3. Proofs of the Main Results

THE ASYMPTOTICAL STABILITY OF A DYNAMIC SYSTEM WITH STRUCTURAL DAMPING

XUEZHANGHOU^∗

∗Mathematics Department, Towson University, Baltimore, Maryland 21252-0001, USA e-mail:xhou@towson.edu

A dynamic system with structural damping described by partial differential equations is investigated. The system is first converted to an abstract evolution equation in an appropriate Hilbert space, and the spectral and semigroup properties of the system operator are discussed. Finally, the well-posedness and the asymptotical stability of the system are obtained by means of a semigroup of linear operators.

Keywords: dynamic system, evolution equation, asymptotic stability

1. Introduction

We shall be concerned with the following system of par- tial differential equations with initial and boundary condi- tions:



































∂²u(x, t)

∂t² + η∂⁵u(x, t)

∂t∂x⁴ + ∂²

∂x²



p(x)∂²u(x, t)

∂x²



= f (t, u),

∂²u(x, t)

∂x²  

_x=0,l= ∂

∂x



p(x)∂²u(x, t)

∂x²

 

_x=0,l= 0,

u(x, 0) = ϕ0(x), ∂u(x, t)

∂t  

_t=0= ψ0(x).

(1)

So far, we have been concerned with clamped beam equa- tions in (Hou and Tsui, 1998; 1999; 2000; 2003), in which the systems are different from (1). The system (1) stands for a typical beam equation with two free ends (Komkov, 1978; Köhne, 1978; Li and Zhu, 1988), where u(x, t) is the transverse displacement of the point x at the time t, l is the length of the beam, p(x) is the bending rigidity at the point x, and f (t, u) represents the controlled moment of the system.

Suppose that p(x) ∈ C²[0, l], and 0 < p0≤ p(x) ≤ p₁ < +∞, where p0 and p₁ are constants. Now, we take L²[0, l] as the state space, with the inner product and norm respectively defined as follows:

hf, gi0= Z l

0

f (x)g(x) dx, f, g ∈ L²[0, l],

kf k²₀= Z l

0

|f (x)|²dx, f ∈ L²[0, l].

Let H1 = span {1, x}. Then L²[0, l] = H1 ⊕ H2, where H2 is the orthogonal complement of H1 in L²[0, l]. Suppose that P1 is the operator of projection onto H1 and I − P1 is the operator of projection onto H2, so that the system (1) can be rewritten as











∂²u(x, t)

∂t² = P₁f (t, u), u(x, 0) = P1ϕ0, ∂u(x, t)

∂t  

_t=0= P1ψ0.

(2)

It is clear that the solution of (2) can be described as u⁽¹⁾(x, t) = a₁+ a₂x + a₃t + a₄tx, (3) where a1, a2, a3 and a4 are determined by P1ϕ0, P1ψ0, and P1f (t, u).

For the system (1) in H2, we have



































∂²(x, t)

∂t² + η∂⁵u(x, t)

∂t∂x⁴ + ∂²

∂x²



p(x)∂²u(x, t

∂x²



= (I − P₁)f (t, u),

∂²u(x, t)

∂x²  

_x=0,l= ∂

∂x



p(x)∂²u(x, t)

∂x²

 

_x=0,l= 0, u(x, 0) = (I − P₁)ϕ₀, ∂u(x, t)

∂t  

_t=0= (I − P₁)ψ₀. (4) If we denote by u⁽²⁾(x, t) the solution of (4), then the solution of the system (1) can be represented as

u(x, t) = u⁽¹⁾(x, t) ⊕ u⁽²⁾(x, t). (5)

It should be noted that the form of u⁽¹⁾(x, t) is given by (3), and u⁽²⁾(x, t) will play a key role in investigat- ing the solution of the system (1).

We now define differential operators A and T as follows:

Aϕ = p(x)ϕ⁰⁰(x)
00

, ϕ ∈ D(A), D(A) =ϕ | ϕ ∈ H2, ϕ⁰⁰(0) = ϕ⁰⁰(l) = 0,

p(x)ϕ⁰⁰
0

_x=0,l= 0, p(x)ϕ⁰⁰(x)
00

∈ H2 , T ϕ = ηϕ⁰⁰⁰⁰(x), ϕ ∈ D(T ), D(T ) = D(A).

From the definitions of A and T it can be seen that H1= span {1, x} is the null space of A, and both A and T are positive definite self-adjoint operators in H2, and there is a greatest positive number λ such that

hAϕ, ϕi0≥ λkϕk²₀, ϕ ∈ D(A). (6) It is easy to show that

p₀

η T ≤ A ≤ p₁

η T. (7)

In fact, integrating by parts and taking account of the def- initions of A and T as well as the boundary conditions, we have

hAϕ, ϕi0 = Z l

0

(p(x)ϕ⁰⁰(x))⁰⁰ϕ(x) dx

= Z l

0

(p(x)ϕ⁰⁰(x))⁰ϕ⁰(x) dx

= Z l

0

p(x)ϕ⁰⁰(x)ϕ⁰⁰(x) dx.

From the inequalities 0 < p0 ≤ p(x) ≤ p1 < ∞ it follows that

p0

Z l 0

ϕ⁰⁰(x)ϕ⁰⁰(x) dx ≤ hAϕ, ϕi0

≤ p1

Z l 0

ϕ⁰⁰(x)ϕ⁰⁰(x) dx.

In other words,

p0kϕ⁰⁰k²₀≤ hAϕ, ϕi0≤ p1kϕ⁰⁰k²₀. Similarly, we have

hT ϕ, ϕi₀= hηϕ⁰⁰⁰⁰, ϕi₀= hηϕ⁰⁰⁰, ϕ⁰i₀

= hηϕ⁰⁰, ϕ⁰⁰i0= ηkϕ⁰⁰k²₀ and

ϕ⁰⁰k²₀= 1

ηhT ϕ, ϕi₀.

Hence Dp

ηT ϕ, ϕE

0≤ hAϕ, ϕi0≤Dp₁ η T ϕ, ϕE

0

and therefore p₀

η T ≤ A ≤ p₁ η T.

In terms of the operators A and T we can rewrite the system (4) as follows:









 d²u

dt² + d

dt(T u) + Au = (I − P1)f (t, u), u(0) = (I − P₁)ϕ₀, du(x, t)

dt  

_t=0= (I − P₁)ψ₀. (8)

Let us now introduce the Hilbert space H = H2× H2 equipped with the general inner product. Set

~ u = u1

u₂



, u1= A¹²u, u2= du dt, A =

"

0

−A¹² A¹²

−T

#

, D(A) = D(A¹²) × D(A),

F (t, ~~ u) =

 0

(I − P₁)f (t, u)

 ,

~u0= u₁(0) u2(0)



=

"

A¹²(I − P1)ϕ0

(I − P1)ψ0

# .

Then the evolution equation (8), or the original sys- tem (1), is equivalent to the following first-order evolution equation:





 d~u(t)

dt = A~u(t) + ~F (t, ~u),

~

u(0) = ~u0,

(9)

and the corresponding equation is given by





 d~u(t)

dt = A~u(t),

~

u(0) = ~u₀.

(10)

2. Main Results

We shall first discuss the semigroup properties of the op- erator A, and then investigate the well-posedness and asymptotical stability of the system (1). The following results will be obtained:

Theorem 1. The linear operator A in the system (9) is the infinitesimal generator of a C₀ semigroup T (t) sat- isfying

kT (t)k ≤ M e^−δt, t ≥ 0, where M and δ are positive constants.

Theorem 2. If f (t, u) : [0, T ] × L²[0, l] → L²[0, l]

is continuous in t for any T > 0 on [0, T ] and uni- formly Lipschitz continuous in u on L²[0, l], then for every ~u0 ∈ H, the evolution equation (9) has a unique weak solution in C([0, T ]; H). Moreover, the mapping

~

u0→ ~u is Lipschitz continuous.

Theorem 3. Suppose that f (t, u) meets the conditions of Theorem 2 with the Lipschitz constant N satisfying

N < δ M

√ λ,

where M and δ are the same as those in Theorem 1, and λ is the same as that in the equality (6). Then the solution of the system (9) (and therefore the solution of the system (1)) is exponentially stable.

3. Proofs of the Main Results

In this section, we shall prove Theorems 1–3. To prove Theorem 1, we shall first prove the following lemmas.

Lemma 1. If λ is a complex number with Re λ ≥ 0, then (λ²+ λT + A)⁻¹ exists and is bounded.

Proof. Clearly, the result is true for λ = 0. If λ 6= 0, then for any x ∈ D(A), we let λ = σ + iτ , σ ≥ 0.

Consequently,



λ + T + 1 λA

x kxk

≥  

D

λ + T +1 λA

x, xE    

σkxk²+ hT x, xi

+ σ

σ²+ τ²hAx, xi + i[τ kxk²

− τ

σ²+ τ²hAx, xi]

≥ hT x, xi ≥ ωkxk², where ω > 0 is the smallest eigenvalue of T .

Since x ∈ D(A), it can be seen that D

λ + T + 1 λA

x, xE

= σkxk²+ hT x, xi + σ

σ²+ τ²hAx, xi + ih

τ kxk²− τ

σ²+ τ²hAx, xii , and

ReD

−

λ + T + 1 λA

x, xE

≤ −hT x, xi ≤ −ωkxk².

It follows that the numerical range of −(λ + T +_λ¹A) has the form

V

−

λ + T +1 λA

=n

−D

λ + T + 1 λA

x, xE

: kxk = 1,

x ∈ D

λ + T +1 λAo

⊆λ | Re λ ≤ −ω . This implies that 0 ∈ ρ(−(λ + T + ¹_λA)) (see (Bal- askrishnan, 1981)), and so 0 ∈ ρ(λ²+ λT + A). Thus, (λ²+ λT + A)⁻¹ exists and is bounded.

Lemma 2. If λ is a complex number with Re λ ≥ 0, λ 6= 0, then (¹_λ + λA⁻¹+ A⁻¹²T A⁻¹²)⁻¹ exists and is bounded.

Proof. First, it should be noted that A⁻¹²T A⁻¹² can be extended to a bounded linear operator on H2, for every x ∈ H₂, λ = σ + iτ , σ ≥ 0. Since

1

λ+ λA⁻¹+ A⁻¹²T A⁻¹² x

kxk

≥  

D1

λ+ λA⁻¹+ A⁻¹²T A⁻¹² x, xE

=   

σ

σ²+ τ²kxk²+ σhA⁻¹x, xi + hA⁻¹²T A⁻¹²x, xi + ih −τ

σ²+ τ²kxk²+ τ hA⁻¹x, xii  

≥ σ

σ²+ τ²kxk²+ σhA⁻¹x, xi + hA⁻¹²T A⁻¹²x, xi

≥ η p1

kxk²,

the operator _λ¹ + λA⁻¹+ A⁻¹²T A⁻¹² is invertible. We also see that its image is dense in H2. In fact, if y0∈ H2, then

D1

λ+ λA⁻¹+ A⁻¹²T A⁻¹² x, y0

E

= 0, x ∈ H2. Noticing that _λ¹+ λA⁻¹+ A⁻¹²T A⁻¹² is self-adjoint, we have

D x,1

λ+ λA⁻¹+ A⁻¹²T A⁻¹² y₀E

= 0, x ∈ H₂. Since _λ¹+ λA⁻¹+ A⁻¹²T A⁻¹² is invertible, y₀= 0, and therefore the range of (_λ¹+ λA⁻¹+ A⁻¹²T A⁻¹²) is dense in H2. Thus, (_λ¹+ λA⁻¹+ A⁻¹²T A⁻¹²)⁻¹ exists and is bounded.

Lemma 3. If λ is a complex number with Re λ ≥ 0, λ 6= 0, then the resolvent of A can be expressed by

R(λ, A) = 1 λ

"

R11, R12

R21, R22

# ,

where

R11= I − 1 λ²

1

λ² + A⁻¹+1

λA⁻¹²T A⁻¹²−1

,

R12= 1 λ

 1

λ² + A⁻¹+1

λA⁻¹²T A⁻¹²−1

A⁻¹²

R₂₁= −1 λA⁻¹²

λ² + A⁻¹+ 1

λA⁻¹²T A⁻¹²−1

,

R22= A⁻¹²1

λ² + A⁻¹+ 1

λA⁻¹²T A⁻¹²−1

A⁻¹². Proof. From Lemma 2 we know that R(λ, A) is a bounded linear operator on H and the expression for R(λ, A) can be obtained by a direct calculation.

Lemma 4. If λ is complex number with Re λ ≥ 0 and λ 6= 0, the family of the operators with the parameter λ

F (λ) = 1 λ

 1

λ² + A⁻¹+1

λA⁻¹²T A⁻¹²−1

=1

λ+ λA⁻¹+ A⁻¹²T A⁻¹²−1

is uniformly bounded.

Proof. Let

Zλ=1

λ+ λA⁻¹+ A⁻¹²T A⁻¹²−1

x, x ∈ H2. Then {kZλk} is bounded for all λ. Otherwise, there is a λ0 such that

lim

λ→λ0

kZλk = +∞.

As regards the inner product of the sequence yλ = Z_λ/kZ_λk with λ = σ + iτ , we have

D1

λ+ λA⁻¹+ A⁻¹²T A⁻¹² y_λ, y_λE

= σ

σ²+ τ² + σhA⁻¹y_λ, y_λi + hA⁻¹²T A⁻¹²y_λ, y_λi + ih −τ

σ²+ τ²+ τ hA⁻¹yλ, yλii

. (11)

Obviously, the real part of the right-hand side (11) is greater than η/p0> 0. On the other hand,

lim

λ→λ0

1

λ+ λA⁻¹+ A⁻¹²T A⁻¹²

yλ= lim

λ→λ0

x kZ_λk = 0, so that a contradiction occurs. Hence {kZλk} is uni- formly bounded for every x ∈ H2, and the result of this lemma follows from the Principle of Uniform Bounded- ness.

Lemma 5. If λ is complex number with Re λ ≥ 0, λ 6=

0, and there is a λ0 > 0 such that if |λ| ≥ λ0, then (_λ¹+ T A⁻¹+ λA⁻¹)⁻¹ is uniformly bounded.

Proof. For every x ∈ H2, it is easy to see that

1

λ+ T A⁻¹+ λA⁻¹ x

2

=D1

λ+ T A⁻¹+ λA⁻¹ x,1

λ+ T A⁻¹+ λA⁻¹ xE

= 1

|λ|²kxk²+λ

λhx, A⁻¹xi +λ

λ¯hA⁻¹x, xi + |λ|²kA⁻¹xk²+ λhT A⁻¹x, A⁻¹xi + +λhA⁻¹x, T A⁻¹xi + 1

λhx, T A⁻¹xi +1

λhT A⁻¹x, xi + kT A⁻¹xk²

≥ 1

λhx, T A⁻¹i + 1

λhT A⁻¹x, xi + kT A⁻¹xk². (12) Since T A⁻¹ is bounded, there is a λ0> 0, such that if |λ| ≥ λ0, the right-hand side of the above inequality has the form

1

λhx, T A⁻¹xi + 1

λhT A⁻¹x, xi + kT A⁻¹xk²

≥ 1

4kT A⁻¹xk²≥1

4δ₀²kxk², (13) where δ0> 0, and the last inequality is due to the invert- ibility of T A⁻¹. This follows from

1

λ+ T A⁻¹+ λA⁻¹ x

≥ 1

2δ0kxk. (14) Hence (_λ¹+ T A⁻¹+ λA⁻¹) is invertible.

Next, we shall show by contradiction that the range of (_λ¹ + T A⁻¹+ λA⁻¹) is dense in H2. If the range of (¹_λ + T A⁻¹ + λA⁻¹) is not dense in H2, there is a y0∈ H2, y06= 0 such that

D1

λ+ T A⁻¹+ λA⁻¹ x, y₀E

= 0, x ∈ H₂. This implies

DA

λ + T + λ y, y0

E

= 0, y ∈ D(A), where y = A⁻¹x.

In view of Lemma 1, (_λ¹A + T + λ)⁻¹ is a bounded linear operator, and its range is dense in H₂. Hence y₀= 0, but this contradicts y0 6= 0. Thus the range of (¹_λ + T A⁻¹+ λA⁻¹) is dense in H2. If |λ| ≥ λ0, Re λ 6= 0, and for a fixed x ∈ H2 we set

Zλ=1

λ+ λA⁻¹+ T A⁻¹−1

x, |λ| ≥ λ0,

then it can be shown that {kZλk} is bounded. Otherwise, there is a sequence {λn} with |λn| ≥ λ0 and Re λn ≥ 0 such that

n→∞lim kZλnk = ∞, and

 1 λn

+λnA⁻¹+T A⁻¹ Zλ_n

kZλ_nk = x

kZλ_nk → 0, n → ∞.

Let yn = Zλ_n/kZλ_nk. From (13) it follows that

 1

λ_n + T A⁻¹+ λ_nA⁻¹ y_n

≥ δ0

2ky_nk = δ0

2 > 0, which contradicts (15). Hence {kZλk} is bounded, for every x ∈ H2. From the Principle of Uniform Bounded- ness it follows that (¹_λ+ λA⁻¹+ T A⁻¹)⁻¹ is uniformly bounded for |λ| ≥ λ0 and Re λ ≥ 0.

Lemma 6. Under the condition of Lemma 5, if |λ| ≥ λ0

and Re λ ≥ 0, the family of operators with λ

A⁻¹² 1

λ² + A⁻¹+1

λA⁻¹²T A⁻¹²−1

A⁻¹²

= 1 λ²A + 1

λT + I−1

is uniformly bounded.

Proof. If |λ| ≥ λ0 and Re λ ≥ 0, from Lemma 5 we have

 1 λ²A +1

λT + I−1

= A⁻¹1 λ² +1

λT A⁻¹+ A⁻¹−1

. Thus, the result of Lemma 6 is concluded by virtue of Lemma 5.

Proof of Theorem 1. Since

A =

"

0 A¹²

−A¹² −T

#

and A and T are positive definite self-adjoint operators, we can easily verify that (iA)^∗ = iA. From the cele- brated Stone Theorem (Pazy, 1983) it follows that A is the infinitesimal generator of a C0 semigroup T (t) on H. On the other hand, we can see that 0 ∈ ρ(A) gives us

A⁻¹=

"

−A⁻¹²T A⁻¹² A¹²

−A⁻¹² 0

# .

If Re λ ≥ 0 and λ 6= 0, we can show that the resolvent R(λ, A) of A satisfies

kR(λ, A)k ≤ M

|λ|, 1 ≤ M < ∞. (15)

In fact, from Lemma 3 we deduce that R(λ, A) is an analytic function of λ on the right-half complex plane. According to the analyticity of R(λ, A), it suf- fices to show that if |λ| ≥ λ0 > 0 and Re λ ≥ 0, then kR(λ, A)k ≤ M1/|λ|. However, this can be easily ob- tained by Lemmas 4–6.

Since ρ(A) ⊃ {λ | Re λ ≥ 0}, ρ(A) being an open set on the complex plane, there is a constant  > 0 such that

σ(A) ⊂λ | Re λ ≤ −

and therefore from the stability theorem of the analytic semigroup (Pazy, 1983) we conclude that there is a con- stant δ > 0 such that

kT (t)k ≤ M e^−δt, t ≥ 0.

The proof of Theorem 1 is thus complete.

Proof of Theorem 2. For F (t, ~~ u) = [0, (I − P₁)f (t, u)]^T ∈ H in (9), we have k ~F (t, ~uk = k(I − P₁)f (t, u)k. Since I − P1 is a bounded linear opera- tor, and f (t, u) is continuous in t on [0, T ] and uni- formly Lipschitz continuous in u on L²[0, l], ~F (t, ~u) has the same properties as f (t, u). Applying Theorem 1.2 of (Pazy, 1983) yields Theorem 2.

In order to prove Theorem 3, we first introduce a continuous function space C[0, +∞) equipped with the norm

kgkm= max

t≥0 |g(t)| < +∞, g ∈ C[0, +∞), and define the linear operator K through

Kg(t) = Z t

0

e^−δ(t−s)g(s) ds, where δ is the same as in Theorem 1.

We see that K is a bounded linear operator on C[0, +∞). In fact,

|Kg(t)| ≤ Z t

0

e^δ(t−s)|g(s)| ds ≤ kgkm

Z t 0

e^−δ(t−s)ds

= kgk_m1

δ(1 − e^−δt) ≤1 δkgk_m for any t ≥ 0. Thus we have

kKgkm= max

t≥0 |Kg(t)| ≤1 δkgkm, and

kKk_m≤ 1

δ. (16)

Proof of Theorem 3. From Theorem 1 and (Pazy, 1983) we know that the evolution equation (9) has a unique solution

~u(x, t), and hence the system (4) has a unique solution

~u(x, t). We now decompose u(x, t) as follows:

~u(x, t) = ~ξ(x, t) + ~η(x, t), where ~ξ(x, t) and ~η(x, t) satisfy























∂²ξ(x, t)~

∂t² + η∂⁵~ξ(x, t)

∂t∂x⁴ + ∂²

∂x²



p(x)∂²~ξ(x, t)

∂x²



= 0,

∂²ξ(x, t)~

∂x²  

_x=0,l = ∂

∂x



p(x)∂²ξ(x, t)~

∂x²

 

_x=0,l= 0,

~ξ(x, 0) = (I − P₁)ϕ₀,∂~ξ(x, t)

∂t  

_t=0= (I − P₁)ψ₀, (17) and































∂²~η(x, t)

∂t² + η∂⁵~η(x, t)

∂t∂x⁴ + ∂²

∂x²



p(x)∂²~η(x, t)

∂x²



= (I − P1)f (t, u(x, t)),

∂²~η(x, t)

∂x²  

_x=0,l= ∂

∂x

p(x)∂²ξ(x, t)~

∂x²

 

_x=0,l = 0,

~

η(x, 0) = 0, ∂~η(x, t)

∂t  

_t=0= 0,

(18) respectively. Here u(x, t) in f (t, u) is the solution of the system (4).

From Theorem 1 it should be noted that the sys- tem (17) in H is equivalent to the system (10), whose solution ~ξ(x, t) = T (t)~ξ₀ satisfies

k~ξ(x, t)k ≤ M e^−δtk~ξ0k. (19) It is obvious that the system (18) in H is equivalent to the system





 d~η(t)

dt = A~η(t) + ~F (t, ~u),

~

η(0) = ~η0= 0, where

~

η = (η1, η2)^T, η1= A¹²η, η2= dη dt and

F (t, ~~ u) =0, (1−P1)f (t, u)T

=0, (I−P1)f (t, ξ+η)T

. Since A generates a C0 semigroup T (t) on H and

~

η0= 0, we have

~ η(x, t) =

Z t 0

T (t − s) ~F s, ~u(s)
ds,

and from Theorem 1 it follows that k~η(x, t)k ≤ M

Z t 0

e^−δ(t−s) k ~F (s, ~u(s)k dx

= M Z t

0

e^−δ(t−s)k(I − P1)f (x, u(s))k0ds

≤ M Z t

0

e^−δ(t−s)kf (s, u(s))k0ds

≤ M N Z t

0

e^−δ(t−s)ku(s)k0ds

≤ M N Z t

0

e^−δ(t−s)(kξ(s)k0

+ kη(s)k0) ds. (20)

By virtue of (6) and the definition of the inner prod- uct described before, we have

k~ξk² = hξ1, ξ1i + hξ2, ξ2i =D

A¹²ξ, A¹²ξE

0+ kξ2k²

= hAξ, ξi₀+ kξ₂k²≥ λkξk²₀+ kξ₂k²≥ λkξk²₀, and so

kξk0≤ 1

√λk~ξk. (21)

Similarly, it can be shown that k~ηk²≥ λkηk²₀ and kηk0≤ 1

√λk~ηk. (22)

Combining (19)–(22) leads to k~η(x, t)k ≤ M N

Z t 0

e^−δ(t−s) kξ(s)k0+ kη(s)k0
ds

≤ M N

√ λ

Z t 0

e^−δ(t−s)k~ξ(s)k ds

+M N

√ λ

Z t 0

e^−δ(t−s)k~η(s)k ds

≤ M²N

√λ e^−δtk~ξ0k

+M N

√ λ

Z t 0

e^−δ(t−s)k~η(s)k ds

= M²N

√

λ te^−δtk~ξ0k + M N

√

λK k~η(s)k
, and therefore

I −M N

√ λ K

k~η(s)k ≤ M²N

√

λ te^−δtk~ξ₀k. (23)

Looking at (21) with the assumption that N < δ√ λ/M and (16), we find

M N√ λ K

_m< 1.

Hence (I −^{M N√}

λK) is invertible, and



I −M N

√

λ K−1

=

∞

X

n=0

M N

√ λ

n

Kⁿ. (24)

Analysing the definition of K, we see that K is a mono- tonically increasing operator on C[0, +∞), and so is (I − ^{M N√}

λK)⁻¹ based on (24). Now, multiply two sides of (23) by (I − ^{M N√}

λK)⁻¹ to find k~ηk ≤ M²N

√λ k~ξ0k

∞

X

n=0

M N

√λ

n

Kⁿ(te^−δt).

Since

K(te^δt) = Z t

0

e^−δ(t−s)se^−δsds

= e^−δt Z t

0

s ds =t² 2!e^−δt, it can be shown by induction that

Kⁿ(te^−δt) = tⁿ⁺¹ (n + 1)!e^−δt. Thus

k~η(x, t)k ≤ M²N

√ λ k~ξ₀k

∞

X

n=0

M N

√ λ

ⁿ tⁿ⁺¹ (n + 1)!e^−δt

=M²N

√ λ

√λ M Nk~ξ₀k

∞

X

n=0

M N

√ λ

ⁿ⁺¹ tⁿ⁺¹ (n+1)!e^−δt

< M k~ξ₀ke^−δte^{M N√λt}= M k~ξ₀ke^{−(δ−M N√λ)t}. Since

N < δ M

√

λ, δ −M N

√ λ > 0, let

α = δ −M N

√λ. Then

k~η(x, t)k ≤ M k~ξ0ke^−αt. (25)

As for ~u(x, t) = ~ξ(x, t) + ~η(x, t), we have

~

u = (u1, u2)^T =

A¹²u,du dt

^T

=

A¹²(ξ + η),dξ dt +dη

dt

T

=

A¹²ξ,dξ dt

T

+

A¹²η,dξ dt

T

= (ξ₁, ξ₂)^T+ (η₁, η₂)^T = ~ξ + ~η and

k~u(x, t)k = k~ξ(x, t) + ~η(x, t)k ≤ k~ξ(x, t)k+k~η(x, t)k.

Combining (19) and (25) gives

k~u(x, t)k ≤ M k~ξ₀ke^−δt+ M k ~ξ₀ke^−αt. Since 0 < α < δ , we have

k~u(x, t)k ≤ 2M k~ξ0ke^−αt, t ≥ 0.

From ~ξ0= ~u0 it follows that

k~u(x, t)k ≤ 2M k~u0ke^−αt, t ≥ 0.

This implies that the solution ~u(x, t) to the evolution equation (9) is exponentially stable, and thus the solution to the original system (1) is also exponentially stable. The proof is complete.

4. Conclusion

The beam equation with two free ends (1) was studied by means of functional analysis and semigroups of linear op- erators. First, the system (1) was converted to an abstract evolution equation (9). Second, the properties of the sys- tem operator A were investigated and a significant result that A generates a C0-semigroup T (t) with exponential decay property that kT (t)k ≤ M e^−δt (M > 0, δ > 0) was derived (Theorem 1). Then, the well-posedness of the system (1) was discussed (Theorem 2) using the semi- group technique. Finally, the exponential stability of the system (1) was proved under appropriate conditions (The- orem 3). In further research, concrete designs of the con- trollers for this system to be asymptotically stable would be quite significant.

Acknowledgement

The author deeply appreciates the valuable comments of the anonymous reviewers.

References

Hou X. and Tsui S.-K. (2000): Control and stability of a tor- sional elastic robot arm. — J. Math. Anal. Appl., Vol. 243, pp. 140–162.

Hou X. and Tsui S.-K. (1999): A mathematical model for flexible robot arms system modelling and optimization, In: Chap- man & Hall / CRC Res. Notes Math. — Boca Raton FL:

Chapman & Hall / CRC, Vol. 396, pp. 391–398.

Hou X. and Tsui S.-K. (1998): A control theory for Cartesian flexible robot arms. — J. Math. Anal. Appl., Vol. 225, pp. 265–288.

Hou X. and Tsui S.-K. (2003): A feedback control and simu- lation for a flexible robot arm. — J. Appl. Math. Comp., Vol. 142, No. 2–3, pp. 389–407.

Komkov V. (1978): Control theory variational principles and optimal design of elastic systems, In: Optimal Control and Differential Equations. — New York: Academic Press, pp. 35–40.

Li S. and Zhu G. (1988): A property of the C0-semigroup corre- sponding to the slender flying vehicle system with struc- tural damp. — J. Syst. Sci. Math. Sci., Vol. 8, No. 3, pp. 219–225.

Köhne M. (1978): The control of vibrating elastic systems in control and systems theory. — Distributed parameter sys- tems, Vol. 6, pp. 388–456.

Pazy A. (1983): Semigroups of Linear Operators and Appli- cations to Partial Differential Equaitons. — New York:

Spinger-Verlag.

Balaskrishnan A.V. (1981): Applied Functional Analysis. — New York: Springer-Verlag.

Received: 20 August 2002 Revised: 23 December 2002