Autonomous Underwater Vehicle (AUV) works autonomously in complex marine environments. After a severe accident, an AUV will lose its power and rely on its small buoyancy to ascend at a slow speed. If the reserved buoyancy is insufficient, when reaching the thermocline, the buoyancy will rapidly decrease to zero. Consequently, the AUV will experience prolonged lateral drift within the thermocline. This study focuses on developing a prediction method for the drift trajectory of an AUV after a long-term power loss accident. The aim is to forecast the potential resurfacing location, providing technical support for surface search and salvage operations of the disabled AUV. To the best of our knowledge, currently, there is no mature and effective method for predicting long-term AUV underwater drift trajectories. In response to this issue, based on real AUV catastrophes, this paper studies the prediction of long-term AUV underwater drift trajectories in the cases of power loss. We propose a three-dimensional trajectory prediction method based on the Lagrange tracking approach. This method takes the AUV's longitudinal velocity, the time taken to reach different depths, and ocean current data at various depths into account. The reason for the AUV's failure to ascend to sea surface lies that the remaining buoyancy is too small to overcome the thermocline. As a result, AUV drifts long time within the thermocline. To address this issue, a method for estimating thermocline currents is proposed, which can be used to predict the lateral drift trajectory of the AUV within the thermocline. Simulation is conducted to compare the results obtained by the proposed method and that in a real accident. The results demonstrate that the proposed approach exhibits small directional and positional errors. This validates the effectiveness of the proposed method.
Citation: Shuwen Zheng, Mingjun Zhang, Jing Zhang, Jitao Li. Lagrange tracking-based long-term drift trajectory prediction method for Autonomous Underwater Vehicle[J]. Mathematical Biosciences and Engineering, 2023, 20(12): 21075-21097. doi: 10.3934/mbe.2023932
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Autonomous Underwater Vehicle (AUV) works autonomously in complex marine environments. After a severe accident, an AUV will lose its power and rely on its small buoyancy to ascend at a slow speed. If the reserved buoyancy is insufficient, when reaching the thermocline, the buoyancy will rapidly decrease to zero. Consequently, the AUV will experience prolonged lateral drift within the thermocline. This study focuses on developing a prediction method for the drift trajectory of an AUV after a long-term power loss accident. The aim is to forecast the potential resurfacing location, providing technical support for surface search and salvage operations of the disabled AUV. To the best of our knowledge, currently, there is no mature and effective method for predicting long-term AUV underwater drift trajectories. In response to this issue, based on real AUV catastrophes, this paper studies the prediction of long-term AUV underwater drift trajectories in the cases of power loss. We propose a three-dimensional trajectory prediction method based on the Lagrange tracking approach. This method takes the AUV's longitudinal velocity, the time taken to reach different depths, and ocean current data at various depths into account. The reason for the AUV's failure to ascend to sea surface lies that the remaining buoyancy is too small to overcome the thermocline. As a result, AUV drifts long time within the thermocline. To address this issue, a method for estimating thermocline currents is proposed, which can be used to predict the lateral drift trajectory of the AUV within the thermocline. Simulation is conducted to compare the results obtained by the proposed method and that in a real accident. The results demonstrate that the proposed approach exhibits small directional and positional errors. This validates the effectiveness of the proposed method.
HIV spreads through either cell-free viral infection or direct transmission from infected to healthy cells (cell-to-cell infection) [1]. It is reported that more than 50% of viral infection is caused by the cell-to-cell infection [2]. Cell-to-cell infection can occur when infected cells encounter healthy cells and form viral synapse [3]. Recently, applying reaction-diffusion equations to model viral dynamics with cell-to-cell infection have been received attentions (see, e.g., [4,5,6]). Ren et al. [5] proposed the following reaction-diffusion equation model with cell-to-cell infection:
∂T(t,x)∂t=∇⋅(d1(x)∇T)+λ(x)−β1(x)TT∗−β2(x)TV−d(x)T, t>0, x∈Ω,∂T∗(t,x)∂t=∇⋅(d2(x)∇T∗)+β1(x)TT∗+β2(x)TV−r(x)T∗, t>0, x∈Ω,∂V(t,x)∂t=∇⋅(d3(x)∇V)+N(x)T∗−e(x)V, t>0, x∈Ω,T(0,x)=T0(x)>0, T∗(0,x)=T∗0(x)≥0, V(0,x)=V0(x)≥0, x∈Ω. | (1.1) |
In the model (1.1), T(t,x), T∗(t,x) and V(t,x) denote densities of healthy cells, infected cells and virus at time t and location x, respectively. The detailed biological meanings of parameters for the model (1.1) can be found in [5]. The well-posedness of the classical solutions for the model (1.1) have been studied. The model (1.1) admits a basic reproduction number R0, which is defined by the spectral radius of the next generation operator [5]. The model (1.1) defines a solution semiflow Ψ(t), which has a global attractor. The model (1.1) admits a unique virus-free steady state E0=(T0(x),0,0), which is globally attractive if R0<1. If R0>1, the model (1.1) admits at least one infection steady state and virus is uniformly persistent [5].
It is a challenging problem to consider the global stability of E0 in the critical case of R0=1. In [6], Wang et al. studied global stability analysis in the critical case by establishing Lyapunov functions. Unfortunately, the method can not be applied for the model consisting of two or more equations with diffusion terms, which was left it as an open problem.
Adopting the idea in [7,8,9], the present study is devoted to solving this open problem and shows that E0 is globally asymptotically stable when R0=1 for the model (1.2). For simplicity, in the following, we assume that the diffusion rates d1(x), d2(x) and d3(x) are positive constants. That is, we consider the following model
∂T(t,x)∂t=d1ΔT+λ(x)−β1(x)TT∗−β2(x)TV−d(x)T, t>0, x∈Ω,∂T∗(t,x)∂t=d2ΔT∗+β1(x)TT∗+β2(x)TV−r(x)T∗, t>0, x∈Ω,∂V(t,x)∂t=d3ΔV+N(x)T∗−e(x)V, t>0, x∈Ω,T(0,x)=T0(x)>0, T∗(0,x)=T∗0(x)≥0, V(0,x)=V0(x)≥0, x∈Ω, | (1.2) |
with the boundary conditions:
∂T(t,x)∂ν=∂T∗(t,x)∂ν=∂V(t,x)∂ν=0, t>0, x∈∂Ω, | (1.3) |
where Ω is the spatial domain and ν is the outward normal to ∂Ω. We assume that all the location-dependent parameters are continuous, strictly positive and uniformly bounded functions on ¯Ω.
Let Y=C(¯Ω,R3) with the supremum norm ∥⋅∥Y, Y+=C(¯Ω,R3+). Then (Y,Y+) is an ordered Banach space. Let T be the semigroup for the system:
∂T∗(t,x)∂t=d2ΔT∗+β1(x)T0(x)T∗+β2(x)T0(x)V−r(x)T∗,∂V(t,x)∂t=d3ΔV+N(x)T∗−e(x)V, |
where T0(x) is the solution of the elliptic problem d1ΔT+λ(x)−d(x)T=0 under the boundary conditions (1.3). Then T has the generator
˜A=(d2Δ+β1(x)T0(x)−r(x)β2(x)T0(x)N(x)d3Δ−e(x)). |
Let us define the exponential growth bound of T as
¯ω=¯ω(T):=limt→+∞ln‖T‖t, |
and define the spectral bound of ˜A by
s(˜A):=sup{Reλ,λ∈σ(˜A)}. |
Theorem 2.1. If R0=1, E0 of the model (1.2) is globally asymptotically stable.
Proof. We first show the local asymptotic stability of E0 of the model (1.2). Suppose ζ>0 and let v0=(T0,T∗0,V0) with ‖v0−E0‖≤ζ. Define m1(t,x)=T(t,x)T0(x)−1 and p(t)=maxx∈¯Ω{m1(t,x),0}. According to d1ΔT0(x)+λ(x)−d(x)T0(x)=0, we have
∂m1∂t−d1Δm1−2d1∇T0(x)∇m1T0(x)+λ(x)T0(x)m1=−β1(x)TT∗T0(x)−β2(x)TVT0(x). |
Let ˜T1(t) be the positive semigroup generated by
d1Δ+2d1∇T0(x)∇T0(x)−λ(x)T0(x) |
associated with (1.3) (see Theorem 4.4.3 in [10]). From Theorem 4.4.3 in [10], we can find q>0 such that ‖˜T1(t)‖≤M1e−qt for some M1>0. Hence, one gets
m1(⋅,t)=˜T1(t)m10−∫∞0˜T1(t−s)[β1(⋅)T(⋅,s)T∗(⋅,s)T0(⋅)+β2(⋅)T(⋅,s)V(⋅,s)T0(⋅)]ds, |
where m10=T0T0(x)−1. In view of the positivity of ˜T1(t), it follows that
p(t)=maxx∈¯Ω{ω1(t,x),0}=maxx∈¯Ω{˜T1(t)m10−∫∞0˜T1(t−s)[β1(⋅)T(⋅,s)T∗(⋅,s)T0(⋅)+β2(⋅)T(⋅,s)V(⋅,s)T0(⋅)]ds,0}≤maxx∈¯Ω{˜T1(t)m10,0}≤‖˜T1(t)m10‖≤M1e−qt‖T0T0(x)−1‖≤ζM1e−qtTm, |
where Tm=minx∈¯Ω{T0(x)}. Note that (T∗,V) satisfies
∂T∗(t,x)∂t=d2ΔT∗+β1(x)T0(x)T∗+β2(x)T0(x)V−r(x)T∗ +β1(x)T0(x)(TT0(x)−1)T∗+β2(x)T0(x)(TT0(x)−1)V,∂V(t,x)∂t=d3ΔV+N(x)T∗−e(x)V. |
It then follows that
(T∗(⋅,t)V(⋅,t))=T(t)(T∗0V0) |
+∫∞0T(t−s)(β1(⋅)T0(⋅)(T(⋅,s)T0(⋅)−1)T∗(⋅,s)+β2(⋅)T0(⋅)(T(⋅,s)T0(⋅)−1)V(⋅,s)0)ds. |
From Theorem 3.5 in [11], we have that s(˜A)=sup{Reλ,λ∈σ(˜A)} has the same sign as R0−1. If R0=1, then s(˜A)=0. Then we easily verify all the conditions of Proposition 4.15 in [12]. It follows from R0=1 and Proposition 4.15 in [12] that we can find M1>0 such that ‖T(t)‖≤M1 for t≥0, where M1 can be chosen as large as needed in the sequel. Since p(s)≤ζM1e−qsTm, one gets
max{‖T∗(⋅,t)‖,‖V(⋅,t)‖}≤M1max{‖T∗0‖,‖V0‖} +M1(‖β1‖+‖β2‖)‖T0‖∫∞0p(s)max{‖T∗(s)‖,‖V(s)‖}ds≤M1ζ+M2ζ∫∞0e−qsmax{‖T∗(s)‖,‖V(s)‖}ds, |
where
M2=M21(‖β1‖+‖β2‖)‖T0‖Tm. |
By using Gronwall's inequality, we get
max{‖T∗(⋅,t)‖,‖V(⋅,t)‖}≤M1ζe∫∞0ζM2e−qsds≤M1ζeζM2q. |
Then ∂T∂t−d1ΔT>λ(x)−d(x)T−M1ζeζM2q(β1(x)+β2(x))T. Let ˆu1 be the solution of the system:
∂ˆu1(t,x)∂t=d1Δˆu1+λ(x)−d(x)ˆu1−M1ζeζM2q(β1(x)+β2(x))ˆu1, x∈Ω, t>0,∂ˆu1(t,x)∂ν=0, x∈∂Ω, t>0,ˆu1(x,0)=T0, x∈¯Ω. | (2.1) |
Then T(t,x)≥ˆu1(t,x) for x∈¯Ω and t≥0. Let Tζ(x) be the positive steady state of the model (2.1) and ˆm(t,x)=ˆu1(t,x)−Tζ(x). Then ˆm(t,x) satisfies
∂ˆm(t,x)∂t=d1Δˆm−[d(x)+M1ζeζM2q(β1(x)+β2(x))]ˆm, x∈Ω, t>0,∂ˆm(t,x)∂ν=0, x∈∂Ω, t>0,ˆm(x,0)=T0−Tζ(x), x∈¯Ω. |
For sufficiently large M1, from Theorem 4.4.3 in [10], we have ‖F1(t)‖≤M1e¯α0t, where ¯α0<0 is a constant and F1(t):C(¯Ω,R)→C(¯Ω,R) is the C0 semigroup of d1Δ−d(⋅) subject to (1.3) [5]. Hence, we have
ˆm(⋅,t)=F1(t)(T0−Tζ(x))−∫∞0F1(t−s)M1ζeζM2q(β1(⋅)+β2(⋅))ˆm(⋅,s)ds,‖ˆm(⋅,t)‖≤M1‖T0−Tζ(x)‖e¯α0t+∫∞0M21e¯α0(t−s)ζeζM2q(‖β1(⋅)‖+‖β2(⋅)‖)‖ˆm(⋅,s)‖ds. |
Let K=M21ζeζM2q(‖β1‖+‖β2‖). By employing Gronwall's inequality, one gets
‖ˆu1(⋅,t)−Tζ(x)‖=‖ˆm(⋅,t)‖≤M1‖T0−Tζ(x)‖eKt+¯α0t. |
Choosing ζ>0 sufficiently small such that K<−¯α02, then
‖ˆu1(⋅,t)−Tζ(x)‖≤M1‖T0−Tζ(x)‖e¯α0t/2, |
and
T(⋅,t)−T0(x)≥ˆu1(⋅,t)−ˆU(x)=ˆu1(⋅,t)−Uπ(x)+Uπ(x)−ˆU(x)≥−ζM1−(M1+1)‖Tζ(x)−T0(x)‖. |
Since p(t)≤ζM1Tm, one gets T(⋅,t)−T0(x)≤ζM1‖T0(x)‖Tm, and hence
‖T(⋅,t)−T0(x)‖=max{ζM1+(M1+1)‖Tζ(x)−T0(x)‖,ζM1‖T0(x)‖Tm}. |
From
limζ→0Tζ(x)=T0(x), |
by choosing ζ small enough, for t>0, there holds ‖T(⋅,t)−T0(x)‖, ‖T∗(⋅,t)‖, ‖V(⋅,t)‖≤ε, which implies the local asymptotic stability of E0.
By Theorem 1 in [5], the solution semiflow Ψ(t):Y+→Y+ of the model (1.2) has a global attractor Π. In the following, we prove the global attractivity of E0. Define
∂Y1={(˜T,~T∗,˜V)∈Y+:~T∗=˜V=0}. |
Claim 1. For v0=(T0,T∗0,V0)∈Π, the omega limit set ω(v0)⊂∂Y1.
Since ∂T∂t≤d1ΔT+λ(x)−d(x)T, T is a subsolution of the problem
∂ˆT(t,x)∂t=d1ΔˆT+λ(x)−d(x)ˆT, x∈Ω, t>0,∂ˆT(t,x)∂ν=0, x∈∂Ω, t>0,ˆT(x,0)=T0(x), x∈¯Ω. | (2.2) |
It is well known that model (2.2) has a unique positive steady state T0(x), which is globally attractive. This together with the comparison theorem implies that
lim supt→+∞T(t,x)≤lim supt→+∞ˆT(t,x)=T0(x), |
uniformly for x∈Ω. Since v0=(T0,T∗0,V0)∈Π, we know T0≤T0. If T∗0=V0=0, the claim easily holds. We assume that either T∗0≠0 or V0≠0. Thus one gets T∗(t,x)>0 and V(t,x)>0 for x∈¯Ω and t>0. Then T(t,x) satisfies
∂T(t,x)∂t<d1ΔT+λ(x)−d(x)T(t,x),x∈Ω, t>0,∂T(t,x)∂ν=0,x∈∂Ω, t>0,T(x,0)≤T0(x), x∈Ω. |
The comparison principle yields T(t,x)<T0(x) for x∈¯Ω and t>0. Following [7], we introduce
h(t,v0):=inf{˜h∈R:T∗(⋅,t)≤˜hϕ2,V(⋅,t)≤˜hϕ3}. |
Then h(t,v0)>0 for t>0. We show that h(t,v0) is strictly decreasing. To this end, we fix t0>0, and let ¯T∗(⋅,t)=h(t0,v0)ϕ2 and ¯V(⋅,t)=h(t0,v0)ϕ3 for t≥t0. Due to T(⋅,t)<T0(x), one gets
∂¯T∗(t,x)∂t>d2Δ¯T∗+β1(x)T¯T∗+β2(x)T¯V−r(x)¯T∗,∂¯V(t,x)∂t=d3Δ¯V+N(x)¯T∗−e(x)¯V,¯T∗(x,t0)≥T∗(x,t0), ¯V(x,t0)≥V(x,t0), x∈Ω. | (2.3) |
Hence (¯T∗(t,x),¯V(t,x))≥(T∗(t,x),V(t,x)) for x∈¯Ω and t≥t0. From the model (2.3), one gets h(t0,v0)ϕ2(x)=¯T∗(t,x)>T∗(t,x) for x∈¯Ω and t>t0. Similarly, we get h(t0,v0)ϕ3(x)=¯V(t,x)>V(t,x) for x∈¯Ω and t>t0. Since t0>0 is arbitrary, h(t,v0) is strictly decreasing. Let h∗=limt→+∞h(t,v0). Then we have h∗=0. Let Q=(Q1,Q2,Q3)∈ω(v0). Then there is {tk} with tk→+∞ such that Ψ(tk)v0→Q. We get h(t,Q)=h∗ for t≥0 due to limt→+∞Ψ(t+tk)v0=Ψ(t)limt→+∞Ψ(tk)v0=Ψ(t)Q. If Q2≠0 and Q3≠0, we repeat the above discussions to illustrate that h(t,Q) is strictly decreasing, which contradicts to h(t,Q)=h∗. Thus, we have Q2=Q3=0.
Claim 2. Π={E0}.
Since {E0} is globally attractive in ∂Y1, {E0} is the only compact invariant subset of the model (1.2). From the invariance of ω(v0) and ω(v0)⊂∂Y1, one gets ω(v0)={E0}. By Lemma 3.11 in [9], we get Π={E0}.
The local asymptotic stability and global attractivity yield the global asymptotic stability of E0.
The research is supported by the NNSF of China (11901360) to W. Wang and supported by the Fundamental Research Funds for the Central Universities and the Research Funds of Renmin University of China (2019030196) to X. Lai. We also want to thank the anonymous referees for their careful reading that helped us to improve the manuscript.
The authors decare no conflicts of interest.
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