![](https://csdnimg.cn/release/download_crawler_static/10928093/bga.jpg)
with the property that for each sequence S
i
1
, S
i
2
, ··· , S
i
j
of positive length, the matrix prod-
uct S
i
j
S
i
j−1
···S
i
1
is SIA. Then, for each infinite sequence S
i
1
, S
i
2
, ···, there exists a column
vector y such that lim
j→∞
S
i
j
S
i
j−1
···S
i
1
= 1
n
y
T
(see [37]). More detailed proof of the main
result of [11] will be shown by the instructor in the class.
Key result from [20]:
(1) Main result: Suppose that A
n
is constant. Algorithm (6) achieves average consensus if
and only if directed graph G
n
is strongly connected and balanced or undirected graph G
n
is connected. Suppose that A
n
(t) is switching. Algorithm (6) achieves average consensus if
directed graph G
n
is strongly connected and balanced at each time instant.
(2) Approach: algebraic graph theory, Lyapunov theory, matrix theory
Key prop erties used: If directed graph G
n
is strongly connected and balanced, then L
n
has a simple zero eigenvalue with an associated right eigenvector 1
n
and an associated left
eigenvector 1
n
and all other eigenvalue have positive real parts. More detailed proof of the
main result of [20] will be shown by the instructor in the class.
Key result from [17]:
(1) Main result: Algorithm (9) achieves consensus if and only if there exists K ≥ 0 such
that the union of directed graphs G
n
(t) has a directed spanning tree across each interval of
length Kh, whereh is the sample time.
(2) Approach: algebraic graph theory, set-valued Lyapunov theory
Key technique used in the proof: A set-valued Lyapunov function V is defined as V (x
1
, . . . , x
n
) =
(conv{x
1
, . . . , x
n
})
n
, where conv{x
1
, . . . , x
n
} denotes the convex hull of {x
1
, . . . , x
n
}, and
X
n
4
= X × ··· × X. More detailed proof of the main result of [17] will be shown by the
instructor in the class.
Key result from [26]:
(1) Main result: Suppose that A
n
is constant. Algorithm (6) achieves consensus if and
only if directed graph G
n
has a directed spanning tree. Suppose that A
n
(t) is switching.
Algorithm (6) achieves consensus if directed graph G
n
(t) has a directed spanning tree jointly.
(2) Approach: algebraic graph theory, matrix theory
Key technique used in the proof: The nonsymmetric Laplacian matrix L
n
has a simple zero
eigenvalue and all other eigenvalues have positive real parts if and only if G
n
has a directed
spanning tree. In addition, there exist 1
n
satisfying L1
n
= 0 and a nonnegative vector
p ∈ R
n
satisfying p
T
L
n
= 0 and p
T
1
n
= 1. That is, 1
n
and p are, respectively, right and
left eigenvectors of L
n
associated with the zero eigenvalue. If G has a directed spanning
tree, then p is unique. More detailed proof of the main result of [26] will be shown by the
instructor in the class.
Key result from [5]:
(1) Main result: Algorithm (9) achieves consensus if and only if there exists K ≥ 0 such that
the composition of directed graphs G
n
(t) has a directed spanning tree across each interval of
length Kh, whereh is the sample time.
(2) Approach: A graph composition op eration is defined which is used to discuss the con-
nectivity of a sequence of graphs.
More detailed proof of the main result of [5] will be shown by the instructor in the class.
10