非线性奇异摄动系统中Tikhonov定理的新进展
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更新于2024-08-28
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"这篇论文主要探讨了非线性奇异摄动系统中的指数稳定性问题，并提出了一种新的条件。文章首先给出了一种非线性系统的通用指数稳定性准则，引入了一种新的特征值，使得指数稳定性可以度量。基于这个准则，论文对非线性奇异摄动系统的Tikhonov定理进行了新的发展。关键词包括：非线性奇异摄动系统、均匀最终有界性和Tikhonov定理。"
非线性奇异摄动系统是数学和控制理论中的一个重要研究领域，它涉及在系统中存在小参数或微小扰动的情况。这些扰动可能导致系统的动态行为发生显著变化，因此理解和分析这类系统的稳定性和行为至关重要。论文的主要贡献在于提出了一种新的指数稳定性判据，这是对非线性系统稳定性的更一般形式的表述。
在传统的稳定性分析中，指数稳定性是指系统状态随着时间的推移以指数速率收敛到平衡点或吸引子。然而，对于非线性系统，尤其是当存在奇异摄动时，确定稳定性可能变得复杂。论文中引入的新特征值概念提供了一种衡量系统指数稳定性的新方法，这有助于简化分析过程并拓宽了应用范围。
Tikhonov定理是常用于分析奇异摄动系统稳定性的一个重要工具。原始的Tikhonov定理通常关注的是线性系统的渐近行为，但在非线性系统中，其应用受到了限制。论文中提到的新发展，即对Tikhonov定理的扩展，意味着可以更有效地处理非线性奇异摄动系统，尤其是在指数稳定性方面。
此外，论文还提到了“均匀最终有界性”这一概念，这是另一种稳定性类型，指的是系统的所有解最终都将被限制在一个固定的有界区域内。这个概念与指数稳定性相结合，可以提供更全面的系统行为理解。
这篇论文为非线性奇异摄动系统的稳定性分析提供了新的理论基础，新提出的特征值和对Tikhonov定理的扩展，为解决实际工程和科学问题提供了新的工具和思路，特别是对于那些受微小扰动影响的复杂系统。这项工作不仅丰富了非线性控制理论，也为后续研究和应用开辟了新的方向。
K. Wang et al. / Nonlinear Analysis 74 (2011) 2869–2879 2871
i.e., v(δ
m
) ∈ Ω. Since v(t) is continuous, there exists a constant β > 0 that v(t) ∈ Ω for all t ∈ (δ
m
− β, δ
m
+ β). Noticing
that v(t) ∈ Ω for all t ∈ [t
0
, δ
m
), we have v(t) ∈ Ω for all t ∈ [t
0
, δ
m
+ β), which contradicts with the definition of δ
m
.
Therefore, δ
m
= +∞.
Inequality (6) shows that v(t) ≡ e if v(t
0
) = e. Substituting it into (2), we have
˙
e = H(e) = 0. So e is an equilibrium
point of (2). Inequality (4) shows that the equilibrium point e is exponentially stable.
Remark 1. (i) The limitation in (3) does exist. In fact, it is easy to prove that for all x, y ∈ R
k
(x = 0), function f (s) =
‖x−sy‖−‖x‖
s‖x‖
is monotonically increasing in (0, +∞), and ∀s ∈ (0, +∞), |f (s)| ≤ ‖x‖
−1
‖y‖. So lim
s→o
+
f (s) exists.
(ii) If system (2) is linear, and that the norm ‖ · ‖ used in (3) is an l
2
norm, then α(H, Ω, e) defined in (3) is a kind of matrix
measure of H [13]. So the characteristic value α(H, Ω, e) is a kind of nonlinear extension of matrix measures. A similar
term with l
1
norm has been introduced in [12] that is named as nonlinear measure. The meaning of introducing index
α(H, Ω, e) is obvious: in a nonlinear system, the exponential stability and its corresponding process of exponential
decay can be described by α(H, Ω, e) quantitatively as we do in the linear system.
Notice that α(H, Ω, e) depends on not only the function H, the domain Ω and the point e, but also the selection of the
norm ‖ · ‖. In fact, the freedom in choosing the norm makes α(H, Ω, e) valuable in applications. When ‖ · ‖ is l
2
-norm, we
have
α(H, Ω, e) = sup
v∈Ω /e
t∈[t
0
,∞)
lim
s→0
+
‖v − e + sH(t, v)‖ − ‖v − e‖
s‖v − e‖
= sup
v∈Ω /e
t∈[t
0
,∞)
lim
s→0
+
‖v − e + sH(t, v)‖
2
− ‖v − e‖
2
s‖v − e‖(‖v − e + sH(t, v)‖ + ‖v − e‖)
= sup
v∈Ω /e
t∈[t
0
,∞)
lim
s→0
+
‖v − e‖
2
+ 2⟨v − e, sH(t, v)⟩ + s
2
‖H(t, v)‖
2
− ‖v − e‖
2
s‖v − e‖(‖v − e + sH(t, v)‖ + ‖v − e‖)
= sup
v∈Ω /e
t∈[t
0
,∞)
lim
s→0
+
2⟨v − e, sH(t, v)⟩ + s
2
‖H(t, v)‖
2
s‖v − e‖(‖v − e + sH(t, v)‖ + ‖v − e‖)
= sup
v∈Ω /e
t∈[t
0
,∞)
⟨v − e, H(t, v)⟩
‖v − e‖
2
. (7)
In the following proposition, we give a result about the boundedness of the solution v(t) to some nonlinear systems.
Without losing generality, here we just consider the situation that e = 0 for convenience.
Proposition 2. Consider the following system:
˙v(t) = H(t, v(t)) + B(t, v(t)), v(t) ∈ R
k
, t ≥ t
0
. (8)
Suppose that H is absolutely continuous and Ω = {v ∈ R
k
: ‖v‖ < r, r > 0}. B(t, v(t)) is bounded in R
+
× Ω such that
b = sup
(t,v)∈R
+
×Ω
‖B(t, v(t))‖,
where ‖ · ‖ is a vector norm. If α(H, Ω, 0) < 0 and −α(H, Ω, 0)
−1
b < r, then the solution v(t) to (8) exponentially converges
to
Γ = {v ∈ R
k
: ‖v‖ ≤ −α(H, Ω, 0)
−1
b}
if v(t
0
) ∈ Ω.
Proof. Suppose that v(t) is a solution to system (8). It is easy to see that ‖v(t)‖ is absolutely continuous in [t
0
, +∞), so that
it is differentiable in [t
0
, +∞) a.e. Similar to the proof of inequality (5), we have
d‖v(t)‖
dt
≤ lim
s→0
+
‖v(t) + s[H(t, v(t)) + B(t, v)]‖ − ‖v(t)‖
s
≤ lim
s→0
+
‖v(t) + sH(t, v(t))‖ − ‖v(t)‖
s
+ ‖B(t, v)‖
≤ α(H, Ω, 0)‖v(t)‖ + b,
which holds at every differentiable point t. By simple integration we can derive that for all t ≥ t
0
,
‖v(t)‖ ≤ e
α(H,Ω ,0)(t−t
0
)
‖v(t
0
)‖ − bα(H, Ω, 0)
−1
1 − e
α(H,Ω ,0)(t−t
0
)
.
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