
The interaction network
.Terrain surface roughness, often described abstractly, poses challenges in quantitative characterization with various descriptors found in the literature. In this study, we compared five commonly used roughness descriptors, exploring correlations among their quantified terrain surface roughness maps across three terrains with distinct spatial variations. Additionally, we investigated the impacts of spatial scales and interpolation methods on these correlations. Dense point cloud data obtained through Light Detection and Ranging technique were used in this study. The findings highlighted both global pattern similarities and local pattern distinctions in the derived roughness maps, emphasizing the significance of incorporating multiple descriptors in studies where local roughness values play a crucial role in subsequent analyses. The spatial scales were found to have a smaller impact on rougher terrain, while interpolation methods had minimal influence on roughness maps derived from different descriptors.
Citation: Lei Fan, Yang Zhao. Comparing roughness maps generated by five typical roughness descriptors for LiDAR-derived digital elevation models[J]. AIMS Geosciences, 2024, 10(2): 228-241. doi: 10.3934/geosci.2024013
[1] | Tingting Zhu . Synchronization of the generalized Kuramoto model with time delay and frustration. Networks and Heterogeneous Media, 2023, 18(4): 1772-1798. doi: 10.3934/nhm.2023077 |
[2] | Xiaoxue Zhao, Zhuchun Li . Synchronization of a Kuramoto-like model for power grids with frustration. Networks and Heterogeneous Media, 2020, 15(3): 543-553. doi: 10.3934/nhm.2020030 |
[3] | Seung-Yeal Ha, Yongduck Kim, Zhuchun Li . Asymptotic synchronous behavior of Kuramoto type models with frustrations. Networks and Heterogeneous Media, 2014, 9(1): 33-64. doi: 10.3934/nhm.2014.9.33 |
[4] | Seung-Yeal Ha, Jaeseung Lee, Zhuchun Li . Emergence of local synchronization in an ensemble of heterogeneous Kuramoto oscillators. Networks and Heterogeneous Media, 2017, 12(1): 1-24. doi: 10.3934/nhm.2017001 |
[5] | Seung-Yeal Ha, Se Eun Noh, Jinyeong Park . Practical synchronization of generalized Kuramoto systems with an intrinsic dynamics. Networks and Heterogeneous Media, 2015, 10(4): 787-807. doi: 10.3934/nhm.2015.10.787 |
[6] | Seung-Yeal Ha, Jeongho Kim, Jinyeong Park, Xiongtao Zhang . Uniform stability and mean-field limit for the augmented Kuramoto model. Networks and Heterogeneous Media, 2018, 13(2): 297-322. doi: 10.3934/nhm.2018013 |
[7] | Seung-Yeal Ha, Hansol Park, Yinglong Zhang . Nonlinear stability of stationary solutions to the Kuramoto-Sakaguchi equation with frustration. Networks and Heterogeneous Media, 2020, 15(3): 427-461. doi: 10.3934/nhm.2020026 |
[8] | Young-Pil Choi, Seung-Yeal Ha, Seok-Bae Yun . Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto--Daido model with inertia. Networks and Heterogeneous Media, 2013, 8(4): 943-968. doi: 10.3934/nhm.2013.8.943 |
[9] | Vladimir Jaćimović, Aladin Crnkić . The General Non-Abelian Kuramoto Model on the 3-sphere. Networks and Heterogeneous Media, 2020, 15(1): 111-124. doi: 10.3934/nhm.2020005 |
[10] | Wenlian Lu, Fatihcan M. Atay, Jürgen Jost . Consensus and synchronization in discrete-time networks of multi-agents with stochastically switching topologies and time delays. Networks and Heterogeneous Media, 2011, 6(2): 329-349. doi: 10.3934/nhm.2011.6.329 |
Terrain surface roughness, often described abstractly, poses challenges in quantitative characterization with various descriptors found in the literature. In this study, we compared five commonly used roughness descriptors, exploring correlations among their quantified terrain surface roughness maps across three terrains with distinct spatial variations. Additionally, we investigated the impacts of spatial scales and interpolation methods on these correlations. Dense point cloud data obtained through Light Detection and Ranging technique were used in this study. The findings highlighted both global pattern similarities and local pattern distinctions in the derived roughness maps, emphasizing the significance of incorporating multiple descriptors in studies where local roughness values play a crucial role in subsequent analyses. The spatial scales were found to have a smaller impact on rougher terrain, while interpolation methods had minimal influence on roughness maps derived from different descriptors.
Synchronized behavior in complex systems is ubiquitous and has been extensively investigated in various academic communities such as physics, biology, engineering [2,7,29,36,37,39,40,42], etc. Recently, sychronization mechanism has been applied in control of robot systems and power systems [12,13,34]. The rigorous mathematical treatment of synchronization phenomena was started by two pioneers Winfree [43] and Kuramoto [27,28] several decades ago, who introduced different types of first-order systems of ordinary differential equations to describe the synchronous behaviors. These models contain rich emergent behaviors such as synchronization, partially phase-lcoking and nonlinear stability, etc., and have been extensively studied in both theoretical and numerical level [3,5,11,14,15,16,17,19,26,32,39].
In this paper, we address the synchronous problem of Kuramoto model on a general graph under the effect of frustration. To fix the idea, we consider a digraph
χij={1if thejth oscillator influences theith oscillator,0otherwise. |
Then, the set of neighbors of
{˙θi(t)=Ωi+K∑j∈Nisin(θj(t)−θi(t)+α),t>0,i∈V,θi(0)=θi0, | (1) |
where
Comparing to the original Kuramoto model, there are two additional structures, i.e., frustration and general digraph. The frustration was introduced by Sakaguchi and Kuramoto [38], due to the observation that a pair of strongly coupled oscillators eventually oscillate with a common frequency that deviates from the average of their natural frequencies. On the other hand, the original all-to-all symmetric network is an ideal setting, thus it is natural to further consider general digraph case. Therefore, the frustration model with general digraph is more realistic in some sense. Moreover, these two structures also lead to richer phenomenon. For instance, the author in [6] observed that the frustration is common in disordered interactions, and the author in [44] found that frustration can induce the desynchronization through varying the value of
However, mathematically, for the Kuramoto model, the frustration and general digraph structures generate a lot of difficulties in rigorous analysis. For instance, the conservation law and gradient flow structure are lost, and thus the asymptotic states and dissipation mechanism become non-trivial. For all-to-all and symmetric case with frustration, in [20], the authors provided sufficient frameworks leading to complete synchronization under the effect of uniform frustration. In their work, they required initial configuration to be confined in half circle. Furthermore, the authors in [31] dealt with the stability and uniqueness of emergent phase-locked states. In particular, the authors in [22] exploited order parameter approach to study the identical Kuramoto oscillators with frustration. They showed that an initial configuration whose order parameter is bounded below will evolve to the complete phase synchronization or the bipolar state exponentially fast. On the other hand, for non-all-to-all case without frustration, the authors in [9] lifted the Kuramoto model to second-order system such that the second-order formulation enjoys several similar mathematical structures to that of Cucker-Smale flocking model [8]. But this method only works when the size of initial phases is less than a quarter circle, as we know the cosine function becomes negative if
Our interest in this paper is studying the system (1) with uniform frustration on a general digraph. As far as the authors know, when the ensemble is distributed in half circle, the dissipation structure of the Kuramoto type model with general digragh is still unclear. The main difficulty comes from the loss of uniform coercive inequality, which is due to the non-all-to-all and non-symmetric interactions. Thus we cannot expect to capture the dissipation from Gronwall-type inequality of phase diameter. For example, the time derivative of the phase diameter may be zero at some time for general digraph case. To this end, we switch to apply the idea of node decomposition in [25] to gain the dissipation through hypo-coercivity. Due to the lack of monotonicity of sine function in half circle, we follow the delicate constructions and estimates of the convex combinations in [45]. Eventually, we have the following main theorem.
Theorem 1.1. Suppose the network topology
D(θ(0))<π. |
Then for sufficient large coupling strength
D(θ(t))≤D∞,∀t∈[t∗,∞), |
and
D(ω(t))≤C1e−C2(t−t∗),t≥t∗, |
where
Remark 1. The first part of Theorem 1.1 claims that all oscillators confined initially in half circle will enter into a region less than quarter cirlce after some finite time. It is natural to ask how large
tanα<1(1+(d+1)ζζ−D(θ(0)))2Ncβd+1D∞[4(2N+1)c]d,D∞+α<π2,1>(1+(d+1)ζζ−D(θ(0)))c[4(2N+1)c]dβd+1D∞(D(Ω)Kcosα+2Nsinαcosα), | (2) |
where
D(θ(0))<ζ<γ<π,η>max{1sinγ,21−ζγ},β=1−2η,c=(N−1∑j=1ηjA(2N,j)+1)γsinγ, | (3) |
where
Remark 2. For
The rest of the paper is organized as follows. In Section 2, we recall some concepts on the network topology and provide a priori local-in-time estimate on the phase diameter of whole ensemble with frustration. In Section 3, we consider a strongly connected ensemble with frustration for which the initial phases are distributed in a half circle. We show that for large coupling strength and small frustration, the phase diameter is uniformly bounded by a value less than
In this section, we first introduce some fundamental concepts such as synchronization, spanning tree and node decomposition of a general network (1). Then, we will provide some necessary notations and a priori estimate that will be frequently used in later sections.
For the Kuramoto-type model, we recall the definition of synchronization as follows.
Definition 2.1. Let
limt→∞|˙θi(t)−˙θj(t)|=0,∀ i≠j. |
Let the network topology be registered by the neighbor set
Definition 2.2. (1)The Kuramoto digraph
(2)A path in
is∈Nis+1for 1≤s≤k−1. |
If there exists a path from
(3)The Kuramoto digraph contains a spanning tree if we can find a vertex such that any other vertex of
In order to guarantee the emergence of synchronization, we will always assume the existence of a spanning tree throughout the paper. Now we recall the concepts of root and general root introduced in [25]. Let
ci≥0,l≤i≤kandk∑i=lci=1. |
For an ensembel of
Lkl(Cl,k):=k∑i=lciθi. |
Note that each
Definition 2.3. (Root and general root)
1. We say
2. We say
Lemma 2.4.[25] The following assertions hold.
1. If the network contains a spanning tree, then there is at most one root.
2. Assume the network contains a spanning tree. If
In this part, we will recall the concept of maximum node introduced in [25]. Then, we can follow node decomposition introduced in [25] to represent the whole graph
Definition 2.5. [25] (Node) Let
Intuitively, a node can be understood through a way that a set of oscillators can be viewed as a "large" oscillator. The concept of node can be exploited to simplify the structure of the digraph, which indeed helps us to catch the attraction effect more clearly in the network topology.
Lemma 2.6.[25] Any digraph
Lemma 2.7.[25] (Node decomposition)Let
1.
2. For any
Proof. As
Remark 3. Lemma 2.7 shows a clear hierarchical structure on a general digraph. For the convenience of later analysis, we make some comments on important notations and properties that are used throughout the paper.
1. From the definition of maximum node, for
G=d⋃i=0Gi, |
where
2.For a given oscillator
In this part, for notational simplicity, we introduce some notations, such as the extreme phase, phase diameter of
θ=(θ1,θ2,…,θN),ω=(ω1,ω2,…,ωN),Ω=(Ω1,Ω2,…,ΩN),θM=max1≤k≤N{θk}=max0≤i≤dmax1≤j≤Ni{θij},θm=min1≤k≤N{θk}=min0≤i≤dmin1≤j≤Ni{θij},D(θ)=θM−θm,Dk(θ)=max0≤i≤kmax1≤j≤Ni{θij}−min0≤i≤kmin1≤j≤Ni{θij},ΩM=max0≤i≤dmax1≤j≤Ni{Ωij},Ωm=min0≤i≤dmin1≤j≤Ni{Ωij},D(Ω)=ΩM−Ωm,Ni=|Gi|,Sk=k∑i=0Ni,0≤k≤d,d∑i=0Ni=N. |
Finally, we provide a priori local-in-time estimate on the phase diameter to finish this section, which states that all oscillators can be confined in half circle in short time.
Lemma 2.8. Let
D(θ(0))<ζ<γ<π, |
where
D(θ(t))<ζ,t∈[0,ˉt)whereˉt=ζ−D(θ(0))D(Ω)+2NKsinα. |
Proof. From system (1), we see that the dynamics of extreme phases is given by the following equations
˙θM(t)=ΩM+K∑j∈NMsin(θj(t)−θM(t)+α),˙θm(t)=Ωm+K∑j∈Nmsin(θj(t)−θm(t)+α), |
where
˙D(θ(t))=˙θM(t)−˙θm(t)=ΩM−Ωm+K∑j∈NMsin(θj−θM+α)−K∑j∈Nmsin(θj−θm+α)≤D(Ω)+K∑j∈NM[sin(θj−θM)cosα+cos(θj−θM)sinα]−K∑j∈Nm[sin(θj−θm)cosα+cos(θj−θm)sinα]=D(Ω)+Kcosα(∑j∈NMsin(θj−θM)−∑j∈Nmsin(θj−θm))+Ksinα(∑j∈NMcos(θj−θM)−∑j∈Nmcos(θj−θm)), | (4) |
where
sin(x+y)=sinxcosy+cosxsiny |
When the phase diameter satisfies
sin(θj−θM)≤0, j∈NMandsin(θj−θm)≥0, j∈Nm. |
Then we see from (4) that
˙D(θ)≤D(Ω)+2NKsinα. | (5) |
That is to say, when
D(θ(t))≤D(θ(0))+(D(Ω)+2NKsinα)t. |
Therefore, it yields that there exists a finite time
D(θ(t))<ζ,∀ t∈[0,ˉt), |
where
ˉt=ζ−D(θ(0))D(Ω)+2NKsinα. |
We will first study the special case, i.e., the network is strongly connected. Without loss of generality, we denote by
Step 1. For any given time
θ01(t)≤θ02(t)≤…≤θ0N0(t). | (6) |
In order to introduce the following steps, we first provide the process of iterations for
ˉLN0k−1(ˉCk−1,N0)=ˉak−1ˉLN0k(ˉCk,N0)+θ0k−1ˉak−1+1. |
L_l+11(C_1,l+1)=a_l+1L_l1(C_1,l)+θ0l+1a_l+1+1 |
Step 2. According to the strong connectivity of
Step 3. Similarly, we know that
It is worth emphasizing that the order of the oscillators may change along time
Lemma 3.1. Let
D0(θ(t))<γ<π,η>max{1sinγ,21−ζγ}, |
where
{N0∑i=n(ηi−nminj∈N0i(0)j≤isin(θ0j−θ0i))≤sin(θ0ˉkn−θ0N0), ˉkn=minj∈∪N0i=nN0i(0)j, 1≤n≤N0,n∑i=1(ηn−imaxj∈N0i(0)j≥isin(θ0j−θ0i))≥sin(θ0k_n−θ01), k_n=maxj∈∪ni=1N0i(0)j, 1≤n≤N0. |
Proof. For the proof of this lemma, please see [45] for details.
Recall the strongly connected ensemble
θ01(t)≤θ02(t)≤…≤θ0N0(t). |
Then we apply the process
ˉLN0k−1(ˉCk−1,N0) with ˉa0N0=0, ˉa0k−1=η(2N0−k+2)(ˉa0k+1),2≤k≤N0,L_k+11(C_1,k+1) with a_01=0, a_0k+1=η(k+1+N0)(a_0k+1),1≤k≤N0−1, | (7) |
where
ˉa0k−1=N0−k+1∑j=1ηjA(2N0−k+2,j),2≤k≤N0,a_0k+1=k∑j=1ηjA(k+1+N0,j),1≤k≤N0−1. |
Note that
ˉθ0k:=ˉLN0k(ˉCk,N0),θ_0k:=L_k1(C_1,k),1≤k≤N0. | (8) |
We define a non-negative quantity
Lemma 3.2. Let
βD0(θ(t))≤Q0(t)≤D0(θ(t)),β=1−2η, |
where
Proof. As we choose the same design for coefficients of convex combination as that in [45], the proof of this lemma is same as that in [45], please see [45] for details.
From Lemma 3.2, we see that the quantity
Lemma 3.3. Let
D0(θ(0))<ζ<γ<π,η>max{1sinγ,21−ζγ},tanα<1(1+ζζ−D(θ(0)))2N0cβD∞,D∞+α<π2,K>(1+ζζ−D0(θ(0))(D(Ω)+2N0Ksinα)ccosα1βD∞, | (9) |
where
D0(θ(t))<γ,t∈[0,+∞), |
and the dynamics of
˙Q0(t)≤D(Ω)+2N0Ksinα−KcosαcQ0(t),t∈[0,+∞). |
Proof. The proof is similar to [45] under the assumption that the frustration
Lemma 3.3 states that the phase diameter of the digraph
Lemma 3.4. Let
D0(θ(t))≤D∞,for t∈[t0,+∞), |
where
t0<ζKcosαcβD∞−(D(Ω)+2N0Ksinα)<ˉt. | (10) |
Remark 4. According to Lemma 2.8, we see that
Proof. From Lemma 3.3, we see that the dynamics of quantity
˙Q0(t)≤D(Ω)+2N0Ksinα−KcosαcQ0(t),t∈[0,+∞). | (11) |
We next show that there exists some time
˙Q0(t)≤D(Ω)+2N0Ksinα−KcosαcQ0(t)≤D(Ω)+2N0Ksinα−KcosαcβD∞<0. | (12) |
That is to say, when
t0=inf{t≥0 | Q0(t)≤βD∞}. |
And based on the definition of
Q0(t0)=βD∞. | (13) |
Moerover, from (12), it is easy to see that the stopping time
t0≤Q0(0)−βD∞KcosαcβD∞−(D(Ω)+2N0Ksinα). | (14) |
Now we study the upper bound of
Q0(t)≤βD∞, t∈[t0,+∞). | (15) |
Q0(t)≤βD∞,t∈[0,+∞). | (16) |
Then in this case, we directly set
Therefore, from (15), (16), and Lemma 3.2, we derive the upper bound of
D0(θ(t))≤Q0(t)β≤D∞,for t∈[t0,+∞). | (17) |
On the other hand, in order to verify (10), we further study
t0<ζKcosαcβD∞−(D(Ω)+2N0Ksinα). | (18) |
Here, we use the truth that
K>(1+ζζ−D0(θ(0))(D(Ω)+2N0Ksinα)ccosα1βD∞, |
it yields the following estimate about
t0<ζ(1+ζζ−D0(θ(0)))(D(Ω)+2N0Ksinα)−(D(Ω)+2N0Ksinα)=ζ−D0(θ(0))D(Ω)+2N0Ksinα=ˉt, | (19) |
where in this special strongly connected case, it's clear that
Thus, combining (17), (18) and (19), we derive the desired results.
In this section, we investigate the general network with a spanning tree structure, and prove our main result Theorem 1.1, which state that synchronization will emerge for Kuramoto model with frustration. According to Definition 2.5 and Lemma 2.6, we see that the digraph
We have studied the situation
From Remark 3 and node decomposition, the graph
G=d⋃k=0Gk,|Gk|=Nk, |
and we denote the oscillators in
θk1(t)≤θk2(t)≤…≤θkNk(t),0≤k≤d. |
For each subdigraph
{ˉLNkl−1(ˉCl−1,Nk) with ˉakNk=0, ˉakl−1=η(2N−l+2)(ˉakl+1),2≤l≤Nk,L_l+11(C_1,l+1) with a_k1=0, a_kl+1=η(l+1+2N−Nk)(a_kl+1),1≤l≤Nk−1. | (20) |
By induction principle, we deduce that
{ˉakl−1=Nk−l+1∑j=1ηjA(2N−l+2,j),2≤l≤Nk,a_kl+1=l∑j=1ηjA(l+1+2N−Nk,j),1≤l≤Nk−1. |
Note that
ˉak1=Nk−1∑j=1(ηjA(2N,j)),ˉak1≤N−1∑j=1(ηjA(2N,j)),0≤k≤d. | (21) |
And we further introduce the following notations,
ˉθkl:=ˉLNkl(ˉCl,Nk),θ_kl:=L_l1(C_1,l),1≤l≤Nk,0≤k≤d, | (22) |
ˉθk:=ˉLNk1(ˉC1,Nk),θ_k:=L_Nk1(C_1,Nk),0≤k≤d, | (23) |
Qk(t):=max0≤i≤k{ˉθi}−min0≤i≤k{θ_i},0≤k≤d. | (24) |
Due to the analyticity of the solution,
Lemma 4.1. Let
βDk(θ(t))≤Qk(t)≤Dk(θ(t)),0≤k≤d,β=1−2η, |
where
Proof. As we adopt the same construction of coefficients of convex combination in [45] which deals with the Kuramoto model without frustration on a general network, thus for the detailed proof of this lemma, please see [45].
Now we are ready to prove our main Theorem 1.1. To this end, we will follow similar arguments in Section 3 to complete the proof. Actually, we will investigate the dynamics of the constructed quantity
Lemma 4.2. Suppose that the network topology contains a spanning tree, and let
D(θ(0))<ζ<γ<π,η>max{1sinγ,21−ζγ}, | (25) |
where
tanα<1(1+(d+1)ζζ−D(θ(0)))2Ncβd+1D∞[4(2N+1)c]d,D∞+α<π2,K>(1+(d+1)ζζ−D(θ(0)))(D(Ω)+2NKsinα)ccosα[4(2N+1)c]dβd+1D∞, | (26) |
where
1. The dynamics of
˙Q0(t)≤D(Ω)+2NKsinα−KcosαcQ0(t),t∈[0,+∞), |
2. there exists time
D0(θ(t))≤βdD∞[4(2N+1)c]d,for t∈[t0,+∞), |
where
t0<ζKcosαcβd+1D∞[4(2N+1)c]d−(D(Ω)+2NKsinα)<ˉt. |
Next, inspiring from Lemma 4.2, we make the following reasonable ansatz for
Ansatz:
1. The dynamics of quantity
˙Qk(t)≤D(Ω)+2NKsinα+(2N+1)KcosαDk−1(θ(t))−KcosαcQk(t), | (27) |
where we assume
2. There exists a finite time
Dk(θ(t))≤βd−kD∞[4(2N+1)c]d−k,∀ t∈[tk,+∞), | (28) |
where
tk<(k+1)ζKcosαcβd+1D∞[4(2N+1)c]d−(D(Ω)+2NKsinα)<ˉt=ζ−D(θ(0))D(Ω)+2NKsinα. | (29) |
In the subsequence, we will split the proof of the ansatz into two lemmas by induction criteria. More precisely, based on the results in Lemma 4.2 as the initial step, we suppose the ansatz holds for
Lemma 4.3. Suppose the assumptions in Lemma 4.2 are fulfilled, and the ansatz in (27), (28) and (29) holds for some
Proof. We will use proof by contradiction criteria to verify the ansatz for
Bk+1={T>0 : Dk+1(θ(t))<γ, ∀ t∈[0,T)}. |
From Lemma 2.8, we see that
Dk+1(θ(t))≤D(θ(t))<ζ<γ,∀ t∈[0,ˉt). |
It is clear that
ˉt≤T∗,Dk+1(θ(t))<γ, ∀ t∈[0,T∗),Dk+1(θ(T∗))=γ. | (30) |
As the solution to system (1) is analytic, in the finite time interval
[0,T∗)=r⋃l=1Jl,Jl=[tl−1,tl). |
such that in each interval
˙Qk+1(t)≤D(Ω)+2NKsinα+(2N+1)KcosαDk(θ(t))−KcosαcQk+1(t). | (31) |
As the proof is slightly different from that in [45] and rather lengthy, we put the detailed proof in Appendix B.
˙Qk+1(t)≤−Kcosαc(Qk+1(t)−(2N+1)cDk(θ(t))−(D(Ω)+2NKsinα)cKcosα), | (32) |
where
c=(∑N−1j=1ηjA(2N,j)+1)γsinγ. | (33) |
For the term
Dk(θ(t))≤βd−kD∞[4(2N+1)c]d−k, t∈[tk,+∞),tk<ˉt. | (34) |
And from the condition (26), it is obvious that
K>(1+(d+1)ζζ−D(θ(0)))(D(Ω)+2NKsinα)ccosα[4(2N+1)c]dβd+1D∞>(D(Ω)+2NKsinα)ccosα[4(2N+1)c]dβd+1D∞. |
This directly yields that
(D(Ω)+2NKsinα)cKcosα<βd+1D∞[4(2N+1)c]d<βd−kD∞4d−k[(2N+1)c]d−k−1, | (35) |
where
(2N+1)cDk(θ(t))+(D(Ω)+2NKsinα)cKcosα≤(2N+1)cβd−kD∞[4(2N+1)c]d−k+βd−kD∞4d−k[(2N+1)c]d−k−1≤βd−kD∞2[4(2N+1)c]d−k−1<βd−kD∞[4(2N+1)c]d−k−1,t∈[tk,+∞). | (36) |
From Lemma 2.8, we have
Qk+1(t)≤max{Qk+1(tk),βd−kD∞[4(2N+1)c]d−k−1}:=Mk+1,t∈[tk,T∗). | (37) |
Suppose not, then there exists some
Ck+1:={tk≤t<˜t:Qk+1(t)≤Mk+1}. |
Since
t∗<˜t,Qk+1(t∗)=Mk+1,Qk+1(t)>Mk+1for t∈(t∗,˜t]. | (38) |
From the construction of
−Kcosαc(Qk+1(t)−(2N+1)cDk(θ(t))−(D(Ω)+2NKsinα)cKcosα)<−Kcosαc(Mk+1−βd−kD∞[4(2N+1)c]d−k−1)≤0. |
Wen apply the above inequality and integrate on both sides of (32) from
Qk+1(˜t)−Mk+1=Qk+1(˜t)−Qk+1(t∗)≤−∫˜tt∗Kcosαc(Qk+1(t)−(2N+1)cDk(θ(t))−(D(Ω)+2NKsinα)cKcosα)dt<0, |
which contradicts to the truth
βd−kD∞[4(2N+1)c]d−k−1<D∞,tk<ˉt,Qk+1(tk)≤Dk+1(θ(tk))≤D(θ(tk))<ζ, |
we directly obtain
Qk+1(t)≤max{Qk+1(tk),βd−kD∞[4(2N+1)c]d−k−1}<max{ζ,D∞}=ζ,t∈[tk,T∗). |
From Lemma 4.1 and the condition (25), it yields that
Dk+1(θ(t))≤Qk+1(t)β<ζβ<γ,t∈[tk,T∗). |
Since
Dk+1(θ(T∗))=limt→(T∗)−Dk+1(θ(t))≤ζβ<γ, |
which obviously contradicts to the assumption
Thus, we combine all above analysis to conclude that
Dk+1(θ(t))<γ,∀ t∈[0,+∞). | (39) |
Then for any finite time
˙Qk+1(t)≤D(Ω)+2NKsinα+(2N+1)KcosαDk(θ(t))−KcosαcQk+1(t). | (40) |
Therefore, we complete the proof of the Ansatz (27) for
Lemma 4.4. Suppose the conditions in Lemma 4.2 are fulfilled, and the ansatz in (27), (28) and (29) holds for some
Proof. From Lemma 4.3, we know the dynamic of
˙Qk+1(t)≤−Kcosαc(Qk+1(t)−(2N+1)cDk(θ(t))−(D(Ω)+2NKsinα)cKcosα), | (41) |
where
˙Qk+1(t)≤−Kcosαc(βd−kD∞[4(2N+1)c]d−k−1−βd−kD∞2[4(2N+1)c]d−k−1)=−Kcosαcβd−kD∞2[4(2N+1)c]d−k−1<0. | (42) |
That is to say, when
tk+1=inf{t≥tk | Qk+1(t)≤βd−kD∞[4(2N+1)c]d−k−1}. |
Then, based on (42) and the definition of
Qk+1(tk+1)=βd−kD∞[4(2N+1)c]d−k−1. | (43) |
Moreover, from (42), it yields that the stopping time
tk+1≤Qk+1(tk)−βd−kD∞[4(2N+1)c]d−k−1Kcosαcβd−kD∞2[4(2N+1)c]d−k−1+tk. | (44) |
Now we study the upper bound of
Qk+1(t)≤βd−kD∞[4(2N+1)c]d−k−1,t∈[tk+1,+∞). | (45) |
On the other hand, in order to verify (29), we further study
Qk+1(tk)≤Dk+1(θ(tk))≤D(θ(tk))<ζ,βd−kD∞2[4(2N+1)c]d−k−1>βd+1D∞[4(2N+1)c]d, |
we have the following estimates
Qk+1(tk)−βd−kD∞[4(2N+1)c]d−k−1Kcosαcβd−kD∞2[4(2N+1)c]d−k−1<ζKcosαcβd+1D∞[4(2N+1)c]d−(D(Ω)+2NKsinα), | (46) |
where the denominator on the right-hand side of above inequality is positive from the conditions about
tk<(k+1)ζKcosαcβd+1D∞[4(2N+1)c]d−(D(Ω)+2NKsinα)<ˉt=ζ−D(θ(0))D(Ω)+2NKsinα. | (47) |
Thus it yields from (44), (46) and (47) that the time
tk+1<(k+2)ζKcosαcβd+1D∞[4(2N+1)c]d−(D(Ω)+2NKsinα). | (48) |
Moreover, from (26), it is easy to see that the coupling strength
K>(1+(d+1)ζζ−D(θ(0)))(D(Ω)+2NKsinα)ccosα[4(2N+1)c]dβd+1D∞≥(1+(k+2)ζζ−D(θ(0)))(D(Ω)+2NKsinα)ccosα[4(2N+1)c]dβd+1D∞,0≤k≤d−1. | (49) |
Thus we combine (48) and (49) to verify the Ansatz (29) for
tk+1<ˉt=ζ−D(θ(0))D(Ω)+2NKsinα. | (50) |
Qk+1(t)≤βd−kD∞[4(2N+1)c]d−k−1,t∈[tk,+∞). | (51) |
In this case, we directly set
Finally, we are ready to verify the ansatz (28) for
Dk+1(θ(t))≤Qk+1(t)β≤βd−k−1D∞[4(2N+1)c]d−k−1,t∈[tk+1,+∞). | (52) |
Then we combine (48), (50) and (52) in Case 1 and similar analysis in Case 2 to conclude that the Ansatz (28) and (29) is true for
Now, we are ready to prove our main result.
Lemma 4.5. Let
D(θ(t))≤D∞,for t∈[t∗,+∞), |
where
Proof. Combining Lemma 4.2, Lemma 4.3 and Lemma 4.4, we apply inductive criteria to conclude that the Ansatz (27) –(29) hold for all
D(θ(t))=Dd(θ(t))≤D∞,for t∈[td,+∞). |
Thus we set
Remark 5. For the Kuramoto model with frustration, in Lemma 4.5, we show the phase diameter of whole ensemble will be uniformly bounded by a value
More precisely, we can introduce phase velocity or frequency
{˙θi(t)=ωi(t),t>0,i=1,2,…,N,˙ωi(t)=K∑j∈Nicos(θj(t)−θi(t)+α)(ωj(t)−ωi(t)),(θi(0),ωi(0))=(θi(0),˙θi(0)). | (53) |
Now for the second-order system (53), we apply the results in [9] for Kuramoto model without frustration on a general digraph and present the frequency synchronization for Kuramoto model with frustrations.
Lemma 4.6. Let
D(ω(t))≤C1e−C2(t−t∗),t≥t∗, |
where
Proof. According to Lemma 4.5 and the condition
D(θ(t))+α≤D∞+α<π2,t∈[t∗,+∞). |
Therefore, we can apply the methods and results in the work of Dong et al. [9] for Kuramoto model without frustration to yield the emergence of exponentially fast synchronization. As the proof is almost the same as that in [9], we omit its details.
Proof of Theorem 1.1: Combining Lemma 4.5 and Lemma 4.6, we ultimately derive the desired result in Theorem 1.1.
In this section, we present several numerical simulations to illustrate the main results in Theorem 1.1, which state that in sufficiently large coupling strength and small frustration regimes, synchronous behavior will emerge for the Kuramoto model with frustration in half circle case.
For the simulation, we use the fourth-order Runge-Kutta method and employ the parameter
(χij)=(1010000000110000000001100000000101001000000110000000001100000001011000001000010100000001100000000011). |
For the fixed digraph in Figure 1, we choose large
In Figure 3, we fix the coupling strength
In Figure 4, we fix the frustration
Moreover, for
In this paper, under the effect of frustration, we provide sufficient frameworks leading to the complete synchronization for the Kuramoto model with general network containing a spanning tree. To this end, we follow a node decomposition introduced in [25] and construct hypo-coercive inequalities through which we can study the upper bounds of phase diameters. When the initial configuration is confined in a half circle, for sufficiently small frustration and sufficiently large coupling strength, we show that the relative differences of Kuramoto oscillators adding a phase shift will be confined into a region of quarter circle in finite time, thus we can directly apply the methods and results in [9] to prove that the complete frequency synchronization emerges exponentially fast. And we provide some numerical simulations to illustrate the main results.
We really appreciate the editors and reviewers for their thorough reviews and insightful suggestions.
We will split the proof into six steps. In the first step, we suppose by contrary that the phase diameter of
B0:={T>0: D0(θ(t))<γ, ∀ t∈[0,T)}. |
From Lemma 2.8 where
D0(θ(t))=D(θ(t))<ζ<γ,∀ t∈[0,ˉt), |
which directly yields that
D0(θ(t))<γ,∀ t∈[0,T∗),D0(θ(T∗))=γ. | (54) |
In particular, we have
[0,T∗)=r⋃l=1Jl,Jl=[tl−1,tl), |
where the end point
θ01(t)≤θ02(t)≤…≤θ0N0(t),t∈Jl. |
˙θ0N0(t)=Ω0N0+K∑j∈N0N0(0)sin(θ0j−θ0N0+α)≤ΩM+K∑j∈N0N0(0)[sin(θ0j−θ0N0)cosα+cos(θ0j−θ0N0)sinα]≤ΩM+N0Ksinα+Kcosαminj∈N0N0(0)sin(θ0j−θ0N0). | (55) |
For the dynamics of
\begin{equation} \begin{aligned} \dot{\bar{\theta}}^0_{N_0-1} & = \frac{d}{dt} \left(\frac{\bar{a}^0_{N_0-1} \theta^0_{N_0} + \theta^0_{N_0-1}}{\bar{a}^0_{N_0-1} +1}\right) = \frac{\bar{a}^0_{N_0-1}}{\bar{a}^0_{N_0-1} + 1}\dot{\theta}^0_{N_0} + \frac{1}{\bar{a}^0_{N_0-1}+1}\dot{\theta}^0_{N_0-1} \\ & \le \frac{\bar{a}^0_{N_0-1}}{\bar{a}^0_{N_0-1} + 1} \left(\Omega_M + N_0K \sin \alpha + K \cos \alpha \min\limits_{j \in \mathcal{N}^0_{N_0}(0)} \sin (\theta^0_j - \theta^0_{N_0}) \right) \\ &+ \frac{1}{\bar{a}^0_{N_0-1}+1} \left(\Omega^0_{N_0-1} + K \sum\limits_{j \in \mathcal{N}^0_{N_0-1}(0)} \sin (\theta^0_j - \theta^0_{N_0-1} + \alpha)\right) \\ &\le \Omega_M + K \cos \alpha \frac{1}{\bar{a}^0_{N_0-1} + 1} \eta (N_0 +2) \min\limits_{j \in \mathcal{N}^0_{N_0}(0)} \sin (\theta^0_j - \theta^0_{N_0}) \\ &+ \frac{\bar{a}^0_{N_0-1}}{\bar{a}^0_{N_0-1} + 1} N_0 K \sin \alpha + K\cos \alpha \frac{1}{\bar{a}^0_{N_0-1} +1} \sum\limits_{j \in \mathcal{N}^0_{N_0-1}(0)} \sin (\theta^0_j - \theta^0_{N_0-1}) \\ &+ K \sin \alpha \frac{1}{\bar{a}^0_{N_0-1} + 1} \sum\limits_{j \in \mathcal{N}^0_{N_0 -1}(0) }\cos (\theta^0_j - \theta^0_{N_0 -1})\\ &\le \Omega_M + K \cos \alpha \frac{1}{\bar{a}^0_{N_0-1} + 1} 2\eta \min\limits_{j \in \mathcal{N}^0_{N_0}(0)} \sin (\theta^0_j - \theta^0_{N_0}) + \frac{\bar{a}^0_{N_0-1}}{\bar{a}^0_{N_0-1} + 1} N_0 K \sin \alpha \\ &+ K\cos \alpha \frac{1}{\bar{a}^0_{N_0-1} + 1} \left(\underset{j \le N_0 -1}{\sum\limits_{j \in \mathcal{N}^0_{N_0-1}(0)}} \sin(\theta^0_j - \theta^0_{N_0-1}) + \sin (\theta^0_{N_0} - \theta^0_{N_0 -1})\right) \\ &+\frac{1}{\bar{a}^0_{N_0-1} + 1} N_0 K \sin \alpha \\ &\le \Omega_M + K \cos \alpha \frac{1}{\bar{a}^0_{N_0-1} + 1} \eta \min\limits_{j \in \mathcal{N}^0_{N_0}(0)} \sin (\theta^0_j - \theta^0_{N_0}) \\ &+K\cos \alpha \frac{1}{\bar{a}^0_{N_0-1} + 1} \underset{j \le N_0 -1}{\min\limits_{j \in \mathcal{N}^0_{N_0-1}(0)}} \sin (\theta^0_j - \theta^0_{N_0-1}) \\ &+ K\cos \alpha \frac{1}{\bar{a}^0_{N_0-1} + 1} \underbrace{\left(\eta \min\limits_{j \in \mathcal{N}^0_{N_0}(0)} \sin (\theta^0_j - \theta^0_{N_0}) + \sin (\theta^0_{N_0} - \theta^0_{N_0 -1})\right)}_{\mathcal{I}_1} + N_0K\sin \alpha, \end{aligned} \end{equation} | (56) |
where we use
\begin{equation*} \begin{aligned} &|\sum\limits_{j \in \mathcal{N}^0_{N_0-1}(0)} \cos (\theta^0_j - \theta^0_{N_0-1})| \le N_0, \ K \cos \alpha \frac{1}{\bar{a}^0_{N_0-1} + 1} \eta N_0 \min\limits_{j \in \mathcal{N}^0_{N_0}(0)} \sin (\theta^0_j - \theta^0_{N_0}) \le 0,\\ &\underset{j \le N_0 -1}{\sum\limits_{j \in \mathcal{N}^0_{N_0-1}(0)}} \sin(\theta^0_j - \theta^0_{N_0-1}) \le \underset{j \le N_0 -1}{\min\limits_{j \in \mathcal{N}^0_{N_0-1}(0)}} \sin (\theta^0_j - \theta^0_{N_0-1}). \end{aligned} \end{equation*} |
Next we show the term
\min\limits_{j \in \mathcal{N}^0_{N_0}(0)} \sin(\theta^0_j - \theta^0_{N_0}) \le \sin(\theta^0_{\bar{k}_{N_0}} - \theta^0_{N_0}) \quad \text{where} \ \bar{k}_{N_0} = \min\limits_{j \in \mathcal{N}^0_{N_0}(0)} j. |
Note that
\begin{equation*} 0 \le \theta^0_{N_0}(t) - \theta^0_{N_0-1}(t) \le \theta^0_{N_0}(t) - \theta^0_{\bar{k}_{N_0}}(t) \le \frac{\pi}{2}, \end{equation*} |
which implies that
\begin{equation*} \mathcal{I}_1 \le \eta \sin(\theta^0_{\bar{k}_{N_0}} - \theta^0_{N_0}) + \sin(\theta^0_{N_0} - \theta^0_{N_0-1}) \le \sin(\theta^0_{\bar{k}_{N_0}} - \theta^0_{N_0}) + \sin(\theta^0_{N_0} - \theta^0_{N_0-1}) \le 0. \end{equation*} |
On the other hand, if
\eta > \frac{1}{\sin \gamma} \quad \mbox{and} \quad \sin(\theta^0_{N_0}(t) - \theta^0_{\bar{k}_{N_0}}(t)) > \sin \gamma, |
to conclude that
\begin{equation*} \mathcal{I}_1 \le \eta \sin(\theta^0_{\bar{k}_{N_0}} - \theta^0_{N_0}) + \sin(\theta^0_{N_0} - \theta^0_{N_0-1}) \le -1 + 1 \le 0. \end{equation*} |
Thus, for
\begin{equation} \mathcal{I}_1 = \eta \min\limits_{j \in \mathcal{N}^0_{N_0}(0)} \sin(\theta^0_j - \theta^0_{N_0}) + \sin(\theta^0_{N_0} - \theta^0_{N_0-1}) \le 0. \end{equation} | (57) |
Then combining (56) and (57), we derive that
\begin{equation} \begin{aligned} \dot{\bar{\theta}}^0_{N_0-1} &\le \Omega_M + N_0 K \sin \alpha \\ & + K \cos \alpha \frac{1}{\bar{a}^0_{N_0-1} + 1} \left(\eta \min\limits_{j \in \mathcal{N}^0_{N_0}(0)} \sin (\theta^0_j - \theta^0_{N_0}) + \underset{j \le N_0 -1}{\min\limits_{j \in \mathcal{N}^0_{N_0-1}(0)}} \sin (\theta^0_j - \theta^0_{N_0-1}) \right). \end{aligned} \end{equation} | (58) |
\begin{equation} \dot{\bar{\theta}}^0_n(t) \le \Omega_M + N_0K\sin \alpha + K \cos \alpha \frac{1}{\bar{a}^0_{n} + 1} \sum\limits_{i = n}^{N_0} \eta^{i-n} \underset{j \le i}{\min\limits_{j \in \mathcal{N}^0_{i}(0)}} \sin (\theta^0_j(t) - \theta^0_{i}(t)). \end{equation} | (59) |
In fact, it is known that (59) already holds for
\begin{equation} \begin{aligned} \dot{\bar{\theta}}^0_{n-1} &\le \Omega_M + N_0K \sin \alpha\\ &+ K \cos \alpha \frac{1}{\bar{a}^0_{n-1} + 1} \eta\sum\limits_{i = n}^{N_0} \eta^{i-n} \underset{j \le i}{\min\limits_{j \in \mathcal{N}^0_{i}(0)}} \sin (\theta^0_j - \theta^0_{i}) \\ &+ K \cos \alpha \frac{1}{\bar{a}^0_{n-1}+1} \underset{j \le n-1}{\min\limits_{j\in \mathcal{N}_{n-1}^0(0)}} \sin (\theta^0_j - \theta^0_{n-1})\\ &+K \cos \alpha \frac{1}{\bar{a}^0_{n-1} + 1} \\ &\times\left(\underbrace{\eta(N_0 - n +1)\sum\limits_{i = n}^{N_0} \eta^{i-n} \underset{j \le i}{\min\limits_{j \in \mathcal{N}^0_{i}(0)}} \sin (\theta^0_j - \theta^0_{i}) + \underset{j > n-1}{\sum\limits_{j \in \mathcal{N}^0_{n-1}(0)}} \sin (\theta^0_j - \theta^0_{n-1})}_{\mathcal{I}_2}\right). \end{aligned} \end{equation} | (60) |
Moreover, we can prove the term
\begin{equation} \begin{aligned} \dot{\bar{\theta}}^0_{1} &\le \Omega_M + N_0 K \sin \alpha + K \cos \alpha \frac{1}{\bar{a}^0_1+1} \sum\limits_{i = 1}^{N_0} \eta^{i-1} \underset{j \le i}{\min\limits_{j \in \mathcal{N}^0_{i}(0)}} \sin (\theta^0_j - \theta^0_{i}) \\ &\le \Omega_M + N_0K \sin \alpha+ K \cos \alpha \frac{1}{\bar{a}^0_1+1} \sin (\theta^0_{\bar{k}_1} - \theta^0_{N_0}) \\ & = \Omega_M + N_0K \sin \alpha+ K \cos \alpha \frac{1}{\bar{a}^0_1+1} \sin (\theta^0_{1} - \theta^0_{N_0}) , \end{aligned} \end{equation} | (61) |
where
\begin{equation} \begin{aligned} \frac{d}{dt} \underline{\theta}^0_{N_0} (t) & \ge \Omega_m - N_0K \sin \alpha + K \cos \alpha \frac{1}{\bar{a}^0_1 + 1} \sin (\theta^0_{N_0} - \theta^0_1). \end{aligned} \end{equation} | (62) |
Then we recall the notations
\begin{equation*} \begin{aligned} \dot{Q}^0(t) & = \frac{d}{dt}(\bar{\theta}_0 - \underline{\theta}_0) \le D(\Omega) + 2N_0K \sin \alpha- K \cos \alpha \frac{2}{\bar{a}^0_1 + 1} \sin (\theta^0_{N_0} - \theta^0_1) \\ &\le D(\Omega) + 2N_0K \sin \alpha - K\cos \alpha \frac{1}{\sum_{j = 1}^{N_0-1} \eta^j A(2N_0,j) +1} \sin (\theta^0_{N_0} - \theta^0_1), \end{aligned} \end{equation*} |
where we use the property
\bar{a}^0_{1} = \sum\limits_{j = 1}^{N_0-1} \eta^j A(2N_0,j) . |
As the function
\sin (\theta^0_{N_0} - \theta^0_1) \ge \frac{\sin \gamma}{\gamma}(\theta^0_{N_0} - \theta^0_1). |
Moreover, due to the fact
\begin{equation} \begin{aligned} \dot{Q}^0(t) &\le D(\Omega) + 2N_0 K \sin \alpha - K\cos \alpha \frac{1}{\sum_{j = 1}^{N_0-1} \eta^j A(2N_0,j) +1} \frac{\sin \gamma}{\gamma}(\theta^0_{N_0} - \theta^0_1) \\ &\le D(\Omega) + 2N_0 K \sin \alpha - K\cos \alpha \frac{1}{\sum_{j = 1}^{N_0-1} \eta^j A(2N_0,j) +1} \frac{\sin \gamma}{\gamma} Q^0(t) , \quad t \in J_l. \end{aligned} \end{equation} | (63) |
Note that the constructed quantity
\begin{equation*} M_0 = \max\left\{Q^0(0), \beta D^\infty\right\}. \end{equation*} |
We claim that
\begin{equation} Q^0(t) \le M_0 \quad \mbox{for all} \ t\in [0,T^*). \end{equation} | (64) |
Suppose not, then there exists some
\mathcal{C}_0 : = \{t < \tilde{t} \ | \ Q^0(t) \le M_0\} . |
Since
\begin{equation} t^* < \tilde{t}, \quad Q^0(t^*) = M_0, \quad Q^0(t) > M_0 \quad \mbox{for} \ t \in (t^*, \tilde{t}]. \end{equation} | (65) |
For the given constant
\begin{equation} K > \left(1+ \frac{\zeta}{\zeta - D(\theta(0)}\right) \frac{(D(\Omega) +2N_0 K\sin \alpha)c}{\cos \alpha} \frac{1}{\beta D^\infty} > \frac{(D(\Omega) +2N_0 K\sin \alpha)c}{\cos \alpha} \frac{1}{\beta D^\infty}, \end{equation} | (66) |
where
\begin{equation*} \begin{aligned} &D(\Omega) + 2N_0 K \sin \alpha - K\cos \alpha \frac{1}{\sum_{j = 1}^{N_0-1} \eta^j A(2N_0,j) +1} \frac{\sin \gamma}{\gamma} Q^0(t) \\ & < D(\Omega) + 2N_0 K \sin \alpha - K\cos \alpha \frac{1}{\sum_{j = 1}^{N_0-1} \eta^j A(2N_0,j) +1} \frac{\sin \gamma}{\gamma} \beta D^\infty < 0. \end{aligned} \end{equation*} |
Then, we apply the above inequality and integrate on the both sides of (63) from
\begin{equation*} \begin{aligned} &Q^0(\tilde{t}) - M_0\\ & = Q^0(\tilde{t}) - Q^0(t^*) \\ &\le \int_{t^*}^{\tilde{t}} \left(D(\Omega) + 2N_0 K \sin \alpha - K\cos \alpha \frac{1}{\sum_{j = 1}^{N_0-1} \eta^j A(2N_0,j) +1} \frac{\sin \gamma}{\gamma} Q^0(t) \right)dt < 0, \end{aligned} \end{equation*} |
which obviously contradicts to the fact
\begin{equation*} Q^0(t) \le M_0 = \max\left\{Q^0(0), \beta D^\infty\right\} < \zeta, \quad t\in [0,T^*). \end{equation*} |
Then we apply the relation
D_0(\theta(t)) \le \frac{Q^0(t)}{\beta} < \frac{\zeta}{\beta} < \gamma, \quad t \in [0,T^*) \quad \mbox{where} \ \beta = 1 - \frac{2}{\eta}. |
As
\begin{equation*} D_0(\theta(T^*)) = \lim\limits_{t \to (T^*)^-}D_0(\theta(t)) \le \frac{\zeta}{\beta} < \gamma, \end{equation*} |
which contradicts to the situation that
\begin{equation} D_0(\theta(t)) < \gamma, \quad \mbox{for all} \ t \in [0, +\infty). \end{equation} | (67) |
Then for any finite time
\begin{equation*} \dot{Q}^0(t) \le D(\Omega) + 2N_0 K \sin \alpha- K\cos \alpha \frac{1}{\sum_{j = 1}^{N_0-1} \eta^j A(2N_0,j) +1} \frac{\sin \gamma}{\gamma} Q^0(t) , \ t \in [0, +\infty). \end{equation*} |
We will show the detailed proof of Step 1 in Lemma 4.3. Now we pick out any interval
Figure 5 shows the four possible relations between
\begin{equation*} \max\limits_{0 \le i\le k+1}\{\bar{\theta}_i\} = \bar{\theta}_{k+1} , \quad \min\limits_{0\le i\le k+1}\{\underline{\theta}_i\} = \underline{\theta}_{k+1} \quad \mbox{for}\;t \in J_l . \end{equation*} |
Without loss of generality, we assume that
\begin{equation*} \theta^{k+1}_1 \le \theta^{k+1}_2 \le \dots \le \theta^{k+1}_{N_{k+1}}, \quad \mbox{for $t\in J_l$}. \end{equation*} |
\begin{equation} \begin{aligned} \frac{d}{dt} \bar{\theta}_n^{k+1}(t) &\le \Omega_M + S_{k+1}K\sin \alpha + S_kK \cos \alpha D_k (\theta(t))\\ &+ K \cos \alpha \frac{1}{\bar{a}^{k+1}_n + 1}\sum\limits_{i = n}^{N_{k+1}}\left( \eta^{i-n} \underset{j \le i}{\min\limits_{j\in \mathcal{N}_{i}^{k+1}(k+1)}} \sin (\theta^{k+1}_j(t) - \theta^{k+1}_{i}(t))\right), \end{aligned} \end{equation} | (68) |
where
\begin{equation} \begin{aligned} &\frac{d}{dt} \bar{\theta}^{k+1}_{N_{k+1}} \\ & = \Omega^{k+1}_{N_{k+1}} + K \cos \alpha \sum\limits_{j \in \mathcal{N}^{k+1}_{N_{k+1}}(k+1)} \sin (\theta^{k+1}_j - \theta^{k+1}_{N_{k+1}}) \\ &+ K \cos \alpha \sum\limits_{l = 0}^k \sum\limits_{j \in \mathcal{N}^{k+1}_{N_{k+1}}(l)} \sin (\theta^l_j - \theta^{k+1}_{N_{k+1}}) +K \sin \alpha \sum\limits_{l = 0}^{k+1} \sum\limits_{j \in \mathcal{N}^{k+1}_{N_{k+1}}(l)} \cos (\theta^l_j - \theta^{k+1}_{N_{k+1}}) \\ &\le \Omega_M + S_{k+1}K \sin \alpha \\ &+K \cos \alpha \underbrace{\sum\limits_{j \in \mathcal{N}^{k+1}_{N_{k+1}}(k+1)} \sin (\theta^{k+1}_j - \theta^{k+1}_{N_{k+1}})}_{\mathcal{I}_{11}} + K \cos \alpha \underbrace{\sum\limits_{l = 0}^k \sum\limits_{j \in \mathcal{N}^{k+1}_{N_{k+1}}(l)} \sin (\theta^l_j - \theta^{k+1}_{N_{k+1}})}_{\mathcal{I}_{12}}, \end{aligned} \end{equation} | (69) |
where we use
|\sum\limits_{l = 0}^{k+1} \sum\limits_{j \in \mathcal{N}^{k+1}_{N_{k+1}}(l)} \cos (\theta^l_j - \theta^{k+1}_{N_{k+1}})| \le \sum\limits_{l = 0}^{k+1} N_l = S_{k+1}. |
\begin{equation*} \sin (\theta^{k+1}_j - \theta^{k+1}_{N_{k+1}}) \le 0, \quad \mbox{for} \ j \in \mathcal{N}_{N_{k+1}}^{k+1}(k+1). \end{equation*} |
Then we immediately have
\begin{equation} \mathcal{I}_{11} = \sum\limits_{j\in \mathcal{N}_{N_{k+1}}^{k+1}(k+1)} \sin (\theta^{k+1}_j - \theta^{k+1}_{N_{k+1}}) \le \min\limits_{j \in \mathcal{N}^{k+1}_{N_{k+1}}(k+1)} \sin (\theta^{k+1}_j - \theta^{k+1}_{N_{k+1}}). \end{equation} | (70) |
If
\begin{equation*} \sin (\theta^{l}_j - \theta^{k+1}_{N_{k+1}}) \le 0. \end{equation*} |
If
\begin{equation*} \label{F-d7} \theta^{i}_{N_i} \ge \bar{\theta}_i \ge \underline{\theta}_i \ge \theta^i_1, \quad 0 \le i \le d, \end{equation*} |
we immediately obtain
\begin{equation} \theta^{k+1}_{N_{k+1}} \ge \bar{\theta}_{k+1} = \max\limits_{0 \le i\le k+1}\{\bar{\theta}_i\} \ge \max\limits_{0 \le i\le k}\{\bar{\theta}_i\} \ge \min\limits_{0 \le i\le k}\{\underline{\theta}_i\} \ge \min\limits_{0 \le i \le k}\min\limits_{1 \le j \le N_i} \{\theta^i_j\}. \end{equation} | (71) |
Thus we use the property of
\begin{equation*} \label{F-d8} \sin (\theta^{l}_j - \theta^{k+1}_{N_{k+1}}) \le \theta^{l}_j - \theta^{k+1}_{N_{k+1}} \le \theta^{l}_j - \min\limits_{0 \le i \le k}\min\limits_{1 \le j \le N_i} \{\theta^i_j\} \le D_k(\theta(t)). \end{equation*} |
Therefore, combining the above discussion, we have
\begin{equation} \mathcal{I}_{12} = \sum\limits_{l = 0}^k \sum\limits_{j \in \mathcal{N}^{k+1}_{N_{k+1}}(l)} \sin (\theta^l_j - \theta^{k+1}_{N_{k+1}}) \le S_k D_k(\theta(t)). \end{equation} | (72) |
From (69), (70) and (72), it yields that (68) holds for
\begin{equation} \begin{aligned} &\dot{\bar{\theta}}^{k+1}_{n-1}\\ &\le \Omega_M + \frac{\bar{a}^{k+1}_{n-1}}{\bar{a}^{k+1}_{n-1}+1} S_{k+1}K\sin \alpha + \frac{\bar{a}^{k+1}_{n-1}}{\bar{a}^{k+1}_{n-1}+1} S_kK \cos \alpha D_k (\theta(t))+ K \sin\alpha \frac{1}{\bar{a}^{k+1}_{n-1}+1}S_{k+1} \\ &+ K\cos \alpha \frac{1}{\bar{a}^{k+1}_{n-1} + 1} \eta (N_{k+1}-n+2+S_k) \sum\limits_{i = n}^{N_{k+1}}\left( \eta^{i-n} \underset{j \le i}{\min\limits_{j\in \mathcal{N}_{i}^{k+1}(k+1)}} \sin (\theta^{k+1}_j - \theta^{k+1}_{i})\right) \\ &+ K \cos\alpha \frac{1}{\bar{a}^{k+1}_{n-1}+1} \underset{j \le n-1}{\min\limits_{j\in \mathcal{N}_{n-1}^{k+1}(k+1)}} \sin (\theta^{k+1}_j - \theta^{k+1}_{n-1}) \\ &+ K \cos\alpha \frac{1}{\bar{a}^{k+1}_{n-1}+1}\underbrace{\underset{j > n-1}{\sum\limits_{j\in \mathcal{N}_{n-1}^{k+1}(k+1)}}\sin (\theta^{k+1}_j - \theta^{k+1}_{n-1})}_{\mathcal{I}_{21}}\\ &+ K \cos\alpha \frac{1}{\bar{a}^{k+1}_{n-1}+1} \underbrace{\sum\limits_{l = 0}^k \sum\limits_{j\in \mathcal{N}_{n-1}^{k+1}(l)} \sin (\theta^{l}_j - \theta^{k+1}_{n-1})}_{\mathcal{I}_{22}}. \end{aligned} \end{equation} | (73) |
Next we do some estimates about the terms
\begin{equation} \sum\limits_{i = n}^{N_{k+1}}\left( \eta^{i-n} \underset{j \le i}{\min\limits_{j\in \mathcal{N}_{i}^{k+1}(k+1)}} \sin (\theta^{k+1}_j(t) - \theta^{k+1}_{i}(t))\right) \le \sin(\theta^{k+1}_{\bar{k}_{n}} - \theta^{k+1}_{N_{k+1}}), \end{equation} | (74) |
where
(i) For the first case that
\begin{equation} 0 \le \theta^{k+1}_{j}(t) - \theta^{k+1}_{n-1}(t) \le \theta^{k+1}_{N_{k+1}}(t) - \theta^{k+1}_{n-1}(t) \le \theta^{k+1}_{N_{k+1}}(t) - \theta^{k+1}_{\bar{k}_{n}}(t) \le \frac{\pi}{2}. \end{equation} | (75) |
Then it yieldst from (74), (75) and
\begin{equation*} \begin{aligned} &\eta(N_{k+1} - n + 1) \sum\limits_{i = n}^{N_{k+1}}\left( \eta^{i-n} \underset{j \le i}{\min\limits_{j\in \mathcal{N}_{i}^{k+1}(k+1)}} \sin (\theta^{k+1}_j(t) - \theta^{k+1}_{i}(t))\right) + \mathcal{I}_{21} \\ & \le \eta(N_{k+1} - n + 1) \sin(\theta^{k+1}_{\bar{k}_{n}} - \theta^{k+1}_{N_{k+1}}) + \underset{j > n-1}{\sum\limits_{j\in \mathcal{N}_{n-1}^{k+1}(k+1)}}\sin (\theta^{k+1}_j - \theta^{k+1}_{n-1}) \\ & \le (N_{k+1} - n + 1) \sin(\theta^{k+1}_{\bar{k}_{n}} - \theta^{k+1}_{N_{k+1}}) + (N_{k+1} - n + 1) \sin (\theta^{k+1}_{N_{k+1}} - \theta^{k+1}_{n-1}) \\ &\le 0. \end{aligned} \end{equation*} |
(ii) For the second case that
\begin{equation} \eta > \frac{1}{\sin \gamma} \quad \mbox{and} \quad \sin(\theta^{k+1}_{N_{k+1}} - \theta^{k+1}_{\bar{k}_{n}}) > \sin \gamma, \end{equation} | (76) |
which yields
\begin{equation*} \begin{aligned} &\eta(N_{k+1} - n + 1) \sum\limits_{i = n}^{N_{k+1}}\left( \eta^{i-n} \underset{j \le i}{\min\limits_{j\in \mathcal{N}_{i}^{k+1}(k+1)}} \sin (\theta^{k+1}_j(t) - \theta^{k+1}_{i}(t))\right) + \mathcal{I}_{21} \\ & \le \eta(N_{k+1} - n + 1) \sin(\theta^{k+1}_{\bar{k}_{n}} - \theta^{k+1}_{N_{k+1}}) + \underset{j > n-1}{\sum\limits_{j\in \mathcal{N}_{n-1}^{k+1}(k+1)}}\sin (\theta^{k+1}_j - \theta^{k+1}_{n-1}) \\ & \le -(N_{k+1} - n + 1) +(N_{k+1} - n + 1) = 0. \end{aligned} \end{equation*} |
Then, we combine the above arguments in (i) and (ii) to obtain
\begin{equation} \eta(N_{k+1} - n + 1) \sum\limits_{i = n}^{N_{k+1}}\left( \eta^{i-n} \underset{j \le i}{\min\limits_{j\in \mathcal{N}_{i}^{k+1}(k+1)}} \sin (\theta^{k+1}_j - \theta^{k+1}_{i})\right) + \mathcal{I}_{21} \le 0. \end{equation} | (77) |
(i) If
(ii) If
(a) For the case that
0 \le \theta^l_j - \theta^{k+1}_{n-1} \le \theta^{k+1}_{N_{k+1}} - \theta^{k+1}_{n-1}\le \theta^{k+1}_{N_{k+1}} - \theta^{k+1}_{\bar{k}_{n}} \le \frac{\pi}{2}. |
Thus from the above inequality and (74), we have
\begin{equation*} \begin{aligned} &\eta \sum\limits_{i = n}^{N_{k+1}}\left( \eta^{i-n} \underset{j \le i}{\min\limits_{j\in \mathcal{N}_{i}^{k+1}(k+1)}} \sin (\theta^{k+1}_j(t) - \theta^{k+1}_{i}(t))\right) + \sin (\theta^l_j - \theta^{k+1}_{n-1}) \\ &\le \eta \sin(\theta^{k+1}_{\bar{k}_{n}} - \theta^{k+1}_{N_{k+1}}) + \sin (\theta^l_j - \theta^{k+1}_{n-1}) \\ &\le \sin(\theta^{k+1}_{\bar{k}_{n}} - \theta^{k+1}_{N_{k+1}}) + \sin(\theta^{k+1}_{N_{k+1}} - \theta^{k+1}_{\bar{k}_{n}}) = 0. \end{aligned} \end{equation*} |
(b) For another case that
\begin{equation*} \begin{aligned} &\eta \sum\limits_{i = n}^{N_{k+1}}\left( \eta^{i-n} \underset{j \le i}{\min\limits_{j\in \mathcal{N}_{i}^{k+1}(k+1)}} \sin (\theta^{k+1}_j(t) - \theta^{k+1}_{i}(t))\right) + \sin (\theta^l_j - \theta^{k+1}_{n-1}) \\ &\le \eta \sin(\theta^{k+1}_{\bar{k}_{n}} - \theta^{k+1}_{N_{k+1}}) + \sin (\theta^l_j - \theta^{k+1}_{n-1}) \\ &\le -1 + 1 = 0. \end{aligned} \end{equation*} |
Hence, combining the above arguments in (a) and (b), we obtain that
\eta \sum\limits_{i = n}^{N_{k+1}}\left( \eta^{i-n} \underset{j \le i}{\min\limits_{j\in \mathcal{N}_{i}^{k+1}(k+1)}} \sin (\theta^{k+1}_j(t) - \theta^{k+1}_{i}(t))\right) + \sin (\theta^l_j - \theta^{k+1}_{n-1}) \le 0. |
(iii) If
\begin{equation} \sin (\theta^l_j - \theta^{k+1}_{n-1}) \le \sin (\theta^l_j - \theta^{k+1}_{N_{k+1}}) + \sin (\theta^{k+1}_{N_{k+1}} - \theta^{k+1}_{n-1}). \end{equation} | (78) |
For the second part on the right-hand side of above inequality (78), we apply the same analysis in (ii) to obtain
\eta \sum\limits_{i = n}^{N_{k+1}}\left( \eta^{i-n} \underset{j \le i}{\min\limits_{j\in \mathcal{N}_{i}^{k+1}(k+1)}} \sin (\theta^{k+1}_j - \theta^{k+1}_{i})\right) + \sin (\theta^{k+1}_{N_{k+1}} - \theta^{k+1}_{n-1}) \le 0. |
For the first part on the right-hand side of (78), the calculation is the same as (72), thus we have
\begin{equation*} \sin (\theta^{l}_j - \theta^{k+1}_{N_{k+1}}) \le \theta^{l}_j - \theta^{k+1}_{N_{k+1}} \le \theta^{l}_j - \min\limits_{0 \le i \le k}\min\limits_{1 \le j \le N_i} \{\theta^i_j\} \le D_k(\theta(t)). \end{equation*} |
Therefore, we combine the above estimates to obtain
\begin{equation} \begin{aligned} &\eta S_k \sum\limits_{i = n}^{N_{k+1}}\left( \eta^{i-n} \underset{j \le i}{\min\limits_{j\in \mathcal{N}_{i}^{k+1}(k+1)}} \sin (\theta^{k+1}_j(t) - \theta^{k+1}_{i}(t))\right) + \mathcal{I}_{22} \\ & \le \eta S_k \sin(\theta^{k+1}_{\bar{k}_{n}} - \theta^{k+1}_{N_{k+1}}) + \sum\limits_{l = 0}^k \sum\limits_{j\in \mathcal{N}_{n-1}^{k+1}(l)} \sin (\theta^{l}_j - \theta^{k+1}_{n-1}) \\ & \le S_k D_k(\theta(t)). \end{aligned} \end{equation} | (79) |
Then from (73), (77), and (79), it yields that
\begin{equation*} \begin{aligned} &\frac{d}{dt} \bar{\theta}^{k+1}_{n-1}\\ &\le \Omega_M + S_{k+1}K \sin\alpha + \frac{\bar{a}^{k+1}_{n-1}}{\bar{a}^{k+1}_{n-1}+1} S_kK \cos \alpha D_k (\theta(t)) + + K\cos\alpha \frac{1}{\bar{a}^{k+1}_{n-1} + 1} S_kD_k(\theta(t))\\ &+ K\cos \alpha \frac{1}{\bar{a}^{k+1}_{n-1} + 1} \eta \sum\limits_{i = n}^{N_{k+1}}\left( \eta^{i-n} \underset{j \le i}{\min\limits_{j\in \mathcal{N}_{i}^{k+1}(k+1)}} \sin (\theta^{k+1}_j - \theta^{k+1}_{i})\right)\\ &+K \cos\alpha \frac{1}{\bar{a}^{k+1}_{n-1}+1} \underset{j \le n-1}{\min\limits_{j\in \mathcal{N}_{n-1}^{k+1}(k+1)}} \sin (\theta^{k+1}_j - \theta^{k+1}_{n-1})\\ &\le \Omega_M + S_{k+1}K\sin\alpha + S_kK\cos\alpha D_k(\theta(t))\\ &+K\cos \alpha \frac{1}{\bar{a}^{k+1}_{n-1} + 1} \sum\limits_{i = n-1}^{N_{k+1}}\left( \eta^{i-(n-1)} \underset{j \le i}{\min\limits_{j\in \mathcal{N}_{i}^{k+1}(k+1)}} \sin (\theta^{k+1}_j - \theta^{k+1}_{i})\right). \end{aligned} \end{equation*} |
This means that the claim (68) does hold for
\begin{equation*} \sum\limits_{i = 1}^{N_{k+1}}\left( \eta^{i-1} \underset{j \le i}{\min\limits_{j\in \mathcal{N}_{i}^{k+1}(k+1)}} \sin (\theta^{k+1}_j - \theta^{k+1}_{i})\right) \le \sin(\theta^{k+1}_1 - \theta^{k+1}_{N_{k+1}}). \end{equation*} |
From the notations in (22) and (23), it is known that
\begin{equation*} \bar{\theta}_1^{k+1} = \bar{\theta}_{k+1}, \quad \underline{\theta}_{N_{k+1}}^{k+1} = \underline{\theta}_{k+1}. \end{equation*} |
Thus, we exploit the above inequality and set
\begin{equation} \begin{aligned} &\frac{d}{dt} \bar{\theta}_{k+1} = \frac{d}{dt} \bar{\theta}^{k+1}_1\\ &\le \Omega_M + S_{k+1}K \sin\alpha + S_kK\cos\alpha D_k(\theta(t))\\ &+ K\cos \alpha \frac{1}{\bar{a}^{k+1}_{1} + 1} \sum\limits_{i = 1}^{N_{k+1}}\left( \eta^{i-1} \underset{j \le i}{\min\limits_{j\in \mathcal{N}_{i}^{k+1}(k+1)}} \sin (\theta^{k+1}_j - \theta^{k+1}_{i})\right) \\ &\le \Omega_M + S_{k+1}K \sin\alpha + S_kK\cos\alpha D_k(\theta(t)) + K\cos \alpha \frac{1}{\bar{a}^{k+1}_{1} + 1} \sin(\theta^{k+1}_1 - \theta^{k+1}_{N_{k+1}}). \end{aligned} \end{equation} | (80) |
We further apply the similar arguments in (80) to derive the differential inequality of
\begin{equation} \frac{d}{dt} \underline{\theta}_{k+1} \ge \Omega_m - S_{k+1}K\sin \alpha - S_kK\cos\alpha D_k(\theta(t)) + K\cos \alpha \frac{1}{\bar{a}^{k+1}_{1} + 1} \sin (\theta^{k+1}_{N_{k+1}} - \theta^{k+1}_1). \end{equation} | (81) |
Due to the monotone decreasing property of
\sin (\theta^{k+1}_{N_{k+1}} - \theta^{k+1}_1) \ge \frac{\sin \gamma}{\gamma}(\theta^{k+1}_{N_{k+1}} - \theta^{k+1}_{1}). |
Then we combine the above inequality, (80), (81) and (21) to get
\begin{equation*} \begin{aligned} \dot{Q}^{k+1}(t) & = \frac{d}{dt} (\bar{\theta}_{k+1} - \underline{\theta}_{k+1}) \\ &\le D(\Omega) + 2S_{k+1}K\sin\alpha + 2S_kK\cos\alpha D_k(\theta(t)) \\ &- K\cos \alpha \frac{2}{\bar{a}^{k+1}_{1} + 1} \sin (\theta^{k+1}_{N_{k+1}} - \theta^{k+1}_1) \\ &\le D(\Omega) + 2S_{k+1}K\sin\alpha + 2S_kK\cos\alpha D_k(\theta(t)) \\ &- K\cos \alpha \frac{1}{\bar{a}^{k+1}_{1} + 1} \frac{\sin \gamma}{\gamma} (\theta^{k+1}_{N_{k+1}} - \theta^{k+1}_1)\\ &\le D(\Omega) + 2NK\sin \alpha + (2N+1)K \cos \alpha D_k(\theta(t)) \\ &- K\cos \alpha \frac{1}{\sum\limits_{j = 1}^{N-1}\eta^jA(2N,j) + 1} \frac{\sin \gamma}{\gamma}Q^{k+1}(t),\qquad t \in J_l, \end{aligned} \end{equation*} |
where we use the fact that
\begin{equation} \begin{aligned} \dot{Q}^{k+1}(t) &\le D(\Omega) + 2NK\sin\alpha + (2N+1)K\cos\alpha D_k(\theta(t)) - \frac{K\cos\alpha}{c}Q^{k+1}(t). \end{aligned} \end{equation} | (82) |
As our analysis does not depend on the choice of
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