Table Of ContentO C
N ONTROLLABILITY OF
L V
INEAR IBRATIONS
O C
N ONTROLLABILITY OF
L V
INEAR IBRATIONS
W. KRABS AND G. M. SKYLAR
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Contents
Preface vii
1 Null-Controllability of Undamped Linear Vibrations 1
1.1 Distributed Control. . . . . . . . . . . . . . . . . . . . . . 1
2
1.2 Boundary H -Control . . . . . . . . . . . . . . . . . . . . 6
1.3 The Problem of Null-Controllability . . . . . . . . . . . . 9
1.4 On the Solution of the Problem of Null-Controllability . . 11
1.5 Boundary Null-Controllability with
1
H -Controls . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.6 Approximate Boundary Null-Controllability
2
with H -Controls . . . . . . . . . . . . . . . . . . . . . . 27
1.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2 Null-Controllability of Damped Linear Vibrations 39
2.1 Distributed Control . . . . . . . . . . . . . . . . . . . . . 39
2.1.1 The Model Equation and its Solution . . . . . . . 39
2.1.2 The Problem of Null-Controllability . . . . . . . . 45
2.2 Boundary Control . . . . . . . . . . . . . . . . . . . . . . 49
2.2.1 The Model Equations . . . . . . . . . . . . . . . . 49
2.2.2 On the Solution of the Model Equations . . . . . . 51
2.2.3 The Problem of Null-Controllability . . . . . . . . 53
2.2.4 Approximate Null-Controllability . . . . . . . . . 56
2.2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . 61
3 On the Controllability of a Rotating Beam 69
3.1 The Model Equations and their Solution . . . . . . . . . . 69
3.2 The Problem of Controllability from Rest to Rest. . . . . 74
3.3 The Asymptotic Behaviour of the Eigenvalues
of the Operator A in (3.3) . . . . . . . . . . . . . . . . . . 76
3.4 On Controllability from Rest to Rest . . . . . . . . . . . . 84
v
vi W. Krabs and G. M. Sklyar
3.5 On Approximate Controllability from Rest to an
Arbitrary Position . . . . . . . . . . . . . . . . . . . . . . 85
3.6 The Limit Case γ2 = EA = ∞ . . . . . . . . . . . . . . . . 93
K
3.7 On Stabilizability . . . . . . . . . . . . . . . . . . . . . . . 94
3.8 On Control and Stabilization with the Aid of the Torque . 106
Appendix 125
A. 1 Linear Operators . . . . . . . . . . . . . . . . . . . . . . . 125
A. 2 Semigroups of Linear Operators . . . . . . . . . . . . . . . 130
A. 2.1 Basic Definitions. . . . . . . . . . . . . . . . . . . . 130
0
A. 2.2 On the Generation of C -Semigroups. . . . . . . . 131
A. 2.3 Application to the Solution of Evolution Equations. 133
A. 3 Strong Stabilizability of Evolution Equations . . . . . . . 135
A. 4 Modeling a Rotating Beam . . . . . . . . . . . . . . . . . 147
A. 5 Bibliographical Remarks and
Further Problems . . . . . . . . . . . . . . . . . . . . . . . 152
Bibliography 159
Index 161
Preface
This book can be considered as a research monograph of the authors.
The first two chapters are devoted to the problem of null-controllability
of undamped and damped linear vibrations. This problem is formulated
with the aid of linear differential equations in Hilbert spaces which gives
exactlytherightabstractframeworkforthedescriptionofvibrationswith
finitely and infinitely many degrees of freedom. In both chapters the two
cases of distributed and boundary control are treated separately and it
is shown how boundary control can be transferred to distributed control,
if certain conditions are satisfied which can be frequently encountered in
special cases.
Chapter3isdevotedtotheproblemofcontrollability ofaTimoshenko
beam which rotates in a horizontal plane. Besides controllability also
stabilizability is studied. The material of this chapter is based on recent
investigations of the authors.
In order to make the book selfcontained an appendix is provided in
whichthebackgroundfromfunctionalanalysisispresentedthatisneeded
in order to develop the theories given in chapters 1 to 3. In addition the
model equations of the rotating beam in chapter 3 are derived. The
appendix ends with bibliographical remarks and further problems.
The authors are indebted to Mrs. A. Garhammer for her competent
writing the manuscript on the computer.
Darmstadt in December 2000.
vii
Chapter 1
Null-Controllability of
Undamped Linear
Vibrations
1.1 Distributed Control
The starting point is an abstract wave equation of the form
y¨(t)+Ay(t) = Cu(t) , t ∈(0,T) , (1.1)
for some T > 0 where y : [0,T] → H with H being a Hilbert space over
C,
1
y˙(t) = lim (y(t+h)−y(t)) for t ∈ (0,T) , (1.2)
h→0 h
A : D(A) → H is a linear operator on a dense subspace D(A) of H,
u : [0,T] → U with U being a subspace of a Hilbert space, C : U → H
is a bounded linear operator. For the beginning we assume the second
derivative y¨(t) to be defined by (1.2) with y replaced by y˙. The second
order wave equation (1.1) can be equivalently rewritten as a system of
first order. For this purposewe introduce the vector function z : [0,T] →
D(A)×H by
z(t) = (y(t),y˙(t))T , t ∈ [0,T] . (1.3)
Further we define linear operators A: D(A)×H → H ×H and C : U →
H ×H by
(cid:2) (cid:3) (cid:2) (cid:3)
0 −I 0
A = , C = where 0 = zero operator
A 0 C
1
2 W. Krabs and G. M. Sklyar
and I = identitiy. Then (1.1) can be rewritten in the form
z˙(t)+Az(t)= Cu(t) , t ∈ (0,T) . (1.4)
In addition to the wave equation (1.1) we consider initial conditions
y(0) = y0 , y˙(0) = y˙0 (1.5)
where y0 ∈ D(A) and y˙0 ∈ H are given.
These initial conditions are equivalent to
T
z(0) = z0 = (y0,y˙0) . (1.6)
At first we investigate under which conditions the initial value problem
(1.4), (1.5) has a unique solution. We begin with the homogeneous case
where C : U → H ×H is the zero operator so that (1.4) reads
z˙(t)+Az(t) = θH×H , t ∈(0,T) . (1.7)
Assumption: Let A :D(A) → H be self adjoint and positive definite .
Then there exists a unique “square root” of A, i.e., a selfadjoint linear
operator A12 : D(A12) → H with D(A) ⊆ D(A12), A12 : D(A) → D(A12),
and A= A12 ◦A12. In particular we have
(cid:7)Av,w(cid:8)H = (cid:7)A12v,A12w(cid:8)H for all v ∈ D(A) and w ∈D(A12) .
Let us put E = D(A12) and H = E×H. Then H is a Hilbert space with
scalar product
(cid:7)z1,z2(cid:8)H = (cid:7)A12v1,A12v2(cid:8)H +(cid:7)w1,w2(cid:8)H
for all z1 = (v1,w1)T and z2 =(v2,w2)T in H.
Further it follows, for every z =(v,w)T ∈ D(A)×E, that
(cid:7)Az,z(cid:8)H = −(cid:7)A12w,A12v(cid:8)H + (cid:4)(cid:7)Av(cid:5),(cid:6)w(cid:8)H(cid:7) = 0 . (1.8)
1 1
(cid:3)A2v,A2w(cid:4)H
Let z1 = (v1,w1)T and z2 = (v2,w2)T in D(A) ×E be given. Then it
follows that
Null-Controllability of Undamped Linear Vibrations 3
(cid:7)Az1,z2(cid:8)H = −(cid:7)A12w1,A12v2(cid:8)H +(cid:7)Av1,w2(cid:8)H
= −(cid:7)w1,Av2(cid:8)H +(cid:7)A12v1,A12w2(cid:8)H = −(cid:7)z1,Az2(cid:8)H .
∗
(1.9 )
From (1.8) forall z ∈D(A)×E weinferthattheoperator A : D(A)×
E → E×H ismonotoneandfrom(1.9∗)forallz1,z2 ∈ D(A)×E itfollows
that A is skew adjoint , i.e., A = −A∗ where A∗ : D(A)×E → E ×H
denotes the adjoint operator . Since D(A)×E = H, it follows that
A is maximal monotone (see Corollary A.1.10). By Corollary A.2.3 this
impliesthat(−A,D(A)×E)generatesaC◦-semigroup{T(t)| 0 ≤ t < ∞}
on H. By Theorem A.2.4 the unique solution z : [0,T] → D(A)×E of
(1.7) and (1.6) for z0 ∈ D(A)×E is given by
z(t) = T(t)z0 , t ∈ [0,T] . (1.9)
If z0 ∈ H, we take the function z : [0,T] → H defined by (1.9) as a mild
solution of (1.7), (1.6). We assume Cu ∈ L2([0,T],H) for every u ∈ U
where L2([0,T],H) consists of all (classes of) functions f : [0,T] → H
such that t → (cid:10)f(t)(cid:10)H, t ∈ [0,T], is measurable and
(cid:8)T
(cid:10)f(t)(cid:10)2 dt < ∞ .
H
0
By Theorem A.2.5 the unique mild solution z : [0,T] → H of (1.4) and
(1.6) for z0 ∈ H is then given by
(cid:8)t
z(t) = T(t)z0 + T(t−s)Cu(s)ds , t ∈ [0,T] . (1.10)
0
This solution formula can be given in a more explicit form. For that
purpose we assume that the operator A : D(A) → H has a complete
orthonormal sequence (ϕj)j∈N of eigenelements ϕj ∈ D(A), j ∈ N, and
correspondingsequenceofeigenvalues λj ∈ R, j ∈ N,offinitemultiplicity
with
0< λ1 ≤ λ2 ≤ ... ≤ λj → ∞ as j → ∞ .
Then it follows that
(cid:9)∞
D(A) ={v ∈H| λ2j |(cid:7)v,ϕj(cid:8)H|2 < ∞}
j=1
