The paper focused on the distributed tracking problem for a specific class of multi-agent systems, characterized by bandwidth constraint and dead zone actuators, where the bandwidth limitations exist in neighbor agents and the dead zone nonlinearity refers to a generalized mathematical model. Initially, a series of event-triggered mechanisms with relative thresholds were established for neighbor agents, ensuring that control signals were transmitted only when necessary. Next, the generalized dead zone models were decomposed into two parts: indefinite terms with control coefficients and disturbance-like terms, resulting in unpredictability and damaging effects. Subsequently, based on the backstepping procedure, final consensus controllers with multiple polynomial compensators were constructed. These controllers offset the coupling coefficients caused by event-triggered mechanisms and dead zone non-smooth. Stability analysis was given to substantiate the theoretical correctness of this method and support the claim of Zeno behavior avoidance. Finally, simulation studies were performed for the feasibility of our proposed methodology.
Citation: Xiaohang Su, Peng Liu, Haoran Jiang, Xinyu Yu. Neighbor event-triggered adaptive distributed control for multiagent systems with dead-zone inputs[J]. AIMS Mathematics, 2024, 9(4): 10031-10049. doi: 10.3934/math.2024491
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The paper focused on the distributed tracking problem for a specific class of multi-agent systems, characterized by bandwidth constraint and dead zone actuators, where the bandwidth limitations exist in neighbor agents and the dead zone nonlinearity refers to a generalized mathematical model. Initially, a series of event-triggered mechanisms with relative thresholds were established for neighbor agents, ensuring that control signals were transmitted only when necessary. Next, the generalized dead zone models were decomposed into two parts: indefinite terms with control coefficients and disturbance-like terms, resulting in unpredictability and damaging effects. Subsequently, based on the backstepping procedure, final consensus controllers with multiple polynomial compensators were constructed. These controllers offset the coupling coefficients caused by event-triggered mechanisms and dead zone non-smooth. Stability analysis was given to substantiate the theoretical correctness of this method and support the claim of Zeno behavior avoidance. Finally, simulation studies were performed for the feasibility of our proposed methodology.
In this paper, we consider the Cauchy problem for the following two-dimensional inhomogeneous incompressible viscoelastic rate-type fluids with stress-diffusion:
{ρt+u⋅∇ρ=0,(t,x)∈R+×R2,ρ(ut+u⋅∇u)−Δu+∇P+σdiv(∇b⊗∇b−12|∇b|2I)=0,bt+u⋅∇b+1ν(e′(b)−σΔb)=0,divu=0,(ρ,u,b)(t,x)|t=0=(ρ0,u0,b0)(x), | (1.1) |
where the unknowns ρ=ρ(x,t), u=(u1(x,t),u2(x,t)) and b=b(x,t) stand for the density, velocity of the fluid and the spherical part of the elastic strain, respectively. P is a scalar pressure function, which guarantees the divergence-free condition of the velocity field. The coefficients ν and σ are two positive constants. In addition, we suppose that e(⋅) is a smooth convex function about b and e(0)≤0, e′(0)=0, e″(b)≤C0, where C0 is a positive constant depending on the initial data. The class of fluids is the elastic response described by a spherical strain [3]. Compared with [3], we have added the divergence-free condition to investigate the effect of density on viscoelastic rate-type fluids, while the divergence-free condition is for computational convenience.
It is easy to observe that for σ=0, the system (1.1) degenerates two distinct systems involving the inhomogeneous Navier-Stokes equation for the fluid and a transport equation with damped e′(b). Numerous researchers have extensively studied the well-posedness concern regarding the inhomogeneous Navier-Stokes equations; see [1,7,8,9,11,14] and elsewhere. However, the transport equation has a greater effect on the regularity of density than on that of velocity. Additionally, due to the presence of the damped term e′(b), the initial elasticity in system (1.1) exhibits higher regularity compared to the initial velocity.
In the case where σ>0, system (1.1) resembles the inhomogeneous magnetohydrodynamic (MHD) equations, with b as a scalar function in (1.1) that does not satisfy the divergence condition found in MHD equations. It is essential to highlight that the system (1.1) represents a simplified model, deviating from standard viscoelastic rate-type fluid models with stress-diffusion to facilitate mathematical calculations. Related studies on system (1.1) can be found in [3,4,15]. In particular, Bulíček, Málek, and Rodriguez in [5] established the well-posedness of a 2D homogeneous system (1.1) in Sobolev space. Our contribution lies in incorporating the density equation into this established framework.
Inspired by [11,18], we initially establish a priori estimates for the system (1.1). Subsequently, by using a Friedrich's method and the compactness argument, we obtain the existence and uniqueness of the solutions. Our main result is as follows:
Theorem 1.1. Let the initial data (ρ0,u0,b0) satisfy
0<m<ρ0(x)<M<∞,(u0,b0)∈H1(R2)×H2(R2),e(b0)∈L1(R2), | (1.2) |
where m,M are two given positive constants with m<M. Then system (1.1) has a global solution (ρ,u,b) such that, for any given T>0, (t,x)∈[0,T)×R2,
m<ρ(t,x)<M,u∈L∞(0,T;H1(R2))∩L2(0,T;H2(R2)),∂tu∈L∞(0,T;L2(R2))∩L2(0,T;H1(R2)),b∈L∞(0,T;H2(R2))∩L2(0,T;H3(R2)),∂tb∈L∞(0,T;H1(R2))∩L2(0,T;H2(R2)). |
Moreover, if ∇ρ0∈L4(R2), then the solution is unique.
Remark 1.1. Compared to the non-homogeneous MHD equations, handling the damping term e′(b) poses a challenge, so that we cannot obtain the time-weighted energy of the velocity field. To explore the uniqueness of the solution, it is necessary to improve the regularity of the initial density data.
The key issue to prove the global existence part of Theorem 1.1 is establishing the a priori L∞(0,T;H1(R2)) estimate on (u,∇b) for any positive time T. We cannot directly estimate the L2 estimate of (u,b), which mainly occurs in the velocity term div(∇b⊗∇b−12|∇b|2I). Therefore, we need to estimate the L2 of the ∇b equation. Afterwards, the L2 estimation of equation b was affected by a damping term e′(b), so we made an L2 estimation of equation e′(b). Finally, to show the L∞(0,T;H1(R2)) of u, we also need an estimate of the second derivative of b. In summary, we found that the initial value of the b equation needs to be one derivative higher than the initial value of the u equation.
Concerning the uniqueness of the strong solutions, a common approach is to consider the difference equations between two solutions and subsequently derive some energy estimates for the resulting system differences based on the fundamental natural energy of the system. However, for system (1.1), the presence of a damping term e′(b) of the equation b and density equation prevents the calculation of the time-weighted energy of the velocity field. To research the solution's uniqueness, we need to enhance the regularity of the initial density data.
The paper is structured as follows: Section 2 presents prior estimates for system (1.1). In Section 3, we establish the existence and uniqueness of Theorem 1.1.
Proposition 2.1. Assume that m,M are two given positive constants and 0<m≤M<∞, the initial data ρ0 satisfies 0<m≤ρ0≤M<+∞, and the initial data (√ρ0u0,∇b0)∈L2(R2)×L2(R2). Let (ρ,u,b) be a smooth solution of system (1.1), then there holds for any t>0,
0<m≤ρ(t)≤M<+∞, | (2.1) |
‖(√ρu,∇b,u)(t)‖2L2+∫t0‖(∇u,∇2b)‖2L2dτ≤C‖(√ρ0u0,∇b0)‖2L2, | (2.2) |
where C is a constant depending only on σ, ν.
Proof. First, any Lebesgue norm of ρ0 is preserved through the evolution, and 0<m≤ρ(t)≤M<+∞.
To prove (2.2), taking the L2 inner product of the second equation of (1.1) with u and integrating by parts, then we obtain
12ddt‖√ρu‖2L2+‖∇u‖2L2=−σ∫R2Δb∇b⋅udx, | (2.3) |
where we used the fact that
div(∇b⊗∇b−12|∇b|2I)=Δb∇b. |
Multiplying the third equation of (1.1) by −σΔb and integrating by parts, we obtain
σ2ddt‖∇b‖2L2+σ2ν‖Δb‖2L2−σ2ν∫R2e′(b)Δbdx=σ∫R2u⋅∇bΔbdx. | (2.4) |
Thanks to the convexity of e(b), we know
−σ2ν∫R2e′(b)Δbdx=σ2ν∫R2∇(e′(b))∇bdx=σ2ν∫R2e″(b)|∇b|2dx≥0. | (2.5) |
By inserting (2.5) into (2.4), combining the result with (2.3), one yields
ddt‖(√ρu,∇b)‖2L2+‖∇u‖2L2+C‖Δb‖2L2≤0. |
Integrating it with respect to time, we have
‖(√ρu,∇b)(t)‖2L2+∫t0‖(∇u,∇2b)‖2L2dτ≤C‖(√ρ0u0,∇b0)‖2L2. | (2.6) |
On the other hand, applying 0<m≤ρ≤M<+∞, which together with (2.6) implies
‖u‖2L∞(L2)≤m−1‖√ρu‖2L∞(L2)≤C‖(√ρ0u0,∇b0)‖2L2, |
which, along with inequality (2.6), yields (2.2).
Proposition 2.2. Under the assumptions of Proposition 2.1, the corresponding solution (ρ, u, b) of the system (1.1) admits the following bound for any t>0:
‖(∇u,∇2b,∇b,b)‖2L2+∫t0‖(∇2u,∇3b,√ρuτ,bτ,uτ)‖2L2dτ≤C, | (2.7) |
where C is a positive constant depending on m, M, u0, ρ0, and ∇b0.
Proof. First, we obtain by taking L2 inner product of (1.1)3 with e′(b) that
ddt∫R2e(b)dx+1ν‖e′(b)‖2L2≤‖u⋅∇b‖L2‖e′(b)‖L2+14ν‖e′(b)‖2L2+C‖Δb‖2L2≤12ν‖e′(b)‖2L2+C‖u‖2L4‖∇b‖2L4+C‖Δb‖2L2≤12ν‖e′(b)‖2L2+C‖u‖L2‖∇u‖L2‖∇b‖L2‖∇2b‖L2+C‖Δb‖2L2≤12ν‖e′(b)‖2L2+C(‖∇u‖2L2+‖∇2b‖2L2). |
Integrating with respect to time, we obtain
‖e(b)‖L∞(L1)+‖e′(b)‖2L2(L2)≤‖(√ρ0u0,∇b0)‖2L2+‖e(b0)‖L1. | (2.8) |
Similarly, multiplying (1.1)3 by b, we have
12ddt‖b‖2L2+‖∇b‖2L2≤C(‖b‖2L2+‖e′(b)‖2L2), |
after using (2.8) and Grönwall's inequality, we obtain
b∈L∞(0,t;L2(R2))∩L2(0,t;H1(R2)). | (2.9) |
In the following, applying Laplace operator Δ to (1.1)3 and multiplying the resulting equation by Δb; additionally, multiplying (1.1)2 by ut and (1.1)3 by bt, respectively, then integrating them on R2 and adding up all these results together, we obtain
12ddt∫R2(|Δb|2+|∇u|2+σν|∇b|2)dx+∫R2(ρ|ut|2+|bt|2+σν|∇3b|2)dx=−∫R2ρu⋅∇u⋅utdx−σ∫R2Δb∇b⋅utdx−∫R2u⋅∇bbtdx−1ν∫R2e′(b)btdx−∫R2Δ(u⋅∇b)⋅Δbdx−1ν∫R2Δe′(b)⋅Δbdx≜6∑j=1Ij. | (2.10) |
Utilizing Gagliardo-Nirenberg's, Hölder's, Young's inequalities (2.2), we estimate the first term as follows:
I1≤‖√ρ‖L∞‖√ρut‖L2‖u‖L4‖∇u‖L4≤116‖√ρut‖2L2+C‖u‖2L4‖∇u‖2L4≤116‖√ρut‖2L2+C‖u‖L2‖∇u‖2L2‖∇2u‖L2≤116‖√ρut‖2L2+116‖∇2u‖2L2+C‖∇u‖4L2. |
Similarly, by direct calculations, the other terms can be bounded as
I2≤116‖√ρut‖2L2+σ8ν‖∇3b‖2L2+C‖∇2b‖4L2,I3≤14‖bt‖2L2+C‖∇u‖2L2+C‖∇b‖2L2‖Δb‖2L2,I4≤14‖bt‖2L2+C‖e′(b)‖2L2,I5≤3σ16ν‖∇3b‖2L2+116‖∇2u‖2L2+C‖∇u‖4L2+C‖∇2b‖4L2,I6≤3σ16ν‖∇3b‖2L2+C‖∇b‖2L2. |
Next, according to the regularity theory of the Stokes system in Eq (1.1)2, it follows that
‖∇2u‖2L2≤‖ρut‖2L2+‖ρu⋅∇u‖2L2+σ‖∇bΔb‖2L2≤‖√ρut‖2L2+C‖∇u‖2L2‖∇2u‖L2+C‖∇2b‖2L2‖∇3b‖L2≤‖√ρut‖2L2+12‖∇2u‖2L2+σ2ν‖∇3b‖2L2+C(‖∇u‖4L2+‖∇2b‖4L2), |
after multiplying by 18, we arrive at
116‖∇2u‖2L2≤18‖ρut‖2L2+σ16ν‖∇3b‖2L2+C(‖∇u‖4L2+‖∇2b‖4L2). | (2.11) |
Substituting the estimates I1−I6 into (2.10) and combining inequality (2.11), we have
ddt(‖(∇u,∇b,∇2b)‖2L2+1)+‖(√ρut,bt,∇3b,∇2u)‖2L2≤C(‖(∇u,∇b,∇2b)‖2L2+1)‖(∇u,∇b,∇2b)‖2L2+C‖e′(b)‖2L2, |
which, along with Grönwall's inequality (2.2), (2.8), and (2.9), leads to
‖(∇u,∇b,∇2b)‖2L2+∫t0‖(√ρuτ,bτ,∇3b,∇2u)‖2L2dτ≤C, | (2.12) |
which completes the proof of Proposition 2.2.
Proposition 2.3. Under the assumptions of Proposition 2.2, there holds
‖(√ρut,bt,∇bt,ut)‖2L2+∫t0‖(∇uτ,∇bτ,Δbτ)‖2L2dτ≤C, | (2.13) |
where C is a positive constant depending on m, M, u0, ρ0 and b0.
Proof. Taking the derivative of Eq (1.1)2 with respect to time t, then multiplying ut on both sides of the resulting equation and integrating by parts gives
12ddt‖√ρut‖2L2+‖∇ut‖2L2=−∫R2ρtut⋅utdx−∫R2ρtu⋅∇u⋅utdx−∫R2ρut⋅∇u⋅utdx−∫R2σΔbt∇b⋅utdx−σ∫R2Δb∇bt⋅utdx. | (2.14) |
Next, we compute each term on the right-hand side of the equation above one by one using estimates (2.2) and (2.7). The bound of the first term has been estimated as
−∫R2ρtut⋅utdx=∫R2 div(ρu)ut⋅utdx=−∫R22ρuut⋅∇utdx≤C‖ρ‖L∞‖u‖L4‖ut‖L4‖∇ut‖L2≤C‖u‖12L2‖∇u‖12L2‖∇ut‖32L2‖ut‖12L2≤110‖∇ut‖2L2+C‖∇u‖2L2‖ut‖2L2. |
By using Gagliardo-Nirenberg's, Hölder's, and Young's inequalities and (2.2), we have
−∫R2ρtu⋅∇u⋅utdx=∫R2∇⋅(ρu)u⋅∇u⋅utdx=−∫R2ρu⋅∇u⋅∇u⋅utdx−∫R2ρu⋅u⋅∇2u⋅utdx−∫R2u⋅∇u⋅ρu⋅∇utdx≤‖ρ‖L∞(‖u‖L6‖∇u‖2L3‖ut‖L6+‖u‖2L6‖∇2u‖L2‖ut‖L6+‖u‖2L6‖∇u‖L6‖∇ut‖L2)≤C‖u‖13L2‖∇u‖2L2‖∇2u‖23L2‖ut‖13L2‖∇ut‖23L2+C‖u‖23L2‖∇u‖43L2‖∇2u‖L2‖ut‖13L2‖∇ut‖23L2 +C‖u‖23L2‖∇u‖53L2‖∇2u‖23L2‖∇ut‖L2≤110‖∇ut‖2L2+C‖∇2u‖2L2+C‖∇u‖2L2‖ut‖2L2. |
Similarly,
−∫R2ρut⋅∇u⋅utdx≤C‖ρ‖L∞‖∇u‖L2‖ut‖2L4≤110‖∇ut‖2L2+C‖∇u‖2L2‖ut‖2L2 |
and
−∫R2σΔbt∇b⋅utdx−σ∫R2Δb∇bt⋅utdx≤C‖Δbt‖L2‖∇b‖L4‖ut‖L4+C‖Δb‖L4‖∇bt‖L2‖ut‖L4≤C‖Δbt‖L2‖∇b‖12L2‖Δb‖12L2‖ut‖12L2‖∇ut‖12L2+C‖Δb‖12L2‖∇3b‖12L2‖∇bt‖L2‖ut‖12L2‖∇ut‖12L2≤σ16ν‖∇bt‖2L2+σ16ν‖Δbt‖2L2+110‖∇ut‖2L2+C(‖∇b‖2L2+‖∇3b‖2L2)‖∇2b‖2L2‖ut‖2L2. |
Inserting these estimates into (2.14), we have
ddt‖√ρut‖2L2+85‖∇ut‖2L2≤σ8ν‖(∇bt,Δbt)‖2L2+C‖∇2u‖2L2+C‖(∇u,∇2b,∇3b)‖2L2‖ut‖2L2. | (2.15) |
Now we turn to the b equation of (1.1). Differentiating (1.1)3 with respect to t, we obtain
btt+ut⋅∇b+u⋅∇bt+1ν(e″(b)bt−σΔbt)=0. |
Multiplying it by bt and −Δbt, integrating the resulting equation, and summing up these results, due to the divergence-free condition divu=0, we obtain
12ddt‖(bt,∇bt)‖2L2+σν‖(∇bt,Δbt)‖2L2+1ν∫R2e″(b)(bt)2dx=∫R2ut⋅b⋅∇btdx+∫R2ut⋅∇b⋅Δbtdx+∫R2u⋅∇bt⋅Δbtdx+1ν∫R2e″(b)bt⋅Δbtdx≤‖ut‖L4‖b‖L4‖∇bt‖L2+‖ut‖L4‖∇b‖L4‖Δbt‖L2+‖u‖L4‖∇bt‖L4‖Δbt‖L2+C‖bt‖L2‖Δbt‖L2≤σ4ν‖∇bt‖2L2+‖ut‖L2‖∇ut‖L2‖b‖L2‖∇b‖L2+σ8ν‖Δbt‖2L2+‖ut‖L2‖∇ut‖L2‖∇b‖L2‖Δb‖L2+σ8ν‖Δbt‖2L2+C‖u‖L2‖∇u‖L2‖∇bt‖L2‖∇2bt‖L2+σ8ν‖Δbt‖2L2+C‖bt‖2L2≤σ4ν‖∇bt‖2L2+σ2ν‖Δbt‖2L2+12‖∇ut‖2L2+C(‖∇b‖2L2+‖Δb‖2L2)‖ut‖2L2+C‖bt‖2L2+C‖∇u‖2L2‖∇bt‖2L2. | (2.16) |
Summing up (2.15) and (2.16) yields that
ddt‖(√ρut,bt,∇bt)‖2L2+‖(∇ut,∇bt,Δbt)‖2L2≤C‖∇2u‖2L2+C‖bt‖2L2+C‖(∇u,∇2b,∇3b,∇b)‖2L2‖(√ρut,bt,∇bt)‖2L2. |
Applying (2.7) and Grönwall's inequality to the above inequality, we obtain
‖(√ρut,bt,∇bt)‖2L2+∫t0‖(∇uτ,∇bτ,Δbτ)‖2L2dτ≤C. |
What's more, by the same argument of ‖u‖L∞(L2) in Proposition 2.1, we have
‖ut‖L∞(L2)≤C, |
which completes the proof of Proposition 2.3.
Proposition 2.4. Under the assumption of Proposition 2.3, it holds that for any t>0:
∫t0‖∇u‖L∞dτ≤Ct23 | (2.17) |
and
supt>0‖∇ρ(t)‖Lp≤C(t). | (2.18) |
Proof. Again, it follows from the regularity of the Stokes system
‖∇2u‖L4+‖∇P‖L4≤‖ρut‖L4+‖ρu⋅∇u‖L4+‖Δb∇b‖L4≤C(‖ut‖L4+‖u‖L∞‖∇u‖L4+‖Δb‖L4‖∇b‖L∞)≤C(‖ut‖12L2‖∇ut‖12L2+‖u‖12L2‖∇2u‖L2‖∇u‖12L2+‖∇2b‖12L2‖∇3b‖L2‖∇b‖12L2). |
By Propositions 2.1–2.3, we obtain
∫t0‖∇2u‖L4dτ+∫t0‖∇P‖L4dτ≤C(∫t0‖∇2u‖2L4dτ)12t12+C(∫t0‖∇P‖2L4dτ)12t12≤C(‖ut‖L2(L2)+‖∇ut‖L2(L2)+‖u‖12L∞(L2)‖∇u‖12L∞(L2)‖∇2u‖L2(L2)+‖∇b‖12L∞(L2)‖∇2b‖12L∞(L2)‖∇3b‖L2(L2))t12≤Ct12, |
and
∫t0‖∇u‖L∞dτ≤∫t0‖∇u‖13L2‖∇2u‖23L4dτ≤C(∫t0‖∇2u‖2L4dτ)13t23≤(‖ut‖23L2(L2)+‖∇ut‖23L2(L2)+‖u‖23L∞(L2)‖∇u‖23L∞(L2)‖∇2u‖23L2(L2)+‖∇b‖23L∞(L2)‖∇2b‖23L∞(L2)‖∇3b‖23L2(L2))t23≤Ct23, |
which leads to (2.17). Finally, we recall that the density ρ satisfies
∂tρ+u⋅∇ρ=0. |
Applying the operator ∇ to both sides of the above equation yields
∂t∇ρ+u⋅∇(∇ρ)=−∇u⋅∇ρ. |
By applying the Lp estimate to the above equation, combined with the divergence free condition implies
ddt‖∇ρ‖Lp≤‖∇u‖L∞‖∇ρ‖Lp. |
The Grönwall's inequality implies
‖∇ρ‖Lp≤‖∇ρ0‖Lpexp∫t0‖∇u‖L∞dτ≤C(t). |
We thus complete the proof of Proposition 2.4.
The section is to prove Theorem 1.1. For any given ρ0 and (u0,b0)∈Hs(R2)×Hs+1(R2), we define the initial data
ρϵ0=ρ0∗ηϵ,uϵ0=u0∗ηϵ,bϵ=b0∗ηϵ, |
where ηϵ is the standard Friedrich's mollifier with ϵ>0. With the initial data (ρϵ0,uϵ0,bϵ0), the system (1.1) has a unique global smooth solution (ρϵ,uϵ,bϵ). From Propositions 2.1 and 2.2, we obtain
m≤ρϵ(x,t)≤M, |
‖(uϵ,bϵ,∇uϵ,∇bϵ,∇2bϵ)‖2L2+∫t0‖(√ρuϵτ,bϵτ,∇3bϵ,∇2uϵ)‖2L2dτ≤C. |
By standard compactness arguments and Lions-Aubin's Lemma, we can obtain a subsequence denoted again by (uϵ,bϵ), that (uϵ,bϵ) converges strongly to (u,b) in L2(R+;Hs1)×L2(R+;Hs2), as ϵ→0, for s1<2 and s2<3. By the definition of ρϵ0 and let ϵ→0, we find that the limit ρ of ρϵ satisfies m≤ρ≤M.
Next, we shall prove the uniqueness of the solutions. Assume that (ρi,ui,bi)(i=1,2) be two solutions of system (1.1), which satisfy the regularity propositions listed in Theorem 1.1. We denote
(˜ρ,˜u,˜b,˜P)def=(ρ2−ρ1,u2−u1,b2−b1,P2−P1). |
Then the system for (˜ρ,˜u,˜b,˜P) reads
{˜ρt+u2⋅∇˜ρ=−˜u⋅∇ρ1,ρ2˜ut+ρ2u2⋅∇˜u−Δ˜u+∇˜P=˜F,˜bt+u2⋅∇˜b+1ν(e′(b2)−e′(b1)−σΔ˜b)=−˜u⋅∇b1,div˜u=0,(˜ρ,˜u,˜b)(t,x)|t=0=(0,0,0), | (3.1) |
where
˜F=−σΔ˜b∇b2−σΔb2∇˜b−˜ρ∂tu1−˜ρu1⋅∇u1−ρ2˜u⋅∇u1. |
Setting ν=σ=1 in what follows.
Step 1: Taking L2 inner product to the second equation of (3.1) with ˜u, we have
12ddt‖√ρ2˜u‖2L2+‖∇˜u‖2L2=−∫R2Δ˜b∇b2⋅˜udx−∫R2Δb2∇˜b⋅˜udx−∫R2˜ρ∂tu1⋅˜udx−∫R2˜ρu1⋅∇u1⋅˜udx−∫R2ρ2˜u⋅∇u1⋅˜udx. | (3.2) |
By Hölder's and interpolation inequalities, we have
−∫R2Δ˜b∇b2⋅˜udx−∫R2Δb1∇˜b⋅˜udx≤C‖Δ˜b‖L2‖∇b2‖L4‖˜u‖L4+C‖Δb1‖L4‖∇˜b‖L2‖˜u‖L4≤C‖Δ˜b‖L2‖∇b2‖12L2‖∇2b2‖12L2‖˜u‖12L2‖∇˜u‖12L2+C‖Δb1‖12L2‖∇3b1‖12L2‖∇˜b‖L2‖˜u‖12L2‖∇˜u‖12L2≤18‖(Δ˜b,∇˜b)‖2L2+18‖∇˜u‖2L2+C(‖∇b2‖2L2‖∇2b2‖2L2+‖∇3b1‖2L2‖∇2b1‖2L2)‖˜u‖2L2. | (3.3) |
Similarly,
−∫R2˜ρ∂tu1⋅˜udx−∫R2˜ρu1⋅∇u1⋅˜udx≤‖˜ρ‖L2(‖∂tu1‖L4+‖u1⋅∇u1‖L4)‖˜u‖L4≤‖˜ρ‖L2(‖∂tu1‖L2+‖∇∂tu1‖L2+‖u1‖L∞‖Δu1‖L2+‖u1‖L∞‖∇u1‖L2)×(‖˜u‖L2+‖∇˜u‖L2)≤18‖∇˜u‖2L2+F1(t)‖˜ρ‖2L2+C‖˜u‖2L2, | (3.4) |
where
F1(t)=‖∂tu1‖2L2+‖∇∂tu1‖2L2+‖u1‖2L∞‖Δu1‖2L2+‖u1‖2L∞‖∇u1‖2L2. |
Hölder's inequality implies
−∫R2ρ2˜u⋅∇u1⋅˜udx≤‖∇u1‖L∞‖√ρ2˜u‖2L2. | (3.5) |
By substituting above estimates (3.3)–(3.5) into (3.2), we have
ddt‖√ρ2˜u‖2L2+‖∇˜u‖2L2≤14‖Δ˜b‖2L2+14‖∇˜b‖2L2+CF2(t)‖˜u‖2L2+F1(t)‖˜ρ‖2L2, | (3.6) |
where
F2(t)=‖∇b2‖2L2‖∇2b2‖2L2+‖∇3b1‖2L2‖∇2b1‖2L2+‖∇u1‖L∞+1. |
Step 2: Taking L2 inner product to the third equation of (3.1) with ˜b−Δ˜b, we obtain
12ddt‖(˜b,∇˜b)‖2L2+‖(∇˜b,Δ˜b)‖2L2+∫R2[e′(b2)−e′(b1)]˜bdx=∫R2u2⋅∇˜b⋅Δ˜bdx−∫R2˜u⋅∇b1⋅(˜b−Δ˜b)dx+∫R2[e′(b2)−e′(b1)]Δ˜bdx. | (3.7) |
We shall estimate each term on the right-hand side of (3.7). For the first term of (3.7), using Hölder's inequality, we have
∫R2u2⋅∇˜b⋅Δ˜bdx≤‖u2‖L∞‖∇˜b‖L2‖Δ˜b‖L2≤18‖Δ˜b‖2L2+C‖u2‖2L∞‖∇˜b‖2L2. | (3.8) |
Meanwhile, we have
∫R2[e′(b2)−e′(b1)]˜bdx=∫R2e″(ξ)˜b2dx>0, | (3.9) |
where ξ is a function between b2 and b1.
Moreover,
−∫R2˜u⋅∇b1⋅(˜b−Δ˜b)dx≤C‖˜u‖L4‖∇b1‖L4(‖˜b‖L2+‖Δ˜b‖L2)≤18‖Δ˜b‖2L2+C‖˜b‖2L2+C‖˜u‖L2‖∇˜u‖L2‖∇b1‖L2‖Δb1‖L2≤18‖Δ˜b‖2L2+C‖˜b‖2L2+18‖∇˜u‖2L2+C‖˜u‖2L2‖∇b1‖2L2‖Δb1‖2L2, | (3.10) |
and
∫R2[e′(b2)−e′(b1)]Δ˜bdx=∫R2e″(ξ)˜bΔ˜bdx≤C0‖˜b‖L2‖Δ˜b‖L2≤14‖Δ˜b‖2L2+C‖˜b‖2L2. | (3.11) |
By inserting (3.8)–(3.11) into (3.7), one yields
12ddt‖(˜b,∇˜b)‖2L2+12‖(∇˜b,Δ˜b)‖2L2≤C‖u2‖2L∞‖∇˜b‖2L2+C‖˜b‖2L2+18‖∇˜u‖2L2+C‖˜u‖2L2‖∇b1‖2L2‖Δb1‖2L2. | (3.12) |
Step 3: We will derive the estimate of ‖˜ρ‖L2 as follows:
12ddt‖˜ρ‖2L2≤‖˜u⋅∇ρ1‖L2‖˜ρ‖L2≤‖˜u‖L4‖∇ρ1‖L4‖˜ρ‖L2≤‖∇˜u‖12L2‖˜u‖12L2‖∇ρ1‖L4‖˜ρ‖L2≤14‖∇˜u‖2L2+C‖∇ρ1‖43L4(‖˜ρ‖2L2+‖˜u‖2L2). | (3.13) |
Step 4: Summing up the above estimates, that is, (3.6), (3.12), and (3.13), we obtain
12ddt‖(˜ρ,√ρ2˜u,˜b,∇˜b)‖2L2+‖(∇˜u,∇˜b,Δ˜b)‖2L2≤CF5(t)‖∇˜b‖2L2+C‖˜b‖2L2+CF4(t)‖˜u‖2L2+F3(t)‖˜ρ‖2L2≤C(1+F3(t)+F4(t)+F5(t))‖(˜ρ,√ρ2˜u,˜b,∇˜b)‖2L2, | (3.14) |
where
F3(t)=‖∂tu1‖2L2+‖∇∂tu1‖2L2+‖u1‖2L∞‖Δu1‖2L2+‖u1‖2L∞‖∇u1‖2L2+‖∇ρ1‖43L4,F4(t)=‖∇b2‖2L2‖∇2b2‖2L2+‖∇3b1‖2L2‖∇2b1‖2L2+‖∇u1‖L∞+‖∇b1‖2L2‖Δb1‖2L2+‖∇ρ1‖43L4+1,F5(t)=‖u2‖2L∞. |
Noticing the fact that ∫t0(1+F3(τ)+F4(τ)+F5(τ))dτ≤Ct+C and that ‖f‖2L∞≤‖f‖L2‖∇2f‖L2, we can obtain that there exists a small ϵ0 such that
‖(˜ρ,√ρ2˜u,˜b,∇˜b)‖L∞(L2)=0, |
for t∈[0,ϵ0]. Therefore, we obtain ˜ρ(t)=˜u(t)=˜b(t)≡0 for any t∈[0,ϵ0]. The uniqueness of such strong solutions on the whole time interval [0,+∞) then follows by a bootstrap argument.
Moreover, the continuity with respect to the initial data may also be obtained by the same argument in the proof of the uniqueness, which ends the proof of Theorem 1.1.
This paper focuses on two-dimensional inhomogeneous incompressible viscoelastic rate-type fluids with stress-diffusion. We have established its global solution, and the uniqueness of the solution in specific situations is also proved in this paper.
Xi Wang and Xueli Ke: Conceptualization, methodology, validation, writing-original draft, writing-review & editing. All authors have read and approved the final version of the manuscript for publication.
The authors would like to thank the anonymous referees for their suggestions which make the paper more readable.
The authors have no relevant financial or non-financial interests to disclose. The authors have no competing interests to declare that are relevant to the content of this article.
[1] |
R. Postoyan, P. Tabuada, D. Nesic, A. Anta, A framework for the event-triggered stabilization of nonlinear systems, IEEE T. AUTOMAT. CONTR., 60 (2015), 982–996. https://doi.org/10.1109/TAC.2014.2363603 doi: 10.1109/TAC.2014.2363603
![]() |
[2] |
S. Liu, B. Niu, G. Zong, X. Zhao, N. Xu, Data-driven-based event-triggered optimal control of unknown nonlinear systems with input constraints, NONLINEAR DYNAM., 109 (2022), 891–909. https://doi.org/10.1007/s11071-022-07459-7 doi: 10.1007/s11071-022-07459-7
![]() |
[3] |
X. Han, X. Zhao, T. Sun, Y. Wu, N. Xu, G. Zong, Event-triggered optimal control for discrete-time switched nonlinear systems with constrained control input, IEEE T. SYST. MAN CY.-S., 51 (2021), 7850–7859. https://doi.org/10.1109/TSMC.2020.2987136 doi: 10.1109/TSMC.2020.2987136
![]() |
[4] |
L. Xing, C. Wen, Z. Liu, H. Su, J. Cai, Adaptive compensation for actuator failures with event-triggered input, AUTOMATICA, 85 (2017), 129–136. https://doi.org/10.1016/j.automatica.2017.07.061 doi: 10.1016/j.automatica.2017.07.061
![]() |
[5] | J. Zhang, D. Yang, H. Zhang, Y. Wang, B. Zhou, Dynamic event-based tracking control of boiler turbine systems with guaranteed performance, IEEE T. AUTOM. SCI. ENG., (2023), to be published. https://doi.org/10.1109/TASE.2023.3294187 |
[6] |
L. Cao, H. Li, G. Dong, R. Lu, Event-triggered control for multiagent systems with sensor faults and input saturation, IEEE T. SYST. MAN CY.-S., 51 (2021), 3855–3866. https://doi.org/10.1109/TSMC.2019.2938216 doi: 10.1109/TSMC.2019.2938216
![]() |
[7] |
J. Huang, W. Wang, C. Wen, G. Li, Adaptive event-triggered control of nonlinear systems with controller and parameter estimator triggering, IEEE T. AUTOMAT. CONTR., 65 (2020), 318–324. https://doi.org/10.1109/TAC.2019.2912517 doi: 10.1109/TAC.2019.2912517
![]() |
[8] |
H. Wang, K. Xu, J. Qiu, Event-triggered adaptive fuzzy fixed-time tracking control for a class of nonstrict-feedback nonlinear systems, IEEE T. CIRCUITS-I, 68 (2021), 3058–3068. https://doi.org/10.1109/TCSI.2021.3073024 doi: 10.1109/TCSI.2021.3073024
![]() |
[9] |
J. Zhang, H. Zhang, S. Sun, Y. Cai, Adaptive time-varying formation tracking control for multiagent systems with nonzero leader input by intermittent communications, IEEE T. CYBERNETICS, 53 (2023), 5706–5715. https://doi.org/10.1109/TCYB.2022.3165212 doi: 10.1109/TCYB.2022.3165212
![]() |
[10] |
X. Li, Z. Sun, Y. Tang, H. R. Karimi, Adaptive event-triggered consensus of multiagent systems on directed graphs, IEEE T. AUTOMAT. CONTR., 66 (2021), 1670–1685. https://doi.org/10.1109/TAC.2020.3000819 doi: 10.1109/TAC.2020.3000819
![]() |
[11] |
W. Hu, C. Yang, T. Huang, W. Gui, A distributed dynamic event-triggered control approach to consensus of linear multiagent systems with directed networks, IEEE T. CYBERNETICS, 50 (2020), 869–874. https://doi.org/10.1109/TCYB.2018.2868778 doi: 10.1109/TCYB.2018.2868778
![]() |
[12] |
H. Zhang, J. Zhang, Y. Cai, S. Sun, J. Sun, Leader-following consensus for a class of nonlinear multiagent systems under event-triggered and edge-event triggered mechanisms, IEEE T. CYBERNETICS, 52 (2022), 7643–7654. https://doi.org/10.1109/TCYB.2020.3035907 doi: 10.1109/TCYB.2020.3035907
![]() |
[13] |
Q. Zhou, W. Wang, H. Ma, H. Li, Event-triggered fuzzy adaptive containment control for nonlinear multiagent systems with unknown bouc-wen hysteresis input, IEEE T. FUZZY SYST., 29 (2021), 731–741. https://doi.org/10.1109/TFUZZ.2019.2961642 doi: 10.1109/TFUZZ.2019.2961642
![]() |
[14] |
Y. Wang, Z. Chen, M. Sun, Q. Sun, A novel implementation of an uncertain dead-zone-input-equipped extended state observer and sign estimator, INFORM. SCIENCES, 626 (2023), 75–93. https://doi.org/10.1016/j.ins.2023.01.060 doi: 10.1016/j.ins.2023.01.060
![]() |
[15] |
Z. Zhao, Z. Tan, Z. Liu, M. O. Efe, C. K. Ahn, Adaptive inverse compensation fault-tolerant control for a flexible manipulator with unknown dead-zone and actuator faults, IEEE T. IND. ELECTRON., 70 (2023), 12698–12707. https://doi.org/10.1109/TIE.2023.3239926 doi: 10.1109/TIE.2023.3239926
![]() |
[16] |
Z. Xi, Y. Wang, H. Zhang, F. Sun, Q. Zheng, Z. Zhu, Research on afterburning control of more electric engine with a nonlinear fuel supply system, PROCEEDINGS OF THE INSTITUTION OF MECHANICAL ENGINEERS PART G-JOURNAL OF AEROSPACE ENGINEERING, 237 (2023), 2647–2664. https://doi.org/10.1177/09544100231155696 doi: 10.1177/09544100231155696
![]() |
[17] |
Z. Wang, X. Wang, Fault-tolerant control for nonlinear systems with a dead zone: Reinforcement learning approach, MATH. BIOSCI. ENG., 20 (2023), 6334–6357. https://doi.org/10.3934/mbe.2023274 doi: 10.3934/mbe.2023274
![]() |
[18] | V.-T. Nguyen, T.-T. Bui, H.-Y. Pham, A finite-time adaptive fault tolerant control method for a robotic manipulator in task-space with dead zone, and actuator faults, INT. J. CONTROL AUTOM. SYST., (2023), to be published. https://doi.org/10.1007/s12555-022-1069-5 |
[19] |
S. Dong, Y. Zhang, Identification modelling and fault-tolerant predictive control for industrial input nonlinear actuator system, MACHINES, 11 (2023), 240. https://doi.org/10.3390/machines11020240 doi: 10.3390/machines11020240
![]() |
[20] |
Y. H. Pham, T. L. Nguyen, T. T. Bui, T. Nguyen, V, Adaptive active fault tolerant control for a wheeled mobile robot under actuator fault and dead zone, IFAC PAPERSONLINE, 55 (2022), 314–319. https://doi.org/10.1016/j.ifacol.2022.11.203 doi: 10.1016/j.ifacol.2022.11.203
![]() |
[21] | G. Shao, X.-F. Wang, R. Wang, A distributed strategy for games in euler-lagrange systems with actuator dead zone, NEUROCOMPUTING, (2023), to be published. https://doi.org/10.1016/j.neucom.2023.126844 |
[22] |
W. Wang, T. Wen, X. He, G. Xu, Path following with prescribed performance for under-actuated autonomous underwater vehicles subjects to unknown actuator dead-zone, IEEE T. INTELL. TRANSP. SYST., 24 (2023), 6257–6267. https://doi.org/10.1109/TITS.2023.3248153 doi: 10.1109/TITS.2023.3248153
![]() |
[23] |
Z. Liu, F. Wang, Y. Zhang, X. Chen, C. L. P. Chen, Adaptive tracking control for a class of nonlinear systems with a fuzzy dead-zone input, IEEE T. FUZZY SYST., 23 (2015), 193–204. https://doi.org/10.1109/TFUZZ.2014.2310491 doi: 10.1109/TFUZZ.2014.2310491
![]() |
[24] |
T. Zhang, R. Bai, Y. Li, Practically predefined-time adaptive fuzzy quantized control for nonlinear stochastic systems with actuator dead zone, IEEE T. FUZZY SYST., 31 (2023), 1240–1253. https://doi.org/10.1109/TFUZZ.2022.3197970 doi: 10.1109/TFUZZ.2022.3197970
![]() |
[25] |
J. Wang, Y. Yan, J. Liu, C. L. P. Chen, Z. Liu, C. Zhang, Nn event-triggered finite-time consensus control for uncertain nonlinear multi-agent systems with dead-zone input and actuator failures, ISA T., 137 (2023), 59–73. https://doi.org/10.1016/j.isatra.2023.01.032 doi: 10.1016/j.isatra.2023.01.032
![]() |
[26] |
Y. Wang, B. Ma, D. Wang, T. Chai, Event-triggered prespecified performance control for steer-by-wire systems with input nonlinearity, IEEE T. INTELL. TRANSP. SYST., 24 (2023), 6922–6931. https://doi.org/10.1109/TITS.2023.3242949 doi: 10.1109/TITS.2023.3242949
![]() |
[27] |
L. Xing, C. Wen, Z. Liu, H. Su, J. Cai, Event-triggered output feedback control for a class of uncertain nonlinear systems, IEEE T. AUTOMAT. CONTR., 64 (2019), 290–297. https://doi.org/10.1109/TAC.2018.2823386 doi: 10.1109/TAC.2018.2823386
![]() |
[28] |
A. Souahi, O. Naifar, A. Ben Makhlouf, M. A. Hammami, Discussion on barbalat lemma extensions for conformable fractional integrals, INT. J. CONTROL, 92 (2019), 234–241. https://doi.org/10.1080/00207179.2017.1350754 doi: 10.1080/00207179.2017.1350754
![]() |
[29] |
Y.-X. Li, G.-H. Yang, S. Tong, Fuzzy adaptive distributed event-triggered consensus control of uncertain nonlinear multiagent systems, IEEE T. SYST. MAN CY.-S., 49 (2019), 1777–1786. https://doi.org/10.1109/TSMC.2018.2812216 doi: 10.1109/TSMC.2018.2812216
![]() |