In this paper, motivated by the advantages of the generalized conformable derivatives, an impulsive conformable Cohen–Grossberg-type neural network model is introduced. The impulses, which can be also considered as a control strategy, are at fixed instants of time. We define the notion of practical stability with respect to manifolds. A Lyapunov-based analysis is conducted, and new criteria are proposed. The case of bidirectional associative memory (BAM) network model is also investigated. Examples are given to demonstrate the effectiveness of the established results.
Citation: Trayan Stamov, Gani Stamov, Ivanka Stamova, Ekaterina Gospodinova. Lyapunov approach to manifolds stability for impulsive Cohen–Grossberg-type conformable neural network models[J]. Mathematical Biosciences and Engineering, 2023, 20(8): 15431-15455. doi: 10.3934/mbe.2023689
[1] | Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding . The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28(4): 1395-1418. doi: 10.3934/era.2020074 |
[2] | Shu Wang, Mengmeng Si, Rong Yang . Dynamics of stochastic 3D Brinkman-Forchheimer equations on unbounded domains. Electronic Research Archive, 2023, 31(2): 904-927. doi: 10.3934/era.2023045 |
[3] | Lingrui Zhang, Xue-zhi Li, Keqin Su . Dynamical behavior of Benjamin-Bona-Mahony system with finite distributed delay in 3D. Electronic Research Archive, 2023, 31(11): 6881-6897. doi: 10.3934/era.2023348 |
[4] | Pan Zhang, Lan Huang, Rui Lu, Xin-Guang Yang . Pullback dynamics of a 3D modified Navier-Stokes equations with double delays. Electronic Research Archive, 2021, 29(6): 4137-4157. doi: 10.3934/era.2021076 |
[5] | Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao . Hybridized weak Galerkin finite element methods for Brinkman equations. Electronic Research Archive, 2021, 29(3): 2489-2516. doi: 10.3934/era.2020126 |
[6] | Wei Shi, Xinguang Yang, Xingjie Yan . Determination of the 3D Navier-Stokes equations with damping. Electronic Research Archive, 2022, 30(10): 3872-3886. doi: 10.3934/era.2022197 |
[7] | Keqin Su, Rong Yang . Pullback dynamics and robustness for the 3D Navier-Stokes-Voigt equations with memory. Electronic Research Archive, 2023, 31(2): 928-946. doi: 10.3934/era.2023046 |
[8] | Nisachon Kumankat, Kanognudge Wuttanachamsri . Well-posedness of generalized Stokes-Brinkman equations modeling moving solid phases. Electronic Research Archive, 2023, 31(3): 1641-1661. doi: 10.3934/era.2023085 |
[9] | Zhiqing Li, Wenbin Zhang, Yuanfei Li . Structural stability for Forchheimer fluid in a semi-infinite pipe. Electronic Research Archive, 2023, 31(3): 1466-1484. doi: 10.3934/era.2023074 |
[10] | Wenlong Sun . The boundedness and upper semicontinuity of the pullback attractors for a 2D micropolar fluid flows with delay. Electronic Research Archive, 2020, 28(3): 1343-1356. doi: 10.3934/era.2020071 |
In this paper, motivated by the advantages of the generalized conformable derivatives, an impulsive conformable Cohen–Grossberg-type neural network model is introduced. The impulses, which can be also considered as a control strategy, are at fixed instants of time. We define the notion of practical stability with respect to manifolds. A Lyapunov-based analysis is conducted, and new criteria are proposed. The case of bidirectional associative memory (BAM) network model is also investigated. Examples are given to demonstrate the effectiveness of the established results.
The delay effect originates from the boundary controllers in engineering. The dynamics of a system with boundary delay could be described mathematically by a differential equation with delay term subject to boundary value condition such as [20]. There are many results available in literatures on the well-posedness and pullback dynamics of fluid flow models with delays especially the 2D Navier-Stokes equations, which can be seen in [1], [2], [8] and references therein. Inspired by these works, in this paper, we study the stability of pullback attractors for 3D Brinkman-Forchheimer (BF) equation with delay, which is also a continuation of our previous work in [6]. The existence and structure of attractors are significant to understand the large time behavior of solutions for non-autonomous evolutionary equations. Furthermore, the asymptotic stability of trajectories inside invariant sets determines many important properties of trajectories. The 3D Brinkman-Forchheimer equation with delay is given below:
{∂u∂t−νΔu+αu+β|u|u+γ|u|2u+∇p=f(t,ut)+g(x,t),∇⋅u=0, u(t,x)|∂Ω=0,u|t=τ=uτ(x), x∈Ω,uτ(θ,x)=u(τ+θ,x)=ϕ(θ), θ∈(−h,0), h>0. | (1) |
Here,
(1). a general delay
or
(2). the special application of
f(t,ut)=F(u(t−ρ(t))) | (2) |
for a smooth function
The BF equation describes the conservation law of fluid flow in a porous medium that obeys the Darcy's law. The physical background of 3D BF model can be seen in [14], [9], [18], [19]. For the dynamic systems of problem
(a) For problem (1) with delay
(b) For problem (1) with special application of
(c) The asymptotic stability of trajectories inside pullback attractors is further research of the results established in [6]. However, the stability of pullback attractors for (1) with infinite delay is still unknown.
In this section, we give some notations and the equivalent abstract form of (1) in this section.
Denoting
By the Helmholz-Leray projection defined above, (1) can be transformed to the abstract equivalent form
{∂u∂t+νAu+P(αu+β|u|u+γ|u|2u)=Pf(t,ut)+Pg(t,x),u|∂Ω=0,u|t=τ=uτ(x),uτ(θ,x)=ϕ(θ,x) for θ∈(−h,0), | (3) |
then we show our results for (3) with
We also define some Banach spaces on delayed interval as
‖ϕ‖CH=supθ∈[−h,0]‖ϕ(θ)‖H, ‖ϕ‖CV=supθ∈[−h,0]‖ϕ(θ)‖V, |
respectively. The Lebesgue integrable spaces on delayed interval can be denoted as
Some assumptions on the external forces and parameters which will be imposed in our main results are the following:
‖f(t,ξ)−f(t,η)‖H≤Lf‖ξ−η‖CH, for ξ,η∈CH. |
∫tτ‖f(r,ur)−f(r,vr)‖2Hdr≤C2f∫tτ−h‖u(r)−v(r)‖2Hdr, for τ≤t. | (4) |
∫t−∞eηs‖g(s,⋅)‖2V′ds<∞. | (5) |
holds for any
Lemma 3.1. (The Gronwall inequality with differential form) Let
ddtm(t)≤v(t)m(t)+h(t), m(t=τ)=mτ, t≥τ. | (6) |
Then
m(t)≤mτe∫tτv(s)ds+∫tτh(s)e∫tsv(σ)dσds, t≥τ. | (7) |
In this part, we shall present some retarded integral inequalities from Li, Liu and Ju [5]. Consider the following retarded integral inequalities:
‖y(t)‖X≤E(t,τ)‖yτ‖X+∫tτK1(t,s)‖ys‖Xds+∫∞tK2(t,s)‖ys‖Xds+ρ, ∀ t≥τ, | (8) |
where
Let
κ(K1,K2)=supt≥τ(∫tτK1(t,s)ds+∫∞tK2(t,s)ds). |
We assume that
limt→+∞E(t+s,s)=0 | (9) |
uniformly with respect to
Lemma 3.2. (The retarded Gronwall inequality) Denoting
(1) If
‖yt‖X<μρ+ε, | (10) |
for
(2) If
‖yt‖X≤M‖y0‖Xe−λt+γρ, t≥τ | (11) |
for all bounded functions
(3) If
Proof. See Li, Liu and Ju [5].
Remark 1. (The special case:
The minimal family of pullback attractors will be stated here in preparation for our main result.
Lemma 3.3. (1) (See [7], [11]) Assume that
(|a|β−2a−|b|β−2b)⋅(a−b)≥γ0|a−b|β, |
where
(2) The following
|xq−yq|≤Cq(|x|q−1+|y|q−1)|x−y| |
for the integer
Theorem 3.4. Assume that the external forces
Proof. Step 1. Existence of local approximate solution.
By the property of the Stokes operator
Awi=λiwi, i=1,2,⋯. | (12) |
Let
{(∂tum,wj)+ν(∇um,∇wj)+(αum+β|um|um+γ|um|2um,wj)=(f(t,umt),wj)+⟨g,wj⟩,um(τ)=Pmuτ=uτm,umτ(θ,x)=Pmϕ(θ)=ϕm(θ) for θ∈[−h,0], | (13) |
Then it is easy to check that (13) is equivalent to an ordinary differential equations with unknown variable function
Step 2. Uniform estimates of approximate solutions.
Multiplying (13) by
12ddt‖um‖2H+ν‖um‖2V+α‖um‖2H+β‖um‖3L3(Ω)+γ‖um‖4L4(Ω)≤|(g(t)+f(s,umt(s)),um)|≤α‖um‖2H+ν2‖um‖2V+12ν‖g(t)‖2V′+14α‖f(t,umt)‖2H. | (14) |
Integrating in time, using the hypotheses on
‖um‖2H+ν∫tτ‖um‖2Vds+2β∫tτ‖um‖3L3(Ω)ds+2γ∫tτ‖um‖4L4(Ω)ds≤‖uτ‖2H+C2f4α∫0−h‖ϕ(s)‖2Hds+12ν∫tτ‖g(s)‖2V′ds+C2f4α∫tτ‖um‖2Hds. | (15) |
Using the Gronwall Lemma of integrable form, we conclude that
{um} is bounded in the spaceL∞(τ,T;H)∩L2(τ−h,T;V)∩L3(τ,T;L3(Ω))∩L4(τ,T;L4(Ω)). |
Step 3. Compact argument and passing to limit for deriving the global weak solutions.
In this step, we shall prove
dumdt=−νAum−αum−β|um|um−γ|um|2um+P(g(t)+f(t,umt) | (16) |
and assumptions
By virtue of the Aubin-Lions Lemma, we obtain that
{um(t)⇀u(t) weakly * in L∞(τ,T;H),um(t)→u(t) stongly in L2(τ,T;H),um(t)⇀u(t) weakly in L2(τ,T;V),dum/dt⇀du/dt weakly in L2(τ,T;V′),f(⋅,um⋅)⇀f(⋅,u⋅) weakly in L2(τ,T;H),um⇀u(t) weakly in L3(τ,T;L3(Ω)),um⇀u(t) weakly in L4(τ,T;L4(Ω)) | (17) |
which coincides with the initial data
For the purpose of passing to limit in (13), denoting
∫Tτ(β|um|um−β|u|u,wj)ds≤Cλ1β‖um‖4L4(τ,T;L4(Ω))‖um−u‖4L4(τ,T;L4(Ω))+Cβ‖um−u‖L∞(τ,T;H)‖u‖2L2(τ−h,T;H) |
and
∫Tτ(γ|um|2um−γ|u|2u,wj)ds≤Cγ‖um‖2L2(τ,T;V)‖um−u‖4L4(τ,T;L4(Ω))+Cγ‖um−u‖4L4(τ,T;L4(Ω))(‖u‖2L2(τ−h,T;V)+‖um‖4L4(τ,T;L4(Ω))) | (18) |
and the convergence of delayed external force
Thus, passing to the limit of (13), we conclude that
Proposition 1. Assume that the external forces
Proof. Taking inner product of (3) with
12ddt‖A1/2u‖2H+ν‖Au‖2H+α‖A1/2u‖2H+β∫Ω|u|u⋅Audx+γ∫Ω|u|2u⋅Audx=(f(t,ut),Au)+(g(t),Au). | (19) |
According to Lemma 3.3, the nonlinear terms have the following estimates
|β(|u|u,Au)|≤ν2‖Au‖2H+β4ν‖u‖4L4 | (20) |
and
γ∫Ω|u|2u⋅Audx=γ2∫Ω|∇(|u|2)|2dx+γ∫Ω|u|2|∇u|2dx | (21) |
and
(f(t,ut),Au)+(g(t),Au)≤12ν‖f(t,ut)‖2H+12ν‖g(t)‖2H+ν2‖Au‖2H, | (22) |
hence, we conclude that
ddt‖A1/2u‖2H+2α‖A1/2u‖2H+γ∫Ω|∇(|u|2)|2dx+2γ∫Ω|u|2|∇u|2dx≤β2ν‖u‖4L4+1ν‖f(t,ut)‖2H+1ν‖g(t)‖2H. | (23) |
Letting
‖A1/2u(t)‖2H+2α∫ts‖A1/2u(r)‖2Hdr≤‖A1/2u(s)‖2H+β2ν∫ts‖u(r)‖4L4dr+2ν∫ts‖f(r,ur)‖2Hdr+2ν∫ts‖g(r)‖2Hdr | (24) |
and
∫ts‖f(r,ur)‖2Hdr≤L2f‖ϕ(θ)‖2L2H+L2f∫ts‖u(r)‖2Hdr. | (25) |
Then integrating with
‖A1/2u(t)‖2H≤∫tt−1‖A1/2u(s)‖2Hds+β2ν∫tt−1‖u(r)‖4L4dr+2L2fν‖ϕ(θ)‖2L2H+2L2fν∫tτ‖u(r)‖2Hdr+2ν∫tt−1‖g(r)‖2Hdr≤C[‖ϕ‖2L2H+‖uτ‖2H]+C∫tτ‖g‖2Hds+2L2fνλ1∫tτ‖u(r)‖2Vdr, | (26) |
which means the uniform boundedness of the global weak solution
Proposition 2. Assume the hypotheses in Theorem 3.4 hold. Then the global weak solution
Proof. Using the same energy estimates as above, we can deduce the uniqueness easily, here we skip the details.
To description of pullback attractors, the functional space
∫tτeηs‖f(s,us)‖2Hds<C2f∫tτ−heηs‖u(s)‖2Hds. | (27) |
for any
Proposition 3. For given
Lemma 3.5. Assume that
‖u(t)‖2H≤e−8ηCfα(t−τ)(‖uτ‖2H+Cf‖ϕ(r)‖2L2H)+e−8ηCfαtν−ηλ−1∫tτeηr‖g(r)‖2V′dr | (28) |
and
ν∫ts‖u(r)‖2Vdr≤‖u(s)‖2H+8Cfα‖us‖2L2H+1ν∫ts‖g(r)‖2V′dr+8Cfα∫ts‖u(r)‖2Hdr, | (29) |
β∫ts‖u(r)‖3L3(Ω)dr≤‖u(s)‖2H+8Cfα‖us‖2L2H+1ν∫ts‖g(r)‖2V′dr+8Cfα∫ts‖u(r)‖2Hdr, | (30) |
γ∫ts‖u(r)‖4L4(Ω)dr≤‖u(s)‖2H+8Cfα‖us‖2L2H+1ν∫ts‖g(r)‖2V′dr+8Cfα∫ts‖u(r)‖2Hdr. | (31) |
Proof. By the energy estimate of (1) and using Young's inequality, we arrive at
ddt‖u‖2H+2ν‖u‖2V+2α‖u‖2H+2β‖u‖3L3(Ω)+2γ‖u‖2L4(Ω)≤1ν−ηλ−1‖g‖2V′+(ν−ηλ−1)‖u‖2V+2α‖u‖2H+8α‖f(t,ut)‖2H, | (32) |
where
Multiplying the above inequality by
ddt(eηt‖u‖2H)+eηtνλ1‖u‖2H+2βeηt‖u‖3L3(Ω)+2γeηt‖u‖2L4(Ω)≤1ν−ηλ−1eηt‖g‖2V′+8Cfαeηt‖f(t,ut)‖2H. |
Thus integrating with respect to time variable, it yields
eηt‖u‖2H+νλ1∫tτeηr‖u(r)‖2Hdr≤eητ(‖uτ‖2H+Cf∫0−h‖ϕ(r)‖2Hdr)+1ν−ηλ−1∫tτeηr‖g(r)‖2V′dr+8Cfα∫tτeηr‖u(r)‖2Hdr | (33) |
and by the Gronwall Lemma, we can derive the estimate in our theorem.
Using the energy estimate of (1) again, we can check that
ddt‖u‖2H+2ν‖u‖2V+2α‖u‖2H+2β‖u‖3L3(Ω)+2γ‖u‖2L4(Ω)≤1ν‖g‖2V′+ν‖u‖2V+2α‖u‖2H+8α‖f(t,ut)‖2H, | (34) |
Integrating from
Based on Lemma 3.5, we can present the pullback dissipation based on the following universes for the tempered dynamics.
Definition 3.6. (Universe). (1) We will denote by
limτ→−∞(eητsup(ξ,ζ)∈D(τ)‖(ξ,ζ)‖2MH)=0. | (35) |
(2)
Remark 2. The universes
Proposition 4. (The
D0(t)=¯BH(0,ρH(t))×(¯BL2V(0,ρL2H(t))∩¯BCH(0,ρCH(t))) |
is the pullback
ρ2H(t)=1+e−8ηCfα(t−h)ν−ηλ−1∫t−∞eηr‖g(r)‖2V′dr,ρ2L2V(t)=1ν[1+‖uτ‖2H+8Cfα‖ϕ‖2L2H+‖g(r)‖2L2(t−h,t;V′)ν+8Cfhαρ2H(t)]. |
Moreover, the pullback
Proof. Using the estimates in Lemma 3.5, choosing any
‖u(t,τ;uτ,ϕ)‖2H≤ρ2H(t)=1+e−8ηCfα(t−h)ν−ηλ−1∫t−∞eηr‖g(r)‖2V′dr | (36) |
holds for any
Theorem 3.7. Assume that
Proof. Step 1. Weak convergence of the sequence
For arbitrary fixed
By using the similar energy estimate in Theorem 3.4 and technique in Proposition 4, there exists a pullback time
‖(un)′‖L2(t−h−1,t;V′)≤ν‖un‖L2(t−h−1,t;V)+αλ−11‖un‖L2(t−h−1,t;V)+β‖un‖L4(t−h−1,t;L4(Ω))+Cλ1,|Ω|γ‖un‖L2(t−h−1,t;V)+Cα‖f(t,unt)‖L2(t−h−1,t;H)+Cν‖g‖L2(t−h−1,t;V′). | (37) |
From the hypotheses
\begin{equation} \left\{ \begin{array}{ll} u^n\rightharpoonup u\ \mbox{weakly * in} \ L^{\infty}(t-3h-1,t;H), \\ u^n\rightharpoonup u\ \mbox{weakly in} \ L^2(t-2h-1,t;V),\\ (u^n)'\rightharpoonup u'\ \mbox{weakly in} \ L^2(t-h-1,t;V'),\\ u_m\rightharpoonup u(t)\ \mbox{weakly in} \ L^3(t-2h-1,t;\mathbb{L}^3(\Omega)),\\ u_m\rightharpoonup u(t)\ \mbox{weakly in} \ L^4(t-2h-1,t;\mathbb{L}^4(\Omega)),\\ u^n\rightarrow u\ \mbox{stongly in} \ L^2(t-h-1,t;H), \\ u^n(s)\rightarrow u(s)\ \mbox{stongly in} \ H,\ \mbox{a.e.}\ s\in (t-h-1,t). \end{array}\right. \end{equation} | (38) |
By Theorem 3.4, from the hypothesis on
\begin{eqnarray} f(\cdot, u^n_{\cdot}) \rightharpoonup f(\cdot,u_{\cdot})\ \mbox{weakly in} \ L^2(t-h-1,t;H). \end{eqnarray} | (39) |
Thus, from (38) and (39), we can conclude that
From the uniform bounded estimate of
\begin{eqnarray} u^n\rightarrow u\ \mbox{strongly in}\ C([t-h-1,t];H). \end{eqnarray} | (40) |
Therefore, we can conclude that
\begin{eqnarray} u^n(s_n) \rightharpoonup u(s)\ \ \mbox{weakly in } H \end{eqnarray} | (41) |
for any
\begin{eqnarray} \liminf\limits_{n\rightarrow\infty}\|u^{n}(s_n)\|_{H}\geq \|u(s)\|_{H}. \end{eqnarray} | (42) |
Step 2. The strong convergence of corresponding sequences via energy equation method:
The asymptotic compactness of sequence
\begin{eqnarray} \|u^{n}(s_n)-u(s)\|_{H}\rightarrow 0\ \mbox{as}\ n\rightarrow+\infty, \end{eqnarray} | (43) |
which is equivalent to prove (42) combining with
\begin{eqnarray} \limsup\limits_{n\rightarrow\infty}\|u^{n}(s_n)\|_H\leq \|u(s)\|_H \end{eqnarray} | (44) |
for a sequence
Using the energy estimate to all
\begin{eqnarray} &&\|u^n(s_2)\|^2_H+\nu \int^{s_2}_{s_1}\|u^n(r)\|^2_Vdr+2\beta \int^{s_2}_{s_1}\|u^n(r)\|^3_{\mathbb{L}^4(\Omega)}dr+2\gamma \int^{s_2}_{s_1}\|u^n(r)\|^4_{\mathbb{L}^4(\Omega)}\\ &\leq& \frac{2C_f^2}{\alpha}\int^{s_2}_{s_2}\|u^n_r\|^2_Hdr+\frac{8}{\nu}\int^{s_2}_{s_1}\|g(r)\|^2_{V'}dr \end{eqnarray} | (45) |
and
\begin{eqnarray} &&\|u(s_2)\|^2_H+\nu \int^{s_2}_{s_1}\|u(r)\|^2_Vdr+2\beta \int^{s_2}_{s_1}\|u(r)\|^3_{\mathbb{L}^4(\Omega)}dr+2\gamma \int^{s_2}_{s_1}\|u(r)\|^4_{\mathbb{L}^4(\Omega)}\\ &\leq& \frac{2C_f^2}{\alpha}\int^{s_2}_{s_2}\|u_r\|^2_Hdr+\frac{8}{\nu}\int^{s_2}_{s_1}\|g(r)\|^2_{V'}dr. \end{eqnarray} | (46) |
Then, we define the functionals
\begin{eqnarray} J_n(s)& = &\frac{1}{2}\|u^n\|^2_H-\int^{s}_{t-h-1}\langle g(r),u^n(r)\rangle dr-\int^{s}_{t-h-1}(f(r,u^n_r),u^n(r))dr \end{eqnarray} | (47) |
and
\begin{eqnarray} J(t)& = &\frac{1}{2}\|u(s)\|^2_H-\int^{s}_{t-h-1}\langle g(r),u(r)\rangle dr-\int^{s}_{t-h-1}(f(r,u_r),u(r))dr. \end{eqnarray} | (48) |
Combining the convergence in (38), observing that
\begin{eqnarray} &&\int^t_{t-h-1}\langle g(r),u^n(r)\rangle dr\rightarrow 2\int^t_{t-h-1}\langle g(r),u(r)\rangle dr \end{eqnarray} | (49) |
and
\begin{eqnarray} &&\int^t_{t-h-1}(f(r,u^n_r),u^n(r))dr\rightarrow 2\int^t_{t-h-1}(f(r,u_r),u(r))dr \end{eqnarray} | (50) |
as
\begin{eqnarray} J_n(s)\rightarrow J(s)\ \ \mbox{a.e.} s\in (t-h-1,t), \end{eqnarray} | (51) |
i.e., for
\begin{eqnarray} |J_n(s_k)-J(s_k)|\leq \frac{\varepsilon}{2}. \end{eqnarray} | (52) |
Since
\begin{eqnarray} |J(s_{k})-J(s)|\leq \frac{\varepsilon}{2}, \end{eqnarray} | (53) |
Choosing
\begin{eqnarray} |J_n(s_n)-J(s)|\leq |J_n(s_n)-J(s_n)|+|J(s_n)-J(s)| < \varepsilon. \end{eqnarray} | (54) |
Therefore, for any
\begin{eqnarray} \limsup\limits_{n\rightarrow\infty}J_n(s_n)\leq J(s), \end{eqnarray} | (55) |
which implies
\begin{eqnarray} \limsup\limits_{n\rightarrow \infty}\|u^n(s_n)\|_H \leq \|u(s)\|_H. \end{eqnarray} | (56) |
we conclude the strong convergence
Step 3. The strong convergence:
Combining the energy estimates in (45) and (46), noting the energy functionals
\begin{eqnarray} \|u^n(s)\|_{L^2(t-h,t;V)}\rightarrow \|u(s)\|_{L^2(t-h,t;V)}. \end{eqnarray} | (57) |
Hence jointing with the weak convergence in (38), we can derive that
Step 4. The
By using the results from Steps 2 to 4 and noting the definition of universe, we can conclude that the processes is
Remark 3. Using the similar technique, we can derive the processes
Theorem 3.8. Assume that
\begin{eqnarray} \mathcal{A}_{\mathcal{D}^{M_H}_{F}}(t) \subset \mathcal{A}_{\mathcal{D}^{M_H}_{\eta}}(t). \end{eqnarray} | (58) |
Proof. From Proposition 3, we observe that the process
Based on the universes defined in Definition 3.6, the relation between
Definition 3.9. The pullback attractors is asymptotically stable if the trajectories inside attractor reduces to a single orbit as
Theorem 3.10. Assume that
\mathit{\mbox{G}}(t)\leq K_0, |
where
\begin{equation} K_0 = \Big\{[\nu^2\lambda_1(2\nu\lambda_1+\alpha)]\Big/\Big[4C_{|\Omega|}\beta \Big(\frac{L_f^2}{\alpha}\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}+\frac{1}{\alpha}\Big)\Big]\Big\}^{1/2},\nonumber \end{equation} |
here
Proof. Let
\begin{eqnarray} u(\tau+\theta)|_{\theta\in [-h,0]} = \phi(\theta),\ \ \ u|_{t = \tau} = u_{\tau} \end{eqnarray} | (59) |
and
\begin{eqnarray} v(\tau+\theta)|_{\theta\in [-h,0]} = \tilde{\phi}(\theta), \ \ v|_{t = \tau} = \tilde{u}_{\tau} \end{eqnarray} | (60) |
respectively. Denoting
\begin{eqnarray} (u,u_t) = U(t,\tau)(u_{\tau},\varphi)\ \ \mbox{and}\ \ (v,v_t) = U(t,\tau)(\tilde{u}_{\tau},\tilde{\varphi}) \end{eqnarray} | (61) |
as two trajectories inside the pullback attractors, letting
\begin{equation} \begin{cases} \frac{\partial w}{\partial t}+\nu A w +P\Big(\alpha w+\beta(|u|u-|v|v)+\gamma (|u|^2u-|v|^2v)\Big)&\\ \quad = P(f(t, u_t)-f(t,v_t)),& \\ w|_{\partial\Omega} = 0,& \\ w(t = \tau) = u_{\tau}-\tilde{u}_{\tau},&\\ w(\tau+\theta) = \phi(\theta)-\tilde{\phi}(\theta),\ \theta\in [-h,0].& \end{cases} \end{equation} | (62) |
Taking inner product of (62) with
\begin{eqnarray} \gamma(|u|^2u-|v|^2v, u-v)\geq \gamma \gamma_0 \|u-v\|^4_{\bf{L}^4} \end{eqnarray} | (63) |
and
\begin{eqnarray} &&\frac{1}{2}\frac{d}{d t}\|w\|^2_H+\nu\|w\|^2_V+\alpha \|w\|_H^2+\gamma \gamma_0 \|w\|^4_{\bf{L}^4}\\ &\leq& \Big|\beta(|u|u-|v|v,w)\Big|+\Big|(f(t,u_t)-f(t,v_t),w)\Big|\\ &\leq&\beta\Big(\int_{\Omega}|u|^2|w|dx+\int_{\Omega}|w||v|^2dx\Big)+\frac{\alpha}{2}\|w\|^2_H+\frac{L_f^2}{2\alpha}\|w_t\|^2_H \end{eqnarray} |
\begin{eqnarray} &\leq& \beta(\|u\|^2_{\bf{L}^4}+\|v\|^2_{\bf{L}^4})\|w\|^2_{H}+\frac{\alpha}{2}\|w\|^2_H+\frac{L_f^2}{2\alpha}\|w_t\|^2_H\\ &\leq& C_{|\Omega|}\beta(\|u\|^2_{V}+\|v\|^2_{V})\|w\|^2_{H}+\frac{\alpha}{2}\|w\|^2_H+\frac{L_f^2}{2\alpha}\|w_t\|^2_H. \end{eqnarray} | (64) |
Using the Poincaré inequality and Lemma 3.1, noting that if
\begin{eqnarray} 2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V) > 0, \end{eqnarray} | (65) |
then we can obtain
\begin{eqnarray} \|w\|^2_H&\leq& e^{\int^t_{\tau}[2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)-(2\nu\lambda_1+\alpha)]ds}\Big[\|u_{\tau}-\tilde{u}_{\tau}\|^2_H+\\ &&+\frac{L_f^2}{\alpha}\int^t_{\tau}e^{-\int^t_{s}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]d\sigma}\|w_t\|^2_Hds\Big]. \end{eqnarray} | (66) |
Denoting
\begin{eqnarray} E(t,\tau) = e^{-\int^t_{\tau}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]ds} \end{eqnarray} | (67) |
and
\begin{eqnarray} K_1(t,s) = \frac{L_f^2}{\alpha}e^{-\int^t_{s}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]d\sigma} \end{eqnarray} | (68) |
and
\begin{eqnarray} \Theta = \sup\limits_{t\geq s\geq \tau}E(t,s),\ \ \ \ \kappa(K_1,0) = \sup\limits_{t\geq\tau}\int^t_{\tau}K_1(t,s)ds, \end{eqnarray} | (69) |
by virtue of Lemma 3.2, choosing
\begin{eqnarray} \|w_t\|^2_H&\leq& M\|u_{\tau}-\tilde{u}_{\tau}\|^2_H e^{-\lambda (t-\tau)}. \end{eqnarray} | (70) |
Substituting (70) into (64), using Lemma 3.1 again, we can conclude the following estimate
\begin{eqnarray} \|w\|^2_H &\leq& \|u_{\tau}-\tilde{u}_{\tau}\|^2_He^{-\int^t_{\tau}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]ds}\\ &&+\frac{L_f^2}{\alpha}M\|u_{\tau}-\tilde{u}_{\tau}\|^2_H e^{-\lambda (t-\tau)}\int^t_{\tau}e^{-\int^t_{s}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]d\sigma}ds.\\ \end{eqnarray} | (71) |
From (70) and (71), if we fixed
\begin{equation} 2\nu\lambda_1+\alpha > 2C_{|\Omega|}\beta \langle \|u\|^2_V+\|v\|^2_V\rangle_{\leq t}, \end{equation} | (72) |
where
\begin{equation} \langle h \rangle_{\leq t} = \limsup\limits_{\tau\rightarrow -\infty}\frac{1}{t-\tau}\int^t_{\tau}h(r)dr. \end{equation} | (73) |
Since
\begin{eqnarray} &&\frac{1}{2}\frac{d}{dt}\|u\|^2_H+\nu\|A^{1/2}u\|^2_H+\alpha\|u\|^2_H+\beta \|u\|^3_{\bf{L}^3}+\gamma \|u\|^4_{\bf{L}^4}\\ &\leq&\alpha \|u\|^2_H+\frac{1}{2\alpha}\Big[\|f(t,u_t)\|^2_H+\|g\|^2_{H}\Big]\\ &\leq&\alpha \|u\|^2_H+\frac{L_f^2}{2\alpha}\|u_t\|^2_H+\frac{1}{2\alpha}\|g\|^2_{H}. \end{eqnarray} | (74) |
Using the Poincaré inequality and Lemma 3.1, then we can obtain
\begin{eqnarray} \|u\|^2_H&\leq& e^{-2\nu\lambda_1(t-\tau)}\|u_{\tau}\|^2_H+\\ &&+\frac{L_f^2}{\alpha}\int^t_{\tau}e^{-2\nu\lambda_1(t-s)}\|u_s\|^2_Hds+\frac{1}{\alpha}\int^t_{\tau}e^{-2\nu\lambda_1(t-s)}\|g\|^2_Hds. \end{eqnarray} | (75) |
Denoting
\begin{eqnarray} E(t,\tau) = e^{-2\nu\lambda_1(t-\tau)} \end{eqnarray} | (76) |
and
\begin{eqnarray} K_1(t,s) = \frac{L_f^2}{\alpha}e^{-2\nu\lambda_1(t-s)} \end{eqnarray} | (77) |
and
\begin{equation} \rho = \frac{1}{\alpha}\int^t_{\tau}e^{-2\nu\lambda_1(t-s)}\|g\|^2_Hds, \end{equation} | (78) |
letting
\begin{eqnarray} \Theta = \sup\limits_{t\geq s\geq \tau}E(t,s),\ \ \ \ \kappa(K_1,0) = \sup\limits_{t\geq\tau}\int^t_{\tau}K_1(t,s)ds, \end{eqnarray} | (79) |
by virtue of Lemma 3.2, choosing
\begin{eqnarray} \|u_t\|^2_H&\leq& \hat{M}\|u_{\tau}\|^2_H e^{-\lambda (t-\tau)}+\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}\int^t_{\tau}e^{-2\nu\lambda_1(t-s)}\|g\|^2_Hds\\ &\leq& \hat{M}\|u_{\tau}\|^2_H e^{-\lambda (t-\tau)}+\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}\int^t_{\tau}\|g\|^2_Hds. \end{eqnarray} | (80) |
Substituting (80) into (75), using Lemma 3.1 again, we can conclude the following estimate
\begin{eqnarray} \|u\|^2_H &\leq& C\|u_{\tau}\|^2_He^{-\lambda (t-\tau)}+\Big(\frac{L_f^2}{\alpha}\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}+\frac{1}{\alpha}\Big)\int^t_{\tau}\|g\|^2_Hds. \end{eqnarray} | (81) |
Integrating (74) from
\begin{eqnarray} &&\|u\|^2_H+2\nu\int^t_{\tau}\|u\|^2_Vds+2\beta\int^t_{\tau} \|u\|^3_{\bf{L}^3}ds+2\gamma \int^t_{\tau}\|u\|^4_{\bf{L}^4}ds\\ &\leq& \Big[\frac{1}{\alpha}\|\phi\|^2_{L^2_H}+\|u_{\tau}\|^2_H\Big]+\frac{L_f^2}{\alpha}\int^{t}_{\tau}\|u_t(s)\|^2_Hds+\frac{1}{\alpha}\int^t_{\tau}\|g\|^2_{H}ds. \end{eqnarray} | (82) |
By the estimate of (80) and (81), we derive
\begin{eqnarray} \int^t_{\tau}\|u(r)\|^4_{\bf{L}^4}dr\leq C\Big[\frac{1}{\alpha}\|\phi\|^2_{L^2_H}+\|u_{\tau}\|^2_H\Big]+\Big(\frac{L_f^2}{\alpha}\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}+\frac{1}{\alpha}\Big)\int^t_{\tau}\|g\|^2_{H}ds \end{eqnarray} | (83) |
and
\begin{eqnarray} \int^t_{\tau}\|u(r)\|^2_{V}dr\leq C\Big[\frac{1}{\alpha}\|\phi\|^2_{L^2_H}+\|u_{\tau}\|^2_H\Big]+\Big(\frac{L_f^2}{\alpha}\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}+\frac{1}{\alpha}\Big)\int^t_{\tau}\|g\|^2_{H}ds. \end{eqnarray} | (84) |
Combining (72), (73) with (84), we conclude that
\begin{eqnarray} && \langle \|u\|^2_V+\|v\|^2_V\rangle|_{\leq t}\leq 2 \Big(\frac{L_f^2}{\alpha}\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}+\frac{1}{\alpha}\Big)\langle \|g\|^2_H\rangle|_{\leq t} \end{eqnarray} | (85) |
and hence the asymptotic stability holds provided that
\begin{eqnarray} 4C_{|\Omega|}\beta \Big(\frac{L_f^2}{\alpha}\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}+\frac{1}{\alpha}\Big)\langle \|g\|^2_H\rangle|_{\leq t} \leq 2\nu\lambda_1+\alpha. \end{eqnarray} | (86) |
If we define the generalized Grashof number as
\begin{eqnarray} G(t)\leq \Big\{(2\nu\lambda_1+\alpha)\Big/\Big[4C_{|\Omega|}\nu^2\beta\lambda_1 \Big(\frac{L_f^2}{\alpha}\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}+\frac{1}{\alpha}\Big)\Big]\Big\}^{1/2} = K_0, \end{eqnarray} | (87) |
which completes the proof for our first result.
Remark 4. Theorem 3.10 is a further research for the existence of pullback attractor in [6].
We first state some hypothesis on the external forces and sub-linear operator.
\Big|\frac{d\rho}{dt}\Big|\leq\rho^{\ast} < 1, \ \ \forall t\geq 0. |
\begin{eqnarray} \|F(y)\|^2_H\leq a(t)\|y\|^2_H+b(t), \ \ \forall t\geq\tau, y\in H. \end{eqnarray} | (88) |
\begin{eqnarray} \|F(u)-F(v)\|_H\leq L(R)\kappa^\frac{1}{2}(t)\|u-v\|_H, \ u,v\in H. \end{eqnarray} | (89) |
holds for
\begin{eqnarray} \int^{t}_{-\infty}e^{ms}\|g(s,\cdot)\|^{2}_Hds < \infty, \ \ \forall t\in\mathbb{R}. \end{eqnarray} | (90) |
\begin{eqnarray} \frac{\nu}{2}-\frac{\|a\|_{L^q_{loc}(\mathbb{R})}}{1-\rho^\ast} > 0. \end{eqnarray} | (91) |
In this part, the well-posedness and pullback attractors for problem (1) with sub-linear operator will be stated for our discussion in sequel.
Assume that the initial date
\begin{equation} \begin{cases} u(t)+\int^t_\tau P(\nu Au+\alpha u+\beta|u|u+\gamma |u|^2u)ds &\\ \quad = u(\tau) +\int^t_\tau P\Big(F\big(u(s-\rho(s))\big)+g(s,x)\Big)ds,& \\ w|_{\partial\Omega} = 0,& \\ u(t = \tau) = u_{\tau},&\\ u(\tau+t) = \phi(t),\ t\in [-h,0],& \end{cases} \end{equation} | (92) |
which possesses a global mild solution as the following theorem.
Theorem 4.1. Assume that the external forces
\begin{eqnarray} &&\|u(t)\|^{2}_H+2\nu\int^t_{\tau}\|u(s)\|^{2}_Vds+2\alpha \int^t_{\tau}\|u(s)\|^{2}_Hds\\ &&+2\beta\int^t_{\tau}\|u(s)\|^{3}_{\bf{L}^3}ds+2\gamma\int^t_{\tau}\|u(s)\|^{4}_{\bf{L}^4}ds\\ & = &\|u_{\tau}\|^{2}_H+2\int^t_{\tau}\Big[\big(F(u(s-\rho(s))),u(s)\big)+2(g(s,x),u(s))\Big]ds. \end{eqnarray} | (93) |
Moreover, we can define a continuous process
Proof. Using the Galerkin method and compact argument as in Section 3.3, we can easily derive the result.
After obtaining the existence of the global well-posedness, we establish the existence of the pullback attractors to (1) with sub-linear operator.
Theorem 4.2. (The pullback attractors in
Proof. Using the similar technique as in Section3.3, we can obtain the existence of pullback attractors, here we skip the details.
Theorem 4.3. We assume that the external forces
Then the trajectories inside pullback attractors
\begin{equation} \mathit{\mbox{G}}(t)\leq \tilde{K}_0, \end{equation} | (94) |
where
\begin{equation} \tilde{K}_0 = \Big\{(2\nu\lambda_1+\alpha)\Big/\Big[2C_{|\Omega|}\beta\nu\lambda_1 (\frac{1}{\alpha^2(1-\rho^*)}+\|\tilde{a}(t)\|_{L^1})\Big]\Big\}^{1/2} > 0,\nonumber \end{equation} |
here
Proof. Step 1. The inequality for asymptotic stability of trajectories.
Let
\begin{eqnarray} u(\theta+\tau)|_{\theta\in [-h,0]} = \phi(\theta)|_{\theta\in[-h,0]},\ \ u|_{t = \tau} = u_{\tau} \end{eqnarray} | (95) |
and
\begin{eqnarray} v(\theta+\tau)|_{\theta\in [-h,0]} = \tilde{\phi}(\theta)|_{\theta\in[-h,0]},\ \ v|_{t = \tau} = \tilde{u}_{\tau} \end{eqnarray} | (96) |
respectively, then
\begin{eqnarray} (u,u_t) = (U(t,\tau)u_{\tau},U(t,\tau)\phi),\ \ (v,v_t) = (U(t,\tau)\tilde{u}_{\tau},U(t,\tau)\tilde{\phi}). \end{eqnarray} | (97) |
If we denote
\begin{equation} \begin{cases} \frac{\partial w}{\partial t}+\nu A w +P\Big(\alpha w+\beta(|u|u-|v|v)+\gamma (|u|^2u-|v|^2v)\Big)&\\ \quad = P\Big(F\big(u(t-\rho(t))\big)-F\big(v(t-\rho(t))\big)\Big),& \\ w|_{\partial\Omega} = 0,& \\ w(t = \tau) = u_{\tau}-\tilde{u}_{\tau},&\\ w(\tau+\theta) = \phi(\theta)-\tilde{\phi}(\theta),\ \theta\in [-h,0].& \end{cases} \end{equation} | (98) |
Multiplying (98) with
\begin{eqnarray} &&\frac{1}{2}\frac{d}{d t}\|w\|^2_H+\nu\|w\|^2_V+\alpha \|w\|_H^2+\gamma \gamma_0 \|w\|^4_{\bf{L}^4}\\ &\leq& |\beta(|u|u-|v|v,w)|+\Big|\Big(F\big(u(t-\rho(t))\big)-F\big(v(t-\rho(t))\big),w\Big)\Big|\\ &\leq& C_{|\Omega|}\beta(\|u\|^2_{V}+\|u\|^2_{V})\|w\|^2_{H}+\frac{\alpha}{2}\|w\|^2_H\\ &&+\frac{1}{\alpha}\|F\big(u(t-\rho(t))\big)-F\big(v(t-\rho(t))\big)\|^2_H\\ &\leq& C_{|\Omega|}\beta(\|u\|^2_{V}+\|u\|^2_{V})\|w\|^2_{H}+\frac{\alpha}{2}\|w\|^2_H\\ &&+\frac{L^2(R)\kappa(t)}{\alpha}\|w(t-\rho(t))\|^2_H. \end{eqnarray} | (99) |
Using the Poincaré inequality and Lemma 3.1, noting that if
\begin{eqnarray} 2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V) > 0, \end{eqnarray} | (100) |
then we can obtain
\begin{eqnarray} \|w\|^2_H&\leq& e^{\int^t_{\tau}[2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)-(2\nu\lambda_1+\alpha)]ds}\Big[\|u_{\tau}-\tilde{u}_{\tau}\|^2_H+\\ &&+\frac{L^2(R)\|\kappa(t)\|_{L^{\infty}}}{\alpha}\int^t_{\tau}e^{-\int^t_{s}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]d\sigma}\\ && \quad \times\|w(t-\rho(t))\|^2_Hds\Big]. \end{eqnarray} | (101) |
Denoting
\begin{eqnarray} E(t,\tau) = e^{-\int^t_{\tau}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]ds} \end{eqnarray} | (102) |
and
\begin{eqnarray} K_1(t,s) = \frac{L^2(R)\|\kappa(t)\|_{L^{\infty}}}{\alpha}e^{-\int^t_{s}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]d\sigma} \end{eqnarray} | (103) |
and
\begin{eqnarray} \Theta = \sup\limits_{t\geq s\geq \tau}E(t,s),\ \ \ \ \kappa(K_1,0) = \sup\limits_{t\geq\tau}\int^t_{\tau}K_1(t,s)ds, \end{eqnarray} | (104) |
by virtue of Lemma 3.2, choosing
\begin{eqnarray} \|w(t-\rho(t))\|^2_H&\leq& \tilde{M}\|u_{\tau}-\tilde{u}_{\tau}\|^2_H e^{-\tilde{\lambda} (t-\tau)}. \end{eqnarray} | (105) |
Substituting (105) into (99), using Lemma 3.1 again, we can conclude the following estimate
\begin{eqnarray} \|w\|^2_H &\leq& \|u_{\tau}-\tilde{u}_{\tau}\|^2_He^{-\int^t_{\tau}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]ds}\\ &&+\frac{L^2(R)\|\kappa(t)\|_{L^{\infty}}}{\alpha}\tilde{M}\|u_{\tau}-\tilde{u}_{\tau}\|^2_H e^{-\tilde{\lambda} (t-\tau)}\\ && \quad \times\int^t_{\tau}e^{-\int^t_{s}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]d\sigma}ds. \end{eqnarray} | (106) |
From the result in last section, we can find that the pullback attractors is asymptotically stable as
\begin{equation} 2\nu\lambda_1+\alpha > 2C_{|\Omega|}\beta \langle \|u\|^2_V+\|v\|^2_V\rangle_{\leq t}. \end{equation} | (107) |
Step 2.Some energy estimate for (1) with sub-linear operator.
Multiplying (3) with
\begin{eqnarray} &&\frac{1}{2}\frac{d}{dt}\|u\|^2_H+\nu\|A^{1/2}u\|^2_H+\alpha\|u\|^2_H+\beta \|u\|^3_{\bf{L}^3}+\gamma \|u\|^4_{\bf{L}^4}\\ &\leq&\alpha \|u\|^2_H+\frac{1}{2\alpha}\Big[\|f\big(t,u(t-\rho(t))\big)\|^2_H+\|g\|^2_{H}\Big]. \end{eqnarray} | (108) |
Moreover, let
\begin{equation} d\theta = (1-\rho'(s))ds,\ a(t)\rightarrow \tilde{a}(\bar{t})\in L^p(\tau,T), \end{equation} | (109) |
which means
\begin{align} &\int^t_\tau\|f(s,u(s-\rho(s)))\|^2_Hds\\ \leq&\int^t_\tau a(s)\|u(s-\rho(s))\|^2_Hds+\int^T_\tau b(s)ds\\ \leq& \ \dfrac{1}{1-\rho^*}\int_{\tau-\rho(\tau)}^{t-\rho(t)} \tilde{a}(s)\|u(s)\|^2_Hds+\int^t_\tau b(s)ds\\ \leq& \ \dfrac{1}{1-\rho^*}\left(\int_{-\rho(\tau)}^{0}\tilde{a}(t+\tau)\|\phi(t)\|^2_Hdt +\int^t_\tau\tilde{a}(s)\|u(s)\|^2_Hds\right)+\int^t_\tau b(s)ds\\ \leq& \dfrac{1}{1-\rho^*}\left(\|\phi(t)\|^2_{L^{2q}_{H}}\|\tilde{a}\|_{L^q(\tau-h,\tau)} +\int^t_\tau\tilde{a}(s)\|u(s)\|^2_Hds\right)+\int^t_\tau b(s)ds, \end{align} | (110) |
Integrating (108) with time variable from
\begin{eqnarray} &&\|u\|^2_H+2\nu\int^t_{\tau}\|u\|^2_Vds+2\beta\int^t_{\tau} \|u\|^3_{\bf{L}^3}ds+2\gamma \int^t_{\tau}\|u\|^4_{\bf{L}^4}ds\\ &\leq& \dfrac{\|\tilde{a}\|_{L^q(\tau-h,\tau)}}{\alpha(1-\rho^*)}\|\phi(t)\|^2_{L^{2q}_{H}}+\|u_{\tau}\|^2_H+\frac{1}{\alpha(1-\rho^*)}\int^{t}_{\tau}\tilde{a}(s)\|u(s)\|^2_Hds\\ &&+\frac{1}{\alpha}\int^t_{\tau}\|g\|^2_{H}ds+\frac{1}{\alpha}\int^t_\tau b(s)ds, \end{eqnarray} | (111) |
then we can achieve that
\begin{eqnarray} \|u(t)\|^2_H&\leq& \Big[\dfrac{\|\tilde{a}\|_{L^q(\tau-h,\tau)}}{\alpha(1-\rho^*)}\|\phi(t)\|^2_{L^{2q}_{H}}+\|u_{\tau}\|^2_H\Big]e^{-\chi_{\sigma}(t,\tau)}\\ &&+\frac{1}{\alpha}\int^t_{\tau}\|g\|^2_{H}e^{-\chi_{\sigma}(t,s)}ds+\frac{1}{\alpha}\int^t_{\tau}b(s)e^{-\chi_{\sigma}(t,s)}ds, \end{eqnarray} | (112) |
where the new variable index
\begin{eqnarray} \chi_{\sigma}(t,s) = (2\nu\lambda_1-\sigma)(t-s)-\frac{1}{\alpha(1-\rho^*)}\int^t_{s}\tilde{a}(r)dr, \end{eqnarray} | (113) |
which satisfies the relations
\begin{eqnarray} \chi_{\sigma}(0,t)-\chi_{\sigma}(0,s) = -\chi_{\sigma}(t,s) \end{eqnarray} | (114) |
and
\begin{eqnarray} \chi_{\sigma}(0,r)\leq \chi_{\sigma}(0,t)+\Big(2\nu\lambda_1-\delta\Big)h,\ \ \mbox{if}\ 2\nu\lambda_1+\alpha-\delta > 0 \end{eqnarray} | (115) |
for
Moreover, using the variable index introduced above, we can conclude that
\begin{eqnarray} &&2\nu\int^t_{\tau}\|u(r)\|^2_{V}dr\\ &\leq& \dfrac{\|\tilde{a}\|_{L^q(\tau-h,\tau)}}{\alpha(1-\rho^*)}\|\phi(t)\|^2_{L^{2q}_{H}}+\|u_{\tau}\|^2_H\\ &&+\frac{1}{\alpha}\int^t_{\tau}\|g\|^2_{H}ds+\frac{1}{\alpha}\int^t_\tau b(s)ds\\ &&+\frac{1}{\alpha(1-\rho^*)}\Big[\dfrac{\|\tilde{a}\|_{L^q(\tau-h,\tau)}}{\alpha(1-\rho^*)}\|\phi(t)\|^2_{L^{2q}_{H}}+\|u_{\tau}\|^2_H\Big]\int^t_{\tau}\tilde{a}(s)e^{-\chi_{\sigma}(s,\tau)}ds\\ &&+\frac{1}{\alpha^2(1-\rho^*)}\int^t_{\tau}\|g(s)\|^2_{H}ds\int^t_{\tau}\tilde{a}(s)ds+\frac{\|b\|_{L^1(\tau,T)}}{\alpha^2(1-\rho^*)}\int^t_{\tau}\tilde{a}(s)ds. \end{eqnarray} | (116) |
Step 3. The sufficient condition for asymptotic stability of trajectories inside pullback attractors.
Combining (107) with (116), we conclude that
\begin{eqnarray} &&2C_{|\Omega|}\beta\langle \|u\|^2_V+\|v\|^2_V\rangle|_{\leq t}\\ &\leq& \frac{2C_{|\Omega|}\beta}{\nu}\Big[\Big(\frac{1}{\alpha^2(1-\rho^*)}+\int^t_{\tau}\tilde{a}(s)ds\Big)\langle \|g(t)\|_H^2\rangle|_{\leq t} \\ &&+\frac{1}{\alpha}\langle \|b(t)\|_{L^1}\rangle|_{\leq t}+\frac{\|b\|_{L^1(\tau,T)}}{\alpha^2(1-\rho^*)}\langle \|\tilde{a}(t)\|_{L^1}\rangle|_{\leq t}\Big]. \end{eqnarray} | (117) |
and hence the asymptotic stability holds provided that
\begin{eqnarray} &&\Big(\frac{1}{\alpha^2(1-\rho^*)}+\|\tilde{a}(t)\|_{L^1}\Big)\langle \|g(t)\|_H^2\rangle|_{\leq t}+\frac{1}{\alpha}\langle \|b(t)\|_{L^1}\rangle|_{\leq t}+\frac{\|b\|_{L^1(\tau,T)}}{\alpha^2(1-\rho^*)}\langle \|\tilde{a}(t)\|_{L^1}\rangle|_{\leq t}\\ &&\leq \frac{\nu(2\nu\lambda_1+\alpha)}{2C_{|\Omega|}\beta}. \end{eqnarray} | (118) |
If we define the generalized Grashof number as
\begin{eqnarray} G(t)\leq \Big\{(2\nu\lambda_1+\alpha)\Big/\Big[2C_{|\Omega|}\beta\nu\lambda_1 (\frac{1}{\alpha^2(1-\rho^*)}+\|\tilde{a}(t)\|_{L^1})\Big]\Big\}^{1/2} = \tilde{K}_0, \end{eqnarray} | (119) |
which completes the proof for our first result.
Remark 5. If we denote
\begin{eqnarray} \limsup\limits_{\tau\rightarrow -\infty}\frac{1}{t-\tau}\int^t_{\tau}b(r)dr = b_0\in [0,+\infty) \end{eqnarray} | (120) |
and
\begin{eqnarray} \limsup\limits_{\tau\rightarrow -\infty}\frac{1}{t-\tau}\int^t_{\tau}\tilde{a}(r)dr = \tilde{a}_0\in [0,+\infty), \end{eqnarray} | (121) |
such that there exists some
\begin{eqnarray} \frac{\nu(2\nu\lambda_1+\alpha)}{2C_{|\Omega|}\beta} > \frac{b_0}{\alpha}+\frac{\|b\|_{L^1(\tau,T)}\tilde{\alpha}_0}{\alpha^2(1-\rho^*)}+\delta \end{eqnarray} | (122) |
holds. Then more precise sufficient condition for the asymptotic stability of pullback attractors is
\begin{eqnarray} G(t)\leq \Big[\frac{\frac{\nu(2\nu\lambda_1+\alpha)}{2C_{|\Omega|}\beta}-\frac{b_0}{\alpha}-\frac{\|b\|_{L^1(\tau,T)}\tilde{\alpha}_0}{\alpha^2(1-\rho^*)}}{\nu^2\lambda_1(\frac{1}{\alpha^2(1-\rho^*)}+\|\tilde{a}(t)\|_{L^1})}\Big]^{1/2} \end{eqnarray} | (123) |
which has smaller upper boundedness than (119).
The structure and stability of 3D BF equations with delay are investigated in this paper. A future research in the pullback dynamics of (1) is to study the geometric property of pullback attractors, such as the fractal dimension.
Xin-Guang Yang was partially supported by the Fund of Young Backbone Teacher in Henan Province (No. 2018GGJS039) and Henan Overseas Expertise Introduction Center for Discipline Innovation (No. CXJD2020003). Xinjie Yan was partly supported by Excellent Innovation Team Project of "Analysis Theory of Partial Differential Equations" in China University of Mining and Technology (No. 2020QN003). Ling Ding was partly supported by NSFC of China (Grant No. 1196302).
The authors want to express their most sincere thanks to refrees for the improvement of this manuscript. The authors also want to thank Professors Tomás Caraballo (Universidad de Sevilla), Desheng Li (Tianjin University) and Shubin Wang (Zhengzhou University) for fruitful discussion on this subject.
[1] | M. A. Arbib, Brains, Machines, and Mathematics, 2nd edition, Springer-Verlag, New York, 1987. |
[2] | S. Haykin, Neural Networks: A Comprehensive Foundation, 2nd edition, Prentice-Hall, Englewood Cliffs, 1998. |
[3] |
M. A. Cohen, S. M. A. Grossberg, Absolute stability of global pattern formation and parallel memory storage by competitive neural networks, IEEE Trans. Syst. Man Cyber., 13 (1983), 815–826. https://doi.org/10.1109/TSMC.1983.6313075 doi: 10.1109/TSMC.1983.6313075
![]() |
[4] |
C. Aouiti, E. A. Assali, Nonlinear Lipschitz measure and adaptive control for stability and synchronization in delayed inertial Cohen–Grossberg-type neural networks, Int. J. Adapt. Control Signal Process., 33 (2019), 1457–1477. https://doi.org/10.1002/acs.3042 doi: 10.1002/acs.3042
![]() |
[5] |
W. Lu, T. Chen, Dynamical behaviors of Cohen–Grossberg neural networks with discontinuous activation functions, Neural Networks, 18 (2015), 231–242. https://doi.org/10.1016/j.neunet.2004.09.004 doi: 10.1016/j.neunet.2004.09.004
![]() |
[6] |
N. Ozcan, Stability analysis of Cohen–Grossberg neural networks of neutral-type: Multiple delays case, Neural Networks, 113 (2019), 20–27. https://doi.org/10.1016/j.neunet.2019.01.017 doi: 10.1016/j.neunet.2019.01.017
![]() |
[7] |
S. Han, C. Hu, J. Yu, H. Jiang, S. Wen, Stabilization of inertial Cohen-Grossberg neural networks with generalized delays: A direct analysis approach, Chaos Solitons Fractals, 142 (2021), 110432. https://doi.org/10.1016/j.chaos.2020.110432 doi: 10.1016/j.chaos.2020.110432
![]() |
[8] |
D. Peng, J. Li, W. Xu, X. Li, Finite-time synchronization of coupled Cohen-Grossberg neural networks with mixed time delays, J. Franklin Inst., 357 (2020), 11349–11367. https://doi.org/10.1016/j.jfranklin.2019.06.025 doi: 10.1016/j.jfranklin.2019.06.025
![]() |
[9] | J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Nat. Acad. Sci., 79 (1982), 2554–2558. https://www.pnas.org/doi/10.1073/pnas.79.8.2554 |
[10] |
L. O. Chua, L. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits Syst., 35 (1988), 1257–1272. https://doi.org/10.1109/31.7600 doi: 10.1109/31.7600
![]() |
[11] |
L. O. Chua, L. Yang, Cellular neural networks: Applications, IIEEE Trans. Circuits Syst., 35 (1988), 1273–1290. https://doi.org/10.1109/31.7601 doi: 10.1109/31.7601
![]() |
[12] | W. M. Haddad, V. S. Chellaboina, S. G. Nersesov, Impulsive and Hybrid Dynamical Systems, Stability, Dissipativity, and Control, 1st edition, Princeton University Press, Princeton, 2006. |
[13] | X. Li, S. Song, Impulsive Systems with Delays: Stability and Control, 1st edition, Science Press & Springer, Singapore, 2022. |
[14] | I. Stamova, G. Stamov, Applied Impulsive Mathematical Models, 1st edition, Springer, Cham, 2016. |
[15] |
C. Aouiti, F. Dridi, New results on impulsive Cohen–Grossberg neural networks, Neural Process. Lett., 49 (2019), 1459–1483. https://doi.org/10.1007/s11063-018-9880-y doi: 10.1007/s11063-018-9880-y
![]() |
[16] |
B. Lisena, Dynamical behavior of impulsive and periodic Cohen–Grossberg neural networks, Nonlinear Anal., 74 (2011), 4511–4519. https://doi.org/10.1016/j.na.2011.04.015 doi: 10.1016/j.na.2011.04.015
![]() |
[17] |
C. Xu, Q. Zhang, On anti–periodic solutions for Cohen–Grossberg shunting inhibitory neural networks with time–varying delays and impulses, Neural Comput., 26 (2014), 2328–2349. https://dl.acm.org/doi/10.1162/NECO_a_00642 doi: 10.1162/NECO_a_00642
![]() |
[18] | T. Yang, Impulsive Control Theory, 1st edition, Springer, Berlin, 2001. |
[19] |
X. Yang, D. Peng, X. Lv, X. Li, Recent progress in impulsive control systems, Math. Comput. Simul., 155 (2019), 244–268. https://doi.org/10.1016/j.matcom.2018.05.003 doi: 10.1016/j.matcom.2018.05.003
![]() |
[20] |
J. Cao, T. Stamov, S. Sotirov, E. Sotirova, I. Stamova, Impulsive control via variable impulsive perturbations on a generalized robust stability for Cohen–Grossberg neural networks with mixed delays, IEEE Access, 8 (2020), 222890–222899. https://doi.org/10.1109/ACCESS.2020.3044191 doi: 10.1109/ACCESS.2020.3044191
![]() |
[21] |
X. Li, Exponential stability of Cohen–Grossberg-type BAM neural networks with time-varying delays via impulsive control, Neurocomputing, 73 (2009), 525–530. https://doi.org/10.1016/j.neucom.2009.04.022 doi: 10.1016/j.neucom.2009.04.022
![]() |
[22] | D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, 1st edition, World Scientific, Singapore, 2012. https://doi.org/10.1100/2012/738423 |
[23] | R. Magin, Fractional Calculus in Bioengineering, 1st edition, Begell House, Redding, 2006. |
[24] | I. Podlubny, Fractional Differential Equations, 1st edition, Academic Press, San Diego, 1999. |
[25] | I. M. Stamova, G. T. Stamov, Functional and Impulsive Differential Equations of Fractional Order: Qualitative Analysis and Applications, 1st edition, CRC Press, 2017. |
[26] |
P. Anbalagan, E. Hincal, R. Ramachandran, D. Baleanu, J. Cao, C. Huang, et al., Delay-coupled fractional order complex Cohen–Grossberg neural networks under parameter uncertainty: Synchronization stability criteria, AIMS Math., 6 (2021), 2844–2873. https://doi.org/10.3934/math.2021172 doi: 10.3934/math.2021172
![]() |
[27] |
A. Pratap, R. Raja, J. Cao, C. P. Lim, O. Bagdasar, Stability and pinning synchronization analysis of fractional order delayed Cohen–Grossberg neural networks with discontinuous activations, Appl. Math. Comput., 359 (2019), 241–260. https://doi.org/10.1016/j.amc.2019.04.062 doi: 10.1016/j.amc.2019.04.062
![]() |
[28] |
C. Rajivganthi, F. A. Rihan, S. Lakshmanan, P. Muthukumar, Finite-time stability analysis for fractional-order Cohen–Grossberg BAM neural networks with time delays, Neural Comput. Appl., 29 (2018), 1309–1320. https://doi.org/10.1007/s00521-016-2641-9 doi: 10.1007/s00521-016-2641-9
![]() |
[29] |
I. Stamova, S. Sotirov, E. Sotirova, G. Stamov, Impulsive fractional Cohen–Grossberg neural networks: Almost periodicity analysis, Fractal Fractional, 5 (2021), 78. https://doi.org/10.3390/fractalfract5030078 doi: 10.3390/fractalfract5030078
![]() |
[30] |
F. Zhang, Z. Zeng, Multiple Mittag-Leffler stability of delayed fractional-order Cohen–Grossberg neural networks via mixed monotone operator pair, IEEE Trans. Cyber., 51 (2021), 6333–6344. https://doi.org/10.1109/TCYB.2019.2963034 doi: 10.1109/TCYB.2019.2963034
![]() |
[31] |
M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fractional Differ. Appl., 1 (2015), 1–13. https://dx.doi.org/10.12785/pfda/010201 doi: 10.12785/pfda/010201
![]() |
[32] |
C. Derbazi, H. Hammouche, Caputo–Hadamard fractional differential equations with nonlocal fractional integro-differential boundary conditions via topological degree theory, AIMS Math., 5 (2020), 2694–2709. https://doi.org/10.3934/math.2020174 doi: 10.3934/math.2020174
![]() |
[33] |
R. Almeida, Caputo–Hadamard fractional derivatives of variable order, Numer. Functional Anal. Optim., 38 (2017), 1–19. https://doi.org/10.1080/01630563.2016.1217880 doi: 10.1080/01630563.2016.1217880
![]() |
[34] |
A. Benkerrouche, M. S. Souid, G. Stamov, I. Stamova, Multiterm impulsive Caputo–Hadamard type differential equations of fractional variable order, Axioms, 11 (2022), 634. https://doi.org/10.3390/axioms11110634 doi: 10.3390/axioms11110634
![]() |
[35] |
T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57–66. https://doi.org/10.1016/j.cam.2014.10.016 doi: 10.1016/j.cam.2014.10.016
![]() |
[36] |
R. Khalil, M. Al-Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70. https://doi.org/10.1016/j.cam.2014.01.002 doi: 10.1016/j.cam.2014.01.002
![]() |
[37] |
H. Kiskinov, M. Petkova, A. Zahariev, M. Veselinova, Some results about conformable derivatives in Banach spaces and an application to the partial differential equations, AIP Conf. Proc., 2333 (2021), 120002. https://doi.org/10.1063/5.0041758 doi: 10.1063/5.0041758
![]() |
[38] | A. A. Martynyuk, I. M. Stamova, Fractional-like derivative of Lyapunov-type functions and applications to the stability analysis of motion, Electron. J. Differ. Equations, 2018 (2018), 1–12. |
[39] | M. Posp{\rm \acute i}{\rm \breve s}il, L. Posp{\rm \acute i}{\rm \breve s}ilova {\rm \breve S}kripkova, Sturm's theorems for conformable fractional differential equation, Math. Commun., 21 (2016), 273–281. |
[40] |
A. Souahi, A. B. Makhlouf, M. A. Hammami, Stability analysis of conformable fractional-order nonlinear systems, Indagationes Math., 28 (2017), 1265–1274. https://doi.org/10.1016/j.indag.2017.09.009 doi: 10.1016/j.indag.2017.09.009
![]() |
[41] |
A. A. Martynyuk, G. Stamov, I. Stamova, Integral estimates of the solutions of fractional-like equations of perturbed motion, Nonlinear Anal. Modell. Control, 24 (2019), 138–149. https://doi.org/10.15388/NA.2019.1.8 doi: 10.15388/NA.2019.1.8
![]() |
[42] | D. Zhao, M. Luo, General conformable fractional derivative and its physical interpretation, Calcolo, 54 (2017), 903–917. https://doi.org/110.1007/s10092-017-0213-8 |
[43] |
M. Bohner, V. F. Hatipoğlu, Cobweb model with conformable fractional derivatives, Math. Methods Appl. Sci., 41 (2018), 9010–9017. https://doi.org/10.1002/mma.4846 doi: 10.1002/mma.4846
![]() |
[44] |
A. Harir, S. Malliani, L. S. Chandli, Solutions of conformable fractional-order SIR epidemic model, Int. J. Differ. Equations, 2021 (2021), 6636686. https://doi.org/10.1155/2021/6636686 doi: 10.1155/2021/6636686
![]() |
[45] |
W. Xie, C. Liu, W. Z. Wu, W. Li, C. Liu, Continuous grey model with conformable fractional derivative, Chaos Solitons Fractals, 139 (2020), 110285. https://doi.org/10.1016/j.chaos.2020.110285 doi: 10.1016/j.chaos.2020.110285
![]() |
[46] |
S. Sitho, S. K. Ntouyas, P. Agarwal, J. Tariboon, Noninstantaneous impulsive inequalities via conformable fractional calculus, J. Inequalities Appl., 2018 (2018), 261. https://doi.org/10.1186/s13660-018-1855-z doi: 10.1186/s13660-018-1855-z
![]() |
[47] |
G. Stamov, A. Martynyuk, I. Stamova, Impulsive fractional-like differential equations: Practical stability and boundedness with respect to h-manifolds, Fractal Fractional, 3 (2019), 50. https://doi.org/10.3390/fractalfract3040050 doi: 10.3390/fractalfract3040050
![]() |
[48] |
J. Tariboon, S. K. Ntouyas, Oscillation of impulsive conformable fractional differential equations, Open Math., 14 (2016), 497–508. https://doi.org/10.1515/math-2016-0044 doi: 10.1515/math-2016-0044
![]() |
[49] |
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
![]() |
[50] |
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
![]() |
[51] |
T. Stamov, I. Stamova, Design of impulsive controllers and impulsive control strategy for the Mittag–Leffler stability behavior of fractional gene regulatory networks, Neurocomputing, 424 (2021), 54–62. https://doi.org/10.1016/j.neucom.2020.10.112 doi: 10.1016/j.neucom.2020.10.112
![]() |
[52] |
I. Stamova, G. Stamov, Impulsive control strategy for the Mittag–Leffler synchronization of fractional-order neural networks with mixed bounded and unbounded delays, AIMS Math., 6 (2021), 2287–2303. https://doi.org/10.3934/math.2021138 doi: 10.3934/math.2021138
![]() |
[53] |
G. Stamov, I. Stamova, Extended stability and control strategies for impulsive and fractional neural networks: A review of the recent results, Fractal Fractional, 7 (2023), 289. https://doi.org/10.3390/fractalfract7040289 doi: 10.3390/fractalfract7040289
![]() |
[54] | G. Ballinger, X Liu, Practical stability of impulsive delay differential equations and applications to control problems, in Optimization Methods and Applications. Applied Optimization (eds. X. Yang, K. L. Teo and L. Caccetta), Springer, (2001), 3–21. |
[55] | V. Lakshmikantham, S. Leela, A. A. Martynyuk, Pract. Stab. Nonlinear Syst., 1st edition, World Scientific, Teaneck, 1990. https://doi.org/10.1142/1192 |
[56] |
T. Stamov, Neural networks in engineering design: Robust practical stability analysis, Cybern. Inf. Technol., 21 (2021), 3–14. https://doi.org/10.2478/cait-2021-0039 doi: 10.2478/cait-2021-0039
![]() |
[57] |
Y. Tian, Y. Sun, Practical stability and stabilisation of switched delay systems with non-vanishing perturbations, IET Control Theory Appl., 13 (2019), 1329–1335. https://doi.org/10.1049/iet-cta.2018.5332 doi: 10.1049/iet-cta.2018.5332
![]() |
[58] |
G. Stamov, I. M. Stamova, X. Li, E. Gospodinova, Practical stability with respect to h-manifolds for impulsive control functional differential equations with variable impulsive perturbations, Mathematics, 7 (2019), 656. https://doi.org/10.3390/math7070656 doi: 10.3390/math7070656
![]() |
[59] |
G. Stamov, E. Gospodinova, I. Stamova, Practical exponential stability with respect to h-manifolds of discontinuous delayed Cohen–Grossberg neural networks with variable impulsive perturbations, Math. Modell. Control, 1 (2021), 26–34. https://doi.org/10.3934/mmc.2021003 doi: 10.3934/mmc.2021003
![]() |
[60] |
A. A. Martynyuk, G. Stamov, I. Stamova, Practical stability analysis with respect to manifolds and boundedness of differential equations with fractional-like derivatives, Rocky Mt. J. Math., 49 (2019), 211–233. https://doi.org/10.1216/RMJ-2019-49-1-211 doi: 10.1216/RMJ-2019-49-1-211
![]() |
[61] |
B. Kosko, Bidirectional associative memories, IEEE Trans. Syst. Cybern., 18 (1988), 49–60. https://doi.org/10.1109/21.87054 doi: 10.1109/21.87054
![]() |
[62] | B. Kosko, Neural Networks and Fuzzy Systems: A Dynamical System Approach to Machine Intelligence, 1st edition, Prentice-Hall, Englewood Cliffs, 1992. |
[63] |
M. Syed Ali, S. Saravanan, M. E. Rani, S. Elakkia, J. Cao, A. Alsaedi, et al., Asymptotic stability of Cohen–Grossberg BAM neutral type neural networks with distributed time varying delays, Neural Process. Lett., 46 (2017), 991–1007. https://doi.org/10.1007/s11063-017-9622-6 doi: 10.1007/s11063-017-9622-6
![]() |
[64] |
H. Jiang, J. Cao, BAM-type Cohen–Grossberg neural networks with time delays, Math. Comput. Modell., 47 (2008), 92–103. https://doi.org/10.1016/j.mcm.2007.02.020 doi: 10.1016/j.mcm.2007.02.020
![]() |
[65] |
X. Li, Existence and global exponential stability of periodic solution for impulsive Cohen–Grossberg-type BAM neural networks with continuously distributed delays, Appl. Math. Comput., 215 (2009), 292–307. https://doi.org/10.1016/j.amc.2009.05.005 doi: 10.1016/j.amc.2009.05.005
![]() |
[66] |
C. Maharajan, R. Raja, J. Cao, G. Rajchakit, A. Alsaedi, Impulsive Cohen–Grossberg BAM neural networks with mixed time-delays: An exponential stability analysis issue, Neurocomputing, 275 (2018), 2588–2602. https://doi.org/10.1016/j.neucom.2017.11.028 doi: 10.1016/j.neucom.2017.11.028
![]() |
[67] |
T. Stamov, Discrete bidirectional associative memory neural networks of the Cohen–Grossberg type for engineering design symmetry related problems: Practical stability of sets analysis, Symmetry, 14 (2022), 216. https://doi.org/10.3390/sym14020216 doi: 10.3390/sym14020216
![]() |
1. | Qiangheng Zhang, Yangrong Li, Regular attractors of asymptotically autonomous stochastic 3D Brinkman-Forchheimer equations with delays, 2021, 20, 1534-0392, 3515, 10.3934/cpaa.2021117 | |
2. | Pan Zhang, Lan Huang, Rui Lu, Xin-Guang Yang, Pullback dynamics of a 3D modified Navier-Stokes equations with double delays, 2021, 29, 2688-1594, 4137, 10.3934/era.2021076 | |
3. | Shu Wang, Mengmeng Si, Rong Yang, Dynamics of stochastic 3D Brinkman-Forchheimer equations on unbounded domains, 2023, 31, 2688-1594, 904, 10.3934/era.2023045 | |
4. | Yang Liu, Chunyou Sun, Inviscid limit for the damped generalized incompressible Navier-Stokes equations on \mathbb{T}^2 , 2021, 14, 1937-1632, 4383, 10.3934/dcdss.2021124 | |
5. | Shu Wang, Mengmeng Si, Rong Yang, Random attractors for non-autonomous stochastic Brinkman-Forchheimer equations on unbounded domains, 2022, 21, 1534-0392, 1621, 10.3934/cpaa.2022034 | |
6. | Wenjing Liu, Rong Yang, Xin-Guang Yang, Dynamics of a 3D Brinkman-Forchheimer equation with infinite delay, 2021, 20, 1553-5258, 1907, 10.3934/cpaa.2021052 | |
7. | Ling-Rui Zhang, Xin-Guang Yang, Ke-Qin Su, Asymptotic Stability for the 2D Navier–Stokes Equations with Multidelays on Lipschitz Domain, 2022, 10, 2227-7390, 4561, 10.3390/math10234561 | |
8. | Xiaona Cui, Wei Shi, Xuezhi Li, Xin‐Guang Yang, Pullback dynamics for the 3‐D incompressible Navier–Stokes equations with damping and delay, 2021, 44, 0170-4214, 7031, 10.1002/mma.7239 | |
9. | Zhengwang Tao, Xin-Guang Yang, Yan Lin, Chunxiao Guo, Determination of Three-Dimensional Brinkman—Forchheimer-Extended Darcy Flow, 2023, 7, 2504-3110, 146, 10.3390/fractalfract7020146 | |
10. | Yonghai Wang, Minhui Hu, Yuming Qin, Upper semicontinuity of pullback attractors for a nonautonomous damped wave equation, 2021, 2021, 1687-2770, 10.1186/s13661-021-01532-7 | |
11. | Xueli SONG, Xi DENG, Baoming QIAO, Dimension Estimate of the Global Attractor for a 3D Brinkman- Forchheimer Equation, 2023, 28, 1007-1202, 1, 10.1051/wujns/2023281001 | |
12. | Songmao He, Xin-Guang Yang, Asymptotic behavior of 3D Ladyzhenskaya-type fluid flow model with delay, 2024, 0, 1937-1632, 0, 10.3934/dcdss.2024135 | |
13. | Lingrui Zhang, Xue-zhi Li, Keqin Su, Dynamical behavior of Benjamin-Bona-Mahony system with finite distributed delay in 3D, 2023, 31, 2688-1594, 6881, 10.3934/era.2023348 | |
14. | Xinfeng Ge, Keqin Su, Stability of thermoelastic Timoshenko system with variable delay in the internal feedback, 2024, 32, 2688-1594, 3457, 10.3934/era.2024160 | |
15. | Keqin Su, Xin-Guang Yang, Alain Miranville, He Yang, Dynamics and robustness for the 2D Navier–Stokes equations with multi-delays in Lipschitz-like domains, 2023, 134, 18758576, 513, 10.3233/ASY-231845 | |
16. | Lingrui Zhang, Xue-zhi Li, Keqin Su, Dynamical behavior of Benjamin-Bona-Mahony system with finite distributed delay in 3D, 2023, 31, 2688-1594, 6881, 10.3934/era.20233348 |