Our main result in the first part of the thesis is contained in this section. We begin with imposing conditions on coupling matrices G and D. We assume that the coupling matrix G satisfies the following:
(i) λ = 0 is a simple eigenvalue of G and e = [1, 1, . . . , 1]T1×m is
its corresponding eigenvector. (3.3a)
(ii) All nonzero eigenvalues of G have negative real part. (3.3b) We further assume that the matrix D is, without loss of generality, of the form
D=
The index k, 1 ≤ k ≤ n, means that the first k components of the subsystem are coupled. If k 6= n, then the system is said to be partial-state coupled. Otherwise, it is said to be full-state coupled.
From time to time, we will refer system (3.1) as coupled system (D, G, F(x, t)).
To study the synchronization of such system, we permute the state variables in the following way:
Then (3.1) can be written as
˙˜x =
˜fi(˜x, t) =
Note that such reformulation is certainly not new (see e.g., [29, 53]). From here on, we will treat ˜ as a function that takes x into ˜xor xi into ˜xi. A transformation of coordi-nates of ˜x is then to be applied to (3.4) so as to decompose the synchronous manifold.
The problem of synchronization of (2.15), and hence (3.5) is then equivalent to proving the asymptotical stability of reduced system (see (3.8)). To study the synchronization of (2.15), we first make a coordinate change to decompose the synchronous subspace.
Let A be an m× m matrix of the form
where e is given as in (3.3a). It is then easy to see that CCT is invertible and that
A−1 =
E(D⊗ G)E−1 = (In⊗ A)(D ⊗ G)(In⊗ A−1) A. Multiplying E to the both side of equation (3.5a), we get
˙˜y =: E ˙˜x = E˜F(˜x, t) + dE(D ⊗ G)E−1y˜
we have that the dynamics of ¯yis satisfied by the following equation
˙¯y = d(D ⊗ ¯G)¯y+ ¯F(¯y, t). (3.8)
Here ¯Fis obtained from E ˜F(E−1y, t) accordingly.˜
The task of obtaining the global synchronization of system (2.15) is now reduced to showing that the origin is globally and asymptotically stable with respect to system (3.8). To this end, the space ¯y is broken into two parts ¯yc, the coupled space, and ¯yu,
Here ¯yc =
. The dynamics on the coupled space with
respect to the linear part is under the influence of ¯G, which is asymptotically stable.
The dynamics of the nonlinear part on coupled space can then be controlled by choosing large coupling strength. In short, this part of the dynamics is easy to contain. In fact, the larger k, the number of state variables being coupled, gives the better chance of the synchronization of the coupled system. On the other hand, the uncoupled space has no stable matrix ¯G to play with. Thus, its corresponding vector field ¯Fu(¯y, t) must have a certain structure to make the trajectory stay closer to the origin as time progresses.
As we shall explain latter.
Now, assume that ¯Fc(¯y, t) satisfies a dual-Lipschitz condition with a dual-Lipschitz constant b1. That is,
k¯Fc(¯y, t)k ≤ b1k¯yk (3.10a)
whenever ¯y in the ball B(m−1)n(α), and for all time t. Since the estimate in the right-hand side of (3.10a) depends on the whole space ¯y, condition (3.10a) is a mild assump-tion provided that coupled system is bounded dissipative. Write ¯Fu(¯y, t) as
F¯u(¯y, t) = U(t)¯yu+ ( ¯Fu(¯y, t)− U(t)¯yu) and kj ∈ N. We assume further that the followings hold.
(i) The matrix measures µi(Uj(t)) are less than −γ for all t and all j,
where γ > 0. (3.10c)
(ii) Let ¯Ru(¯y, t) = dual-Lipschitz condition with a strong dual-Lipschitz constant b2. Specifically, let
¯
, written in accordance with the block structure of U(t). Then we
assume that
Specifically, we break the vector field ¯Fu into (time dependent) linear part U(t)¯yu and nonlinear part ¯Ru(¯y, t). We will further break U(t) into certain block diagonal form if necessary. Note that the form (3.10b) can always be achieved. Since the remainder term ¯Rstill depends on the whole space ¯y. To take control of the dynamics on the linear part, we assume that the matrix measure of each diagonal block Uj(t) is negative. As to contain corresponding dynamics on the nonlinear part, we assume that the (3.10d) holds. Note that though the nonlinear terms Ruj(¯y, t) could possibly depend on the whole space, their norm estimate are required to depend only on the coupled space and uncoupled subspaces with their indexes proceeding j. In this set up, the nonlinear dynamics on uncoupled space can be iteratively controlled by choosing large coupling strength. We also remark that if (3.10c) and (3.10d) are satisfied for l, the number of diagonal blocks, being one, then we do not need to further break U(t).
Such further breaking is needed only if (3.10c) and (3.10d) are not satisfied. The proof in the following theorem gives exactly how the above strategy can be realized.
Theorem 3.2.1. Let G and D be given as in (3.3). Assume that ¯F satisfies (3.10a-d), and system (3.8) is uniformly bounded dissipative with respect to α. Let λ1 =
max{λj|λj ∈ Re(σ( ¯G))}. If
Proof. Since system (3.8) is uniformly bounded dissipative with respect to α, without loss of generality, we may assume that k¯y(t)k ≤ α for all time t ≥ t0. Using (3.10b), we write (3.8) as
Applying the variation of constant formula to (3.12a) on ¯yc, we get
¯
whenever t ≥ t0,1 for some t0,1 > 0. We then apply Theorem 3.1.3 on ¯yu1 and the
It then follows from (3.10c-d) and (3.13a) that
k¯yu1(t)k ≤ αe−γ(t−t0,1)+α
|ν|δl+1, we see that the contraction factor h is strictly less than 1, andk¯y(t)k contracts as time progresses. To complete the proof of the theorem, we note that δ > 1 can be made arbitrary close to 1. Consequently, if d >
1 + (bγ2)22l
cb1
|ν|, then h can still be made to be less than 1.
Remark 3.2.2. (i) In case that ¯G is symmetric, then c and ǫ can be chosen to be one and zero, respectively. (ii) b1 and b2 could possibly depend on α. (iii) If system (3.8) is only bounded dissipative, then the estimate in (3.11) is still valid. However, in this case, b1 and b2 depend not only on α but also on x0.
Corollary 3.2.3. Suppose ¯Fand G are given as in Theorem 3.2.1. Let
D= ¯Dk×k 0
0 0
n×n,
where Re( σ( ¯D) ) > 0. (3.14a) Assume, in addition, that either σ(G) or σ( ¯D) has no complex eigenvalue.
Then assertions in Theorem 3.2.1 still hold true, except dc needs to be replaced by
dc = c b1
Proof. Assumption on D is to ensure that (3.29b) is still valid. Other parts of the proof are similar to those in Theorem 3.2.1 and are thus omitted.
We next turn our attention to finding conditions on the nonlinearities fi(u, t), i = 1, . . . , n, u ∈ Rn, so that assumptions (3.10a-d) are satisfied. To this end, we need the following notations. Let ˜xi and ˜xbe given as in (3.4). Define
[˜xi]−=
We then break ˜Fas given in (3.5a) into two parts so that the breaking is in consistent with ¯y in (3.9). Specifically, we shall write
F(˜˜ x, t) = ˜Fc(˜x, t) F˜u(˜x, t)
.
(3.16) We are now in the position to state the following propositions.
Proposition 3.2.4. Suppose that fi(x, t), i = 1, 2, . . . , k satisfy a Lipschitz condition
we conclude that (3.10a) holds.
From the above proposition, we see that the nonlinearities on the corresponding coupled space are only assumed to be Lipchitz. The following proposition is very useful in the sense that by checking how each component fi of the nonlinearity f is formed, one would then be able to conclude whether (3.10c-d) are satisfied.
Proposition 3.2.5. Let u = (u1, . . . , un)T and v = (v1, . . . , vn)T be vectors in Bn(α2).
fwp−1+i(u, t)− fwp−1+i(v, t)
=
kp
X
j=1
qwp−1+i,wp−1+j(u, v, t)(uwp−1+j− vwp−1+j) + rwp−1+i(u, v, t).
(3.19a)
We further assume that the followings are true.
(i) For p = 1, . . . , l, let Qu,v,p = (qwp−1+i,wp−1+j(u, v, t)), where 1 ≤ i, j ≤ kp. Then µ∗(Vp) < −γ for all p, u, v in Bn(α2) and all time t, where ∗ = 1, 2, ∞.
(3.19b)
(ii) Let rp = rwp−1+1(u, v, t), . . . , rwp(u, v, t)T
. We have that
krpk ≤ b2k
u1− v1
... uwp−1 − vwp−1
k (3.19c)
for all p, u, v in Bn(α2) and all time t.
Then (3.10c) and (3.10d) hold true for ∗ = 1, 2, ∞.
Proof. Since ri(u, v, t) depend on whole space, fi(u, t)− fi(v, t) can always be written as the form in (3.19a). Using (3.19a) and (3.18), we have that the matrices Up(t) in the linear part of ¯Fu(¯y, t) take the form
Up(t) =
mX−1 w=1
Qxw,xw+1,p(t)⊗ Dw, (3.20)
where xw are given as in (1.2), and
(Dw)ij =
1 i = j = w,
0 otherwise, 1≤ i, j ≤ m − 1.
It then follows from (2.9-2.11), and (3.20) that µ∗(Up(t)) < −γ for ∗ = 1 or ∞. For
∗ = 2, we have that
m[−1 w=1
σ{Qxw,xw+1,p(t) + Qxw,xw+1,p(t)T
}
= σ (m−1
X
w=1
Qxw,xw+1,p(t)⊗ Dw+ Qxw,xw+1,p(t)T
⊗ Dw)
= σ Up(t) + UTp(t) ,
where σ(A) is the spectrum of A. We remark that the first equality above can be verified by the definition of eigenvalues due to the structure of Up(t). It then follows from (2.11) that µ2(Up(t)) < −γ. The remainder of the proof is similar to that of Proposition 3.2.4, and is thus omitted.
Remark 3.2.6. The upshot of Proposition 3.2.5 is that by only checking the ”structure”
of the vector field f of the single oscillator, one should be able to determine if our main result can be applied. To be precise, we begin with saving notations by setting f as f = f(x, t) = (f1(x, t), . . . , fn(x, t))T. We then check the form of the difference of ”uncoupled” part of dynamics. That is, we write fi(u, t)− fi(v, t) in the form of (3.19a) with i = k + 1, . . . , n. If (3.19b, c) can be satisfied, then l = 1 gets the job done. Otherwise, we further break the uncoupled states into a set of smaller pieces to see if the resulting (3.19b, c) are satisfied.
We are now ready to state the main theorems of the paper.
Theorem 3.2.7. Assume that system (2.15) is (resp., uniformly) bounded dissipative.
Let the coupling matrices G and D satisfy (3.3) and the nonlinearities fi(x, t), i = 1, 2, . . . , n, satisfy (3.17) and (3.19). Suppose d is greater than dc, as given in (3.11).
Then system (2.15) is (resp., uniformly,) globally synchronized.
Proof. The proof is direct consequences of Propositions 3.2.4 and 3.2.5, and Theorem 3.2.1.
Remark 3.2.8. From here on, we will refer the assumptions in Theorem 3.2.7 as synchronization hypotheses.
Theorem 3.2.9. The coupled system (D, G, F(x, t)), given as in Corollary 3.2.3, is also (resp., uniformly,) globally synchronized provided that its coupled system is (resp., uniformly) bounded dissipative and that d is greater than dc. Here dc is given in (3.14b).