Open-loop stabilizability of infinite-dimensional systems

Math. Control Signals Systems (1998) 11:129-160 �9 1998 Springer-Verlag London Limited Mathematics of Control,

Signals, and Systems

Open-Loop Stabilizability of Infinite-Dimensional Systems*

R . R e b a r b e r ~ f a n d H . Zwar t :~

Abstract. In this paper we study open-loop stabilizability, a general notion of stabilizability for linear differential equations .~ = Ax + Bu in an infinite- dimensional state space. This notion is sufficiently general to be implied by exact controllability, by optimizability, and by various general definitions of closed- loop stabilizability. Here, A is the generator of a strongly continuous semigroup, and we make very few a priori restrictions on the class of controls u. Our results hinge upon the control operator B being smoothly left-invertible, which is a very mild restriction when the input space is finite-dimensional. Since open-loop stabilizability is a weak concept, lack of open-loop stability is quite strong. A focus of this paper is to give necessary conditions for open-loop stabilizability, thus iden- tifying classes of systems which are not open-loop stabilizable.

First we give useful frequency domain conditions that are equivalent to our definitions of open-loop stabilizability, and lead to a version of the Hautus test for open-loop stabilizability. When the input space is finite-dimensional, we give necessary conditions for open-loop stabilizability which involve spectral properties of A. We show that these results are not true if the conditions on B are weakened. We obtain analogous results for discrete-time systems. We show that, for a class of systems without spectrum determined growth, optimizability is impossible. Finally, we show that a system is open-loop stabilizable with a class of controls u if and only if the system with the same A but a more bounded B is open-loop stabilizable with a larger class of controls.

Key words. Distributed parameter systems, Operator semigroups, Discrete-time systems, Stabilizability, Hautus test, Optimizability.

1. I n t r o d u c t i o n

I n th i s p a p e r w e s t u d y a g e n e r a l n o t i o n o f s t a b i l i z a b i l i t y fo r a b s t r a c t l i n e a r d i f f e r -

en t i a l e q u a t i o n s o f t h e f o r m

Yc(t) = A x ( t ) + Su( t ) , x ( 0 ) = x0, (1 .1)

* Date received: September 2, 1996. Date revised: March 20, 1998. This work was partially sup- ported by NSF Grant DMS-9623392.

t Department of Mathematics and Statistics, University of Nebraska, 836 Oldfather Hall, Lincoln, Nebraska 68588-0323, U.S.A. [email protected].

:l: Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands. [email protected].

129

130 R. Rebarber and H. Zwart

where A generates a strongly continuous semigroup T(t) on the state space X, and u(t) ~ U, the control space. We assume that X and U are Banach spaces. We refer to this system as Z(A, B). If X is finite-dimensional and A and B are matrices, then most aspects of stabilizability are well understood. In this case, we say that E(A, B) is stabilizable if there exists a matrix F such that e (a+sr)l is exponentially stable, which is equivalent to the eigenvalues of A + B F being in the open left half- plane. The system X:(A, B) is stabilizable if and only if the Hautus test is satisfied:

r a n k [ s / - A[B] = dim X for all s with nonnegative real part.

Using the Hautus test, it is very easy to prove that the following condition is equivalent to stabilizability when X is finite-dimensional: for any initial condition x0, there exists some control u such that the state trajectory x is square integrable. This result motivates our definition of open-loop stabilizabilty.

When the state space X is infinite-dimensional, there are many distinct deft- nitions of stability. E(A, B) is closed-loop stabilizable when the stabilizing input is obtained by a feedback u(t) = Fx(t) for some operator F, and the resulting closed-loop operator A + B F generates a strongly continuous exponentially stable semigroup. Typically this definition assumes that F is a bounded operator from X into U, but other definitions are possible--see [15], [12], or [17] for general definitions, and many papers on partial differential equations (for instance, [11]) for particular examples. ~(A, B) is open-loop stabilizable if for every initial state there is a distribution u such that the solution of (1.1) is a negative exponential times an L2([0, co); X) function--see Definition 2.1 for a precise statement. In order to be as general as possible, we allow the controls to be distributions, but other input classes are possible, see Definition 2.7. If the controls u are restricted to L2([0, co); U), this is usually known as optimizability or the finite cost condition. Since open-loop stabilizability is a fairly weak notion, lack of open-loop stabilizability is quite strong. If a system is not open-loop stabilizable, then it is not exactly controllable, it is not optimizable, and it is not closed-loop exponentially stabilizable in any useful sense; for instance, if a system is closed-loop stabilizable in the sense of [15] or [12], then it is open-loop stabilizable. One of our aims is to give necessary conditions for open-loop stabilizability, and to identify classes of systems which are not open-loop stabilizable.

The relationship between closed-loop and open-loop stabilizability for distributed parameter systems has not been studied deeply, but if X is a Hilbert space, then some results are available. The most well known is the result by Datko [3], stating that if B is a bounded operator, and if the input u that makes the state square integrable is also square integrable, then there exists a stabilizing (bounded) feedback. Other results linking open- and closed-loop stabilizability can be found in [5], [23], and [24]. In general, however, it is very easy to find systems which are open-loop stabilizable but not closed-loop stabilizable by a bounded feedback. In Examples 2.14 and 2.17 we show that this is possible even when A and B are bounded operators and the state space is a Hilbert space.

We now outline the contents of this paper: In Section 2 we define open-loop stabilizability. To ensure as much generality

as possible, we make no a priori boundedness restrictions on B. Boundary con-

Open-Loop Stabilizability of Infinite-Dimensional Systems 131

trolled partial differential equations are typically modeled with a B that is not in ~ ( U , X). For such a system a larger state space X could of course be choosen, so that B e . ~ ( U , X ) , but it is usually preferable to take the state space to be the space of maximum regularity, that is, the smallest space for which B is admissible; see [20] for details about admissibility, and see several papers by Lasiecka and Triggiani, for instance, [11], for examples of the space of maximum regularity. Sometimes the state space is chosen so that B is not admissible, i.e., there exist inputs u e L2([0, T); U) which do not result in state trajectories in L2([0, T); X ) - - a well-known example of this is the beam equation with control equal to the bending moment at one end, see [1]. Our results hinge on B being smoothly left-invertible (see Definition 2.2), which is satisfied by injective B on tinite- dimensional U, but is quite restrictive when U is infinite-dimensional. In Theorem 2.5 and Lemma 2.8 we give useful frequency domain conditions that are equivalent to our definition of open-loop stabilizability, and lead to a version of the Hautus test for open-loop stabilizability. Furthermore, in Theorem 2.5 and Lemma 2.8 it is shown that if a system is open-loop stabilizable, then the controls u can be chosen to be in a "small" class of distributions, and Bu(t) is in the space X-1 (see (2.1)).

In Section 2.1 we restrict our attention to finite-dimensional input spaces, and obtain stronger necessary conditions. The following result follows from Theorem 2.11:

Theorem 1.1. Suppose that E(A, B) is open-loop stabilizable, and U is -finite - dimensional Then there exists ~ > O, such that the set tr+_6(A) := {2 e o(A)IRe ;t > -6} contains no -finite accumulation point and consists only of a point spectrum of finite multiplicity.

If B is admissible, then there are necessary conditions for E(A, B) to be closed- loop stabilizable by a compact feedback operator which are similar to the conditions stated in Theorem 1.1 (see [13]). The most important difference is that in the closed-loop case tr+_a(A) must consist of only finitely many elements, while in the open-loop case we have not ruled out the possibility that a+_6(A) has infinitely many elements. In fact, there are many simple examples (for instance, a wave equation in one space variable with boundary control) of systems which have infinitely many unstable modes that can be open-loop stabilized, but cannot be closed-loop stabilized by a compact feedback.

In Section 2.2 we give examples which show that the assumption of smooth left-invertiblity of B is necessary for these results to be true. We also show that the Hautus test no longer follows from open-loop stabilizability if B is not smoothly left-invertible.

In Section 2.3 we show that for discrete-time systems with smoothly left- invertible B, open-loop stabilizability is equivalent to closed-loop stabilizability. We give an example to show that this is not true when B is not smoothly left- invertible.

In Sections 3 and 4 we identify systems which are not open-loop stabilizable. Unlike other conditions for lack of stabilizability (see, for instance, [13] or [24]),


these conditions are not about "the boundedness, admissibility, or compactness of the input operator. Let 9b be the growth bound of T(t), and let 9~ = sup{Re(2)12 e a(A)}. Let B be a one-dimensional input operator of the form Bu = bu, for some element b, such that ( s I - A ) - l b is uniformly bounded on some right half-plane--this happens, for instance, when b is an admissible input element. Then Corollary 3.2 implies the following result:

Theorem 1.2. Suppose X is a Hilbert space and 9~ < 9b. Then fo r any e > 0 there exists xo ~ X such that

Ji ~ e-Z<9b-~)'(llx(t)ll2 + Ilu(t)ll 2) d t = oo

f o r any u E L 2 E(O, oo). y b -

This means that if the spectral bound of A is less than the growth bound of T(t) , then it is impossible to improve the stability of the system with a one-dimensional input operator and an L~b_~(0, ~ ) control.

In Section 4 we consider again finite-dimensional operators B, and we prove that a system is open-loop stabilizable with a class of controls u is and only if the system with the same A but a more bounded (unbounded) B is open-loop stabilizable with a larger (smaller) class of u's. The relationship between an increase in unboundedness of B and the corresponding decrease in the class of controls required for open-loop stabilizability is very simple and natural. This result motivates the identification of classes of systems which cannot be open-loop stabilized. For instance, consider the diagonal system with eigenvalues {in}n~z and one- dimensional input operator {b,}, with

b, = 1 for n even,

bn = n 1/4 for n odd.

Then this system is not open-loop stabilizable, even though the moment problem for the control problem is solvable with u ~ L2[0, oo)--the difficulty is that the control which solves the moment problem does not keep x( t ) in the state space. This example shows that the main result of [25] is wrong.

In Section 4 we also show that if a system with a one-dimensional input operator and a Riesz basis of eigenfunctions is open-loop stabilizable, then its unstable spectrum must satisfy the gap condition

inf 12, - 2ml > 0, n ~ m

where J-n, Am are arbitrary elements of the unstable point spectrum. For instance, a wave equation in a region in R", n _> 2, never satisfies this (see p. 395 of [18]), hence is never open-loop stabilizable (or closed-loop stabilizable, or exactly con- tronable) using one-dimensional inputs; see [19] for results of this type for controllability and closed-loop stabilizability of this wave equation and other systems with unitary groups and finite-dimensional inputs.


2. Open-Loop Stabilizability; General Results

We begin this section by introducing the notation we use in this paper. Unless otherwise noted, we assume that the state space is a Banach space X with norm II �9 lit. When we want the state space to be a separable Hilbert space, we denote it by Z with norm I1" IIz- The input space will be the Banach space U. The class of bounded linear operators from the Banach space XI to the Banach space X2 will be denoted by ~(X1, X2), with the usual operator norm. If it is clear on which space we are working, then we write the norm without a subscript. We denote by T(t) a Co-semigroup on At. The infinitesimal generator of T(t) is denoted by A.

For k e N, we define by X-k the Banach space obtained by completing X in the n o r m

Ilxllx_~ := I I ( , t I - A)-kxllx, (2.1)

for some 2 e p(A), the resolvent set of A. The following results can be found in [20]: X-k is independent of the particular choice of 2;

X-k+l '---' X-k, k = 1 ,2 , . . . , (2.2)

where by '--* we denote continuous and dense injection; T(t) extends to a C0- semigroup on X-k, whose generator is an extension of A with domain X-k+I. We denote these extensions again by T(t) and A, respectively.

Following is a list of other symbols that we use:

W' if W is a normed linear space, then this denotes the dual space of W;

if W is a linear operator, then this denotes the adjoint operator of W;

(y , x ) for y e X', and x e X, this denotes the action of the functional y on x;

for x, y ~ Z, this denotes the inner product; r {z ~ ClRe(z) > ~};

C ~176 functions with compact support; ~ ' class of distributions; ~ class of distributions with support in [0, 09); ~ ( W ) class of W-valued distributions with support in [0,09), see

Chapter 8 of [4]; D | w for D e ~ ' and w e W, a Banach space, this denotes the 'distri-

bution in ~ ( W ) given by .[[D| (fD~odt)w, see [4];

Er(t) e n for t > O, and zero elsewhere; cr(A) spectrum of A;

+(A) ~(A) ~ r co + H (C, ;X) Banach space of bounded holomorphic functions from ~+ to

x; H2(ff+;Z) Hardy space of holomorphic Z-valued functions on C + that

satisfy supr>, J~_,~ I[f(r + iy)ll2dy < 09; L02([0, 09); X) { f lEof e L2([0, 09); X)}.


Open-loop stabilizability is a very general notion of stabilizability. We consider as large a class of control operators B as possible and as rich a class o f controls u(.) as possible. The only a priori restriction we place on the input opera tor is that B e .W(U, X-k) for some k e hi. The only a priori restriction we put on the class of controls is that u(-) e ~ ( U ) . We say that x(-) e ~ [ ( X - k ) satisfies (1.1) if the following equality holds int the sense of distributions:

x(t) = T(t)xo + [T(.)B �9 u(.)](t), (2.3)

where �9 denotes the convolution product between distribution, see Section 8.2 of [4].

Since T(.)B need not be measurable in the uniform topology when B is not finite rank, it is not directly clear that it defines a distribution in ' ~o(_~( u, X-k)). We show this by showing that it is the derivative of a continuous function. For u e U and - 2 E p(A), we have that eatT(t)Bu is continuous, and, for t > s > O,

Ji e~'T(~)Budr = + - + eatZ(t)(2I A )-I Bu eaSZ(s)(2I A)-I Bu.

From this we see that

IleZT(t)(2I + A)-lBu - eZ~T(s)(2I + A)-IBu[l:c_, < IleaTT(r)Bullx_, dr

Jl 2T oJt < e Me IIBII drllull - ~ ( u , x _ ~ )

By letting s---, t on the right side of this equation, we see that the function carT(t)(21 + A ) - t B - ( 2 I + A ) - I B for t > 0 (and zero for t < O) is uniformly continuous as a function in .La(U,X_k). Hence it defines a distribution in ~ ( . Z ' ( U , X _ k ) ) via the usual identification. Since eatT(t)B is the distributional derivative of a function in ~(.L~'( U, X-k)) , it is also in ~ ( . Z ' ( U , X-k)).

Definition 2.1. Z(A, B) is g-open-loop stabilizable if, for every x0 E X, we can find u e ~ ( U ) such that the solution of (1.1) satisfies

E_gx E L2([0, oo); X). (2.4)

We call a system open-loop stabilizable if it is - e -open- loop stabilizable for some e > 0 .

Since X'--,X-k for k ~ Z +, it is easy to see that L2([0, oo); X) is a subset of ~9~(X_k) for k E Z +, via the usual identification, which is used in (2.4).

Definition 2.2. The input operator Be-oq ' (U,X_k) is smoothly left-invertible if there exists an operator G ~ &"(X-k, U) such that GB = It/and

IIGaxll t, <- gllxllx_~ (2.5) for all x e X-k+l.


Note that (2.5) implies that G can be extended to a bounded operator on X-k-l . This definition of course depends upon k; typically one chooses the smallest k so that B e .s X-k). We prove below that if U is finite-dimensional, then B is smoothly left-invertible if and only if it is left-invertible. However, if U is infinite- dimensional, then Definition 2.2 is restrictive: this is because if B is smoothly left- invertible, then it is easy to show that there exists M1 such that

Ilnul[x_~ <- M111nullx+, for all u ~ U. (2.6)

Since there trivially exists M2 such that

Ilnullx_~_, <- g2llaul[x_~ for all u e U,

this means that if B is smoothly left-invertible, then X-k and X-k-i induce the same topology on the range of B. If the range of B is finite-dimensional, then this is automatically satisfied. However, if, for instance, the range of B equals X-k, then this implies that A is bounded. It is of course possible to have an infinite rank B which is smoothly left-invertible when A is unbounded; for instance, it is well known that the generator for a Kelvin-Voigt beam is bounded on an infinite- dimensional subspace of the state space. Despite the restrictiveness of this definition, we see in Section 2.2 that the main results of Section 2.1 do not hold if the condition that B is smoothly left-invertible is removed, so this is the appropriate condition.

When the input operator B is one-dimensional with range in X-k, it is of the form Bu = bu for some b ~ X-k, and we write E(A, B) as Z(A, b). It is easy to show that scalar input operators always have a smooth left-inverse, provided that b is nonzero. To see this, let Y denote the space X-k, and let Dr,(A') denote the domain of the adjoint operator of A on the space yr. Since Dr,(A') is weak-. dense in Y', there exists ayb ~ Dr,(A ~) such that (yb ,b) = 1. If we now take G as G = (Yb, "), then it is easy to see that it is a smooth left-inverse. With a similar argument one can show that if U is finite-dimensional and B is injective, then B is smoothly left-invertible.

In Theorem 2.4 we show that if E(A,B) is open-loop stabilizable, then the stabilizing input must be, roughly speaking, the distributional derivative of an L 2- function, provided that B is smoothly left-invertible. For the proof we need the following simple lemma.

Lemma 2.3. Let W be a Banach space, and let f e L20([0 , oo); W). Then

[Erlw *f ] e LZ_g([O, oo); W)

for every r less than 9.

Proof. Let f e Lz_o([O, oo); W). Then

Ji I E_ (t)[Erlw , f ] ( t ) = E-o ( t ) d r = de. o

Since f e L2_o([0, oo); W), and r - 9 < O, we see that the last equality is a con-


volution of an Li-function with an L2-function. From standard results, we know that this convolution is in L2, finishing the proof. �9

Theorem 2.4. Assume that the system Y.(A, B) is O-open-loop stabilizable, and that B is smoothly left-invertible. Let r be a real number less than g. Then for every input u e ~o(U) such that x e L2_g([0, or); X), we have that fir ~ L2_g([0, oo); U), where fir = E J u * u.

Proof. Let u e ~ ( U ) be such that (2.4) holds, and let G be the smooth left- inverse. Applying G to (2.3) gives

Gx(t) = GT(t)xo + [GT(.)B �9 u(.)](t), (2.7)

where the equality is in the sense of distributions with values in U. Since Erlu is a distribution in ~)(U), we have that

[E, Iv * Gx] = [Er(')Iv * GT(.)xo] + [Erl~: * [GT(.)B * u(.)]]

= [E,(.)Iu * GT(.)xo] + [GT(.)B * [E, Iu * u]], (2.8)

by the associative property of the convolution product, see equation (8.2.7) of [4]. It is easy to see, using that G is a smooth left-inverse and that xo e X c X-k, that the distributional derivatives of GT(t)xo and GT(t)B are given by GAT(t)xo + 6(t) | Gxo and GAT(t)B + 6(t) | Iv , respectively. From the distribution theory in [4], we know that the product rule for differentiation of scalar distributions also holds for vector-valued distributions, so using the notation f0 ) for the first distributional derivative of f ,

( f , 9)(U = f ( O , 9 = f * g(O.

Using the first equality on the left-hand side of (2.8), and using the second equality on the right-hand side, we see that

[[rErIv + It: | * Gx] = [Erlv * [GAT(.)xo + Gxo | g(')]]

+ [[GA T(.)B + Iv5(-)] * [ErIu, u]]. (2.9)

Using the fact that the g distribution is the identity in the convolution product, we obtain

[rErlu * Gx] + Gx = [E,(')Iv * GAT(.)xo] + E, Gxo

+ [GAT(.)B �9 [E,Z. �9 u](')] + [ErZ. �9 u]

= [E, Iu * [GAT(.)xo + G A T ( . ) B , u(.)]]

+ ErGxo + [E, Iu * u]. (2.10)

Using (2.7) we see that

[rE, Iv �9 Gx] + Gx - [E, Iu * GAx] - E, Gxo = [E, Iv * u]. (2.1 I)

Since G and GA are bounded operators and x satisfies (2.4), we have that Gx and


G A x are both in L2o([0, oo); U). Since r < 0, we have that Erlu is an element of L2_o([0 , oo); U) as well. Now using Lemma 2.3 and equality (2.11), we see that [erIv * u] E L2 o([0, oo); u). �9

Theorem 2.5. Let the system Y.(A,B) be O-open-loop stabilizable, let B be smoothly left-invertible, and let r be a real number less than O. Then, for all xo ~ X , there exist holomorphic functions ~ and co such that

xo = (sl - A)~(s) - Bco(s) (2.12)

for all s E lE+. The functions r and co(.)/(. - r) are elements o f H~176 X) and H ~176 (lE+.~; U), respectively, for all c5 > O. Furthermore, the ranoe o f Bco(s) (over all s ~ lE+ and xo e X) is contained in X - l .

2 + I f U is a separable Hilbert space, then co(.)/(. - r) ~ H (lE0; U), and i f X is a separable Hilbert space, then ~(-) ~ H2(lE+; X).

Proof. Let u be as in Definition 2.1, and let fi, = [Erlv * u]. From Theorem 2.4 we know that fir e L2_g([0, oo); U). This implies that, for any y > 0, fir is in the class of weighted tempered distributions 6t~+r(U), see [4]. Since Er(t) lu has the

convolution inverse 6(t)(t) | I v -cS(t) | r l v ~ 5a~+r(-oq'~ we have that

U : [6 (1) ~ ] U ~ (~ rlu] * fir ~ 5#~+y(U).

Hence u is Laplace transformable, and we denote its Laplace transform by co. Using the results from Section 8.2 in [4], we conclude that co is holomorphic in

+ for every > 0, and hence co is holomorphic in lEg. lEg+y y + Furthermore, it is easy to see that the Laplace transform of fir is co(s)/(s - r). Since fir e L2_g([0, oo); U), we know that fir e Ll_g_6([0, oo); U) for 6 > 0. Hence the Laplace transform of the

function fir is in oo + . H (IE0+ a, U) for all c~ > 0. Since x ~ L2_0([0, oo); X), we can show in a similar way that the Laplace trans-

form ~ of x satisfies ~ e H ~176 + �9 (lEg+a, X) for all 6 > 0. Therefore we have that all distributions in (2.3) are Laplace transformable. Taking the Laplace tratlsform of (2.3), we obtain

~(s) = (sI - a ) - l x 0 + (sI - A ) - I Bco(s) (2.13)

on some right half-plane. We rewrite (2.13) on this half-plane as (2.12). Since and co are holomorphic in lE+, we see that (2.12) must also hold on lE+.

Since x0 ~ X and ~(s) E X , xo - (sI - A)~(s) takes its values in X-l . By (2.12), we conclude that Bco(s) takes its values in X-l for any x0 e X and s e lE+ '0"

If X is the separable Hilbert space Z, then the Laplace transform of x is an 2 + H (lEo ; Z) function, see Theorem A.6.21 of [2]. If U is a Hilbert space, then

the Laplace transform of fir is in 2 + H (lE0 ; U), and hence co(s)/(s - r) E 2 + H (lEo; U). �9

We refer to (2.12) as a (~, co)-representation of Y.(A, B). This was introduced for finite-dimensional systems in [7]. Thus open-loop stabilizability implies the existence of such a representation for every x0 ~ X, provided that B is smoothly left-


invertible. For A �9 .W(X-k+t, X-k), B �9 La(U, X-k), and [x, u] r �9 X-k+l (9 U, define

( s I - A , ~ ) [ x , u] r = ( s I - A)x + B,,.

A direct consequence of the (~, co)-representation is the Hautus test:

Corollary 2.6. Let the systdm Z(A, B) be g-open-loop stabilizable, and let B be smoothly left-invertible. Then

r a n ( s l - A , B ) D X foral l s � 9 + (2.14) 9"

l f B �9 .oq'(U, X), then we have equality in (2.14), re . ,

ran(sI : A, B) = X for all s �9 lE+. (2.15)

Proof. If Z(A,B) is open-loop stabilizable, (2.12) is true for all x0 �9 X, so (2.14) follows. Equation (2.15) follows easily as well, since if k = 0, then ( s I - A, B) maps into X. �9

We see from Theorem 2.5 that when Z(A, B) is open-loop stabilizable, Boo(s) �9 X_~. For one-dimensional input operators, it is shown in Section 2.1 that if Z(A, b) is open-loop stabilizable but not stable, then it is necessary that b lies in X-I.

Theorem 2.5 motivates a frequency domain definition for open-loop stabilizability.

Definition 2.7. Suppose U and Z are separable Hilbert spaces and B �9 -~(U,Z_I) . Let g �9 F,, and let f~ be a subset of the U-valued functions which are

+ holomorphic on leg. Then E(A, B) is g-open-loop stabilizable with co �9 f~ if, for

2 + every x0 �9 Z, (2.12) holds for some ~ �9 H (lE9 ;Z) and 09 �9 t~.

We say that E(A, B) is open-loop stabilizable with co �9 [2 if it is -e-open-loop stabilizable with co �9 f~ for some e > 0.

Note that if t~ is H2(lE~-; U), then open-loop stabilizability is equivalent to optimizability, or the finite-cost condition, of the cost functional ~(llx(t)ll2+ Ilu(t)ll) dt, see [2].

In the next lemma we show that the (~, co)-representation in Theorem 2.5 can be chosen to be continuously dependent on the initial state x0, provided that U is a separable Hilbert space.

Lemma 2.8. Assume that U is a separable Hilbert space and B �9 .W(U, X-I ) is smoothly left-invertible. I f Y.(A, B) is g-open-loop stabilizable, then for every 3 > 0 and for every xo �9 X there exists a (~, og)-representation satisfying

II~ll.==(c;.;x) ~ MIIxoll and .c~ r H'r ~ MIIxoll, (2.16)


for some M independent o f xo. Furthermore, i f X is a separable Hilbert space, then

II~llH2<q;x) --< Mllx011. (2.17)

Proof. We prove this by using the Baire Category Theorem.

Step 1. F o r N = l , 2 , . . . l e t

VN = Ix0 6 X I there exists a(~, co)-representation with

< N and co('---~) < N'~. II~llH~(%,,x) �9 - - r H I ( ( ~ + ; U ) J

Since the system is 9-open-loop stabilizable, we have by Theorem 2�9 that

~) vN = x. (2.18) N = I

In order to apply the Baire Category Theorem we need to show that VN is a closed subset of X for every N > 0. We do this in Steps 2 and 3.

Step 2. Let {x ,} ,~ l be a sequence in VN which converges to x in X. Denote by (~,, co,) a (~, co)-representation of x, which satisfies the earlier estimates. Since

2 + H (tE a ; U) is a Hilbert space, there exists a weakly convergent subsequence of {co,(.)/(. - r)},~ t with limit co(.)/(. - r). Therefore, without loss of generality we

Y oo assume that {co, ( - ) / ( . - )},=) is weakly convergent�9 Since weak convergence implies weak-pointwise convergence,

fim<u, co , (s )>=(u, co(s)> foral l s~ lE + and u ~ U . (2.19)

Furthermore,

co(') tt2(r v) < l iminf o9,(.____)). - r /~2(r -< N. (2.20)

Step 3. For s ~ lE+ n p(A) define the function

~(s) = (sI - A ) - l x + (sI - A) - l Bco(s). (2.21)

Recall that

~,(s) = (sl - A ) - I x , + (sI - A)-1BCOn(s), s ~ lEa+.~ r ip(A) .

Since x, is convergent and co,(s) is weakly convergent, it is easy to see that

l i m ( y , ~ , ( s ) ) = ( y , ~ ( s ) ) foral l se lE++anp(A) and y ~ X ' . (2�9 . ~ O 0

For y e X ' , {(y ,~ , ( . ) )} is a sequence in oo + . H (lEg-~,lE) which is uniformly bounded by NllYlI. Hence it contains a subsequence which is pointwise convergent

+ We denote the limit function by ~y, and it is to a holomorphic function on lE0+a"

140 R. Rcbarbcr and H. Zwart

easy to see that

II~yll -< NllYlI. (2.23)

Furthermore, from (2.22) it follows that

~y(s) = <y,~(s)> for all s � 9 IE+.~c~p(A). (2.24)

Since for every y �9 X' the function ~y(-) is holomorphic on C9+~ and equals <y, ~(.)> on Cg+.~ c~p(A), we see that ~ has a holomorphic continuation to Cg++a

+ and (2.24) is true on IEg_~. From (2.23) and (2.24) it follows that

I I~ l I .~<c~j ) -< N. (2.25)

Furthermore, it is easy to see from (2.21) that + x = (sl - A)~(s) - Bog(s) for all s �9 Cg+~.

Combining this with (2.20) and (2.25) shows that VN is closed.

Step 4. By the Baire Category Theorem, there exists No such that VN0 contains an open ball B(~, p). Let x0 be an arbitrary element of X, and let z be defined as

z = ~ 2 + ~ x0.

Then it is clear that z �9 B(~,p) c VN0. Hence z has a (~, o9~) representation as in (2.12) with II~lIM~(~.~,x) -< No and Ilco~(.)/(. - r)llH~(r -< No. For ~ we can construct a (~i, og~)-representation satisfying the same bounds. Since the system is linear, it is easy to see that x0 has a (~x0, oJx0) representation with

Go(s) = [G(s) - ~ ( s ) ] 21Ix011 and P

Hence since z and .~ are in B(~, p) c VN0,

II~x~174 < ~-~llxoll and

O~xo(S) = [ o ~ z ( s ) - o ~ ( s ) ] - - 211xoll

P

4N0 ~O~o(.) - l ix011 . ( - - r) H2(C;;U) p

x0 is arbitrary, and so we have proved (2.16).

Step 5. If the state space is a Hilbert space Z, then H2(IE+; Z) is a Hilbert space as well, and the proof is quite a bit easier, where the H ~~ norm in the definition of V, is replaced by the H 2 norm. In particular, showing that ~, converges pointwise is now a direct consequence of the fact that every bounded sequence of the Hilbert space H2(C~-; Z) contains a weakly converging subsequence. �9

Using a similar proof, we can prove the following variation of Lemma 2.8.

Lemma 2.9. Assume that U is a separable Hilbert space and B �9 .~(U, Z - l ) is smoothly left-invertible. Assume further that ]~(A, B) is g-open-loop stabilizable with co �9 H~176 U). Let r > 0 and r < g. Then for every xo there exists a (~,oJ)


satisfying ~(s) = (sI - A)-Ix0 + (sI - A)-l Bco(s),

II~ilH~(C;+~,X) ~ MIIx011 and IlwllH~(C;,u) ~ MIIx011,

for some M independent of xo.

(2.26)

(2.27)

2.1. Finite-Dimensional Input Spaces

In this subsection we investigate open-loop stabilizability when the input operator has finite rank. If the input operator has finite rank, then it is easy to see that the input space U can be decomposed into a finite-dimensional part on which B is injective and another part contained in the kernel of B. Hence, without loss of generality, we may assume that U = C" and that B is injective. As explained after Definition 2.2, any injective B is smoothly left-invertible. If the semigroup T(t) is stable, then it is trivially open-loop stabilizable. Hence we assume here that T(t) is not stable. If Y.(A, B) is open-loop stabilizable, then there must exist an x0 e X for which w in the (~, w) representation is nonzero. Hence the range of Bw(s) is nonzero. By Theorem 2.5, we know that this range is contained in X-l . If U is one-dimensional, then this implies immediately that b ~ X-i . If U = (E", m > 2, then it might happen that the range of B lies outside X-1. However, since Bw(s) lies in X-i for all co in the (~,co) representations, we can always find a B ~ ~((E '~, X-i) for some th < m such that B is injective, and/~co(s) = Bco(s) for all such co. Hence without loss of generality we assume that if Z(A, B) is open-loop stabilizable, then B e 5e((U', X_ 1).

Before we can state the main theorem of this section, we need to introduce some operators and some additional notation. Assume that F:. is a simple closed contour in C, oriented counterclockwise, with only one point 2 e a(A) in its interior and no points of a(A) on F~. We denote all functions that are holomorphic on the interior of F~ and continuous on F~ by f~ , which is a Banach space with the supremum norm. For co e Oh we define the operator

1 I (sI-A)-lBw(s)ds" a~(co) = T ; i r,

We define the spectral projection operator by

P~x= 2~i lr (sI - A) -'xds.

(2.28)

(2.29)

Lemma 2.10. Suppose that B has finite-dimensional range. Then G~ is a compact operator from ~ into X, and the range of Gz is contained in the range of Pz.

Proof. This proof is similar to the proof of Theorem 4.1.5 in [2]. The main step is to approximate Ga uniformly by a sequence of finite rank operators. For sim- plicity we write F for Fx, G for Gx, and ~ for f~x.

Let 1 be the length of F and let (~i}i=l,...,N be a sequence of points on F such that the distance on F between Yi and Yi-l is I/N and for every s e F there exists a Yi such that the distance on F between s and y; is less than or equal to I/N. Define

142 R. R e b a r b e r a n d H. Z w a r t

GN �9 .~([2, X) by

I N v, GN(~ = ~i~i i~__l ( ' i l - A)-l B I,,_ w(s) ds,

where the integral is along F. and we define ?0 = YN- Clearly, since U is finite- dimensional, GN is a compact operator. Furthermore,

N 1 I y' A)-IB]co(s) ds = ~ [(yi I - A ) - 'B - (sl - IIG~(o~) - G(co ) l lx ~ y,-, x

1 N~ 1 I r' ( s - - y i ) ( y i I - A ) - l ( s I - A ) - I B w ( s ) d s x <-- ~ = r~-~

1 N 1 l < ~-~ ~--~ ~ ~ sup II(y,.I - A ) - l ( s I - A ) - l g l l ~ ( ~ , x ) l l o J I I n

7-~ ?~,s �9 F "m

_< g IIo~lln,

where the last inequality follows from the fact that since ( s I - A ) -I and (sI - A) - tB are holomorphic X-valued functions for s �9 p(A), they are uniformly bounded on the compact set F.

Since G is the uniform limit of a sequence of compact operators, it is compact. The proof that Pa is a projection (see Lemma 2.5.7 of [2]) can be easily modified to show that

It then follows immediately that ran G~ ~ ran P~. �9

We now formulate the main theorem of this subsection.

T h e o r e m 2.11. Assume that B has finite-dimensional range, and let Y.(A, B) be g- open-loop stabilizable. Then the following conditions hold."

(1) The spectral subset a~(A) contains no finite accumulation point. (2) a~'(A) consists of only point spectra with finite multiplicity. (3) The system Z(A[v~,x, P~B) is controllable for all 2 �9 a+(A). (4) For every xo �9 X, the (~, w)-representation (2.12) satisfies

Pxxo = -Ga(w) (2.30)

for all 2 �9 a+(A). (5) Let 2 �9 tr+(A) and xo �9 X. l f ~1, o91 are holomorphic functions on some open

set �9 containing ;t and they satisfy

xo = (sl - A)~I (s) - Bw! (s) for all s �9 ~,

then Ga(co) = Ga(ogl) for all (~, og)-representations o f xo.


Remark 2.12. Theorem 2.11 does not rule out the possibility that tr+(A) contains a sequence that converges to a (finite) point with real part equal to g.

Proof. (1) We assume that Z(A,B) is open-loop stabilizable, and write B as B = [bl,. . . ,bm]. Since A is the infinitesimal generator of a C0-semigroup, there exists an ct �9 ~+ c~p(A) . Using xi = (~I - A ) - l b i �9 X in (2.12), there exists ~,i and co~,i as in Theorem 2.5 such that

(aI - A ) - I bi = (s l - A ) ~ , i ( s ) - Bog~,i(s). (2.31)

Defining E~(s) = [~,l ( s ) , . . . , ~,m] and fl~(s) -- [a~,l ( s ) , . . . , a ~ ] , we see that

(~I - A ) - I B = (s l - A)E~(s) - BG~(s). (2.32)

Let A a denote the boundary of a + ( A ) in ~E +. First we prove that, for any g e Ag, det(f~(g) - Im/(g - ct)) = 0, where/,1 is the identity on ~ ' . With this property, we show that Ag consists of isolated points, and then show that the same holds for ~+(A).

We prove that de t ( f l (g ) - I m / ( ~ - a)) ---0 on the boundary by contradiction. Suppose that there exists g �9 Ag and det(D.~(g) - I m / ( g - ~)) ~: 0. There exists a sequence { s , } , ~ l = p ( A ) n ~E + such that s, converges to g. Since ~ is holomorphic on cEg ~, there exists a neighborhood ~ of g, contained in C~', such that de t (~ ( s ) - l , , / ( s - a)) # 0 for all s �9 ~ . Premultiplying (2.32) by (sI - A) - l and using the resolvent identity, we obtain that

( s l - A ) - t B = [E~(s) - (~I - A ) - ' B sI~__ ~] [ - I m + [ . s -~ ~ ( s ) ] - ' (2.33)

for s �9 p (A) n Y/'. Le t xo be an arbitrary element of X, so it follows that there exists { and co as in Theorem 2.5 such that

(sI - A)-lx0 = ~(s) - (sI - A ) - l Bog(s) for s �9 C + n p (A) .

For sufficiently large n, s, E ~ , so using (2.33) we have

(2.34)

Since s, converges to .i and det(f~.(g) - I m / ( g - ~)) :~ 0, we have by the holomorphicity of ~, E,, co, and f~, that the ri~ht-hand side of (2.34) converges. Therefore for every x0 the limit of ( s , I - A ) - xo exists as n ~ oo. By the Princi- ple of Uniform Boundedness, { (s ,1 - A ) -l }~=l is a bounded set. However, from Theorem 7.5-3 of [10], since {s,} approaches a boundary point of the resolvent set, I I ( s J - . '1) -~ II - - ' oo , providing a contradiction. Thus we have shown that if g �9 Aa, then de t (~(g) - Im/(g - ct)) = O.

Suppose that Ag has a finite accumulation point contained in C +. Then by the holomorphicity of f~a, the function det(f2~(s)- l m / ( s - ct)) equals zero every-


+ Since where on lE0"

det ( f ~ ( s ) - s/_-~ ) = det((s-~)f~,(s) - I,,) det (s/--~),_

this implies that det((s - ~)f~(s) - Ira) =- 0 on lE+. Taking s = ~, we obtain a contradiction, and thus the boundary of a+(A) in 112+ does not have a finite accumulation point in IE0 +. Now it is easy to see that this boundary contains only isolated points in lEO+. By standard topology, we conclude that this boundary equals the whole set.

(2), (4), (5) Let 2 be an element of a~(A). Since the points in o'S(A) are isolated, we can find a simple closed contour with 2 being the only point of a~(A) in the interior, and no points of a + (A) on F~. Hence the operators Pa and G~ given in (2.29) and (2.28) are well defined for all 2 e a+(A). From Lemma 2.10, we know that the range of Ga is contained in the range of Pa. We now show that both ranges are equal. Given the (~, co)-representation for x0,

1 J (sI- A)-Ixods P~xo = ~ i r~

1 I (~(s) - ( s I - A)-ZBco(s)) ds = ~ g i Fa

= -~(o~),

by (2.12)

(2.35)

since ~ is holomorphic on lE+. This proves (4) and (5), and shows that the range of P~ and Gx are equal.

Since the range of a projection is always closed, we conclude that G~ is a compact operator with closed range. This implies that the range of Gx is finite- dimensional. Hence PxX is finite-dimensional, and 2 is an eigenvalue with finite multiplicity.

(3) Applying Pa to (2.12), and noting that the spectral projection Pa commutes with the generator A, we see that

(2.36) Paxo = (sI - A)le~xPa~(s) - P;tBco(s).

Since P~ is finite-dimensional, the system E(AIe~x,P~B ) is finite-dimensional. Furthermore, the spectrum of A[eax is only 2. The function Pal is holomorphic at 2, and so (2.36) implies that the rank of ((21 - A)lp~x, PxB) equals the dimension of P~X. Now the Hautus test proves the assertion in part (3). �9

As an example, consider the plate equation in the square of side length 1, considered in [9], where the system is shown to be exactly controllable with an infinite-dimensional input space. In [19] it is shown that this system is not exactly controllable when B has finite rank. We can easily obtain this lack of exact controllability as follows: In this case the eigenvalues of A are { + i ~ ( j 2 +k2) l j , k e Z+}. These eigenvalues do not have bounded multiplicity, so if B is finite- dimensional, E(Alezx,P~B ) cannot be controllable for all 2 e or(A). Hence, by


part (3) of Theorem 2.11, this system cannot be open-loop stabilizable (or closed- loop stabilizable, or controllable) when B is finite-dimensional.

It is easy to see that in the special case when A is the infinitesimal generator o f a holomorphic semigroup, open-loop stabilizability implies closed-loop stabilizability: Since the spectrum of such a generator is contained in a wedge in a left half-plane, if E(A, B) is open-loop stabilizable, conditions (1) and (2) in Theorem 2.11 imply that A can have only finitely many unstable eigenvalues of finite multiplicity. This, combined with condition (3) in Theorem 2.11 is sufficient for closed-loop stabilizability by a bounded feedback.

2.2. B Not Left-lnvertible

In this subsection we present examples which show that Theorem 2.5 does not hold when the condition that B is smoothly left-invertible is removed, and that Theorem 2.11 does not hold if the condition that U is finite-dimensional is removed.

Example 2.13. Let Z = r U = r A = 0, and B = d i a g ( l , � 8 9 Note that while B does not have finite-dimensional range, it is compact and satisfies (2.6). Furthermore, it is easy to see that B is not (smoothly) left-invertible. We first show that E(A, B) is - 1-open-loop stabilizable.

Let z ~ = (z ~ ~ r be the initial conditions for (I.1), so (1.1) is equivalent to the infinite set o f equations

- - _ 0 kn(t) 1 un(t), zn(O) = z n, n = 1 , . . . . (2.37) --n

Choose N1 such that

~-]~ izO[ 2 e 2 - 1 ~ �89 IIz~ 2. (2.38) n=Nt+l

Note that for the one-dimensional system

Yc(t) = bu(t), x(O) = xo, b ~ O, (2.39)

we can always find u e LI(0, 1) such that x(1) = 0 and j'0 I [e*x(z)[2dz is arbitrarily small. Therefore, we can choose u, e LI(0, 1), n = 1 , . . . ,Nl , such that zn(1) = 0 and

jl [etzn(t)l 2 dt < �89 lie0112. (2.40) n=l 0

For n > Ni, choose u, = 0 on [0, 1], so z~(1) = 0 for 1 < n < N1 and z , ( l ) = z ~ for n > NI. Hence, using (2.38)

Ilz(1)ll 2 < �89 2. (2.41)


F u r t h e r m o r e ,

0 n=l n=Nt+l

= Z letz,(t) I: dt + le'z~ 2 dt n=l n=Nt +1

_< } IIz~ 2 + �89 IIz~ 2, (2.42) where we have used (2.38) and (2.40).

Choosing the increasing sequence Nk, with Nk satisfying

I712 eZ~ - e2k-2 < 2-kllz~ (2.43) 2 n=Nk+l

one can similarly as for k = 1 construct an input u e L l ([0, k); r such that the solution z(t) oo [0, k], satisfies = { Z n ( t ) } n = i , t

1. z,(k) = O for l < n < Nk; 2. z,(t) = z ~ f o r n > Nk and t ~ [0,k]; 3. IIz(k)ll 2 _< 2-kllz(0)l12; and 4. f~o Ile'z(t)ll 2 dt <- Ilz(~ 2-s

Using this, we see that we can find an input u e L]or co); ~'2) such that the corresponding state trajectory satisfies

Ile'z(t)ii z dt < Z 2-qlz~ 2 = 211z~ 2" i=O

In other words, the system is - l - o p e n - l o o p stabilizable. The Hautus test (2.15) does not hold in this example, since r a n B 4: Z, and so,

for s = 0, r a n ( s / - A, B) = ran B 4: Z.

Since the Hautus test is a direct consequence of the existence of a (~,og)- representation for every initial condition, see Corollary 2.6, we can conclude that Theorem 2.5 is no longer valid if the smoothly left-invertible is removed from the hypotheses. Furthermore, this example shows that property (2) of Theorem 2.11 does not hold either, since the only spectrum of A = 0 is the eigenvalue 0 with infinite multiplicity.

In the above example we have shown that open-loop stabilizability does not necessarily imply property (2) of Theorem 2.11 when U is infinite-dimensional and B is compact . In the next example, we show that open-loop stabilizability of Y.(A, B) does not necessarily imply property (I) of Theorem 2.11 either.

Example 2.14. Consider the state linear system Y.(A, B) on Z = ~,2 with A = B diag(1,1, . . . , I / n , . . . ) . It is easy to see that z and u satisfy

i(t) = Az(t) + Bu(t), z(O) = zo,


if and only if (z, v) = (z, u - Fz) satisfies

~(t) = (A + BF)z(t) + By(t), z(O) = zo,

where F is an arbitrary operator in A"(E2). If we now choose F = - I , then the second system is the same as in Example 2.13. Since that system is - l -open- loop stabilizable, so is the system Z(A, B), but it is easy to see that this A does not satisfy property (1) of Theorem 2.11.

We can also modify Example 2.13 to obtain an example which is open-loop stabilized by Coo[0, oo) controls but not L2[0, oo) controls: In Example 2.13, we had to construct in each time interval finitely many components of the input function that steered a part of the state to zero. It is well known that these components can be chosen to be Coo-functions. Since the other input components are zero, we see that we could have chosen the input function {un}~=l~~ to be C ~176 With this Coo input function, it is easy to see that the state is also C ~ . Thus, we see that in both examples we could have chosen the stabilizing input functions to be Coo- functions. On the other hand, if follows easily from optimal control (see [3]) that we cannot choose the input trajectories which open-loop stabilize the system to be in L2[0, oo), since otherwise this system would be stabilizable by a bounded feedback law.

2.3. Open-Loop Stabilization for Discrete-Time Systems

We now investigate open-loop stabilizability for discrete-time systems. We consider systems of the form

x(n + 1) = Ax(n) + Bu(n), x(O) = xo, n e N, (2.44)

where ,Y and U are Banach spaces, A e .s and B e .s U, X). We refer to this system as Xd (A, B). Our definition of open-loop stabilizability in the discrete-time case is analogous to that for the continuous-time case:

Definition 2.15. The discrete-time system (2.44) is open-loop stabilizable if there exists an r > 1 such that for any initial condition x0 there exists an input u(.) with values in U such that the solution x(n) of (2.44) satisfies

oo

IIr~x(n)ll 2 < oo. (2.45) n=l

Note that in this definition there are no restrictions on what kind of space the sequence {u(n)} is in.

Theorem 2.16. f f Z and U are Hilbert spaces and B is smoothly lefi-invertible, then the system (2.44) is open-loop stabilizable i f and only i f there exists an F E .~(X, U) such that A + BF is power stable.

Proof. The " i f " part of the theorem is obvious, so we assume that (2.44) is open-loop stabilizable. I f we can show that for every x0 there exists u such


that co

)--~(llx(n)ll 2 + Ilu(n)ll 2) < oo, (2.46) n = l

then there exists F e .L/'(X, U) such that A + BF is stable [21]. We show that open-loop stabilizability implies (2.46) when B is left-invertible.

By the definition of left-invertibility, there exists L ~ ~ ( Z , U) such that LB = It:. Applying L to (2.44) gives

u(n) = Lx(n + 1) - LAx(n). (2.47)

I f u(.) is chosen so that x(-) is square summable, this shows that u(.) is square summable as well. This verifies (2.46), finishing the proof. �9

Note that if U is finite-dimensional, and B is injective, then it is always smoothly left-invertible. Hence for this case we have shown that open-loop stabilizability is equivalent to closed-loop stability.

In the next example, we show that Theorem 2.16 no longer holds if the condition that B is smoothly left-invertible is removed.

Example 2.17. Consider the discrete-time system Ed(A,B) with X = U = :2, A = d i a g ( 3 , 2 + � 8 9 and B = A - 2 I , so B is the same as in Examples 2.13 and 2.14.

Let Pk denote the or thogonal projection on s p a n { e l , . . . , ek}, where ei is the ith basis vector of :2. Note that, for any k, B-Ipk is a bounded opera tor on :2, w h e r e B - I denotes the algebraic inverse of B, which is well defined on the range of Pk. It is easy to see that if we choose

u(n) = - B - I PkAx(n), (2.48)

then the first k components of x(n + 1) are zero. Furthermore, due to the form of A, the nonzero components of x(n + 1) are at most three times the corresponding components of x(n). Hence,

o9 CO

IIx(n + 1)112 = ~ Ixt(n + 1)12 < ~ 9lxt(n)l 2. I = k + l I = k + l

Therefore we can find a k, which is a function of x(n), such that

IIx(n + 1)112 -< �89 2

Since we can do this for every n, we have shown the following: for every initial condition the input {u( j )=--B-lpk(x( j ) )Ax(j)}~.=l results in a state x which satisfies

IIx(n)ll 2 _< 2-"llx(0)ll 2.

From this it follows easily that the system is open-loop stabilizable. Since B is compact , BF is compact for every bounded feedback F. Furthermore,

since A has an essential singularity at 2, and BF is compact , A + BF has an


essential singularity at 2. Thus the system Ed(A, B) is never stabilizable by a bounded feedback. Note that (2.48) can be seen as a nonlinear feedback. Since in this example A and B are bounded operators, on could expect that if Ed(A, B) is stabilizable by some feedback, then it would be also be stabilizable by a bounded linear operator. This example shows that Theorem 2.16 is not true in general for discrete-time systems.

Note further that as in the continuous-time case (see Example 2.13) the Hautus test is not satisfied, since for s = 2 it is easy to see that

r a n ( s / - A, B) #- Z. �9

3. Lack of Stabilizability When the Semigroup Growth Is Greater than the Spectral Bound

In this and subsequent sections we assume that the state space is a Hilbert space, unless otherwise noted. Let 9b be the growth bound for the semigroup T(t), i.e.,

lim (1 lnllT(t)ll) = 9b, t -~ \ t

and let go be the spectral bound for A, so

9o = sup{Re212 ~ a(A)}.

It is well known that 9~ < gb (see, for instance, Equation 5.6 of [2]). We will see that if the growth bound is not equal to the spectral bound, then under very general conditions on b and u, it is impossible to 9-open-loop stabilize E(A, b) for any g < 9b. There are many cases where the growth bound is strictly greater than the spectral bound. This can happen, for instance, when the spectrum consists solely of eigenvalues, but the multiplicity of the eigenvalues is not bounded (see [22]), or for certain shift semigroups (see [6] or Example 5.1.4 of [2]).

Theorem 3.1, Suppose (A, b) satisfies the followin9 conditions."

(i) 9 b > 0 a n d 9 o < 0 - (ii) (.I - A)- Ib ~ H~ Z) for some y e IR.

Then there exists zo ~ Z for which there is no co ~ H~~ •) and ~ ~ H~176 such that

~(s ) = ( s I - A ) - ' z o + (sir - A ) - ' b o g ( s ) (3.1)

for Re(s) > 0.

Proof. In this proof we write H ~ for H~(C~'; IE) and H~176 for H~176 Z). Suppose that, for every z0 E Z, there exist co ~ H ~176 and ~ e H~176 such that (3.1) is satisfied. We show that this leads to a contradiction.

Let g E (0, gb). We can assume without loss of generality that 7 in (ii) is greater than gb +& It follows from Huang [8] that if II(sl- A)-II/ is bounded in C~-, then T(t) would have growth bound less than or equal to g. Therefore, there exists


{~ . }2~ ~ c ~ such that

t im I I ( s j - ~ ) - ~ 1 1 = oo. ~1---*o0

By the principle of uniform boundedness, there exists z0 ~ Z such that

lira II(s.I - A)-lz01l = o o . ( 3 . 2 ) ,.---~OO

By assumption, for this z0 there exist co s H ~~ and ~ E H~176 such that (3.1) is satisfied. In particular,

~(Sn) = (SnI -- A)-lz0 + (s . I - A ) - l bco(sn).

Since by assumption {ll~(s,)ll In e N} and {Ico(s,)l In e N} are bounded sets, it follows from (3.2) that

lira I I ( s J - A)-~bll = 0o. (3.3) ,---400

Let z. = ((s. + ~/)I - A ) - Ib . (3.4)

Since Re(s. + },) > },, by condition (ii) there exists Mi e IR such that

IIz.II -< M, . (3.5)

By assumption, for each z,, there exist co,. ~ H ~176 and ~, ~ H ~176 (Z) such that

~,(s) = (s l - A ) - l z , + (sI - A)- lbco, (s ) , Re(s) > 0. (3.6)

Using (3.5) Lemma 2.9 shows that there exists )1,/2 ~ R such that

[{~.[[H~{Z) < g2 . (3.7)

Define ~ . ( s ) = co.(s + s.) .

Since Re(s.) _> 6, {&. In e N} is a uniformly bounded set in H~~ {12). Hence there exists a subsequence {&.l} of {cb.} which converges uniformly on compact subsets of IE~ to a holomorphic function rb s H~176 IE). We rename this sub-

{con}n=I sequence - co .

By (3.3) and the principle of uniform boundedness, there exists y in the dual space of Z (which we identify with Z), such that

lim [((&I - A ) - l b , y)[ = 00. ,---*O0

Let e a (0,6/2) and B(sn;6) = {s e aZlls - s.I -< e}, and let

fn(s) = ( ( s I - A)-Ib,Y)ln{~.,.},

the restriction of ( ( s I - A ) - l b , y ) to B(sn;e). Since B(sn;e)elF.+, fn is holomorphic on B(s.; e). Thus by the Maximum Principle there exists 0.(e) e [0, 2n)

Open-Loop Stabilizabifity of Infinite-Dimensional Systems 151

such that l im [A(sn + eei~ > l'ma I f . ( s . ) l = oo. (3.8)

go For every e �9 [0,J/2] there exists a subsequence of {0.( )}.=l which converges to 0(e) �9 [0, 27r]. Again we rename the subsequence {0.(e)}.~ I.

Using (3.4), (3.6), and the resolvent identity, we see that

eeiO.(~) ~j = ((s, + 7)I -- A) - lb ~.(s. + eeiO"(e) - - 7

1 + ((s, + eei~ A)-lb[con(sn + ~e i~ ee,O.(~)_ y].

Therefore,

f 1 ] + f ,(s , + ee i~ [co,(s, + ee i~ ~eiO.(~) _ ~J.

The left side of this equation is bounded, using (3.7). The first term on the right side is bounded, using (3.5) and (3.4). Using (3.8) we see that this can only happen if

1 li~n[&,(te ;~ teiO.(~)_7 ] =0 .

Since {&, },~l converges uniformly on compact subsets of C_+~ to & �9 H ~ (C+a; C) and ~ {0.(t)},=~ converges to 0(e), we see that, for every e �9 (0,6/2),

1 &( tei~ ) _ tetO(, ) _ ~, "

The set

{eei~ �9 (0 ,~ ) }

has a limit point 0 in C_+6, so by the uniqueness properties of holomorphic functions,

1 ~ ( s ) = - - , s �9 r \~'. $ - - ) '

This contradicts the fact that d~ �9176 which finishes the proof of Theorem 3.1. �9

Corollary 3.2. Suppose (A, b) satisfies the following conditions."

(i) g~ < oh. (ii) (.I - A) - Ib �9 l -F (C+;Z) for some ~, �9 It.

Then Y.(A, b) cannot be g-open-loop stabilized by 09 �9 H2(~+; C) for any g < gb.


2 + Proof. Suppose Y.(A, b) can be g-open-loop stabilized by co �9 H (Cg ; C) for some 2 + H ( r g < gb, that is, for every z0 � 9 there exist c o � 9 (r162 and ~ � 9 2 +

such that (3.1) holds for Re(s) > g. By condition (i), we can assume without loss of generality that g~ < g. Let p~ be such that g < p~ < 9b and let A = A - pl I , so ,,I generates the C0-semigroup T ( t ) e -p~t. This semigroup has growth bound gb - Pl > 0, and the spectral bound for ,4 is g~ - Pl < 0. Therefore (,~, b) satisfies the conditions of Theorem 3.1.

Let &(s) = co(s + P l ) and ~(s) = ~(s + Pl). Then we see from (3.1) that

~(x) = (s l - .4)-1z0 + (s l - A ) - l b t S ( s )

for R e ( s ) > g - p l with &�9 and ~ �9 Since H2(Cg_p, ; . ) c:: H~176 .), this contradicts the conclusion of Theorem 3.1. �9

Remark 3.3. A result related to Corollary 3.2 appears in [26] for a system known as the Zabczyk example [22]. In particular, suppose Z = ~,2, A is block diagonal with eigenvalues on the imaginary axis, and the multiplicity of the eigenvalues is unbounded. It is shown in [22] that the growth bound of the semigroup generated by A is 1. Assume that the input element b �9 ~,oo, which does not necessarily satisfy condition (ii) in Corollary 3.2. Then it is shown in [26] that Z(A, b) cannot be open-loop stabilized by co �9 H 2. While the class ~o of input vectors is larger than those considered in this paper, the conclusion in [26] is weaker than that in Theorem 3.1, since it only proves the lack of open-loop stabilizability, whereas Theorem 3.1 applied to this example proves the lack of l-open-loop stabilizability.

4. Classes of Related Inputs and Controls

Suppose (A, B) is open-loop stabilizable with co �9 ~ (see Definition 2.7), then it is reasonable to expect that if B is replaced by a more (less) unbounded input operator B, then (A, B) would be stabilizable with co in a smaller (larger) space f2. We show in Theorem 4.2 that this is true; furthermore, we show that the relationship between an increase in unboundedness of B and the corresponding decrease in size of the f~ required for open-loop stabilizability is very simple and natural.

In this section we restrict attention to infinite matrices A which are block diagonal with Jordan blocks. In particular, we assume that A can be written as

A = diagJk(; tk , nk)),

where Jk is an nk x nk matrix with 2k on the diagonal and 1 on the superdiagonal. We assume that there exists an integer N such that nk < N for all k. We further assume that A generates a group on Z = ~2, and that the input vector b is in Z-I . The set of generators which are isomorphic to these A is quite general, including many generators for hyperbolic partial differential equations.

Let f b e holomorphic in a region containing the spectrum of A. Then if Fk is a closed contour in this region with winding number 1 around 2k, f ( A ) can be defined by

f ( A ) = d i a g ( f ( J k ( 2 k , n k ) ) ) , (4.1)


where

Choose

1 I f ( z ) ( z I - 4 ) - ' dz. (4.2) f ( J I , ) = ~ r~

r < inf{Re(;tk)}. (4.3)

I f p e [-1, 1] and f ( z ) is the principal branch of zp, then we can define (A - r l ) p. Let U be any Hilbert space, p ~ [-1,1], and ~ < ~. We define the following

spaces, which we think of as Laplace transforms of controls:

2 + s~ U) := {og(s) = (s - ~)P~(s) l ,~ E n (c~ ; u ) } .

It is easy to verify that this set is independent of the choice of ~ < co. For p �9 [ - 1,0], define the holomorphic function

Z - - Z I + p

f ( z ) = 1 - z ' p e [ - 1 , 0 ] , ~ ( f ) = C \ { z ~ l R l z < O }. (4.4)

Denote the kth derivative of f by f(k).

Lenuna 4.1. There exis ts M l < oo such that Iz~f<")(z)l ___ Ml f o r all z ~ ~ ( f ) and n = 1 , . . . N .

We omit the proof of this lemma. The next theorem shows that if the input operator is made more (less)

unbounded, it requires a less (more) unbounded control to do the same amount of open-loop stabilization in the same state space. This is the open-loop analogue to the closed-loop stabilizability results Theorem 1 of [14].

Theorem 4.2. Suppose B ~ .o~'((E m, Z _ l) and p �9 [ - I , 1] tv such that (A - r I )PB �9

�9 .oq'((E m, Z - l ). Then Y~(A, B) is g-open-loop stabilizable with o9 e H2(r •m) if and

only ifE(A, ( A - r l )PB) is g-open-loop stabilizable with o9 �9 s-pH2(r ; (Era).

Proof. We write sPH2((E+; (]7 m) as ~~ Choose r < O satisfying (4.3). For + z0 �9 Z, o9 �9 H2(C+), and s e (Eg we can write, at least formally,

(s l -- A) -I (z0 + BOg(s)) = (s] - A) -I [z0 + ( A - rl)PBOg(s)(s - r) -p]

4- ( s l - A) -I [1 - (A - r l )P(s - r)-O]Bog(s). (4.5)

We see from this that if the last term on the right side of (4.5) is in 2 + H (Ca ;Z), then s is O-open-loop stabilizable by ogeH2(C +) if and only if E(A, (A - r l )PB) is 0-open-loop stabilizable with co �9 s -PH2(C+) .

Case 1: p �9 [-1,0]. For such p the condition (A - r I )PB �9 .~((E m, Z - l ) is automatically satisfied. We can write the last term of (4.5) as

(sI - A) -I (A - r I ) [ l - (A - rI)O(s - r)-P](A - r I ) - I Bog(s).

Since B e . ~ ' ( C " , Z - I ) , (A - r l ) - I B e La(cm, Z). Since co E H2(~+), the last term

154 R . R e b a r b e r a n d H . Z w a r t

on the right side of (4.5) is in 2" + H (IEg ; Z) if

( s I - A) -I (A - r I ) [ I - (A - r I )P(s - r) -p]

is a bounded operator on H2((E+; Z). By the block diagonal structure of A, this is true if there exists M < oo independent of s ~ (E+_, and k > 1, such that

II(st - Jk) - l (Jk "-- rZk)[Zk -- (Jk - r lk)P(s -- r)-P]ll~k -< M, (4.6)

where lk is the identity matrix in (E"*. Letting

r l ( s , k ) = (s - r ) - l ( J k - - rlk) ,

and noting that

(Slk -- J k ) ( J k -- r lk) -1 = (s -- r ) (Jk -- r lk) -1 -- Ik,

we see that (4.6) is true if

+ and all II(q(s,k) - l - z k ) - I ( I k - q ( s , k ) p ) l l e , k < M fo ra l l s E ( E o k > l .

(4.7)

Since

where ~r162 s, k) is the nk x nk matrix

ot - 1

0

M ( z , s , k ) = : ". .

O ...

0 .... 0 I - 1 .

"i "'. ". 1 �9 . . 1

0

I i 0~-2 ~r ... ot-nk I 0C - 1 0~ - 2 . . . a, - n j , + l

~ - l ( z , s , k ) = ".. ".. ". . �9

".. ".. Ot-I Or-2 . . . . . . 0 Ot -1

= z ( s - r ) + ( r - ,~k).

Equation (4.7) is equivalent to

I l f ( q ( s , k ) ) l l < M fo ra l l s e ( E + and k > l , (4.8)

w h e r e f i s given by (4.4). To verify (4.8), we let Fk have a winding number 1 around (2k -- r ) / ( s -- r), and

use formula (4.2) to see that

f ( q ( s , k ) ) = f ( ( s - r) -] (Jk -- r lk ) )

1 J f ( Z ) ( Z l k -- (S -- r ) - l ( J k - r[k))-l dz - - ~ Fk

i

_ s - r [ f ( z ) . . C r '


(4.8) is true if there exists M2 < oo such that

I I i s - r f ( z ) (z (s - r) + (r - 2k)) n < 3'/2 (4.9)

for all n = 1 , . . . nk, k = 1 , 2 . . . , and s ~ leg +. Since s - r and 2k - r are both in le~-, we see that

r - - r ~ ( = k s - r ] E ~ ( f ) .

The term on the left-hand side of (4.9) can then be rewritten as

( 2 k -- r~ "-I 1 f ( z ) I(X, 1--r)"-I t~-r) ~ Jr, (z-(J.,-~l(s-r))" dz

1 ( . - l f < . - l ) ( f f ) . = (n - 1 ) ! ( ; tk - r ) " - i ( 4 . 1 0 )

Furthermore, by the choice of r, i2k - r[ > m for some m > 0 and all k, so using L e m m a 4.1 wee see that (4.10) is bounded for all n = 1 , . . . nk , k = 1 , 2 , . . . , and ( e ~ ( f ) . This verifies (4.9) and finishes the p roof in Case 1.

Case 2: p ~ [0, 1]. The condition (A - r I )PB ~ .Sf(le m, Z - I ) can be written as

(A - r I ) - i + P S ~ .s Z) . (4.11)

The last term in (4.5) can be written as

(sS - A ) - i [ ( s - r) p - (A - r l ) P ] B ( s - r)-Po9(s)

= (sS - A) -I (A - rS)[(s - r)P(A - rS) -p - l ] (s - r ) - n ( A - r I ) - l+PBo9(s) .

Using (4.11), o9 ~ H2(Ie+), and the fact that (s - r) -p is bounded for s E le~-, we see that (4.5) is in H 2 ( C + ; Z ) if

( s l - A) - i (A - rI)[(s - r)P(A - r l ) -p - I]

is a bounded operator on H2(le+; Z). The p roof is now almost identical to the p roof in Case 1. �9

In the case where p = 1 we obtain a more general result. In particular, we do not need the restrictions that A is block diagonal and the state space is a Hilbert space.

Theorem 4.3. L e t A be a genera tor o f a Co-semigroup on a Banach space X, and suppose B e .L~'(lem,x) and Ix~ [ -1 , 1]. Then ~ . ( A , B ) is g-open- loop s tabi l izable with co ~ ~ H 2 ( l e + ; lem) t f and only i f Y.(A, (A - r I ) B ) is g-open- loop s tabi l i zable with o9 ~ ~ - I H 2 ( I e + ; lem).

Proof. Following the beginning of the p roof of Theorem 4.2, we see that Theorem 4.3 is true if

( s I - A ) - i [ I - ( A - r l ) ( s - r ) - I l B o g ( s )


is in H2( r for co e sUH2(~+;r Since o)(s)= (s-r)~&(s), where r e H2(C+; cm), we can simplify this to

(s - r) ' - I BrS(s),

which is clearly in 2 + H (C o ;X).

Example 4.4. We illustrate Theorem 4.2 with the following simple example. Assume Z = ~2 and A is a diagonal operator with simple eigenvalues )-k. Suppose A generates a group, and there exists ~t e ~, such that, for n e N,

Im(2,) = 0tn + O(1), (4.12)

where in this context O(1) is a bounded function of n. Suppose b e Z-I and there exists m, M e F, such that

0 < m < Ib.I < M.

It is well known that there are one-dimensional wave equations with boundary control which can be represented in this form, and that there are feedback controls which closed-loop, and hence open-loop, stabilize this system with controls in L2[0, oo). Therefore Z(A, b) is open-loop stabilizable with 09 e H2((E+~; ~) for some e > 0.

Now assume that b e Z_1 has components bk, and there exist m, M e F, and p e [-1, 1] such that

m < IAf bkl < g for k # 0. (4.13)

Then Theorem 4.2 implies that Z(A,b) is open-loop stabilizable with 09 ~~162 (E). In particular, we can open-loop stabilize Z(A, b) even if b e l 2, as long as we allow controls u which are in a larger space than L 2 [0, ~ ) .

Assume now that, for all k, ReAk > 0. Define the class ~p c Z- i as all b ~ Z- i such that (4.13) is satisfied. We see that if b e ~p for somep e [-1, 1], then Z(A,b) is open-loop stabilizable. It is a surprising fact that if b can be decomposed into finitely many component vectors, two of which have infinite length and are in dif- ferent classes ~p, then Z(A, b) is not open-loop stabilizable. This is a special case of Theorem 4.8 below. �9

For the remainder of this section we work with diagonal systems with one- dimensional input spaces. In particular, we make the following assumptions: A generates a strongly continuous semigroup on the Hilbert space Z, b e Z- l , A has eigenvalues {'~k}keN, with associated one-dimensional eigenspaces generated by the eigenvectors {(Pk}kElq, which form a Riesz basis of Z. Since {~k}k~N is a Riesz basis of Z, it has a biorthogonal set {~bj}j~N, so (~k, ~'j)= ~jk. We can represent b and a given initial state z0 by

b : ~ bk~k, bk : (b, ~k>, k=0

o o

Zo : ~_~ Yk~k, k=0

y~ = (z0, 0 D .


oo r oo g2 Here {Yk}k=O ~ ~2 and {bk/(2k - - )}k=0 E~ for some r such that 2 k - - r is never zero. Let 3 be the index set

.II = { j ~ NIRe(2j) > 0}. (4.14)

We now give some necessary conditions for E(A, b) to be open-loop stabilizable. These results are conditions on the density of the set of eigenvalues. These results can be compared with the proofs in [19], where the density of the eigenvalues leads to a lack o f exact controllability.

T h e o r e m 4.5 .

and

I f Z (A,b) is open-loop stabilizable, then

inf [2, - 2" I > 0 "e$ ,n~N,n~"

(4.15)

[ bn q sup (4.16) , .~a , .~ , .#m (2" ---2.)b" < oo.

Proof. The proof here is similar to Theorem 3.2 in [16], which dealt with the case where A was the generator o f a unitary group and b ~ Z. Assume that E(A, b) is open-loop stabilizable, then there exists an e > 0 such that E(A, b) is -e -open- loop stabilizable. In particular, ~ and co are holomorphic at 2k for k e $.

First note that if 2, = 2m for some n :~ m, then E(A, b) is not open-loop stabilizable, since part three of Theorem 2.11 is violated. I f we take s = 2k, k ~ $ in (2.12), and take the inner product o f (2.12) with ffk, using the fact that (2k -- A*)~kk = 0, we get Yk = --bkto(2k). Therefore 09 given in Theorem 2.5 solves the interpolation problem

cO(2k) -- Yk k ~ $. (4.17) bk '

I f we take z0 = tpm, we refer to ~ and 09 in Theorem 2.5 by ~" and tom, respectively. Then with s = 2,, n e .ll, in (2.12), where n # m, we use (4.17) to get

r = ( 2 j - A)~"(2.) - bto"(2.) = (2.1 - A)~"(2.) .

Taking the inner product with ~,",

1 = ( ~ , . , ~k"> = ( ( 2 f l - A)~"(2n), ~lm) : ( 2 n - - 2")(~"(2n), ~i"). Hence

~_1 = 1<~"(2.), r MIII~"(2.)IIz M2ll~"llmlr _< < M~, (4.18)

for some Ml, ME, M3 e IR independent of n and m, where we use Lemma 2.8 for the last estimate. This proves (4.15)

For m e $, take z0 = 4" and s = 2,, in (2.12) and using (4.17), we get

1 q~m = ( 2 " I -- A)~"(2m) - b to" (2" ) = ( 2 " I - A ) r "b b b"~m"


Letting n 4= m and taking the inner product with ~b,, we get

b, o = (qJ,., ~o.) = )' , .(~m()'m), ~0.) -- ) , . (~ . , ( ) ," ) , q . . ) + ~ .

Hence I 'b, [

(),,. - -2 , )b , . = I(~,.() ' , .) ' ~k">l < M4

for s o m e M4 E ~ independent of n ~ IN and m E $, where the last inequality follows f rom the same reasoning as in (4.18). This proves (4.16). �9

Now assume that the index set $ is IN. In this ordering, assume that there exists e R such that (4.12) holds. We now give an easily checked condition on b e Z - l

which guarantees that Y.(A, b) is not open-loop stabilizable. This condition differs considerably from other conditions which guarantee a lack of stabil izabil i ty--see, for instance, [13] for conditions on the lack of closed-loop stabilizability. Most other such conditions are that the input operator is " too s m a l l " - - f o r example, compact , or with range in the domain of A - - t o allow stabilizability of infinitely many unstable modes.

Theorem 4.6. Let b ~ Z-1 and let A generate a Co-group and satisfy (4.12). Sup- pose .II = $1 u $2, where $1 and $2 are infinite sets, and there exists Pl, P2, Pl v~ P2, ml, m2, m3, m4 ~ IR such that

b. = I),.IP'c,, 0 < mt < Ic, I _< m2 for n ~ $1, (4.19)

b. = I),.IP2c., 0 < m3 < Ic.I <- ma for n e $2. (4.20)

Then Z(A, b) is not open-loop stabilizable.

Remark 4.7. I f we refer to the vector defined by (4.19) as b I and the vector defined by (4.20) as b 2, we see that b I e ~pt and b 2 e ~p2, where ~'a is defined in Example 4.4. Thus we see that if b can be decomposed into two infinite vectors, one in hot and the other in ~ with P l r P2, then Z(A, b) is not open-loop stabilizable. Note that Example 4.4 shows that this is true despite the fact that if b is in either ~pt or ~p2, Z(A, b) is open-loop stabilizable.

Proof. Suppose that P2 > PI- Let $3 be the set of all n such that ),. e ,lll and )',+l e $2. Since $1 and $2 are both infinite, $3 is also infinite. Using (4.19) and (4.20), for n e JI3,

b.+x m3 )'.+1 (4.21) ()'. - ),.+t)b. > ()'. - - - P' - - ) ' . + 1 ) ) ' .

Now note that

I)', - - )`.+11 z = (Re(),.) - Re() ' .+l))2 + (Ira(),.) - Ira()`,+1))2. (4.22)

Since A is a generator o f a C0-group and $3 c Jl, (4.14) implies that the first term


on the r ight o f (4.22) is bounded . By (4.12), the second term on the r ight o f (4.22) is b o u n d e d . Hence there exists M > 0 such that (4.21) is greater t han or equal to

MI

Since P2 > Pl, this goes to +oo as n --* oo. I f V~(A, b) is open- loop stabil izable, this cont radic ts (4.16), which completes the p ro o f if P2 > Pl. The p r o o f for Pl > P2 is a lmos t identical . �9

A s imilar p roo f can be used if b can be decomposed into finitely m a n y classes �9 ~p,:

Theorem 4.8. Let b ~ Z - i and let (4.12) hold. Suppose that, for some n ~ IN, $ = Uk=t Sk, where at least two $k are infinite sets, and there exists {Pk}~=l c IR (Pk V~pTfor jv~k) , {mk}k=l ~ ,and{Mk}k=l ~ s u c h t h a t , f o r a l l k = I , . . n,

bi = ]2ilPkci, 0 < mk < [ci[ < Mk for i ~ $k.

Then E(A, b) is not open-loop stabilizable.

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[26] H. Zwart; A note on applicalions of interpolation theory to control problems of infinite-dimensional systems, Appl. Math. Comput. Sci, 6 (1996), 5-14.

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Open-loop stabilizability of infinite-dimensional systems

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