Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 245, pp. 1–9.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
ASYMPTOTIC STABILITY OF SOLUTIONS TO ELASTIC
SYSTEMS WITH STRUCTURAL DAMPING
HONGXIA FAN, FEI GAO
Abstract. In this article, we study the asymptotic stability of solutions for
the initial value problems of second order evolution equations in Banach spaces,
which can model elastic systems with structural damping. The discussion is
based on exponentially stable semigroups theory. Applications to the vibration
equation of elastic beams with structural damping are also considered.
1. Introduction
The study of elastic systems with damping seems to have been initiated by Chen
and Russell [1] in 1981. They considered the linear elastic systems with structural
damping,
ü(t) + B u̇(t) + Au(t) = 0,
(1.1)
u(0) = x0 , u̇(0) = y0
in a Hilbert space H with inner product (·, ·), where A (the elastic operator) and
B (the damping operator) are unbounded positive definite self-adjoint operators in
H. Let x1 = A1/2 u, x2 = u̇, we get the equivalent first-order linear systems
d x1
0
A1/2
x1
x1
=
=
L
,
B
1/2
x
x
x2
−A
−B
dt
2
2
x1 (0) = A1/2 x0 ,
x2 (0) = y0 .
Chen and Russell [1] proved that
LB =
0
−A1/2
A1/2
−B
generates an analytic semigroup on W = H ⊕ H, if some additional conditions are
satisfied. In the same paper, they pose the following conjecture proved by Huang
[9, 10]: Let D(B) ⊃ D(A1/2 ); then either of the following conditions (1) and (2)
implies that LB generates an analytic semigroup on W:
(1) ρ1 (A1/2 x, x) ≤ (Bx, x) ≤ ρ2 (A1/2 x, x) for all x ∈ D(A1/2 ) or (not, in
general, equivalent)
(2) ρ1 (Ax, x) ≤ (B 2 x, x) ≤ ρ2 (Ax, x) for all x ∈ D(A)
2000 Mathematics Subject Classification. 35B40, 35G25, 47D03.
Key words and phrases. Asymptotic stability; elastic systems; structural damping;
exponential stability; sectorial operator.
c
2014
Texas State University - San Marcos.
Submitted June 25, 2014. Published November 20, 2014.
1
2
H. FAN, F. GAO
EJDE-2014/245
for some ρ1 , ρ2 > 0 with ρ1 ≤ ρ2 . In addition, the semigroup generated by LB is
exponentially stable.
But these results do not contain the case B = ρA, which could possibly appear
in engineering applications. For this situation, Massatt [16] shows that if B = ρA
with ρ > 0, then
0
1
Aρ =
−A −ρA
generates an analytic semigroup which is exponentially stable.
Huang [11] investigated the more widely used linear elastic systems (1.1) with
damping B related in various ways to Aα ( 12 ≤ α ≤ 1), so that the C0 -semigroups associated with them are analytic and exponentially stable. Meanwhile, the spectral
property and some fundamental results for the analytic property and the exponential stability of the semigroups associated with the systems were discussed. Then
other sufficient conditions for LB generates an analytic semigroup were discussed
in [5, 6, 8, 9, 10, 11, 12] and the references therein.
Recently, the present authors [5] studied the linear second-order evolution equation
ü(t) + ρA u̇(t) + A 2 u(t) = 0, t > 0,
(1.2)
u(0) = x0 , u̇(0) = y0 ,
in a frame of Banach spaces, which can model the elastic systems with structural
damping. New forms of the corresponding first-order evolution equations were
introduced and sufficient conditions for analyticity and exponential stability of the
associated semigroups were given.
In [7] and [6], existence results of mild solutions for the elastic systems with structural damping were established by the fixed point theorems and monotone iterative
technique in the presence of lower and upper solutions, respectively. However, the
theory of the elastic systems with structural damping remains to be developed.
In this paper, we concentrate on the asymptotic behavior of solutions for the
linear elastic systems with structural damping
ü(t) + ρA u̇(t) + A 2 u(t) = h(t),
u(0) = x0 ,
t > 0,
u̇(0) = y0
(1.3)
and the semilinear elastic systems with structural damping
ü(t) + ρA u̇(t) + A 2 u(t) = f (t, u(t)),
u(0) = x0 ,
u̇(0) = y0 ,
t > 0,
(1.4)
in a Banach space X, where “·” means d/dt, ρ is the damping coefficient; A :
D(A ) ⊂ X → X is a sectorial operator and −A generates an analytic and exponentially stable semigroup S(t)(t ≥ 0) on X; f ∈ C(J × X, X), x0 ∈ D(A ),
y0 ∈ X.
2. Preliminaries
Definition 2.1 ([4]). A semigroup T (t)(t ≥ 0) on a Banach space X is called
exponentially stable if there exist constants δ > 0, M ≥ 1 such that
kT (t)k ≤ M e−δt ,
t ≥ 0.
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ASYMPTOTIC STABILITY OF SOLUTIONS
3
First we present a simple result on the asymptotic behavior of mild solutions for
the inhomogeneous initial value problem of the first-order linear evolution equation
u0 (t) = Au(t) + h(t),
t > 0,
(2.1)
u(0) = x.
Lemma 2.2 ([17, Page 119, Theorem 4.4]). Let µ > 0 and let A be the infinitesimal
generator of a C0 -semigroup T (t)(t ≥ 0) satisfying kT (t)k ≤ M e−µt . Let h be
bounded and measurable on [0, +∞). If
lim h(t) = b,
t→+∞
then, u(t), the mild solution of (2.1) satisfies
lim u(t) = −A−1 b.
t→+∞
Next we recall some basic facts and conclusions on the elastic systems (1.3) and
(1.4), which can be found in [5, 7] in order to prove our main results.
Since A is a sectorial operator on X. It follows from the definition that there
exist α ∈ (0, π2 ) and K > 0 satisfying
π
Σα := {λ|| arg λ| < + α} ⊂ ρ(−A ),
(2.2)
2
K
k(λI + A )−1 k ≤
, λ ∈ Σα .
(2.3)
1 + |λ|
For the second-order equation
ü(t) + ρA u̇(t) + A 2 u(t) = h(t),
t > 0,
it has the decomposition
∂
∂
+ σ1 A
+ σ2 A u = h(t),
∂t
∂t
t > 0.
Let
∂u
+ σ2 A u = v(t),
∂t
which means v(0) = y0 + σ2 A x0 := v0 . Then the elastic systems (1.3) can be
transformed into the following two abstract Cauchy problems in X:
∂v
+ σ1 A v = h(t),
∂t
v(0) = v0
t > 0,
(2.4)
and
∂u
+ σ2 A u = v(t),
∂t
u(0) = x0 ,
t > 0,
(2.5)
where
σ1 + σ2 = ρ,
σ1 σ2 = 1.
(2.6)
Lemma 2.3 ([5]). Let A : D(A ) ⊂ X → X be a sectorial operator, if the damping
coefficient ρ > 2 cos α, then −σ1 A , −σ2 A generate analytic and exponentially
stable semigroups on X, where α is defined in (2.2) and σ1 , σ2 are specified in
(2.6).
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H. FAN, F. GAO
EJDE-2014/245
For the convenience of the reader, throughout this paper we assume that −σ1 A
and −σ2 A generate analytic and exponentially stable semigroups S1 (t)(t ≥ 0) and
S2 (t)(t ≥ 0) on X, respectively. By Definition 2.1, there exist constants δ1 > 0, δ2 >
0 and M1 ≥ 1, M2 ≥ 1 such that
kS1 (t)k ≤ M1 e−δ1 t ,
kS2 (t)k ≤ M2 e−δ2 t ,
t ≥ 0.
(2.7)
Definition 2.4 ([7]). Let A : D(A ) ⊂ X → X be a sectorial operator, ρ > 2 cos α,
and f : J × X → X be a continuous function, x0 ∈ D(A ), y0 ∈ X. A continuous
solution of the integral equation
Z t
u(t) = S2 (t)x0 +
S2 (t − s)S1 (s)v0 ds
0
Z tZ s
+
S2 (t − s)S1 (s − τ )f (τ, u(τ ))dτ ds
0
0
is said to be a mild solution of the initial-value problem (1.4), where α is defined
in (2.2).
3. Main results
In this section it is our aim to introduce the asymptotic behavior of solutions for
the elastic systems (1.3) and (1.4), which can be given by the following theorems.
Theorem 3.1. Let A : D(A ) ⊂ X → X be a sectorial operator, the damping
coefficient ρ > 2 cos α, where α is defined in (2.2), x0 ∈ D(A ), y0 ∈ X, h :
[0, +∞) → X is continuous. If
lim h(t) = b,
t→+∞
then, the mild solution u(t) of the initial value problem (1.3) satisfies
lim u(t) = A −2 b.
t→+∞
Proof. Since S1 (t) (t ≥ 0) is exponentially stable on X. By Definition 2.1 and
Lemma 2.2, the mild solution v(t) of the initial-value problem (2.4) satisfies
lim v(t) = (σ1 A )−1 b.
t→+∞
(3.1)
Similarly, since S2 (t)(t ≥ 0) is also exponentially stable on X. By Definition 2.1,
Lemma 2.2 and (3.1), the mild solution u(t) of the initial value problem (2.5)
satisfies
lim u(t) = (σ2 A )−1 lim v(t)
t→+∞
t→+∞
= (σ2 A ) (σ1 A )−1 b
1
A −2 b.
=
σ1 σ2
Combining this fact with (2.6), it follows that limt→+∞ u(t) = A −2 b.
−1
(3.2)
We now show that if the semigroups S1 (t)(t ≥ 0) and S2 (t)(t ≥ 0) are exponentially stable on X, then, we can choose the constants δ1 , δ2 in (2.7) satisfying
0 < δ1 < δ2 . If, on the contrary, let δ1 = δ2 := δ and let δ = δ 0 + δ 00 , where δ 0 > 0,
δ 00 > 0, then for all t ≥ 0, we have
kS2 (t)k ≤ M2 e−δt ,
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ASYMPTOTIC STABILITY OF SOLUTIONS
0
kS1 (t)k ≤ M1 e−δt = M1 e−(δ +δ
00
)t
0
= M1 e−δ t e−δ
00
t
5
0
≤ M1 e−δ t .
It is evident that δ > δ 0 > 0. Hence, in what follows, we always assume that the
constants δ1 and δ2 in (2.7) satisfying 0 < δ1 < δ2 .
Next we establish the globally asymptotic stability result of the zero solution for
the initial value problem (1.4).
Theorem 3.2. Let A : D(A ) ⊂ X → X be a sectorial operator, the damping
coefficient ρ > 2 cos α, where α is defined in (2.2), x0 ∈ D(A ), y0 ∈ X, f :
[0, +∞) × X → X is continuous and satisfies the following conditions:
(H1) There exists L > 0, such that
kf (t, u2 ) − f (t, u1 )k ≤ Lku2 − u1 k,
t ∈ [0, +∞), u1 , u2 ∈ X.
(H2) f (t, θ) = θ (θ is the zero element of X) for t ≥ 0.
(δ2 −δ1 )
(H3) 0 < L < δ1M
.
1 M2
Then the mild solution u(t) of the initial value problem (1.4) satisfies
lim u(t) = θ.
t→+∞
Proof. By assumption (H1) and [7, Theorem 4], the initial value problem (1.4) has
a unique global mild solution u(t), then by the semigroup representation of the mild
solution, u(t) satisfies the integral equation
Z t
u(t) = S2 (t)x0 +
S2 (t − s)S1 (s)v0 ds
0
Z tZ s
+
S2 (t − s)S1 (s − τ )f (τ, u(τ ))dτ ds,
0
(3.3)
t ≥ 0.
0
Using this, we conclude that
Z
t
ku(t)k ≤ kS2 (t)x0 k +
S2 (t − s)S1 (s)v0 ds
0
Z Z
t s
+
S2 (t − s)S1 (s − τ )f (τ, u(τ ))dτ ds,
0
(3.4)
t ≥ 0.
0
From the inequality (2.7), it follows that
kS2 (t)x0 k ≤ kS2 (t)kkx0 k ≤ M2 e−δ2 t kx0 k ≤ M2 e−δ1 t kx0 k
(3.5)
and
Z
0
t
S2 (t − s)S1 (s)v0 ds ≤
Z
t
kS2 (t − s)kkS1 (s)kkv0 k ds
Z t
≤ M1 M2 e−δ2 t kv0 k
e(δ2 −δ1 )s ds
0
0
≤
M1 M2 kv0 k −δ1 t
e
.
δ2 − δ1
(3.6)
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H. FAN, F. GAO
EJDE-2014/245
By assumption (H2) and (2.7), we have
Z Z
t s
S2 (t − s)S1 (s − τ )f (τ, u(τ ))dτ ds
0
0
Z tZ
s
kS2 (t − s)kkS1 (s − τ )kkf (τ, u(τ ))kdτ ds
≤
0
0
Z tZ
s
kS2 (t − s)kkS1 (s − τ )kkf (τ, u(τ )) − f (τ, θ)kdτ ds
≤
(3.7)
0
tZ s
0
Z
kS2 (t − s)kkS1 (s − τ )kkf (τ, θ)kdτ ds
Z t
Z s
≤ LM1 M2 e−δ2 t
e(δ2 −δ1 )s
eδ1 τ ku(τ )kdτ ds.
+
0
0
0
0
From integration by parts, we get
Z t
Z s
e(δ2 −δ1 )s
eδ1 τ ku(τ )kdτ ds
0
0
Z t
Z s
1
=
eδ1 τ ku(τ )kdτ
d[e(δ2 −δ1 )s ]
δ2 − δ1 0
0
(3.8)
Z
Z t
i
1 h (δ2 −δ1 )t t δ1 s
δ2 s
e
=
e ku(s)k ds −
e ku(s)k ds
δ2 − δ1
0
0
Z t
1
≤
e(δ2 −δ1 )t
eδ1 s ku(s)k ds,
δ2 − δ1
0
and therefore
Z
Z Z
t s
LM1 M2 −δ t t δ s
e 1
e 1 ku(s)k ds. (3.9)
S2 (t−s)S1 (s−τ )f (τ, u(τ ))dτ ds ≤
δ
−
δ
2
1
0
0
0
Together with (3.4), (3.5), (3.6) and (3.9) this gives
ku(t)k ≤ M2 e
−δ1 t
M1 M2 kv0 k −δ1 t LM1 M2 −δ1 t
e
+
e
kx0 k +
δ2 − δ1
δ2 − δ1
Z
t
eδ1 s ku(s)k ds,
0
t ≥ 0. Hence
Z
M1 M2 kv0 k LM1 M2 t δ1 s
+
e ku(s)k ds, t ≥ 0.
δ2 − δ1
δ2 − δ1 0
According to Growall inequality, we obtain that
1 M2
M1 M2 kv0 k LM
eδ1 t ku(t)k ≤ M2 kx0 k +
e δ2 −δ1 t , t ≥ 0.
δ2 − δ1
Which means
“
”
1 M2 −δ
M1 M2 kv0 k LM
t
ku(t)k ≤ M2 kx0 k +
e δ2 −δ1 1 , t ≥ 0.
δ2 − δ1
By the assumption (H3), we have
LM1 M2
− δ1 < 0.
(3.10)
δ2 − δ1
This implies u(t) → θ as t → +∞. Consequently, the zero solution is globally
asymptotically stable and it exponentially attracts every mild solution of the initialvalue problem (1.4).
eδ1 t ku(t)k ≤ M2 kx0 k +
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ASYMPTOTIC STABILITY OF SOLUTIONS
7
4. Applications
In this section, we will apply the abstract results in Section 3 to the vibration
equation of elastic beams with structural damping, to obtain the results of asymptotic stability of mild solutions.
The vibration state of an elastic beam with structural damping, whose two
ends are simply supported, can be described by the initial-boundary value problem
(IBVP)
utt − 4uxxt + uxxxx = h(x, t), x ∈ (0, 1), t > 0,
u(0, t) = u(1, t) = 0,
t > 0,
uxx (0, t) = uxx (1, t) = 0,
u(x, 0) = ϕ(x),
t > 0,
(4.1)
x ∈ (0, 1),
ut (x, 0) = ψ(x),
where uxxxx denotes the elastic effect, uxxt is the damping term, ρ = 4 is the
damping coefficient and the non-homogeneous term h(x, t) be defined by
( 2x2 t2
1+3x2 t2 , x ∈ (0, 1), t ≥ 0,
(4.2)
h(x, t) =
2/3,
x = 0, t ≥ 0.
Let I = [0, 1] and choose X = Lp (I)(2 ≤ p < +∞). Define a linear operator
A : D(A ) ⊂ X → X by
D(A ) = W 2,p (I) ∩ W01,p (I),
A u = −∆u,
(4.3)
where ∆ is the Laplace operator acting on functions on the interval I. Choosing
P
α = arccos 2/5 ∈ (0, π/2), by [5], A is a sectorial operator for the region
α
defined by (2.2).
Let h(t) = h(·, t), then the problem (4.1) can be rewritten into the abstract form
ü(t) + 4A u̇(t) + A 2 u(t) = h(t),
u(0) = ϕ,
t > 0,
u̇(0) = ψ.
(4.4)
Theorem 4.1. Let 2 ≤ p < +∞, for every ϕ ∈ W 2,p (I) ∩ W01,p and ψ ∈ Lp (I),
the mild solution u(t) of the equation (4.1) satisfying limt→+∞ u(t) = ∆−2 32 .
Proof. By setting ρ = 4 and α = arccos 2/5, it is easy to verify that the damping
coefficient ρ satisfies ρ > 2 cos α. From (4.2), it follows that h(t) is continuous on
[0, +∞) and limt→+∞ h(t) = 23 . Hence by Theorem 3.1, the mild solution u(t) of
the equation (4.1) satisfying limt→+∞ u(t) = ∆−2 23 .
In what follows, we consider the nonlinear vibration equation of elastic beams
with structural damping, namely the following initial-boundary value problem
1
sin u(x, t), x ∈ (0, 1), t > 0,
2
u(0, t) = u(1, t) = 0, t > 0,
utt − 4uxxt + uxxxx =
uxx (0, t) = uxx (1, t) = 0,
u(x, 0) = ϕ(x),
ut (x, 0) = ψ(x),
(4.5)
t > 0,
x ∈ (0, 1),
Let u(t) = u(·, t), f (t, u(t)) = 21 sin u(·, t). Then the initial-boundary value problem
(4.5) can be rewritten to the Cauchy problem of the second order evolution equation
8
H. FAN, F. GAO
EJDE-2014/245
in the Banach space X
ü(t) + 4A u̇(t) + A 2 u(t) = f (t, u(t)),
u(0) = ϕ,
t > 0,
u̇(0) = ψ,
(4.6)
where A is defined in (4.3) and A is a sectorial operator for the region Σα (α =
arccos 2/5) defined by (2.2). We assume that ϕ ∈ D(A ) and ψ ∈ X, Then the
equation (4.6) has the following decomposition form
∂
∂
+ σ1 A )( + σ2 A )u = f (t, u(t)), t > 0,
∂t
∂t
(4.7)
u(0) = ϕ, u̇(0) = ψ,
√
√
where σ1 = 2 − 3, σ2 = 2 + 3 are defined by (2.6).
It is well-known [8, 17], −A generates an analytic and exponentially stable
semigroup S(t)(t ≥ 0) satisfying
(
kS(t)k ≤ e−t ,
t ≥ 0.
By Lemma 2.3 and the characterization of the infinitesimal generators of C0 semigroups, −σ1 A and −σ2 A generate analytic and exponentially stable semigroups S1 (t)(t ≥ 0) and S2 (t)(t ≥ 0) respectively, which satisfy
kSi (t)k = kS(σi t)k ≤ e−σi t , t ≥ 0, i = 1, 2.
√
√
Now take M1 = M2 = 1, δ1 = σ1 = 2 − 3 and δ2 = σ2 = 2 + 3, we obtain that
√
1
δ1 (δ2 − δ1 )
<
= 4 3 − 6.
(4.8)
2
M1 M2
Theorem 4.2. Let 2 ≤ p < +∞, for every ϕ ∈ W 2,p (I) ∩ W01,p and ψ ∈ Lp (I),
the mild solution u(t) of the equation (4.5) satisfying ku(t)kp → 0 as t → ∞.
Proof. By ρ = 4 and α = arccos 2/5, we can easily obtain that the damping coefficient ρ satisfies ρ > 2 cos α. Since f (x, t, u(x, t)) = 21 sin u(x, t) is continuous on
[0, 1] × [0, +∞) × X and satisfying
1
1
| cos u(x, t)| ≤ ,
2
2
f (x, t, 0) = sin 0 = 0,
|fu0 (x, t, u)| =
(x, t, u) ∈ [0, 1] × [0, +∞) × X;
(x, t) ∈ [0, 1] × [0, +∞).
(4.9)
(4.10)
From (4.9), for u1 , u2 ∈ X, we have
|f (x, t, u2 ) − f (x, t, u1 )| ≤
1
|u2 − u1 |,
2
(x, t) ∈ [0, 1] × [0, +∞).
(4.11)
Which implies
1
ku2 − u1 kp , t ∈ [0, +∞), u1 , u2 ∈ X.
(4.12)
2
Then assumptions (H1) and (H2) hold. According to (4.8) and (4.12), we obtain
that (H3) is satisfied. Hence by Theorem 3.2, we conclude that the mild solution
u(t) of (4.5) satisfying limt→+∞ u(t) = 0, which implies ku(t)kp → 0 as t →
+∞.
kf (t, u2 ) − f (t, u1 )kp ≤
Acknowledgments. This work was supported by the National Natural Science
Foundation of China (No. 11361032) and the Youth Science Foundation of Lanzhou
Jiaotong University (No. 2013025).
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ASYMPTOTIC STABILITY OF SOLUTIONS
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Hongxia Fan
Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, 730070, China
E-mail address: lzfanhongxia@163.com
Fei Gao
Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, 730070, China
E-mail address: 11l111@sina.cn