Introduction
Generalized Results
Further Improved Results
LaSalle-Type Theorems for Stochastic
Differential Delay Equations
Xuerong Mao
Department of Statistics and Modelling Science
University of Strathclyde
Glasgow, G1 1XH
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Outline
1
Introduction
2
Generalized Results
Notation
Useful Corollaries
Linear SDDEs
3
Further Improved Results
Motivated example
Improved theorem
Useful Corollaries
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Outline
1
Introduction
2
Generalized Results
Notation
Useful Corollaries
Linear SDDEs
3
Further Improved Results
Motivated example
Improved theorem
Useful Corollaries
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Outline
1
Introduction
2
Generalized Results
Notation
Useful Corollaries
Linear SDDEs
3
Further Improved Results
Motivated example
Improved theorem
Useful Corollaries
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
SDDEs
Consider the n-dimensional SDDE
dx(t) = f (x(t), x(t − τ ), t)dt + g(x(t), x(t − τ ), t)dB(t)
(1.1)
on t ≥ 0 with initial data
{x(s) : −τ ≤ s ≤ 0} = ξ ∈ CFb 0 ([−τ, 0]; R n ). Here
B(t) = (B1 (t), · · · , Bm (t))T is an m-dimensional Brownian
motion,
f : Rn × Rn × R+ → Rn
and g : Rn × Rn × R+ → R n×m .
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Hypothesis (H1)
Both f and g satisfy the local Lipschitz condition and the linear
growth condition. That is, for each k = 1, 2, · · · , there is a
ck > 0 such that
|f (x, y , t)−f (x̄, ȳ , t)|∨|g(x, y , t)−g(x̄, ȳ , t)| ≤ ck (|x − x̄|+|y − ȳ |)
for all t ≥ 0 and those x, y , x̄, ȳ ∈ Rn with
|x| ∨ |y | ∨ |x̄| ∨ |ȳ | ≤ k , and there is moreover a c > 0 such that
|f (x, y , t)| ∨ |g(x, y , t)| ≤ c(1 + |x| + |y |)
for all (x, y , t) ∈ Rn × Rn × R+ .
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Existence and Uniqueness of a Solution
Under hypothesis (H1), it is well-known that for any initial data
{x(s) : −τ ≤ s ≤ 0} = ξ ∈ CFb 0 ([−τ, 0]; Rn ), equation (1.1) has
a unique solution that is denoted by x(t; ξ) on t ≥ −τ .
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Diffusion operator
Let C 2,1 (Rn × R+ ; R+ ) denote the family of all nonnegative
functions V (x, t) on Rn × R+ which are continuously twice
differentiable in x and once differentiable in t. For
V ∈ C 2,1 (Rn × R+ ; R+ ), define LV : Rn × Rn × R+ → R by
LV (x, y , t) := Vt (x, t) + Vx (x, t)f (x, y , t)
1
+
trace g T (x, y , t)Vxx (x, t)g(x, y , t) .
2
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Theorem
(Mao (1999) Let (H1) hold. Assume that there are functions
V ∈ C 2,1 (Rn × R+ ; R+ ), γ ∈ L1 (R+ ; R+ ) and
w1 , w2 ∈ C(Rn ; R+ ) such that
LV (x, y , t) ≤ γ(t) − w1 (x) + w2 (y )
(1.2)
for (x, y , t) ∈ Rn × Rn × R+ ,
x ∈ Rn ,
(1.3)
inf V (x, t) = ∞.
(1.4)
w1 (x) ≥ w2 (x),
and
lim
|x|→∞ 0≤t<∞
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Theorem
Assume also that for each initial data ξ ∈ CFb 0 ([−τ, 0]; Rn ) there
is a p > 2 such that
sup
E|x(t; ξ)|p < ∞.
(1.5)
−τ ≤t<∞
Then, for every ξ ∈ CFb 0 ([−τ, 0]; Rn )
lim w1 (x(t; ξ)) − w2 (x(t; ξ)) = 0
t→∞
a.s.
(1.6)
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Questions:
(1) The condition of the bounded pth moment is restrictive. Can
we remove it?
(2) Can hypothesis (H1) also be replaced by a weaker one in
order to cover much more general SDDEs?
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Notation
Useful Corollaries
Linear SDDEs
Outline
1
Introduction
2
Generalized Results
Notation
Useful Corollaries
Linear SDDEs
3
Further Improved Results
Motivated example
Improved theorem
Useful Corollaries
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Notation
Useful Corollaries
Linear SDDEs
CFb 0 ([−τ, 0]; Rn ) denotes the family of all F0 -measurable
bounded C([−τ, 0]; Rn )-valued random variables
ξ = {ξ(s) : −τ ≤ s ≤ 0}.
If K is a subset of Rn , denote by d(x, K ) the Haussdorf
semi-distance between x ∈ Rn and the set K , that is,
d(x, K ) = infy ∈K |x − y |.
If w is a real-valued function defined on Rn , then its kernel is
denoted by Ker (w), namely Ker (w) = {x ∈ Rn : w(x) = 0}.
Denote byRL1 (R+ ; R+ ) the family of all functions γ : R+ → R+
∞
such that 0 γ(t)dt < ∞.
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Notation
Useful Corollaries
Linear SDDEs
Hypothesis (H2)
Given any initial data
{x(s) : −τ ≤ s ≤ 0} = ξ ∈ CFb 0 ([−τ, 0]; Rn ) equation (1.1) has a
unique solution denoted by x(t; ξ) on t ≥ 0. Moreover, both
f (x, y , t) and g(x, y , t) are locally bounded in (x, y ) while
uniformly in t. That is, for any h > 0 there is a Kh > 0 such that
|f (x, y , t)| ∨ |g(x, y , t)| ≤ Kh
for all t ≥ 0 and x, y ∈ Rn with |x| ∨ |y | ≤ h.
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Notation
Useful Corollaries
Linear SDDEs
Hypothesis (H2) covers much more stochastic differential delay
equations than (H1). For example, consider a 1-dimensional
stochastic differential delay equation
dx(t) = [−x 3 (t) + x(t − τ )]dt + sin(x(t − τ ))dB(t),
where B(t) is a 1-dimensional Brownian motion. This equation
does not satisfy hypothesis (H1) but it satisfies hypothesis (H2)
by the generalized Hasminskii-type theorems.
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Notation
Useful Corollaries
Linear SDDEs
Theorem
Let (H2) hold. Assume that there are functions
V ∈ C 2,1 (Rn × R+ ; R+ ), γ ∈ L1 (R+ ; R+ ) and
w1 , w2 ∈ C(Rn ; R+ ) such that
LV (x, y , t) ≤ γ(t) − w1 (x) + w2 (y ),
w1 (x) ≥ w2 (x),
(x, y , t) ∈ Rn × Rn × R+ ,
(2.1)
x ∈ Rn ,
(2.2)
and
lim
inf V (x, t) = ∞.
(2.3)
|x|→∞ 0≤t<∞
Then Ker (w1 − w2 ) 6= ∅ and for every ξ ∈ CFb 0 ([−τ, 0]; Rn ),
lim d(x(t; ξ), Ker (w1 − w2 )) = 0
t→∞
Xuerong Mao
LaSalle-Type Theorems
a.s.
(2.4)
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Introduction
Generalized Results
Further Improved Results
Notation
Useful Corollaries
Linear SDDEs
Key steps of the proof
lim supt→∞ V (x(t), t) < ∞ a.s.
R∞
0 [w1 (x(t)) − w2 (x(t))]dt < ∞ a.s.
limt→∞ [w1 (x(t)) − w2 (x(t))] = 0 a.s
Ker (w1 − w2 ) 6= ∅.
limt→∞ d(x(t), Ker (w1 − w2 )) = 0 a.s.
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Notation
Useful Corollaries
Linear SDDEs
Key steps of the proof
lim supt→∞ V (x(t), t) < ∞ a.s.
R∞
0 [w1 (x(t)) − w2 (x(t))]dt < ∞ a.s.
limt→∞ [w1 (x(t)) − w2 (x(t))] = 0 a.s
Ker (w1 − w2 ) 6= ∅.
limt→∞ d(x(t), Ker (w1 − w2 )) = 0 a.s.
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Notation
Useful Corollaries
Linear SDDEs
Key steps of the proof
lim supt→∞ V (x(t), t) < ∞ a.s.
R∞
0 [w1 (x(t)) − w2 (x(t))]dt < ∞ a.s.
limt→∞ [w1 (x(t)) − w2 (x(t))] = 0 a.s
Ker (w1 − w2 ) 6= ∅.
limt→∞ d(x(t), Ker (w1 − w2 )) = 0 a.s.
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Notation
Useful Corollaries
Linear SDDEs
Key steps of the proof
lim supt→∞ V (x(t), t) < ∞ a.s.
R∞
0 [w1 (x(t)) − w2 (x(t))]dt < ∞ a.s.
limt→∞ [w1 (x(t)) − w2 (x(t))] = 0 a.s
Ker (w1 − w2 ) 6= ∅.
limt→∞ d(x(t), Ker (w1 − w2 )) = 0 a.s.
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Notation
Useful Corollaries
Linear SDDEs
Key steps of the proof
lim supt→∞ V (x(t), t) < ∞ a.s.
R∞
0 [w1 (x(t)) − w2 (x(t))]dt < ∞ a.s.
limt→∞ [w1 (x(t)) − w2 (x(t))] = 0 a.s
Ker (w1 − w2 ) 6= ∅.
limt→∞ d(x(t), Ker (w1 − w2 )) = 0 a.s.
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Notation
Useful Corollaries
Linear SDDEs
Outline
1
Introduction
2
Generalized Results
Notation
Useful Corollaries
Linear SDDEs
3
Further Improved Results
Motivated example
Improved theorem
Useful Corollaries
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Notation
Useful Corollaries
Linear SDDEs
Corollary
Let (H2) hold. Assume that there are functions
V ∈ C 2,1 (Rn × R+ ; R+ ), γ ∈ L1 (R+ ; R+ ) and
w1 , w2 ∈ C(Rn ; R+ ) such that
LV (x, y , t) ≤ γ(t) − w1 (x) + w2 (y ),
w1 (x) > w2 (x),
(x, y , t) ∈ Rn × Rn × R+ ,
∀x 6= 0,
(2.5)
and
lim
inf V (x, t) = ∞.
|x|→∞ 0≤t<∞
Then
lim x(t; ξ) = 0 a.s.
t→∞
for every ξ ∈ CFb 0 ([−τ, 0]; Rn ).
Xuerong Mao
(2.6)
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LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Notation
Useful Corollaries
Linear SDDEs
Theorem 3 can also be used to establish the criteria on the
partially asymptotic stability. Let 1 ≤ n̂ < n and
1 ≤ i1 < i2 < · · · < in̂ ≤ n be integers. Let x̂ = (xi1 , xi2 , · · · , xin̂ )
be the partial coordinates
of x, which can be regarded as in Rn̂
q
with the norm |x̂| = xi21 + xi22 + · · · + xi2 . Moreover, let K
n̂
denote the class of continuous (strictly) increasing functions µ
from R+ to itself with µ(0) = 0.
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Notation
Useful Corollaries
Linear SDDEs
Corollary
Let (H2) hold. Assume that there are functions
V ∈ C 2,1 (Rn × R+ ; R+ ), γ ∈ L1 (R+ ; R+ ), w1 , w2 ∈ C(Rn ; R+ )
and µ ∈ K such that
LV (x, y , t) ≤ γ(t) − w1 (x) + w2 (y ),
(x, y , t) ∈ Rn × Rn × R+ ,
w1 (x) − w2 (x) ≥ µ(|x̂|),
x ∈ Rn
(2.7)
and
lim
inf V (x, t) = ∞.
|x|→∞ 0≤t<∞
Then
lim x̂(t; ξ) = 0 a.s.
t→∞
for every ξ ∈ CFb 0 ([−τ, 0]; Rn ).
Xuerong Mao
(2.8)
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LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Notation
Useful Corollaries
Linear SDDEs
Corollary
Let all the assumptions of Theorem 3 hold. If Ker (w1 − w2 ) is
bounded, then for every ξ ∈ CFb 0 ([−τ, 0]; Rn ),
lim |x(t; ξ)| ≤ C
t→∞
a.s.
(2.9)
where C = sup{|x| : x ∈ Ker (w1 − w2 )}.
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Notation
Useful Corollaries
Linear SDDEs
Outline
1
Introduction
2
Generalized Results
Notation
Useful Corollaries
Linear SDDEs
3
Further Improved Results
Motivated example
Improved theorem
Useful Corollaries
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Notation
Useful Corollaries
Linear SDDEs
Consider the linear SDDE
dx(t) = [A0 x(t) + D0 x(t − τ )]dt +
m
X
[Ai x(t) + Di x(t − τ )]dBi (t),
i=1
(2.10)
where Ai ’s and Di ’s are all n × n matrices.
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Notation
Useful Corollaries
Linear SDDEs
Corollary
If there is a symmetric positive-definite n × n matrix Q such that
M := −QA0 − AT0 Q − Q − 2
m
X
ATi QAi − D0T QD0 − 2
i=1
m
X
DiT QDi
i=1
(2.11)
is positive-definite, then the solutions of the linear SDDE will
tend to zero asymptotically with probability one. However, if the
matrix M is only nonnegative-definite, then the solutions will
approach the linear span of v1 , · · · , vk asymptotically with
probability one, where v1 , · · · , vk are the all orthogonal
eigen-vectors of M corresponding to the eigen-value 0.
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Motivated example
Improved theorem
Useful Corollaries
Outline
1
Introduction
2
Generalized Results
Notation
Useful Corollaries
Linear SDDEs
3
Further Improved Results
Motivated example
Improved theorem
Useful Corollaries
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Motivated example
Improved theorem
Useful Corollaries
Consider a simple 1-dimensional SDDE
dx(t) = [−4 cos2 (t)x 3 (t) + 2 cos(t) cos(t − τ )x(t)x 2 (t − τ )]dt
+ | cos(t − τ )|x 2 (t − τ )dB(t),
where B(t) is a scalar Brownian motion. Let V (x, t) = x 2 and
compute
LV (x, y , t) = −8 cos2 (t)x 4 + 4 cos(t) cos(t − τ )x 2 y 2
+ cos2 (t − τ )y 4
≤ −4 cos2 (t)x 4 + 2 cos2 (t − τ )y 4 .
Unfortunately, we can not apply the existing theorem.
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Motivated example
Improved theorem
Useful Corollaries
Denote by Ψ(R+ ; R+ ) the family of all continuous functions
ψ : R+ → R+ with the property that for any δ > 0 such that
Z
t+δ
ψ(s)ds > 0.
lim inf
t→∞
t
There are many functions belong to Ψ(R+ ; R+ ), for example,
| sin t|, cos2 t etc.
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Motivated example
Improved theorem
Useful Corollaries
Denote by D(R+ ; R+ ) the family of all continuous functions
η : R+ → R+ such that for all I ∈ R+ with mesI = ∞
Z
η(t)dt = ∞,
I
where mesI denotes the measure of set I.
Denote by Φ(R+ ; R+ ) the family of all continuous functions
φ : R+ → R+ such that for any δ > 0 and any increasing
sequence {tk }k ≥1 (tk → ∞ as k → ∞) such that
∞ Z
X
tk +δ
φ(s)ds = ∞.
k =1 tk
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Motivated example
Improved theorem
Useful Corollaries
D(R+ ; R+ ) ⊂ Ψ(R+ ; R+ ) and Ψ(R+ ; R+ ) = Φ(R+ ; R+ )
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Motivated example
Improved theorem
Useful Corollaries
Outline
1
Introduction
2
Generalized Results
Notation
Useful Corollaries
Linear SDDEs
3
Further Improved Results
Motivated example
Improved theorem
Useful Corollaries
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Motivated example
Improved theorem
Useful Corollaries
Theorem
Let the initial data ξ ∈ CFb 0 ([−τ, 0]; Rn ) and hypothesis (H2)
hold. Assume that there are functions V ∈ C 2,1 (Rn × R+ ; R+ ),
γ ∈ L1 (R+ ; R+ ), u1 ∈ C(Rn × R+ ; R+ ),
u2 ∈ C(Rn × [−τ, ∞); R+ ), a ∈ Ψ(R+ ; R+ ) and w ∈ C(Rn ; R+ )
such that
LV (x, y , t) ≤ γ(t)−u1 (x, t)+u2 (y , t−τ ),
u1 (x, t) − u2 (x, t) ≥ a(t)w(x),
lim
(x, y , t) ∈ Rn ×Rn ×R+ ,
x ∈ Rn , t ∈ R+ ,
inf V (x, t) = ∞.
|x|→∞ 0≤t<∞
Then Ker(w) 6= ∅ and
lim d(x(t, ξ), Ker(w)) = 0 a.s.
t→∞
Xuerong Mao
LaSalle-Type Theorems
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Introduction
Generalized Results
Further Improved Results
Motivated example
Improved theorem
Useful Corollaries
Let us return to the motivated example. Define
u1 (x, t) = 4 cos2 (t)x 4 ,
u2 (x, t) = 2 cos2 (t)x 4 .
Then
LV (x, y , t) ≤ −u1 (x, t) + u2 (y , t − τ ),
and
u1 (x, t) − u2 (x, t) = 2 cos2 (t)x 4 .
But 2 cos2 (·) ∈ Ψ(R+ ; R+ ) and Ker(x 4 ) = {0}. We can then
conclude that for any initial data ξ ∈ CFb 0 ([−τ, 0]; R), the solution
obeys
lim x(t, ξ) = 0 a.s.
t→∞
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Motivated example
Improved theorem
Useful Corollaries
Outline
1
Introduction
2
Generalized Results
Notation
Useful Corollaries
Linear SDDEs
3
Further Improved Results
Motivated example
Improved theorem
Useful Corollaries
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Xuerong Mao
LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Motivated example
Improved theorem
Useful Corollaries
Corollary
Let (H2) hold. Assume that there are functions
V ∈ C 2,1 (Rn × R+ ; R+ ), γ ∈ L1 (R+ ; R+ ), a1 ∈ Ψ(R+ ; R+ ),
a2 ∈ C([−τ, ∞), R+ ) and w1 , w2 ∈ C(Rn ; R+ ) such that
LV (x, y , t) ≤ γ(t) − a1 (t)w1 (x) + a2 (t − τ )w2 (y ),
a1 (t) ≥ a2 (t), w1 (x) ≥ w2 (x),
lim
inf V (x, t) = ∞.
|x|→∞ 0≤t<∞
Then Ker(w1 − w2 ) 6= ∅ and
lim d(x(t, ξ), Ker(w1 − w2 )) = 0 a.s.
t→∞
for every ξ ∈ CFb 0 ([−τ, 0]; R n ).
Xuerong Mao
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LaSalle-Type Theorems
Introduction
Generalized Results
Further Improved Results
Motivated example
Improved theorem
Useful Corollaries
Corollary
Let (H2) hold. Assume that there are functions
V ∈ C 2,1 (Rn × R+ ; R+ ), γ ∈ L1 (R+ ; R+ ), a1 ∈ C(R+ ; R+ ),
a2 ∈ C([−τ, ∞); R+ ), (a1 − a2 ) ∈ Ψ(R+ ; R+ ), and
w1 , w2 ∈ C(R n ; R+ ) such that
LV (x, y , t) ≤ γ(t) − a1 (t)w1 (x) + a2 (t − τ )w2 (y ),
w1 (x) ≥ w2 (x),
lim
inf V (x, t) = ∞.
|x|→∞ 0≤t<∞
Then Ker (w1 − w2 ) ∩ Ker (w2 ) 6= ∅ and for every
ξ ∈ CFb 0 ([−τ, 0]; R n ),
lim d(x(t, ξ), Ker (w1 − w2 ) ∩ Ker (w2 )) = 0 a.s.
t→∞
Xuerong Mao
LaSalle-Type Theorems
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Introduction
Generalized Results
Further Improved Results
Motivated example
Improved theorem
Useful Corollaries
Corollary
Let the initial data ξ ∈ CFb 0 ([−τ, 0]; Rn ) and hypothesis (H2)
hold. Assume that there are functions V ∈ C 2,1 (Rn × R+ ; R+ ),
γ, b ∈ L1 (R+ ; R+ ), u1 ∈ C(Rn × R+ ; R+ ),
u2 ∈ C(Rn × [−τ, ∞); R+ ), a ∈ Ψ(R+ ; R+ ) and w ∈ C(Rn ; R+ )
such that
LV (x, y , t) ≤ γ(t) − u1 (x, t) + u2 (y , t − τ ) + b(t)V (x, t),
u1 (x, t) − u2 (x, t) ≥ a(t)w(x),
lim
inf V (x, t) = ∞.
|x|→∞ 0≤t<∞
Then Ker(w) 6= ∅ and
lim d(x(t, ξ), Ker(w)) = 0 a.s.
t→∞
Xuerong Mao
LaSalle-Type Theorems
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