and vi(x)=0 if Ii is an external arc, the above equality reduces to

    vj(Nμ)=0 ;

    then vj(x)=0 for all xIj.

    The previous result implies that stationary solutions must have the form (u,0,ϕ), where u and ϕ have to verify the system

    {λiuix=uiϕix,Diϕixx+biϕi=aiui,xIi, iM, t>0,  (3.1)

    with the boundary condition at each outer point ej, jJ,

    ϕi(j)x(ej,t)=0t>0, (3.2)

    and the transmission conditions, at each inner node Nν,

    jMνKνij(uj(Nν)ui(Nν))=0,iMν, (3.3)
    Diϕix(Nν)=jMνανij(ϕj(Nν)ϕi(Nν)),iIν,Diϕix(Nν)=jMνανij(ϕj(Nν)ϕi(Nν)),iOν. (3.4)

    For each fixed inner node Nν, let kMν be the index in condition (2.8) and let consider the transmission relations, for iMν, ik,

    0=jMν,jiKνij(uj(Nν)ui(Nν))=jMν,ji,kKνij(uj(Nν)uk(Nν))(jMν,jiKνij)(ui(Nν)uk(Nν)) ; (3.5)

    the assumptions on Kνkj in (2.8) ensure that the matrix of the coefficients of this linear system in the unknowns (uj(Nν)uk(Nν)), jk, is non singular (if k=1 it is immediate to check that it has strictly dominant diagonal). Then the condition (3.3) can be rewritten as

    uj(Nν)=uk(Nν) for all jMν.

    Now we fix μ00 and we look for stationary solutions such that

    iMIiui(x)dx=μ0; (3.6)

    notice that for the evolution problem, the quantity iMIiui(x,t)dx is preserved for all t0, thanks to the transmission conditions (2.6).

    Integrating the first equation in (3.1) we can rewrite problem (3.1)-(3.6) as the following elliptic problem on network:

    Find C=(C1,C2,...,Cm) and ϕH2(A) such that

    {Diϕixx+biϕi=aiuixIi,iM,ui(x)=Ciexp(ϕi(x)λi)xIi,iM,ϕi(j)x(ej)=0,jJ,Diϕix(Nν)=jMνανij(ϕj(Nν)ϕi(Nν)),iIν,νP,Diϕix(Nν)=jMνανij(ϕj(Nν)ϕi(Nν)),iOν,νP,Cjexp(ϕj(Nν)λj)=Ciexp(ϕi(Nν)λi),i,jMν,νP,iMCiIiexp(ϕi(x)λi)dx=μ0. (3.7)

    We consider the linear operator A:D(A)L2(A),

    D(A)={ϕH2(A):(3.2),(3.4) hold },A(ϕ)={Diϕixx+biϕi}iM ; (3.8)

    then the equation in (3.7) and the boundary and transmission conditions for ϕ can be written as

    Aϕ=F(ϕ,C), (3.9)

    where, for iM, Fi(ϕ(x),C)=aiCiexp(ϕi(x)λi).

    We are going to prove the existence and uniqueness of solutions to the problem (3.7) by using the Banach Fixed Point Theorem; in order to do this we need some preliminary results about the linear equation

    Aϕ=F(f,Cf), (3.10)

    where fH2(A) is a given function, Cf=(Cf1,...,Cfm) and Cfi are non-negative given real constants.

    The existence and uniqueness of the solution ϕH2(A) to the above problem (for a general FL2(A) and a general network) is showed in [12], by Lax-Milgram theorem, in the proof of Proposition 4.1; here, we need to prove some properties holding for the solution in the case of acyclic graphs, under suitable assumptions on f and Cfi.

    The transmission conditions (2.5) imply the following equality which will be useful in the next proofs:

    iMIiDi(ϕi(x)ϕix(x))x dx=νP(iIνDiϕi(Nν)ϕix(Nν)iOνDiϕi(Nν))ϕix(Nν))=νP ijMνανijϕi(Nν)(ϕj(Nν)ϕi(Nν)) =12νP ijMνανij(ϕj(Nν)ϕi(Nν))2 . (3.11)

    Let |A|:=iM|Ii| and g:=max{gi,iM}.

    Lemma 3.1. Let G=(V,A) be an acyclic network, let fH2(A) and let Cfi be non-negative real numbers, for iM. Then the solution ϕ to problem (3.8), (3.10) is non-negative. Moreover, if

    iMCfiIiexp(fi(x)λi)dx=μ0, (3.12)

    then

    ϕx2max{ai}iMmin{Di}iMμ0; (3.13)

    if (3.12) holds and

    fx2max{ai}iMmin{Di}iMμ0, (3.14)

    then there exists a quantity Kμ0=Kμ0(ai,bi,Di,λi,|A|,μ0), depending only on the parameters appearing in brackets, infinitesimal when μ0 goes to zero, such that

    ϕW2,1(A),ϕH2(A)Kμ0 . (3.15)

    Proof. Let consider a function ΓC1(R), strictly increasing in (0,+), and let Γ(y)=0 for y0; following standard methods for the proofs of the maximum principle for elliptic equations and setting Fi(x)=Fi(f(x),Cf), we obtain

    iMIi(Di(ϕix(x)Γ(ϕi(x)))xDiΓ(ϕi(x))ϕ2ix(x)+biϕi(x)Γ(ϕi(x))Fi(x)Γ(ϕi(x))) dx=0 .

    As regard to the first term, we can argue as in (3.11), taking into account the properties of Γ,

    iMIiDi(Γ(ϕi)ϕix)x=12νP ijMνανij(ϕj(Nν)ϕi(Nν))(Γ(ϕj(Nν))Γ(ϕi(Nν)))0 ; (3.16)

    the above inequality and the non-negativity of Fi imply that

    iMbiIiϕi(x)Γ(ϕi(x))dx0,

    so that, thanks to the properties of Γ, we can conclude that ϕi(x)0 for all iM.

    By integration of the equation (3.10), taking into account (3.4) and (3.2), we obtain

    iMbiIiϕi(x)dx=iMIiFi(f(x),Cf)dx (3.17)

    which implies

    ϕ1max{ai}min{bi}μ0 . (3.18)

    In order to obtain (3.13), first we notice that, if Ij is an external arc, then the following inequality holds

    |Djϕjx(x)|IjDj|ϕjyy(y)|dyIj(bjϕj(y)+Cfjajexp(fj(y)λj))dy.

    Then we consider an internal arc Ij and its initial node Nμ and the sets

    Q={νP:Nν is linked to Nμ by a path not covering Ij},
    S={iM:Ii is incident with Nl for some lQ} 

    (see Fig. 1: if, for example, j=9, then μ=4, Q={1,2,3,5}, S={1,2,3,4,5,6,8, 10,11,12}); at each node the conservation of the flux, stated in Section 2 as a consequence of the transmission conditions, holds; then

    νQ{μ}(iIνDiϕix(Nν)iOνDiϕix(Nν))=0 .

    Let x be a point on the arc Ij (see Fig. 1 for j=9, μ=4) and Ixj be the part of Ij which connects Nμ and x; then, using the above equality and the boundary conditions (2.4), we have

    |Djϕjx(x)|=|Djϕjx(x)+νQ{μ}(iIνDiϕix(Nν)iOνDiϕix(Nν))||iSIiDiϕiyy(y)dy+IxjDjϕjyy(y)dy||iSIi(biϕi(y)Cfiaiexp(fi(y)λi))dy+Ixj(bjϕj(y)Cfjajexp(fj(y)λj))dy|. (3.19)

    Then Djϕjx2μ0max{ai}iM for all jM and we obtain (3.13) which implies

    ϕx12max{ai}iMmin{Di}iM|A|μ0, (3.20)

    and

    ϕx22max{ai}iMmin{Di}iM|A|12μ0 ; (3.21)

    moreover, by Sobolev embedding theorem, we obtain

    ϕK1μ0, (3.22)

    where K1=K1(ai,bi,Di,|A|) is a suitable constant.

    The estimates for the function ϕxx follow by using the equation (3.10); first, using (3.12) and (3.17), we obtain

    ϕxx12max{ai}iMmin{Di}iMμ0 ; (3.23)

    then, using (3.11), we have

    iMD2ibiIiϕixx2(x) dxiMFibiIiFi(x) dxmax{aiFi}min{bi}μ0 

    and the embedding of W1,1(Ii) in L(Ii) gives

    iMIiϕixx2(x) dxK2(1+fx)μ20, (3.24)

    where K2=K2(ai,bi,Di,λi) is a suitable constant.

    Finally, the inequalities (3.18), (3.20)-(3.24) imply the inequalities (3.15) in the claim.

    Now we can prove the following theorem.

    Theorem 3.1. Let G=(V,A) be an acyclic network. There exists ϵ>0 such that, if 0μ0ϵ, then problem (2.1)-(2.8) has a unique stationary solution satisfying (3.6); the solution has the form

    (Ciexp(ϕi(x)λi),0,ϕi(x))  iM,

    where ϕi(x)0 and Ci are nonnegative constants such that uj(Nν)=ui(Nν) for all νP, i,jMν.

    Proof. First we notice that, if a stationary solution (u,v,ϕ) satisfying (3.6) exists, then u is non-negative, since the costants Ci in (3.7) must have the same sign, so that they have to be non-negative to satisfy the condition μ00; arguing as in the proof of Lemma 3.1 we prove that ϕ is non-negative too. If μ0>0 then u and ϕ are positive functions.

    We are going to use a fixed point technique. Given ϕ0D(A), we want to define a function u0(x) on the network, such that, for iM,

    u0i(x)=Cϕ0iexp(ϕ0i(x)λi),

    where the constants Cϕ0i satisfy the following linear system composed by the last conditions in (3.7)

    Cϕ0jexp(ϕ0j(Nν)λj)=Cϕ0iexp(ϕ0i(Nν)λi),i,jMν,νP, (3.25)
    iMCϕ0iIiexp(ϕ0i(x)λi)dx=μ0 . (3.26)

    The system (3.25), (3.26) has a unique solution; actually, since the network has no cycles, the system (3.25) has 1 solutions Cϕ0,α=(α,αδ2,αδ3,...,αδm), αR, where δi are suitable coefficients, and the condition (3.26) determines the value of α.

    In order to give an explicit expression for the coefficients δi we consider an arc, I1, and we define

    u01(x):=αexp(ϕ01(x)λ1) .

    Let Nμ one of the extreme points of I1, then we define the function u0 on the other arcs which are incident with Nμ in such a way to verify the equalities in (3.25) for the node Nμ,

    u0j(x):=αexp(ϕ01(Nμ)λ1)exp(ϕ0j(Nμ)λj)exp(ϕ0j(x)λj)  for all jMμ,j1;

    i.e. we set Cϕ0j=αexp(ϕ01(Nμ)λ1)exp(ϕ0j(Nμ)λj), jMμ,j1.

    This procedure can be iterated at each node reached by one of the arcs Ij, jMμ, and at the other extreme point of I1, if it is an internal arc, and so on, defining in this way the function u0 on each arc of the network. Notice that this construction is possible since there are no cycles in the graph. The function u can be expressed, on each arc of the network, as follows (if it is the case, renumbering in suitable way the arcs and the nodes): let consider the path from the outer point e1 to an inner node Nh1, composed from the arcs Ii, i=1,...,h1, (passing through the vertexes Ni, i=1,...,h1), and let Ih be an arc incident with the node Nh1, not belonging to the path (see Fig. 2 where h = 5 and the highlighted arcs form the path); following the procedure described before, after setting

    Eh(ϕ0):=Πi=1,...,h1exp(ϕ0i(Ni)λi)Πi=1,...,h1exp(ϕ0i+1(Ni)λi+1),
    Figure 2. 

    Example: the highlighted arcs form the path from the outer point e1 to the inner node N4 and I5 is an arc incident with N4, not belonghing to the path.

    .

    we define

    u0h(x):=αEh(ϕ0)exp(ϕ0h(x)λh) .

    The quantity α is fixed in such a way to verify the last condition in (3.7),

    αiMEi(ϕ0)Iiexp(ϕ0i(x)λi)dx=μ0,

    so that, for all iM,

    u0i(x)=Cϕ0iexp(ϕ0i(x)λi),  Cϕ0i:=μ0 Ei(ϕ0) jMEj(ϕ0)Ijexp(ϕ0j(x)λj)dx . (3.27)

    Let G be the operator defined in D(A) such that ϕ1:=G(ϕ0) is the solution of problem (3.10) where f=ϕ0 and Cfi=Cϕ0i for iM,

    Aϕ1=F(Cϕ0,ϕ0) ;

    let Kμ0=Kμ0(ai,bi,Di,λi,|A|,μ0) be the quantity in Lemma 3.1, and let

    Bμ0:={ϕD(A):ϕ0,ϕx2max{ai}iMmin{Di}iMμ0,ϕH2Kμ0 }

    equipped with the distance d generated by norm of H2(A); (Bμ0,d) is a complete metric space. From the lemma we know that solutions to problem (3.8)-(3.12) have to belong to Bμ0, then G(Bμ0)Bμ0; next we are proving that, if μ0 is small enough, then G is a contraction in Bμ0.

    We consider ϕ0,¯ϕ0Bμ0 and the corresponding u0,¯u0 and ϕ1,¯ϕ1; using the equation satisfied by ϕ0 and ¯ϕ0, for all iM we can write

    biIi(ϕ1i(x)¯ϕ1i(x))2,dx+DiIi(ϕ1ix(x)¯ϕ1ix(x))2,dxDiIi((ϕ1ix(x)¯ϕ1ix(x))(ϕ1i(x)¯ϕ1i(x)))xdx=aiIi(u0i(x)¯u0i(x))(ϕ1i(x)¯ϕ1i(x))dx ; (3.28)

    using (3.11), from (3.28) we infer that

    iMϕi1¯ϕi1H2K(ai,bi,Di)iMu0i¯u0i2, (3.29)

    We set

    Jϕ0i:=Iiexp(ϕ0i(x)λi)dx,Eϕ0i(x):=exp(ϕ0i(x)λi)  ;

    we have

    |u0i(x)ˉu0i(x)|=μ0|εi(ϕ0)Eϕ0i(x)jMεj(ϕ0)Jϕ0jεi(ˉϕ0)Eˉϕ0i(x)jMεj(ˉϕ0)Jˉϕ0j|. (3.30)

    In order to treat the above quantity we have to consider that, for all gBμ0, Egi(x)1, Jgi|Ii| and there exists a constant K6=K6(Kμ0,λi), increasing with μ0, such that, for all iM

    maxIiEgi(x)K6,JgiK6|Ii|,
    |Eϕ0i(x)Eˉϕ0i(x)|K6|ϕ0i(x)ˉϕ0i(x)|,
    |Jϕ0iJˉϕ0i|K6Ii|ϕ0i(x)ˉϕ0i(x)|dx.

    The above inequalities can be used in (3.30) so that (3.29) implies

    iMϕi1¯ϕi1H2μ0K7(ai,bi,Di,Kμ0,|A|)iMϕ0i¯ϕ0iH1, (3.31)

    where K7 increases with μ0; hence, for μ0 small enough, G is a contraction on Bμ0 and we can use the Banach Fixed Point Theorem.

    Let ϕ be the unique fixed point of G in Bμ0 and let Cϕ=(Cϕ1,Cϕ2,...,Cϕm) where Cϕi, for iM, are computed as in (3.27); then (ϕ,Cϕ) is the unique solution to Problem (3.7) and the claim is proved.

    For any constant U0, the triple (U,0,aibiU) satisfies the equations in (2.1) on the arc Ii. Let Q be a real non-negative number; if aibi=Q for all iM, then the same triple (U,0,QU) satisfies the equations on each arc Ii and it is a stationary solution to the problem (2.1)-(2.8). Then, as a consequence of the previous theorem, we have the following proposition, with ϵ as in the theorem.

    Proposition 3.1. Let G=(V,A) be an acyclic network. If aibi=Q for all iM and 0μ0ϵ, then the unique stationary solution to problem (2.1)-(2.8), (3.6) is the constant solution (μ0|A|,0,Qμ0|A|).

    Remark 3.1. For general networks, when the value of aibi=Q on each arc, the stationary solution of Proposition 3.1 always exists. More precisely, if aibi=Q, in the class of the functions (u,v,ϕ) which are constant on each arc, the stationary solution (μ0|A|,0,Qμ0|A|) is the unique stationary solution with mass μ0; this fact is true without any restrictions on the value of μ0 and on the structure of the network. Actually, if we assume that u is constant on each arc, then, using the equations, we infer that, on each arc, ϕx(x) is constant too, hence ϕxx=0 and ϕ(x) is constant. Then v(x)=0 on each arc; hence, arguing as at the beginning of this section, we obtain that u is continuous on the network.

    In the next proposition we are going to prove that, in a set of small solutions, such stationary solution is the unique one with fixed mass μ0.

    Proposition 3.2. Let aibi=Q for all iM and let (u,v,ϕ)(H1(A))2×H2(A) be a stationary solution of problem (2.1)-(2.8), (3.6). There exists ϵ0>0, depending on λi,ai,bi,Di,βi,|A|, such that, if uH1ϵ0, then (u,v,ϕ)=(μ0|A|,0,Qμ0|A|).

    Proof. We set H:=uH1. The transmission conditions (2.6) imply that

    νP(iIνλiui(Nν)vi(Nν)iOνλiui(Nν)vi(Nν))0,

    so, by using the first two equations in (2.1), we obtain

    2iMβiIiv2i(x)dxiMuiIi(v2i(x)+ϕ2ix(x)) dx

    and

    iMλiIiu2ix(x) dxiMuiIi(u2ix(x)+ϕ2ix(x)) dx +iMβ2iλiIiv2i(x) dx ;

    the above inequalities implies the following one

    v22+ux22K0H(ϕx22+v22+ux22), (3.32)

    where K0 is a positive constant depending on the parameters λi,βi and the Sobolev embedding costant.

    The transmission conditions (2.5) imply that

    νP(iIνDiϕi(Nν)ϕix(Nν)iOνDiϕi(Nν)ϕix(Nν))0 ;

    moreover, the assumption (2.8) imply that, for each νP, for suitable coefficients θνij and suitable kMν,

    uj(Nν)=uk(Nν)+iMν,ikθνijvi(Nν) for all jMν,

    (see Lemma 5.9 in [12]); then, by the last equation in (2.1), arguing as in the proof of Proposition 5.8 in [12], we obtain

    ϕx22+ϕxx22K1(v22+ux22), (3.33)

    where K1 is a positive constant depending on the parameters Di,ai,bi,θνij.

    By inequalities (3.32) and (3.33) we deduce the following one

    v22+ux22K0(1+K1)H(v22+ux22),

    which, for small H, implies v2,ux2=0.

    In the cases when aibi depends on the arc in consideration, stationary solutions with the component u constant on each arc, can be inadmissible. As we showed before, v should be zero, u should be constant on the whole network and ϕ should be constant on each arc,

    ui(x)=μ0|A|,   ϕi(x)=aibiμ0|A|,iM .

    Therefore the transmission conditions, for each νP,

    jMνανijμ0|A|(ajbjaibi)=0,iMν,

    are constraints on the relations between the parameters of the problem which have to hold if the constant stationary solution exists.

    For example, in the case of two arcs, if b2a2b1a1 (and 0<μ0ϵ), the stationary solution can not be constant on the arcs, since the trasmission condition at the node,

    α11μ0|A|(b2a2b1a1)=0,

    cannot be satisfied.

    Hence, in the cases when aibi depends on the arc in consideration, if (u,v,ϕ) is the stationary solution in Theorem 3.1, then u is a continuous function on all the network but it is not constant on each arc.

    In this section we are going to show that the constant stationary solutions previously introduced, provide the asymptotic profiles for a class of solutions to problem (2.1)-(2.8). We recall that existence and uniqueness of global solutions

    u,vC([0,+);H1(A))C1([0,+);L2(A)),ϕC([0,+);H2(A))C1([0,+);L2(A)),ϕxH1(A×(0,+)), (4.1)

    to such problem is proved in [12], when the initial data are sufficiently small in (H1(A))2×H2(A) norm and the following condition holds

    aibi=Q  for all iM ; (4.2)

    in particular it is proved that the functional F defined by

    F2T(u,v,ϕ):=iM(supt[0,T]ui(t)2H1+supt[0,T]vi(t)2H1+supt[0,T]ϕix(t)2H1)+T0(ux(t)22+v(t)2H1+vt(t)22+ϕx(t)2H1+ϕxt(t)22) dt (4.3)

    is uniformly bounded for T>0.

    Here and below we use the notations

    fi(t)2:=fi(,t)L2(Ii),  fi(t)Hs:=fi(,t)Hs(Ii) .

    Now we assume (4.2), we fix ¯μ0 and we consider the constant stationary solution, (¯u,0,¯ϕ), to problem (2.1)-(2.8), such that ¯u|A|=¯μ; moreover let (˜u0,˜v0,˜ϕ0)(H1(A))2×H2(A) be a small perturbation of (¯u,0,¯ϕ), i.e., setting u0:=˜u0¯u,v0:=˜v0,ϕ0:=˜ϕ0¯ϕ , the (H1(A))2×H2(A) norm of (u0,v0,ϕ0) is bounded by a suitable small ϵ0>0.

    If (˜u,˜v,˜ϕ) is the solution to problem (2.1)-(2.8) with initial data (˜u0,˜v0,˜ϕ0) and u:=˜u¯u,v:=˜v,ϕ:=˜ϕ¯ϕ , then (u,v,ϕ) is solution to the system

    {tui+λixvi=0tvi+λixui=(ui+¯u)xϕiβivixIi,t0,iM,tϕi=Dixxϕi+aiuibiϕi, (4.4)

    complemented with the conditions (2.2)-(2.8) and initial data (u0,v0,ϕ0) defined above.

    The existence and uniqueness of local solutions to this problem can be achieved by means of semigroup theory, following the method used in[12], with little modifications.

    On the other hand, if we assume that ¯u is suitably small, the method used in that paper to obtain the global existence result in the case of small initial data can be used here too, modifying the estimates in order to treat the further term in the second equation and then using the smallness of ¯u.

    Below we list a priori estimates holding for the solutions to problem (4.4), (2.2)-(2.7); we don't give the proofs since they are equal to those in [12], in Section 5, except for easy added calculations to treat the term ¯uϕix.

    Proposition 4.1. Let (u,v,ϕ) be a local solution to problem (4.4), (2.2)-(2.7),

    u,vC([0,T];H1(A))C1([0,T];L2(A)),ϕC([0,T];H2(A))C1([0,T];L2(A)), ϕxH1(A.×(0,T)) ;

    then

    a)

    iM(sup[0,T]ui(t)22+sup[0,T]vi(t)22+βiT0vi(t)22dt)CiM(u0i22+v0i22)+CiM(sup[0,T]ui(t)H1+ˉu)To(ϕix(t)22+vi(t)22)dt;

    b)

    iM(sup[0,T]vix(t)22+sup[0,T]vit(t)22+T0vit(t)22,dt)C(v02H1+u02H1ϕ02H2)+CiM(sup[0,T]ui(t)H1+ˉu)T0(ϕixt(t)22+vit(t)22) dt+CiMsup[0,T]ϕx(t)H1T0(vit(t)22+vi(t)2H1)dt ;

    c)

    iMsup[0,T]uix(t)22CiM(sup[0,T]vit(t)22+sup[0,T]vi(t)22)+CiM(sup[0,T]ui(t)H1+ˉu)(sup[0,T]uix(t)22+sup[0,T]ϕix(t)22) ;

    d)

    iMT0uix(t)22,dtCiMT0(vit(t)22+vi(t)22)dt+CiM(sup[0,T]ui(t)H1+ˉu)T0(uix(t)22+ϕix(t)22)dt ;

    e)

    iMT0vix(t)22,dtCiM(v0i22+u0i2H1(1+ϕ0i2H1))+CiM(T0vit(t)22,dt+sup[0,T]vit(t)22)+CiM(sup[0,T]ui(t)H1+sup[0,T]ϕix(t)H1+ˉu)×T0(vi(t)2H1+ϕixt(t)22)dt ;

    f)

    iM(sup[0,T]ϕit(t)22+T0(ϕit(t)22+ϕitx(t)22) dt)CiM(ϕ0i2H2+u0i22+T0uit(t)22) ;

    g)

    iM(sup[0,T]ϕixx(t)22+sup[0,T]ϕix(t)22)CiM(sup[0,T]ϕit(t)22+sup[0,T]ui(t)22) ; 

    h) if (2.8) and (4.2) hold, then

    \begin{array}{*{35}{l}} {} & \sum\limits_{i\in \mathcal{M}}{\int_{0}^{T}{\left( \|{{\phi }_{{{i}_{x}}}}(t)\left. {} \right\|_{2}^{2}+\|{{\phi }_{{{i}_{xx}}}}(t)\left. {} \right\|_{2}^{2} \right)}}\ dt \\ {} & {} \\ \le & C\sum\limits_{i\in \mathcal{M}}{\int_{0}^{T}{\left( \|{{u}_{{{i}_{x}}}}(t)\left. {} \right\|_{2}^{2}+\|{{v}_{i}}(t)\left. {} \right\|_{{{H}^{1}}}^{2}+\|{{\phi }_{{{i}_{t}}}}(t)\left. {} \right\|_{2}^{2} \right)}}\ dt, \\ \end{array}

    for suitable costants C.

    The estimates in the previous proposition allow to prove the following theorem about the existence of global solutions to problem (4.4), (2.2)-(2.8).

    Let F_T(u, v, \phi) be the functional defined in (4.3).

    Theorem 4.1. Let (4.2) hold. There exists \epsilon_0, \epsilon_1>0 such that, if

    \overline {u}\leq \epsilon_1, \ \ \Vert u_0\Vert_{H^1}, \Vert v_0\Vert_{H^1}, \Vert \phi_{0}\Vert_{H^2}\leq \epsilon_0 ,

    then there exists a unique global solution (u, v, \phi) to problem (4.4), (2.2)-(2.8),

    u, v \in C([0, +\infty);H^1(\mathcal{A}))\cap C^1([0, +\infty);L^2(\mathcal{A})) ,
    \phi\in C([0, +\infty);H^2(\mathcal{A}))\cap C^1([0, +\infty);L^2(\mathcal{A})), \ \phi_{x}\in H^1\left( \mathcal A\times (0, +\infty)\right) \ .

    Moreover, F_T(u, v, \phi) is bounded, uniformly in T.

    Proof. It is sufficient to prove that the functional F_T(u, v, \phi) is bounded, uniformly in T.

    We notice that each term in F_T^2(u, v, \phi) is in the left hand side of one of the estimates in Proposition 4.1, therefore, arranging all the estimates, we can prove the following inequality

    F_T^2(u, v, \phi)\leq c_1 F_0^2(u, v, \phi) +c_2 \overline {u} F^2_T(u, v, \phi) +c_3 F_T^3(u, v, \phi),

    taking into account also that, on the right hand side of the estimates, the quadratic terms (not involving initial data) which have not the coefficient \overline {u}, can be bounded by means of cubic ones.

    If \overline {u} is sufficiently small, the previous inequality implies

    F_T^2(u, v, \phi)\leq c_4 F_0^2(u, v, \phi) +c_5 F_T^3(u, v, \phi)

    for suitable positive constants c_i.

    It is easy to verify that, if y_0 is a sufficiently small positive real number and h(y) = c_5 y^3-y^2 +c_4 y_0 then there exists 0<\overline y<\frac 2 {3 c_5} such that h(y)>0 in [0, \overline y) and h(y)<0 in (\overline y, \frac 2 {3 c_5}].

    Then we can conclude that, if F_0 is suitably small, then F_T remains bounded for all T>0.

    The above result, in particular the uniform, in time, boundedness of the functional F_T, allow us to prove the theorem below.

    Let (4.2) hold and let (\overline {u}, 0, \overline {\phi}) be the constant stationary solution to problem (2.1)-(2.8) such that \overline {u} |\mathcal{A}| = \overline \mu; moreover, let \mathcal C(\mathcal{A}) be the set of the funcions f such that f_i\in\mathcal C(\overline I_i) for i\in \mathcal{M}.

    Theorem 4.2. Let (4.2) hold. There exist \epsilon_0, \epsilon_2>0 such that, if

    \overline {u} \leq \epsilon_2, \;\;\;\sum\limits_{i \in \mathcal{M}}\int_{{I_i}} u_0(x) = \overline \mu, \;\;\;\Vert( u_0-\overline {u}, v_0, \phi_0-\overline {\phi})\Vert_{(H^1)^2\times H^2} \leq \epsilon_0,

    then problem (2.1)-(2.8) has a unique global solution (u, v, \phi),

    u, v \in C([0, +\infty);H^1(\mathcal{A}))\cap C^1([0, +\infty);L^2(\mathcal{A})) ,
    \phi\in C([0, +\infty);H^2(\mathcal{A}))\cap C^1([0, +\infty);L^2(\mathcal{A})),

    and, for all i\in\mathcal{M},

    \lim\limits_{t\to+\infty}\Vert {u_i}(\cdot, t)-\overline {u}\Vert_{C(\overline I_i)}, \lim\limits_{t\to+\infty}\Vert {v_i}(\cdot, t)\Vert_{C(\overline I_i)}, \lim\limits_{t\to+\infty}\Vert {\phi _i}(\cdot, t)-\overline {\phi}\Vert_{C^1(\overline I_i)} = 0\ .

    Proof. Let (u, v, \phi) be the local solution to problem (2.1)-(2.8) and let

    \hat u: = u - \bar u,\;\hat v: = v,\;\hat \phi : = \phi - \bar \phi ;

    we already noticed that (\hat u, \hat v, \hat \phi) is the local solution to system (4.4) complemented by the initial condition (u_0-\overline {u}, v_0, \phi_0-\overline {\phi}) and the same boundary and transmission condition given by (2.3)-(2.8) for system (2.1).

    For suitable \epsilon_0, \epsilon_2 the assumptions of Theorem 4.1 are satisfied, then we obtain the uniform boundedness of the functional F_T(\hat u, \hat v, \hat \phi), for T\in [0, +\infty). Hence the set \{\hat u(t), \hat v(t), \hat \phi(t)\}_{t\in[0, +\infty)} is uniformly bounded in (H^1(\mathcal{A}))^2\times H^2(\mathcal{A}); thus, if we call E_s the set of accumulation points of \{\hat u(t), \hat v(t), \hat \phi(t)\}_{t\geq s} in (C(\mathcal{A}))^2\times C^1(\mathcal{A}), then E_s is not empty and E: = \cap_{s\geq 0} E_s \neq \emptyset.

    Let (U(x), V(x), \Phi(x)) be such that, for a sequence t_n\to +\infty,

    \begin{array}{ll} \lim\limits_{n\to +\infty} \sum\limits_{i \in \mathcal{M}}\Vert \hat u_i(\cdot, t_n)- U_i(\cdot)\Vert_{C(\overline I_i)} = 0, \\ \\ \lim\limits_{n\to +\infty} \sum\limits_{i \in \mathcal{M}}\Vert \hat v_i(\cdot, t_n)- V_i(\cdot)\Vert_{C(\overline I_i)} = 0, \\ \\ \lim\limits_{n\to +\infty}\sum\limits_{i \in \mathcal{M}} \Vert \hat \phi_i(\cdot, t_n)- \Phi_i(\cdot)\Vert_{C^1(\overline I_i)} = 0\ . \end{array} (4.5)

    In order to identify the limit functions we notice that \sum\limits_{i \in \mathcal{M}} \int_{{I_i}} U_i(x) dx = 0, since \sum\limits_{i \in \mathcal{M}} \int_{{I_i}} \hat u(x, t_n), dx = 0 for all t_n.

    Moreover, since \hat v_i\in H^1(I_i\times (0, +\infty)) for all i\in\mathcal{M}, if we set

    \omega_i(t): = \Vert \hat v_i(t, \cdot)\Vert_{L^2(I_i)}

    then \omega_i\in H^1((0, +\infty)) and, as a consequence, \lim\limits_{t\to+\infty} \omega_i(t) = 0.

    As \lim\limits_{n\to +\infty}\Vert \hat v_i(\cdot, t_n)\Vert_2 = \Vert V _i(\cdot)\Vert_2, we obtain \Vert V\Vert_2 = 0.

    The same argument can be applied to the functions \hat \phi_{i_x}, for all i\in\mathcal{M}, since they belongs to H^1(I_i\times (0, +\infty)). Finally, it can be applied to the functions a_i\hat u_i-b_i\hat \phi_i since \hat \phi_{i_t}, \hat \phi_{i_{xx}}, \hat u_{i_x}, \hat \phi_{i_x}\in L^2(I_i\times (0, +\infty)), thanks to the uniform boundedness of F_T(\hat u, \hat v, \hat \phi) and to estimate f) in Proposition 4.1.

    As a consequence we have that

    V_i(x) = 0, \;\;\;\; a_i U_i(x)- b_i \Phi_i(x) = 0, \;\;\;\; \Phi_i(x) = \overline \Phi_i, \;\;\;\; \ x\in I_i,

    where \overline \Phi_i are real numbers, so that the limit function is given by (\frac {b_i}{a_i} \Phi_i, 0, \Phi_i) in each interval I_i, for all i\in\mathcal{M}. It is easily seen that such function is a stationary solution to problem (2.1)-(2.8), which is constant in each arc I_i; in particular it verifies the transmission conditions since (\hat u, \hat v, \hat \phi) verifies them and the convergence result (4.5) holds.

    The condition \sum\limits_{i \in \mathcal{M}} \int_{{I_i}} U_i(x) dx = 0 and Remark 3.1 imply that \Phi_i = 0 for all i\in\mathcal{M}, so that we can conclude that the unique possible limit function is (U(x), V(x), \Phi(x)) = (0, 0, 0); this fact proves the claimed convergence results.

    The main features of the present work are: a) the proof of the existence and uniqueness of stationary solutions with fixed small mass to problem (2.1)-(2.6) considered on acyclic networks; b) the proof of the stability of particular stationary solutions, the constant states on the whole network, when the their masses are small and the quantity \frac {a_i} {b_i} does not vary with the index i, for general networks. We can conclude that, in this range of parameters, although the transmission conditions do not impose continuity of the densities at the internal nodes, for suitable initial data the asymptotic profiles of the solutions are continuous functions, constant on the network.

    These results are useful in describing the large time behaviour of small masses of individuals moving on networks driven by chemotaxis.

    For the future, our aim is approaching the same questions when the system (2.1) is complemented by non-null fluxes conditions at the boundaries, which provide models for different situations at the outer nodes, in order to describe the features of the behaviour of cells moving along the arcs searching food. We notice that the condition v_i(x) = 0 for x\in{I_i}, for all i\in \mathcal{M}, which prevents the presence of jumps for the density u at the inner nodes, in this case is no longer necessary for stationary solutions on acyclic networks.



    [1] J. Buckley, Y. Hayashi, Fuzzy neural networks: a survey, Fuzzy Sets Syst., 66 (1994), 1–13. https://doi.org/10.1016/0165-0114(94)90297-6 doi: 10.1016/0165-0114(94)90297-6
    [2] X. Wang, Y. Yu, S. Zhong, K. Shi, N. Yang, D. Zhang, et al., Novel heterogeneous mode-dependent impulsive synchronization for piecewise T-S fuzzy probabilistic coupled delayed neural networks, IEEE Trans. Fuzzy Syst., 30 (2021), 2142–2156. https://doi.org/10.1109/TFUZZ.2021.3076525 doi: 10.1109/TFUZZ.2021.3076525
    [3] X. Wang, Y. Yu, K. Shi, H. Chen, S. Zhong, X. Yang, et al., Membership-mismatched impulsive exponential stabilization for fuzzy unconstrained multilayer neural networks with node-dependent delays, IEEE Trans. Fuzzy Syst., 31 (2022), 1214–1228. https://doi.org/10.1109/TFUZZ.2022.3197925 doi: 10.1109/TFUZZ.2022.3197925
    [4] X. Wang, Y. Yu, J. Cai, N. Yang, K. Shi, S. Zhong, et al., Multiple mismatched synchronization for coupled memristive neural networks with topology-based probability impulsive mechanism on time scales, IEEE Trans. Cybern., 53 (2021), 1485–1498. https://doi.org/10.1109/TCYB.2021.3104345 doi: 10.1109/TCYB.2021.3104345
    [5] A. Iwata, Y. Nagasaka, N. Suzumura, Data compression of the ECG using neural network for digital Holter monitor, IEEE Eng. Med. Biol. Mag., 9 (1990), 53–57. https://doi.org/10.1109/51.59214 doi: 10.1109/51.59214
    [6] M. Patricia, D. Sánchez, Multi-objective optimization for modular granular neural networks applied to pattern recognition, Inf. Sci., 460-461 (2018), 594–610. https://doi.org/10.1016/j.ins.2017.09.031 doi: 10.1016/j.ins.2017.09.031
    [7] Y. Zhao, X. He, T. Huang, J. Huang, P. Li, A smoothing neural network for minimization l_1-l_p in sparse signal reconstruction with measurement noises, Neural Networks, 122 (2020), 40–53. https://doi.org/10.1016/j.neunet.2019.10.006 doi: 10.1016/j.neunet.2019.10.006
    [8] M. R. G. Meireles, P. E. Almeida, M. G. Simes, A comprehensive review for industrial applicability of artificial neural networks, IEEE Trans. Ind. Electron., 50 (2003), 585–601. https://doi.org/10.1109/TIE.2003.812470 doi: 10.1109/TIE.2003.812470
    [9] G. A. Carpenter, Neural network models for pattern recognition and associative memory, Neural Networks, 2 (1989), 243–257. https://doi.org/10.1016/0893-6080(89)90035-X doi: 10.1016/0893-6080(89)90035-X
    [10] W. He, Y. Dong, Adaptive fuzzy neural network control for a constrained robot using impedance learning, IEEE Trans. Neural Networks Learn. Syst., 29 (2017), 1174–1186. https://doi.org/10.1109/TNNLS.2017.2665581 doi: 10.1109/TNNLS.2017.2665581
    [11] Q. Song, W. Yu, J. Cao, F. Liu, Reaching synchronization in networked harmonic oscillators with outdated position data, IEEE Trans. Cybern., 46 (2016), 1566–1578. https://doi.org/10.1109/TCYB.2015.2451651 doi: 10.1109/TCYB.2015.2451651
    [12] T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Syst. Man Cybern., 15 (1993), 116–132. https://doi.org/10.1109/TSMC.1985.6313399 doi: 10.1109/TSMC.1985.6313399
    [13] S. Selcuk, New stability results for Takagi-Sugeno fuzzy Cohen-Grossberg neural networks with multiple delays, Neural Networks, 114 (2019), 60–66. https://doi.org/10.1016/j.neunet.2019.02.010 doi: 10.1016/j.neunet.2019.02.010
    [14] A. Wu, Z. Zeng, Boundedness, Mittag-Leffler stability and asymptotical periodicity of fractional order fuzzy neural networks, Neural Networks, 74 (2016), 73–84. https://doi.org/10.1016/j.neunet.2015.11.003 doi: 10.1016/j.neunet.2015.11.003
    [15] Y. Xu, J. Li, R. Lu, C. Liu, Y. Wu, Finite-horizon l_2-l_\infty synchronization for time-varying Markovian jump neural networks under mixed-type attacks: observer-based case, IEEE Trans. Neural Networks Learn. Syst., 30 (2018), 1695–1704. https://doi.org/10.1109/TNNLS.2018.2873163 doi: 10.1109/TNNLS.2018.2873163
    [16] X. Yang, G. Feng, C. He, J. Cao, Event-triggered dynamic output quantization control of switched T-S fuzzy systems with unstable modes, IEEE Trans. Fuzzy Syst., 30 (2022), 4201–4210. https://doi.org/10.1109/TFUZZ.2022.3145808 doi: 10.1109/TFUZZ.2022.3145808
    [17] J. Li, B. Zhang, R. Lu, Y. Xu, T. Huang, Distributed H_{\infty} state estimator design for time-delay periodic systems over scheduling sensor networks, IEEE Trans. Cybern., 51 (2019), 462–472. https://doi.org/10.1109/TCYB.2019.2894392 doi: 10.1109/TCYB.2019.2894392
    [18] Y. Xu, J. Dong, R. Lu, L. Xie, Stability of continuous-time positive switched linear systems: a weak common copositive Lyapunov functions approach, Automatica, 97 (2018), 278–285. https://doi.org/10.1016/j.automatica.2018.04.037 doi: 10.1016/j.automatica.2018.04.037
    [19] X. Xie, T. Wei, X. Li, Hybrid event-triggered approach for quasi-consensus of uncertain multi-agent systems with impulsive protocols, IEEE Trans. Circuits Syst., 69 (2022), 872–883. https://doi.org/10.1109/TCSI.2021.3119065 doi: 10.1109/TCSI.2021.3119065
    [20] H. Trentelman, K. Takaba, N. Monshizadeh, Robust synchronization of uncertain linear multi-agent systems, IEEE Trans. Autom. Control, 58 (2013), 1511–1523. https://doi.org/10.1109/TAC.2013.2239011 doi: 10.1109/TAC.2013.2239011
    [21] T. Li, M. Fu, L. Xie, J. Zhang, Distributed consensus with limited communication data rate, IEEE Trans. Autom. Control, 56 (2010), 279–292. https://doi.org/10.1109/TAC.2010.2052384 doi: 10.1109/TAC.2010.2052384
    [22] T. Li, F. Wu, J. Zhang, Multi-agent consensus with relative-state-dependent measurement noises, IEEE Trans. Autom. Control, 59 (2014), 2463–2468. https://doi.org/10.1109/TAC.2014.2304368 doi: 10.1109/TAC.2014.2304368
    [23] B. Liu, X. Liu, G. Chen, H. Wang, Robust impulsive synchronization of uncertain dynamical networks, IEEE Trans. Circuits Syst. I, 52 (2005), 1431–1441. https://doi.org/10.1109/TCSI.2005.851708 doi: 10.1109/TCSI.2005.851708
    [24] S. Yang, Z. Guo, J. Wang, Robust synchronization of multiple memristive neural networks with uncertain parameters via nonlinear coupling, IEEE Trans. Syst. Man Cybern., 45 (2015), 1077–1086. https://doi.org/10.1109/TSMC.2014.2388199 doi: 10.1109/TSMC.2014.2388199
    [25] T. Chen, X. Liu, W. Lu, Pinning complex networks by a single controller, IEEE Trans. Circuits Syst. I, 54 (2007), 1317–1326. https://doi.org/10.1109/TCSI.2007.895383 doi: 10.1109/TCSI.2007.895383
    [26] W. Yu, G. Chen, J. Lü, On pinning synchronization of complex dynamical networks, Automatica, 45 (2009), 429–435. https://doi.org/10.1016/j.automatica.2008.07.016 doi: 10.1016/j.automatica.2008.07.016
    [27] 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
    [28] X. Yang, Y. Liu, J. Cao, L. Rutkowski, Synchronization of coupled time-delay neural networks with mode-dependent average dwell time switching, IEEE Trans. Neural Networks Learn. Syst., 31 (2020), 5483–5496. https://doi.org/10.1109/TNNLS.2020.2968342 doi: 10.1109/TNNLS.2020.2968342
    [29] J. Lu, C. Ding, J. Lou, J. Cao, Outer synchronization of partially coupled dynamical networks via pinning impulsive controllers, J. Franklin Inst., 352 (2015), 5024–5041. https://doi.org/10.1016/j.jfranklin.2015.08.016 doi: 10.1016/j.jfranklin.2015.08.016
    [30] X. Li, S. Song, J. Wu, Exponential stability of nonlinear systems with delayed impulses and applications, IEEE Trans. Automa. Control, 64 (2019), 4024–4034. https://doi.org/10.1109/TAC.2019.2905271 doi: 10.1109/TAC.2019.2905271
    [31] X. Li, D. Ho, J. Cao, Finite-time stability and settling-time estimation of nonlinear impulsive systems, Automatica, 99 (2019), 361–368. https://doi.org/10.1016/j.automatica.2018.10.024 doi: 10.1016/j.automatica.2018.10.024
    [32] 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
    [33] 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
    [34] 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 (2018), 3599–3609. https://doi.org/10.1109/TNNLS.2017.2731865 doi: 10.1109/TNNLS.2017.2731865
    [35] S. Ding, Z. Wang, Event-triggered synchronization of discrete-time neural networks: a switching approach, Neural Networks, 125 (2020), 31–40. https://doi.org/10.1016/j.neunet.2020.01.024 doi: 10.1016/j.neunet.2020.01.024
    [36] H. Rao, F. Liu, H. Peng, Y. Xu, R. Lu, Observer-based impulsive synchronization for neural networks with uncertain exchanging information, IEEE Trans. Neural Networks Learn. Syst., 31 (2020), 3777–3787. https://doi.org/10.1109/TNNLS.2019.2946151 doi: 10.1109/TNNLS.2019.2946151
    [37] P. Chen, J. Wang, S. He, X. Luan, F. Liu, Observer-based asynchronous fault detection for conic-type nonlinear jumping systems and its application to separately excited DC motor, IEEE Trans. Circuits Syst. I, 67 (2019), 951–962. https://doi.org/10.1109/TCSI.2019.2949368 doi: 10.1109/TCSI.2019.2949368
    [38] M. de Magistris, M. di Bernardo, E. di Tucci, S. Manfredi, Synchronization of networks of non-identical Chua's circuits: analysis and experiments, IEEE Trans. Circuits Syst. I, 59 (2012), 1029–1041. https://doi.org/10.1109/TCSI.2012.2185279 doi: 10.1109/TCSI.2012.2185279
    [39] F. Liu, C. Liu, R. Rao, Y. Xu, T. Wang, Reliable impulsive synchronization for fuzzy neural networks with mixed controllers, Neural Networks, 143 (2021), 759–766. https://doi.org/10.1016/j.neunet.2021.08.013 doi: 10.1016/j.neunet.2021.08.013
    [40] P. Chen, S. He, V. Stojanovic, X. Luan, F. Liu, Fuzzy fault detection for Markov jump systems with partly accessible hidden information: an event-triggered approach, IEEE Trans. Cybern., 52 (2021), 7352–7361. https://doi.org/10.1109/TCYB.2021.3050209 doi: 10.1109/TCYB.2021.3050209
    [41] H. Rao, F. Liu, H. Peng, Y. Xu, R. Lu, Observer-based impulsive synchronization for neural networks with uncertain exchanging information, IEEE Trans. Neural Networks Learn. Syst., 31 (2020), 3777–3787. https://doi.org/10.1109/TNNLS.2019.2946151 doi: 10.1109/TNNLS.2019.2946151
    [42] L. Wang, Z. Wang, Q. Han, G. Wei, Synchronization control for a class of discrete-time dynamical networks with packet dropouts: a coding-decoding-based approach, IEEE Trans. Cybern., 48 (2018), 2437–2448. https://doi.org/10.1109/TCYB.2017.2740309 doi: 10.1109/TCYB.2017.2740309
    [43] L. Xie, Output feedback H_{\infty} control of systems with parameter uncertainty, Int. J. Control, 63 (1996), 741–750. https://doi.org/10.1080/00207179608921866 doi: 10.1080/00207179608921866
    [44] H. Rao, F. Liu, H. Peng, Y. Xu, R. Lu, Observer-based impulsive synchronization for neural networks with uncertain exchanging information, IEEE Trans. Neural Networks Learn. Syst., 31 (2020), 3777–3787. https://doi.org/10.1109/TNNLS.2019.2946151 doi: 10.1109/TNNLS.2019.2946151
  • This article has been cited by:

    1. Rayed AlGhamdi, Mitotic Nuclei Segmentation and Classification Using Chaotic Butterfly Optimization Algorithm with Deep Learning on Histopathology Images, 2023, 8, 2313-7673, 474, 10.3390/biomimetics8060474
    2. Turki Althaqafi, Mathematical modeling of a Hybrid Mutated Tunicate Swarm Algorithm for Feature Selection and Global Optimization, 2024, 9, 2473-6988, 24336, 10.3934/math.20241184
    3. Mahmoud Ragab, Hybrid firefly particle swarm optimisation algorithm for feature selection problems, 2024, 41, 0266-4720, 10.1111/exsy.13363
    4. Mobina Fathi, Kimia Vakili, Ramtin Hajibeygi, Ashkan Bahrami, Shima Behzad, Armin Tafazolimoghadam, Hadiseh Aghabozorgi, Reza Eshraghi, Vivek Bhatt, Ali Gholamrezanezhad, Cultivating diagnostic clarity: The importance of reporting artificial intelligence confidence levels in radiologic diagnoses, 2024, 08997071, 110356, 10.1016/j.clinimag.2024.110356
    5. Rayed AlGhamdi, Turky Omar Asar, Fatmah Y. Assiri, Rasha A. Mansouri, Mahmoud Ragab, Al-Biruni Earth Radius Optimization with Transfer Learning Based Histopathological Image Analysis for Lung and Colon Cancer Detection, 2023, 15, 2072-6694, 3300, 10.3390/cancers15133300
    6. Abdulkream A. Alsulami, Aishah Albarakati, Abdullah AL-Malaise AL-Ghamdi, Mahmoud Ragab, Identification of Anomalies in Lung and Colon Cancer Using Computer Vision-Based Swin Transformer with Ensemble Model on Histopathological Images, 2024, 11, 2306-5354, 978, 10.3390/bioengineering11100978
    7. Moneerah Alotaibi, Amal Alshardan, Mashael Maashi, Mashael M. Asiri, Sultan Refa Alotaibi, Ayman Yafoz, Raed Alsini, Alaa O. Khadidos, Exploiting histopathological imaging for early detection of lung and colon cancer via ensemble deep learning model, 2024, 14, 2045-2322, 10.1038/s41598-024-71302-9
    8. Amal Alshardan, Nazir Ahmad, Achraf Ben Miled, Asma Alshuhail, Yazeed Alzahrani, Ahmed Mahmud, Transferable deep learning with coati optimization algorithm based mitotic nuclei segmentation and classification model, 2024, 14, 2045-2322, 10.1038/s41598-024-80002-3
    9. Xueping Tan, Dinghui Wu, Hao Wang, Zihao Zhao, Yuxi Ge, Shudong Hu, MMCAF: A Survival Status Prediction Method Based on Cross‐Attention Fusion of Multimodal Colorectal Cancer Data, 2025, 35, 0899-9457, 10.1002/ima.70051
    10. Shuihua Wang, Yudong Zhang, Grad-CAM: Understanding AI Models, 2023, 76, 1546-2226, 1321, 10.32604/cmc.2023.041419
    11. Rong Zheng, Abdelazim G. Hussien, Anas Bouaouda, Rui Zhong, Gang Hu, A Comprehensive Review of the Tunicate Swarm Algorithm: Variations, Applications, and Results, 2025, 1134-3060, 10.1007/s11831-025-10228-5
    12. JAMAL ALSAMRI, FATIMA QUIAM, MASHAEL MAASHI, ABDULLAH M. ALASHJAEE, MOHAMMAD ALAMGEER, AHMED S. SALAMA, PREDICTIVE MODELING OF REAL-TIME COLORECTAL CANCER VIA HYPERPARAMETER CONFIGURATION WITH DEEP LEARNING USING PUBLIC HEALTH INDICATOR ANALYSIS, 2025, 33, 0218-348X, 10.1142/S0218348X25400018
    13. Haseebullah Jumakhan, Sana Abouelnour, Aneesa Al Redhaei, Sharif Naser Makhadmeh, Mohammed Azmi Al-Betar, Recent Versions and Applications of Tunicate Swarm Algorithm, 2025, 1134-3060, 10.1007/s11831-025-10287-8
    14. Burak Gülmez, Deep learning based colorectal cancer detection in medical images: A comprehensive analysis of datasets, methods, and future directions, 2025, 125, 08997071, 110542, 10.1016/j.clinimag.2025.110542
  • Reader Comments
  • © 2024 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(1354) PDF downloads(156) Cited by(4)

Figures and Tables

Figures(4)  /  Tables(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog