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Open Loop Stabilizability, a Research Note
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OPEN LOOP STABILIZABILITY, A RESEARCH NOTE H. Zwart /)('/)(1111111'111
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Abstract. In this paper we study the stability of the infinite-dimensi onal system x-Ax+bu, with an one dimensional input operator. The main resul t is that if this system is open loop stabilizable i.e. x(.) is square inte grable for some u(.), then the unstabl e part of the point spectrum of A consists of only point-spectrum with finite multiplicity, Furthermore there exis ts a feedback that moves all these poles. keywords: Distribured parameter Stability, Optimal control
1.
systems,
Linear
INTRODUCTION,
2,
In this paper we shall study the stabilizability of a single input, infinite dimensional, linear and
x(t) -Ax(t)+bu(c) x(O)-x o
where Xo is an element of the separable Hilbert space H, A is an infinisimal generator of the semi group T(c) on H, b is an element of H, b~O, and u(.) is locally square int egrable function with values in IR (X~oc([O,oo);IR», By the solution of (1.1) we mean the mild solution, see (Curtai.n and Pritchard 1978).
such
a
sys tern
(1.1)
Theorem 2.2, For the system (A,b) the following assertions are equivalent i) The system is closed loop stabilizable. ii) There exists a Q
( 1. 2)
and
(1.2)
by
(A,b).
The stability and stabilizability of this system is one
of
the
main
topics
in
infinite-dimen s ional
system theory see e.g. Pritchard & Zabczyk (1981), Russell (1978) and the references therein. In the next section we shall recapulate the recent results on closed loop stabilizability, meaning under what conditions on (A,b) does there exists a FE X(H,IR) such that the solution of the closed loop system
x(t)-(A+bF)x(t) x(O)-x o
CLOSED-LOOP STABILlZABILlTY,
We remark that throughout this paper' we only consider this type of stability, for other definitions of stability, see e.g, Arendt & Batty (1988) and Pritchard & Zabczyk (1981).
t
denote
analys is,
Definition 2.1: Closed Loop Stabilizability,l The system (A,b) is closed loop stabilizable if there exists a bounded operator F, such that the semigroup generated by t,+bF, TF(t) is exponen t ially stable, i.e, IITF(t)llsHe t; (J
(1.1)
x(t)-T(C)xo+f T(c-s)bu(s)ds o
System
We shall start by giving a formal definition of the concept of closed loop stabilizablity.
time -invariant system, which we represent as
and we
systems,
nu
a,+
flS
restricted to Hu is controllable. iii) For every XoE H, there exists a XZ( [ O, oo» input u(.) such tzhat the solution of (1 . 1) is an element of X ([O,oo),H). Or, equivalently for every Xo the followiW; cost criterium can be made finite; J(Xo,u)-f Ilx(t)llz+llu(t)fdt.
(1. 3)
Remark:
This
theorem
also
holds
if
the
input
space is not one, but finite dimens ional .
converges to zero exponentially, for every XoE H. We shall see that the c lo sed loop stabilizability implies that in the unstable part of the spectrum of A there are finitely many eigenvalues with finite multiplicity. In the third section we shall introduce a weaker concept of stabilizability, namely the concept of open-loop stabilizability, which states XZ behaviour of x(c) for some input u(t), depending on xo · As in the closed loop case the solvability of this problem implies that the unstable part of the spectrum of A consists of only eigenvalues with finite multiplicity. In the last section we shall investigate the relation between open loop stabilizability and closed loop stabilizability,
Remark: This result is one of the most important results in the theory of infinite dimensional systems, The implication ii) to i) is known to hold for a long time, see (Triggiani, 1975). Of the other implication it was known that the spectrum of A restricted to Hu ' consists of only point spectrum wiht finite multiplicity. These results can be eaSily deduced from the the Weinstein-Aronszajn determinants, (Kato, 1984, pp. 244- 250), Notice that the finite dimensionality of Hu was unknown and so was the decay of T(t)I ' The complete
Hs
implication i) to i i ) is proved independently of each other by Jacobson & Nett (1988) and Nefedov & Sholokhovich (1986). The proofs in these papers are totally different and only treat the Hilbert space case. Curtain (1988) extended the results of J.
This notion is also called exponentially stabilizability or just stabilizability.
III
112
H. Z\\,art
Nefedov &. Sholokhovich to a class of unbounded finite rank input operators . She also treats the Hilbert space case only. The implication i) to ii) follows also directly from theorem 4 of (Desch &. Schappacher, 1985). This result is stated in a general Banach space and also incorporates a class of unbounded finite rank input operators. So not only was it published before (Jacobson &. Nett , 1988) and (Nefedov &. Sholokhovich , 1986), it is also more general . Furthermore we remark that the proof as presented by Desch and Schappacher differs from that giv en in (Curtain, 1988) , (Jacobson &. Nett, 1988) and (Nefedov &. Sholokhovich , 1986) Furthermore (Curtain, 1988) and (Desch &. Schappacher, 1985) treat different classes of unbounded input operators , yet most examples will be in both the classes. The implication iii) to i) was already poved by Datko (1971), and gives a relation between open and closed loop behaviour of the system (A,b).
3. If
the
OPEN LOOP ST ABILlZABILlTY.
system operator A
neccessary conditions
does
not
satisfies
of theorem 2.2.11) ,
the
that
is
the unstable part of the spectrum does not c ,mtain finitely many points. Then would it be possible to stabilize this system by means of e.g. an unb ,mnded feedback law , or a time dependent compensat(lr? In order to answer these questions we shall
int(oduce
the concept of open loop stabilizability. This concept is weaker than the concept of stabilizing the system (A,b) by a bounded fe " dback law,
that we considered in 2.
We shall start stabilizability.
with
defining
our
concept
of
Definition 3.1: Open loop stabi l izability. The s y stem (A, b) is open loop stabilizable if for every XoE I( there exists an i nput u(.)e r~=( [ O, oo );~) such that the mild 2 (1.2) of (1.1) satifies X(.) E r ([O , oo);I().
solution
So roughly speaking we have tha t the sys tem (A, b) is open loop stabilizable if for every initial condition there exists an input such that the solution of (1.1) is a r2 function.
xo- (s-A)E(s)-bw(s)
( 3.2)
ii)
0a .(A) contains no finite accumulation point fo'r all a>O. iii) In the half-plane IL. the spectrum of the operator A consits of only point spectrum with finite multiplicity. Proof: See (Zwart , 1988b). Remark: A representation of the form (3.2) is called a (E ,w)-representation of x o , and was introduced for the finite dimensional cas e by Hautus, (1980). Remark: One can also show that open-loop stabilizability implies that the system is modally controllable on IL., and this will imply the results of the theorem too, see Pandolfi (1987). Under stronger asswnptions a similar theorem was found by Fattorini (1975) and Voigt (1980). In the next remarks we shall explain what this result tells us. So from ii) and iii) we have that in every compact subset of [ contained in [+ there are only finitely many points of the spectrum of A, counted with multiplicity. Furthermore properties ii) and iii) give that even with arbitrary inputs one cannot open loop stabilize a system which has an acummulation point in the unstable part of its spectrum and one cannot remove the residual part, or the continuous part of
the spectrum or eigenvalues with in f inite multiplicity. So if A is the identity operat:or on an infinite dimensional Hilbert space, then it is not open-loop stabilizable . The proof of theorem 3.2 is based on the conc e pt of (E,w)-representation, see (3.2). This representation is essentially the Laplace transform of (1. 2) and it has as advantage that instead of 2 working with functions from r ( [ O,oo);I() or r~o , ( [ O,oo) ;~) one works with functions which are holomorphic on some right half plane. Since a lot is known about holomorphic functions and they have very
nice
properties
we
using this (E,w) (Zwart 1988a , 1989a).
In this section we are interested in the properties
4-.
of open-loop stablizable systems (A,b). In order to formulate the main theorem of this section we shall
can
prove
5
trong
representation,
THE RELATION BETWEEN OPEN LOOP ST ABILlZABILlTY
resul ts
see
also
AND
CLOSED
need some notation
IL.,-:~(ZE
I llez>a)
for aE
~
IL I llez5a) for aE
~
If •.• : - (ZE IL
In this section we shall stabilizability implies stabilizability. First properties of the w(. ) in
IL.:~ o ,.
Lemma 4,1, If Xo has a (E,w)-representation with E(.)E H2(1() , then w(s) is an element of H2 for every .\ with
o(A) is the spectrum of A o a,' (A) - o (A)('j[ a " o a , _ (A) - o (A)('j[ a,-
s-.\
(3.1)
I(IL • • ;Z) denotes the set of all ' holomorphic functions on ILa. wi th values in a Hilbert space Z . '
lle(.\)
-.!.
is
finite,
(Rosenblum
xo~(s-A)E(s)-bw(s)
D(A') such that
see &.
From the assumption we have that Xo be be
written as
H2(1() will denote the Hardy space of all functions f(. )el(IL.;I() such that
2 sup ( 1I f(x+iy) 11 dy x>O (Duren, 1970) and Rovnyak, 1985) .
investigate i f open loop a form of closed loop we shall prove extra equation (3.2) .
~
<.Yb,b>~l, then
«s-A)Us) ,h> -
-
«s-A)E(s)'Yb> - w(s) Hence
1
1
1
~O'Yb>~(S-A+'\-A)E(S)'Yb>-~(S)
H2:_H2(1L) .
Or Theorem 3,2 , Assume that the system (A,b) is open loop stabilizable, then i) for every Xo in I( there exitst E ( , )E H2(1() , with E(S)E D(A) for sE IL., and w(.) E I(t.;if ) such that on IL. the following representation holds
For .\ in the open-left half plane th~ right hand side of (4.1) is an element of H2. So ~(s) is an element of H2.
Opell Loop Stabilizability, A Research Note corollary 4.2. If bED (A) , then open loop and closed loop stabilizability are equivalent. Proof: Let>. be a negative real number. th,m
(>.-A)b-(s - A)b+(>.- s)b
(4.2)
xo-{s-A) [
113
U~~~xo]
From lemma 4.1 we have that w(s) E H2 (~,w) - representation
Taking the (4.6) yields
b___l__ (>. _A)b + (s_A)[ __l__ b] >.-5 5->'
( 4.3)
From theorem 3.2 we know that for every XoE JI the following representation holds xo-(s-A)E (s)- .)w(s); 2 with E(.)E H (JI). Combining this with equation (4.3) gives xo- (s - A) [US) Is=>.b] So every Xo in JI has a (E system (A, (>' - A)b) with
'" ), (s)_w(s)_>.l are in -5
(>. - A)b{w(s)>.=s)
( 4 . 4)
,w) - representatio~
E>. (s)-E (5)
+
f,)r the 5_>.1> and
and H2 respectivel." .
H2(JI)
see
the previous lemma. By the Paley- Wiener theoJ:em we have that the inverse Laplace transform of a H2 is a f.2 function. So if we take the inverse laplace transform of equation (4.4) yields that for Xo there exists an input trajectory in f.2([O,~», namely the inverse Laplace transform of w),(.), such that the transform
state trajectory, the invese Laplace of E),(.), is in f.2([O,~);I(). A simple
c alculation further shows that the inverse Laplace transforms of these trajectory satisfy equation (1.2) i.e. t
L-l(E),) (c)
+
f T(C-v) ()'_A)bL-l(w),) (v)dv.
o Since Xo is arbitrary we have by theorem 2.2 that the system (A,(),-A)b) is closed loop stabilizable. In order to prove that (A,b) is closed-loop stabilizable we only have to show that the system (A,b) restricted to Jl u is controllable (theorem 2.2.ii). This follows easily from the controllability of (A, (),-A)b) on Jl u ' Corollary 4.3 If A is a bounded operator, then open-loop and closed loop stabilizability are equivalent. Proof: This follows easily from corollary 4.2, since if A is bounded, then D(A) is JI. From theorem 2.2 we see that closed loop stabilizability implies the finiteness of a cost criterium. The next lemma will show that open loop stabilizability implies a similar but weaker property.
L-
l
L-
the
with X(t)-T(C)Xo+f T(t-s)bu(s)ds. o Furthermore D(A)CI!,. Proof: Let JI, be the subset of I( containing all initial conditions XoE JI such that the cost criterium (4.5) is finite for some input fun-otion. is the norm on f.2 and the :;ystem
o
0.1) is linear, we obvious ly that JI, is a linear subspace. So it remains to show that this subspace Contains D(A). Let XoE D(A) and let ), be a negative real n'lIDber, then from theorem (3.2) we have
(),-A)xo-(s-A)E(s)-bw(s) 2
with E(.)E H (JI). So
(),-s)xo-(s-A){E(s)-xo} - bw(s)
Paley-Wiener
theorem
l
[
we in
the
have
that
f.2([O,~) ;JI) and input L-l [W~~~J
the cost criterium (4.5) is finite. Since the domain of A is dense in JI, the same holds for JI,. In the next lemma we shall use the cost criterium (4.5) and the above lemma to construct a feedback law F such that A+bF has no poles in the righ t -half plane. Therefore we shall first prove that if the cost criterium can be made finite then there exists a unique input which minimizes the cost, lemma 4.5. After proving some technical lemmas concerning this unique input, lemma 4.7, 4.9, and 4.11, we shall construct this feedback and show that it has the desired properties, theorem 4.12. I
Lemma 4.5. If the cost criterium J(xo'u) is finite, exists an unique U such that
then there
J (xo ' u):5J (xo ' u)
for all U(.)E f.2( [ O,~» . Proof: Let the pair minimizing input, state criteriwn T JT(Xo,U)-f Ilx(c)
(UT,X T) be trajectory
the unique of the cost
2 2 11 +llu(t)11 dC
see (Curtain and Pritchard, 1978) for details. If we se~ uT(t) - O ,for t>T, and xT(C) idem, then (uT,XT)E f. ([O,~»(!)f. ([O , ~);I() for every T>O. Furthermore since J(xo,u) is finite, the se 'luence (uT,XT) is bounded, so since f.2([O,~»(!)f.2([O,~);JI) is a Hilbert space we have the existence of a pair (u,x) in f.2([O,~»(!)f.2([O,~);I() which is the weak limit of (UT,XT), and x(C)-T(t)x o +
f'T(C - s)bU(s)ds. Since ~
f have
t
(),-s+s-A)xo-(s-A)E(s) - bw(s)
element of transform of
U~~~xo] (C)-T(C)Xo+tT(C-V)bL-t~~~] (v)dv
U~~~xp] and L-t~~~] are f.2( [O,~» resecptively. So with
we
o
f~IIU(t)112dC
is an Laplace
T
f
2 Ilx (c) 112+llu (c) 11 dt"
Ilx (c) 112+ Ilu (c) fdC"
( 4.5)
J(X o ' u) - (1IX(t) 112+llu (t) 11 dt
Since
[
which inverse
o By
Lemma 4.4. If a system (A,b) is open loop stabilizable, then there exists a dense subspace 1(£ such that for every XoE JI, the following cost criterium can be made finite for some input function 2
So Xo has a
),-5
H (JI)(!)H2.
So
b[w~~~]
-
~ ~
( 4.6)
that
lim JT(XO'U T)
5
T_
Kato (1984; p. 253, lemma 1.4) this implies that (UT,x T) converges strongly to (u,x). Furthermore since (uT,XT) converges weakly to (u,x) we have that lim JT(xO'u T) " J(xo,u).
T_
So
lim JT(XO,UT)-J(XO'U),
and
this
implies
that
T_
J(xo,u)-inf J(xo'u) . It is this input is unique.
a
standard result
that
If we have that J(xo'u) is finite for some u(.), then we can take the Laplace transform of equation (1.2) which gives that Xo has a ~E ,w) representation with (E,w) an element of H2(JI)®H . The space ~(JI)(!)H2 is a Hi1bert space with the inproduct given by «E I ,w, ) , (E 2 ,w2 »
- J~
(4.7)
+ _loow,(ir)w 2 (ir)dr
The pair (E,~) has the minimal norm among all 2 2 (E,w) -representations of Xo in H (KJ0H if and onl~ 2 if ,~) 1. (Eo ,wo) for any pair (E ,wo) in H (K)0H satisfying
(E
O_(s_A)Eo(s)_bwo(s)
(4.8)
Lemma 4 . 9 . The subspaces for all Proof:
s"
E
2
A pair of functions (E,w) E H2(K)0H2 is said to be a minimal pair if there exists a XoE K such that
and it has the satisfying (4.9).
minimal
norm
among
all
pairs
Wz
USo)~(s-A)
[US) -Us o)]
(so-s+s -A )E (so)-bw(so)( s-A)E (so) - (so-s)E (s o) -bw (so)
but
E2 (s 2 )~(s 2 -s 1 )
l[US)-Uso)] , W(S)-W(so)]] 5 0 -5
representation of E(so)' and is analytic [[
1
5 0 -5
is
(E ,w) is in we have that
is in H2 (K)0H 2 too.
f f
E(s,)
may
-
E,.
conclude
1
)~(s-A)
[El (S)-US1)] 5
1
_b[W l
-5
(s)-u.,~] 5
1
-5
2
(S)-U S2)] -b [W2 (S)-W(S2)] 5
-5
2
5
2
-5
52- 5
5 1 -5
r-;oo ~,
see (Doetsch, 1974). So · [Wl (r)-W(Sl)] W ( 51 ) = 1 lm r 1
ooWO(ir) [W(ir)-W(SO)]dr
r-700
s l -r
lim r[W2 (r)-W(S2)] sz-r
so-lr
1<~ °
we
-so'
for all sE IC+. For a H2 function f(.) we have that lim f(r)-O for rE
[E(ir) -E(s o)]> dr + 'so-lr K
_00
~
2
symmetry
of
Since E,(s,) - E2 (s2) and both representations are minimal we have by the uniqueness that Wl (S)-w(s, )] _ [W2 (S)-W(S2)] (4.10) [
Thus it remains to prove that this pair is minimal. 2 2 Let (Eo ,wo) be an arbitrary pair in H (K)0H satisfying (4.8). Then -00
element
SI -5 2
by
EZ (s 2 )~(s-A) [E
(E ,w)-
a
5 0 -5
oo
an
S
and
(so-s)E(s o)
Since in So
E(S) -Us o)] , [W(S) - W(so)]]
defintion
[U 2) -E(Sl)]+US)
and
,
E (s
5 0 -5
2 2 H (K)0H ,
by
Definition 4 . 10: _ for any sE IC+.
(s-A)E(s)-bw(s) ~ (s-A)Uso)-(so-s)E(so)-bw(so'> =>
So
+ w(s)
equality.
So
~E
-5
Lemma 4.11 Let (E"w,) and (E 2 ,w2 ) both be minimal pairs. I f E,(s,)~E2(s2)' then w,(s,)-w 2 (s2)' Proof : From lemma 4.6 we have that the m lnimal pairs for E,(s,) and E2 (s2) are respectively given by;
-
(s-A)
1
1
is
3 52
5
( S ) -_ ( 52-51 ) [W(S)-W(Sl)] 5 -5
E2 (s2)
::;:s ,C
+ Us) and
[US)-US1)]
::;: :~ 3"
5 0 -5
xo~(so-A)E (so)-bw(so)
Then there
_ b [ W(S)-W(SJJl]
5 0 -5
Proof:
be an has a
(s)~(s 2-s 1 )
Now
So Lemma 4.7. Let (E ,w) be an minimal pair and let So arbitrary element of IC+, then E(so) also minimal pair and this pair is given by
~sl-Es2
i. e.
are independent of s.
exists a minimal pair for a x, such that E(s,)-E,. Define x 2 as (s2- s ,)E,+x, , then by the previous lemma and the linear dependence of the optimal pair upon the initial state we have that the minimal pair for x 2 is given by
Definition 4 . 6.Hinimal Pair.
(4.9)
~s
S2 in IC+. Let E, be an element of 3 5 , '
r-)oo
~
w (s2) 2
00
-
(ir) ,E(ir) - E(so»K dr +
;:~~:){W(ir)-W(So)}dr
_loo
rOO<~O(ir) ,E(ir»Kdr
-~
+
so+ir
OO
DC)
f ~ ° . f0
__
Ir w(ir)dr + _ so+ir
'-'V
00
+
Theorem 4.12 Suppose that the system (A, b) is open loop stabilizable. Then there exists a linear operator F from 3 to IC such that i) I f (E ,w) is a minimal pair, then w(s)~FUs), for all sE IC+ . ii) The operator s-A-bF : 3 ~ Kf is injective for all sE IC+ . Proof:i) Let Eo be an arbitrary element of 3. Then there exists a minimal pair (Eo(s),wo(s » and a So such that Eo(so)~Eo' Now define
so+.lr
lOO
EO(ir)
<-_------,E(sO»K dr -s o-ir
W (Ir)--
( 4.11)
~(so)dr.
00
-S oalr
Now (~ (s) ,~ (s) 5 0 +5
2
is an element of H (K)0H
2
and it
5 0 +5
satisfies (4.8). So since (E , w) is a minimal pair the sum of the first two term ofo the last equation is
zero.
Furthermore
since
analytic function in theorem, (Rosenblum ( EO (ir) ,!, <-_--.-'UsO»K dr -
0
is
an
O. The same holds for the w
-So-lr
term. Definition 4 . 8 . 3 5 By 3 5 we shall denote the following subspace of K. 3 5 0 - ( EoE K I :3 Xo and a minimal pair (E ,w) such that
E(so)~Eo
and
xo~(s-A)E(s)-bw(s)
FE(s) ~ w(s)
-5
we have b y the Cauchy that Rovnyak, 1985) ~
By lemma 4.11 this operator is well defined, for if (E,(s),w,(s» would be another minimal pair with E, (s,)~Eo' then w, (s,)-wo(so) . Now it is obvious that if (E(.),w(.» is a minimal pair, then (4.12)
for all sE IC+.
The linearity of F follows easily from the linearity of the minimal pairs i.e. if (E"w 1 ) and (E 2 ,w2 ) are both minimal pairs, then (E,+E 2 ,w,+w2 ) is a minmal pair too. ii) Suppose that SoE IC., and let EoE 3 satisfy (4.13)
Since pair
a
XoE Kf that
and
a
Eo(so)-E o '
mlnimal So
for
Open Loop Stabilizability, A Research Note some XoE Kf we have that xo~(s-A)Eo(s)-bwo(s),
Eo(so)-E o
(4. 14)
By part i) we have that FEo(')-wo(')' So xo~ (so-AYE 0 (so)-bw o (so)-(so-A)E o-bFE 0- 0
Thus xo-O and since (Eo(')'w o ('» is the mlnimal pair for Xo we have that EO(.)EO and wo(.).,O. In particular Eo-Eo(so)~O. So we can defined the formal inverse of (s-.~-bF), and for xoE Kf we have E(s)-(s-A-bF)-lx o ' If He can extend the operator (s-A-bF)-l to the whole state space, then we have moved all unstable poles ~o the left-half plane. Then it remains to show thal: A+bF generates a stable semigroup. This is one of the many open problems. Another problem is to show the relation between this feedback and the Ricatti equation. Formally one would expect that F~-b"P, where P is the positive solution of the Ricatti equation AP
+ PA" - Pbb"P + I ~
0
Roughly speaking we may conclude that if the system (1.1) is open loop stabilizable, then it is also closed loop stabilizable. This stabilizing feedback law is linear and time invariant, but it may be unbounded.
REFERENCES
Arendt,W. and Batty,C.J.K. (1988). Tauherian Theorems and Stability of One-Parameter Semigroups. Trans. ~ Math. ~ 306, 837-852. Curtain,R.F. (1988). Equivalence of Input-Output Stability and Exponential Stability for Ma them., tical Infinite-Dimensional Systems. System Theory 21, 19-48. Curtain,R. F and Pritchard, A.J. (1978). Infinite Dimens ional Linear Sys tern Theory. in Balakrishnan and Thoma (Eds), Lecture No~es in Control
and
Information
Sciences.
vol.
8
Springer Verlag, Berlin. Datko,R. (1971). A Linear Control Problem in an Abstract Hilbert Space. L Diff . fuL .2" 346-359. Desch,W. and Schappacher,W. (1985). Spectral Properties of Finite-Dimensional Perturbed Linear Semigroups. L Diff. ~ ~ 80-102, Doesch,G. (1974). Introduction to the Theory and Application of the Laplace Transform. Springer Verlag, Berlin. Duren,P. (1970). Theory of Hp Spaces. Academic Press, New York. Fattorini,H.O. (1975). Exact Controllability of Linear Systems in Infinite Dimensional Spaces. in Partial Differential Equations and Related Topics (Program, Tulane Univ., New Orleans, L.A., 1974), Lecture Notes in Mathematics vol. 446. Springer Verlag, Berlin, 166-183. Hautus,M.L.J. (1980). (A,B)-invariant and stabilizability subspace, a frequency domain description. Automatica 16, 703-707. Jacobson,C.A. and Nett, C.N. (1988). Linear State-Space Systems in Infinite Dimensional Space: The Role and Characterization of joint Stabilizability/Detectability. IEEE Trans. on ~ 33, 541-549. Kato,T. (1984). Perturbation Theory for Linear Operators. in Grundlehren der Mathematischen Wissenschaften, 132. Springer Verlag, Berlin. Nefedov,S.A. and Sholokhovich,F.A. (1986). A Criterium for the Stabilizability of Dynamical Systems with Finite-Dimensional Input. Differentsial'nye Uraveneniya 21, 163-166, New york, Plenum. Pandolfi,L., (1987). Some Properties of the Frequency Domain Description of Boundary Control Systems. Rapporto Interno 1,
li S
Polytecnico di Torino, Dipartimento di Matematica, Torino, Italy . Pritchard ,A. J . and Zabczyk,J. (1981). Stabili:y and Stabilizability of Infinite Dimensional Systems. SIAM Review 23, 25-52. Rosenblum, M. and Rovnyak, J.(1985). Hardy ~lasses and Operator Theory . Oxford Univ'Hsity Press, New-York Rudin,W. (1966). Real and Complex Analysis. in McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Company. Russell,D.J. (1978). Controllability and Stabilizability Theory for Linear Pa rtial Differential Equations: recent progress and open problems. SIAM Review. ~ 639-739. Triggiani,R. (1975). On the Stability Problem in Banach space. L Ma th. An. and AP.P.L.... 52, 383-403. Voigt,J. (1980). A Perturbation Theorem for the Essential Spectral Radius of Strongly Continuous Semigroups. Monathefte fur Mathematik. 90, 153-161. Zwart,H.J. (1988a). Equivalence between Open loop and Closed Loop invariance for Infinite Dimensional Systems: A Frequency Domain Approach. SIAM Journal on Control and Optimization 26, 1175-1199. Zwart,H.J. (1988b). Some Remarks on Open and Closed Loop Stabilizability for Infinite Dimensional Systems. presented at the fourth conference on control of distributed parameter systems, Vorau, Austria, 1988. Zwart,H.J. (1989a). Geometric Theory for Infinite Dimensional Systems. in Thoma and Wyner (Eds), Lecture Notes in Control and Information Science, vol 105 , Springer Verlag, Berlin. Zwart,H.J. (1989b). On Stability of Infinite Dimensional Systems. preprint.
C:OI)\Tj~hl I F.- \C COIlI rol 01 Dj'l rjhllll'd Parallleter S\"Slt: IIl S. Pc:rpigl1;lIl. Frall(,(,. I ~IH~I
OPEN LOOP STABILIZABILITY, A RESEARCH NOTE H. Zwart /)('/)(1111111'111
uf' A/)/J/in/
,1/(////1'1//(//1(.1, L'lIh'l'l'.Il/y
of'
h/,n//I',
P.(). N()'\ 21/,
/5IJIJ AI:' l :·II.1l'hl'r/I', Thl' .\'dhnlrlll!l,
Abstract. In this paper we study the stability of the infinite-dimensi onal system x-Ax+bu, with an one dimensional input operator. The main resul t is that if this system is open loop stabilizable i.e. x(.) is square inte grable for some u(.), then the unstabl e part of the point spectrum of A consists of only point-spectrum with finite multiplicity, Furthermore there exis ts a feedback that moves all these poles. keywords: Distribured parameter Stability, Optimal control
1.
systems,
Linear
INTRODUCTION,
2,
In this paper we shall study the stabilizability of a single input, infinite dimensional, linear and
x(t) -Ax(t)+bu(c) x(O)-x o
where Xo is an element of the separable Hilbert space H, A is an infinisimal generator of the semi group T(c) on H, b is an element of H, b~O, and u(.) is locally square int egrable function with values in IR (X~oc([O,oo);IR», By the solution of (1.1) we mean the mild solution, see (Curtai.n and Pritchard 1978).
such
a
sys tern
(1.1)
Theorem 2.2, For the system (A,b) the following assertions are equivalent i) The system is closed loop stabilizable. ii) There exists a Q
( 1. 2)
and
(1.2)
by
(A,b).
The stability and stabilizability of this system is one
of
the
main
topics
in
infinite-dimen s ional
system theory see e.g. Pritchard & Zabczyk (1981), Russell (1978) and the references therein. In the next section we shall recapulate the recent results on closed loop stabilizability, meaning under what conditions on (A,b) does there exists a FE X(H,IR) such that the solution of the closed loop system
x(t)-(A+bF)x(t) x(O)-x o
CLOSED-LOOP STABILlZABILlTY,
We remark that throughout this paper' we only consider this type of stability, for other definitions of stability, see e.g, Arendt & Batty (1988) and Pritchard & Zabczyk (1981).
t
denote
analys is,
Definition 2.1: Closed Loop Stabilizability,l The system (A,b) is closed loop stabilizable if there exists a bounded operator F, such that the semigroup generated by t,+bF, TF(t) is exponen t ially stable, i.e, IITF(t)llsHe t; (J
(1.1)
x(t)-T(C)xo+f T(c-s)bu(s)ds o
System
We shall start by giving a formal definition of the concept of closed loop stabilizablity.
time -invariant system, which we represent as
and we
systems,
nu
a,+
flS
restricted to Hu is controllable. iii) For every XoE H, there exists a XZ( [ O, oo» input u(.) such tzhat the solution of (1 . 1) is an element of X ([O,oo),H). Or, equivalently for every Xo the followiW; cost criterium can be made finite; J(Xo,u)-f Ilx(t)llz+llu(t)fdt.
(1. 3)
Remark:
This
theorem
also
holds
if
the
input
space is not one, but finite dimens ional .
converges to zero exponentially, for every XoE H. We shall see that the c lo sed loop stabilizability implies that in the unstable part of the spectrum of A there are finitely many eigenvalues with finite multiplicity. In the third section we shall introduce a weaker concept of stabilizability, namely the concept of open-loop stabilizability, which states XZ behaviour of x(c) for some input u(t), depending on xo · As in the closed loop case the solvability of this problem implies that the unstable part of the spectrum of A consists of only eigenvalues with finite multiplicity. In the last section we shall investigate the relation between open loop stabilizability and closed loop stabilizability,
Remark: This result is one of the most important results in the theory of infinite dimensional systems, The implication ii) to i) is known to hold for a long time, see (Triggiani, 1975). Of the other implication it was known that the spectrum of A restricted to Hu ' consists of only point spectrum wiht finite multiplicity. These results can be eaSily deduced from the the Weinstein-Aronszajn determinants, (Kato, 1984, pp. 244- 250), Notice that the finite dimensionality of Hu was unknown and so was the decay of T(t)I ' The complete
Hs
implication i) to i i ) is proved independently of each other by Jacobson & Nett (1988) and Nefedov & Sholokhovich (1986). The proofs in these papers are totally different and only treat the Hilbert space case. Curtain (1988) extended the results of J.
This notion is also called exponentially stabilizability or just stabilizability.
III
112
H. Z\\,art
Nefedov &. Sholokhovich to a class of unbounded finite rank input operators . She also treats the Hilbert space case only. The implication i) to ii) follows also directly from theorem 4 of (Desch &. Schappacher, 1985). This result is stated in a general Banach space and also incorporates a class of unbounded finite rank input operators. So not only was it published before (Jacobson &. Nett , 1988) and (Nefedov &. Sholokhovich , 1986), it is also more general . Furthermore we remark that the proof as presented by Desch and Schappacher differs from that giv en in (Curtain, 1988) , (Jacobson &. Nett, 1988) and (Nefedov &. Sholokhovich , 1986) Furthermore (Curtain, 1988) and (Desch &. Schappacher, 1985) treat different classes of unbounded input operators , yet most examples will be in both the classes. The implication iii) to i) was already poved by Datko (1971), and gives a relation between open and closed loop behaviour of the system (A,b).
3. If
the
OPEN LOOP ST ABILlZABILlTY.
system operator A
neccessary conditions
does
not
satisfies
of theorem 2.2.11) ,
the
that
is
the unstable part of the spectrum does not c ,mtain finitely many points. Then would it be possible to stabilize this system by means of e.g. an unb ,mnded feedback law , or a time dependent compensat(lr? In order to answer these questions we shall
int(oduce
the concept of open loop stabilizability. This concept is weaker than the concept of stabilizing the system (A,b) by a bounded fe " dback law,
that we considered in 2.
We shall start stabilizability.
with
defining
our
concept
of
Definition 3.1: Open loop stabi l izability. The s y stem (A, b) is open loop stabilizable if for every XoE I( there exists an i nput u(.)e r~=( [ O, oo );~) such that the mild 2 (1.2) of (1.1) satifies X(.) E r ([O , oo);I().
solution
So roughly speaking we have tha t the sys tem (A, b) is open loop stabilizable if for every initial condition there exists an input such that the solution of (1.1) is a r2 function.
xo- (s-A)E(s)-bw(s)
( 3.2)
ii)
0a .(A) contains no finite accumulation point fo'r all a>O. iii) In the half-plane IL. the spectrum of the operator A consits of only point spectrum with finite multiplicity. Proof: See (Zwart , 1988b). Remark: A representation of the form (3.2) is called a (E ,w)-representation of x o , and was introduced for the finite dimensional cas e by Hautus, (1980). Remark: One can also show that open-loop stabilizability implies that the system is modally controllable on IL., and this will imply the results of the theorem too, see Pandolfi (1987). Under stronger asswnptions a similar theorem was found by Fattorini (1975) and Voigt (1980). In the next remarks we shall explain what this result tells us. So from ii) and iii) we have that in every compact subset of [ contained in [+ there are only finitely many points of the spectrum of A, counted with multiplicity. Furthermore properties ii) and iii) give that even with arbitrary inputs one cannot open loop stabilize a system which has an acummulation point in the unstable part of its spectrum and one cannot remove the residual part, or the continuous part of
the spectrum or eigenvalues with in f inite multiplicity. So if A is the identity operat:or on an infinite dimensional Hilbert space, then it is not open-loop stabilizable . The proof of theorem 3.2 is based on the conc e pt of (E,w)-representation, see (3.2). This representation is essentially the Laplace transform of (1. 2) and it has as advantage that instead of 2 working with functions from r ( [ O,oo);I() or r~o , ( [ O,oo) ;~) one works with functions which are holomorphic on some right half plane. Since a lot is known about holomorphic functions and they have very
nice
properties
we
using this (E,w) (Zwart 1988a , 1989a).
In this section we are interested in the properties
4-.
of open-loop stablizable systems (A,b). In order to formulate the main theorem of this section we shall
can
prove
5
trong
representation,
THE RELATION BETWEEN OPEN LOOP ST ABILlZABILlTY
resul ts
see
also
AND
CLOSED
need some notation
IL.,-:~(ZE
I llez>a)
for aE
~
IL I llez5a) for aE
~
If •.• : - (ZE IL
In this section we shall stabilizability implies stabilizability. First properties of the w(. ) in
IL.:~ o ,.
Lemma 4,1, If Xo has a (E,w)-representation with E(.)E H2(1() , then w(s) is an element of H2 for every .\ with
o(A) is the spectrum of A o a,' (A) - o (A)('j[ a " o a , _ (A) - o (A)('j[ a,-
s-.\
(3.1)
I(IL • • ;Z) denotes the set of all ' holomorphic functions on ILa. wi th values in a Hilbert space Z . '
lle(.\)
-.!.
is
finite,
(Rosenblum
xo~(s-A)E(s)-bw(s)
D(A') such that
see &.
From the assumption we have that Xo be be
written as
H2(1() will denote the Hardy space of all functions f(. )el(IL.;I() such that
2 sup ( 1I f(x+iy) 11 dy x>O (Duren, 1970) and Rovnyak, 1985) .
investigate i f open loop a form of closed loop we shall prove extra equation (3.2) .
~
<.Yb,b>~l, then
«s-A)Us) ,h> -
«s-A)E(s)'Yb> - w(s) Hence
1
1
1
~O'Yb>~(S-A+'\-A)E(S)'Yb>-~(S)
H2:_H2(1L) .
Or Theorem 3,2 , Assume that the system (A,b) is open loop stabilizable, then i) for every Xo in I( there exitst E ( , )E H2(1() , with E(S)E D(A) for sE IL., and w(.) E I(t.;if ) such that on IL. the following representation holds
For .\ in the open-left half plane th~ right hand side of (4.1) is an element of H2. So ~(s) is an element of H2.
Opell Loop Stabilizability, A Research Note corollary 4.2. If bED (A) , then open loop and closed loop stabilizability are equivalent. Proof: Let>. be a negative real number. th,m
(>.-A)b-(s - A)b+(>.- s)b
(4.2)
xo-{s-A) [
113
U~~~xo]
From lemma 4.1 we have that w(s) E H2 (~,w) - representation
Taking the (4.6) yields
b___l__ (>. _A)b + (s_A)[ __l__ b] >.-5 5->'
( 4.3)
From theorem 3.2 we know that for every XoE JI the following representation holds xo-(s-A)E (s)- .)w(s); 2 with E(.)E H (JI). Combining this with equation (4.3) gives xo- (s - A) [US) Is=>.b] So every Xo in JI has a (E system (A, (>' - A)b) with
'" ), (s)_w(s)_>.l are in -5
(>. - A)b{w(s)>.=s)
( 4 . 4)
,w) - representatio~
E>. (s)-E (5)
+
f,)r the 5_>.1> and
and H2 respectivel." .
H2(JI)
see
the previous lemma. By the Paley- Wiener theoJ:em we have that the inverse Laplace transform of a H2 is a f.2 function. So if we take the inverse laplace transform of equation (4.4) yields that for Xo there exists an input trajectory in f.2([O,~», namely the inverse Laplace transform of w),(.), such that the transform
state trajectory, the invese Laplace of E),(.), is in f.2([O,~);I(). A simple
c alculation further shows that the inverse Laplace transforms of these trajectory satisfy equation (1.2) i.e. t
L-l(E),) (c)
+
f T(C-v) ()'_A)bL-l(w),) (v)dv.
o Since Xo is arbitrary we have by theorem 2.2 that the system (A,(),-A)b) is closed loop stabilizable. In order to prove that (A,b) is closed-loop stabilizable we only have to show that the system (A,b) restricted to Jl u is controllable (theorem 2.2.ii). This follows easily from the controllability of (A, (),-A)b) on Jl u ' Corollary 4.3 If A is a bounded operator, then open-loop and closed loop stabilizability are equivalent. Proof: This follows easily from corollary 4.2, since if A is bounded, then D(A) is JI. From theorem 2.2 we see that closed loop stabilizability implies the finiteness of a cost criterium. The next lemma will show that open loop stabilizability implies a similar but weaker property.
L-
l
L-
the
with X(t)-T(C)Xo+f T(t-s)bu(s)ds. o Furthermore D(A)CI!,. Proof: Let JI, be the subset of I( containing all initial conditions XoE JI such that the cost criterium (4.5) is finite for some input fun-otion. is the norm on f.2 and the :;ystem
o
0.1) is linear, we obvious ly that JI, is a linear subspace. So it remains to show that this subspace Contains D(A). Let XoE D(A) and let ), be a negative real n'lIDber, then from theorem (3.2) we have
(),-A)xo-(s-A)E(s)-bw(s) 2
with E(.)E H (JI). So
(),-s)xo-(s-A){E(s)-xo} - bw(s)
Paley-Wiener
theorem
l
[
we in
the
have
that
f.2([O,~) ;JI) and input L-l [W~~~J
the cost criterium (4.5) is finite. Since the domain of A is dense in JI, the same holds for JI,. In the next lemma we shall use the cost criterium (4.5) and the above lemma to construct a feedback law F such that A+bF has no poles in the righ t -half plane. Therefore we shall first prove that if the cost criterium can be made finite then there exists a unique input which minimizes the cost, lemma 4.5. After proving some technical lemmas concerning this unique input, lemma 4.7, 4.9, and 4.11, we shall construct this feedback and show that it has the desired properties, theorem 4.12. I
Lemma 4.5. If the cost criterium J(xo'u) is finite, exists an unique U such that
then there
J (xo ' u):5J (xo ' u)
for all U(.)E f.2( [ O,~» . Proof: Let the pair minimizing input, state criteriwn T JT(Xo,U)-f Ilx(c)
(UT,X T) be trajectory
the unique of the cost
2 2 11 +llu(t)11 dC
see (Curtain and Pritchard, 1978) for details. If we se~ uT(t) - O ,for t>T, and xT(C) idem, then (uT,XT)E f. ([O,~»(!)f. ([O , ~);I() for every T>O. Furthermore since J(xo,u) is finite, the se 'luence (uT,XT) is bounded, so since f.2([O,~»(!)f.2([O,~);JI) is a Hilbert space we have the existence of a pair (u,x) in f.2([O,~»(!)f.2([O,~);I() which is the weak limit of (UT,XT), and x(C)-T(t)x o +
f'T(C - s)bU(s)ds. Since ~
f have
t
(),-s+s-A)xo-(s-A)E(s) - bw(s)
element of transform of
U~~~xo] (C)-T(C)Xo+tT(C-V)bL-t~~~] (v)dv
U~~~xp] and L-t~~~] are f.2( [O,~» resecptively. So with
we
o
f~IIU(t)112dC
is an Laplace
T
f
2 Ilx (c) 112+llu (c) 11 dt"
Ilx (c) 112+ Ilu (c) fdC"
( 4.5)
J(X o ' u) - (1IX(t) 112+llu (t) 11 dt
Since
[
which inverse
o By
Lemma 4.4. If a system (A,b) is open loop stabilizable, then there exists a dense subspace 1(£ such that for every XoE JI, the following cost criterium can be made finite for some input function 2
So Xo has a
),-5
H (JI)(!)H2.
So
b[w~~~]
-
~ ~
( 4.6)
that
lim JT(XO'U T)
5
T_
Kato (1984; p. 253, lemma 1.4) this implies that (UT,x T) converges strongly to (u,x). Furthermore since (uT,XT) converges weakly to (u,x) we have that lim JT(xO'u T) " J(xo,u).
T_
So
lim JT(XO,UT)-J(XO'U),
and
this
implies
that
T_
J(xo,u)-inf J(xo'u) . It is this input is unique.
a
standard result
that
If we have that J(xo'u) is finite for some u(.), then we can take the Laplace transform of equation (1.2) which gives that Xo has a ~E ,w) representation with (E,w) an element of H2(JI)®H . The space ~(JI)(!)H2 is a Hi1bert space with the inproduct given by «E I ,w, ) , (E 2 ,w2 »
- J~
(4.7)
+ _loow,(ir)w 2 (ir)dr
The pair (E,~) has the minimal norm among all 2 2 (E,w) -representations of Xo in H (KJ0H if and onl~ 2 if ,~) 1. (Eo ,wo) for any pair (E ,wo) in H (K)0H satisfying
(E
O_(s_A)Eo(s)_bwo(s)
(4.8)
Lemma 4 . 9 . The subspaces for all Proof:
s"
E
2
A pair of functions (E,w) E H2(K)0H2 is said to be a minimal pair if there exists a XoE K such that
and it has the satisfying (4.9).
minimal
norm
among
all
pairs
Wz
USo)~(s-A)
[US) -Us o)]
(so-s+s -A )E (so)-bw(so)( s-A)E (so) - (so-s)E (s o) -bw (so)
but
E2 (s 2 )~(s 2 -s 1 )
l[US)-Uso)] , W(S)-W(so)]] 5 0 -5
representation of E(so)' and is analytic [[
1
5 0 -5
is
(E ,w) is in we have that
is in H2 (K)0H 2 too.
f f
E(s,)
may
-
E,.
conclude
1
)~(s-A)
[El (S)-US1)] 5
1
_b[W l
-5
(s)-u.,~] 5
1
-5
2
(S)-U S2)] -b [W2 (S)-W(S2)] 5
-5
2
5
2
-5
52- 5
5 1 -5
r-;oo ~,
see (Doetsch, 1974). So · [Wl (r)-W(Sl)] W ( 51 ) = 1 lm r 1
ooWO(ir) [W(ir)-W(SO)]dr
r-700
s l -r
lim r[W2 (r)-W(S2)] sz-r
so-lr
1<~ °
we
-so'
for all sE IC+. For a H2 function f(.) we have that lim f(r)-O for rE
[E(ir) -E(s o)]> dr + 'so-lr K
_00
~
2
symmetry
of
Since E,(s,) - E2 (s2) and both representations are minimal we have by the uniqueness that Wl (S)-w(s, )] _ [W2 (S)-W(S2)] (4.10) [
Thus it remains to prove that this pair is minimal. 2 2 Let (Eo ,wo) be an arbitrary pair in H (K)0H satisfying (4.8). Then -00
element
SI -5 2
by
EZ (s 2 )~(s-A) [E
(E ,w)-
a
5 0 -5
oo
an
S
and
(so-s)E(s o)
Since in So
E(S) -Us o)] , [W(S) - W(so)]]
defintion
[U 2) -E(Sl)]+US)
and
,
E (s
5 0 -5
2 2 H (K)0H ,
by
Definition 4 . 10: _ for any sE IC+.
(s-A)E(s)-bw(s) ~ (s-A)Uso)-(so-s)E(so)-bw(so'> =>
So
+ w(s)
equality.
So
~E
-5
Lemma 4.11 Let (E"w,) and (E 2 ,w2 ) both be minimal pairs. I f E,(s,)~E2(s2)' then w,(s,)-w 2 (s2)' Proof : From lemma 4.6 we have that the m lnimal pairs for E,(s,) and E2 (s2) are respectively given by;
-
(s-A)
1
1
is
3 52
5
( S ) -_ ( 52-51 ) [W(S)-W(Sl)] 5 -5
E2 (s2)
::;:s ,C
+ Us) and
[US)-US1)]
::;: :~ 3"
5 0 -5
xo~(so-A)E (so)-bw(so)
Then there
_ b [ W(S)-W(SJJl]
5 0 -5
Proof:
be an has a
(s)~(s 2-s 1 )
Now
So Lemma 4.7. Let (E ,w) be an minimal pair and let So arbitrary element of IC+, then E(so) also minimal pair and this pair is given by
~sl-Es2
i. e.
are independent of s.
exists a minimal pair for a x, such that E(s,)-E,. Define x 2 as (s2- s ,)E,+x, , then by the previous lemma and the linear dependence of the optimal pair upon the initial state we have that the minimal pair for x 2 is given by
Definition 4 . 6.Hinimal Pair.
(4.9)
~s
S2 in IC+. Let E, be an element of 3 5 , '
r-)oo
~
w (s2) 2
00
-
(ir) ,E(ir) - E(so»K dr +
;:~~:){W(ir)-W(So)}dr
_loo
rOO<~O(ir) ,E(ir»Kdr
-~
+
so+ir
OO
DC)
f ~ ° . f0
__
Ir w(ir)dr + _ so+ir
'-'V
00
+
Theorem 4.12 Suppose that the system (A, b) is open loop stabilizable. Then there exists a linear operator F from 3 to IC such that i) I f (E ,w) is a minimal pair, then w(s)~FUs), for all sE IC+ . ii) The operator s-A-bF : 3 ~ Kf is injective for all sE IC+ . Proof:i) Let Eo be an arbitrary element of 3. Then there exists a minimal pair (Eo(s),wo(s » and a So such that Eo(so)~Eo' Now define
so+.lr
lOO
EO(ir)
<-_------,E(sO»K dr -s o-ir
W (Ir)--
( 4.11)
~(so)dr.
00
-S oalr
Now (~ (s) ,~ (s) 5 0 +5
2
is an element of H (K)0H
2
and it
5 0 +5
satisfies (4.8). So since (E , w) is a minimal pair the sum of the first two term ofo the last equation is
zero.
Furthermore
since
analytic function in theorem, (Rosenblum ( EO (ir) ,!, <-_--.-'UsO»K dr -
0
is
an
O. The same holds for the w
-So-lr
term. Definition 4 . 8 . 3 5 By 3 5 we shall denote the following subspace of K. 3 5 0 - ( EoE K I :3 Xo and a minimal pair (E ,w) such that
E(so)~Eo
and
xo~(s-A)E(s)-bw(s)
FE(s) ~ w(s)
-5
we have b y the Cauchy that Rovnyak, 1985) ~
By lemma 4.11 this operator is well defined, for if (E,(s),w,(s» would be another minimal pair with E, (s,)~Eo' then w, (s,)-wo(so) . Now it is obvious that if (E(.),w(.» is a minimal pair, then (4.12)
for all sE IC+.
The linearity of F follows easily from the linearity of the minimal pairs i.e. if (E"w 1 ) and (E 2 ,w2 ) are both minimal pairs, then (E,+E 2 ,w,+w2 ) is a minmal pair too. ii) Suppose that SoE IC., and let EoE 3 satisfy (4.13)
Since pair
a
XoE Kf that
and
a
Eo(so)-E o '
mlnimal So
for
Open Loop Stabilizability, A Research Note some XoE Kf we have that xo~(s-A)Eo(s)-bwo(s),
Eo(so)-E o
(4. 14)
By part i) we have that FEo(')-wo(')' So xo~ (so-AYE 0 (so)-bw o (so)-(so-A)E o-bFE 0- 0
Thus xo-O and since (Eo(')'w o ('» is the mlnimal pair for Xo we have that EO(.)EO and wo(.).,O. In particular Eo-Eo(so)~O. So we can defined the formal inverse of (s-.~-bF), and for xoE Kf we have E(s)-(s-A-bF)-lx o ' If He can extend the operator (s-A-bF)-l to the whole state space, then we have moved all unstable poles ~o the left-half plane. Then it remains to show thal: A+bF generates a stable semigroup. This is one of the many open problems. Another problem is to show the relation between this feedback and the Ricatti equation. Formally one would expect that F~-b"P, where P is the positive solution of the Ricatti equation AP
+ PA" - Pbb"P + I ~
0
Roughly speaking we may conclude that if the system (1.1) is open loop stabilizable, then it is also closed loop stabilizable. This stabilizing feedback law is linear and time invariant, but it may be unbounded.
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