CEJM 2 (2003) 141 156
On stabilizability of e v o l u t i o n s y s t e m s of partial differential e q u a t i o n s on IR~ × [0, +oc) by t i m e - d e l a y e d f e e d b a c k controls L.V. Fardigola*
1 Depar'trnent of Mathematical Analysis, Khar'kiv National Univer'sity, 4, Liber'ty sqr'., 61077 Khar'kiv, Ukraine 2 Mathematical Division, Institute .[or"Low Ternper'atur'e Physics ~4 Engineering of the National Academy of Sciences of the Ukraine, 47, Lenin Ave., 61103 Khar'kiv, Ukraine
Received 2 October 2002; revised 17 J a n u a r y 2003 Abstract: In this work we obtain sufficient conditions for stabilizability by time-delayed feedback controls for the system
0,~(x, t) ot
A (Dx) ,~(x, t) - B (Dx) u(x, t),
~ ~ R 'r~, t > h,
where D~ (-iO/Oxl,...,-iO/Ox.r~), A(u) and B(u) are polynomial matrices (rn x rn), P(D~)w(., t - h) is a control, h > 0. det B(~r) ~ 0 on R '~, w is an unknown function, u(., t) Here P is an infinite differentiable matrix (rn x rn), and the norm of each of its derivatives does not exceed F(1 + 1~12)~ for some F , 7 E R depending on the order of this derivative. Necessary conditions for stabilizability of this system are also obtained. In particular, we study the stabilizability problem for the systems corresponding to the telegraph equation, the wave equation, the heat equation, the SchrSdinger equation and another model equation. To obtain these results we use the Fourier transform method, the Lojasiewicz inequality and the Tarski Seidenberg theorem and its corollaries. To choose an appropriate P and stabilize this system, we also prove some estimates of the real parts of the zeros of the quasipolynomial det { IA - A(~r) + B(~r)P(~r)e -ha } . @ Central European Science Journals. All rights reserved.
Keywords: stabilizability, feedback control, delay, partial differential equation, Fourier transform MSC (2000): 93D15, 35B37, 35A22
* E-maih
[email protected] 142
1
L.V. Fardigola / Central European Journal of Mathematics 2 (2003) 141 156
Introduction
One of the most generally accepted ways to study control systems with distributed parameters is their i n t e r ) r e t a t i o n in the form dw dt
- Aw + Bu,
t > 0,
(1)
where w : (0, +oo) ?-{ is an unknown function; ~ : (0, +oo) are Banach spaces; A is an infinitesimal operator in 7-(; B : H
, H is a control; V-t, H > 7-( is a linear b o u n d e d
operator (see, e.g., [2], [3], [10], [12], [13] [16], [18], [19]). An important advantage of this approach is a possibility to employ the ideas and techniques of semigroup operator theory. At the same time it should be noticed that the results on operator sernigroups t h a t are most substantial and important for applications deal with the case when the semigroup generator A has a discrete spectrum and may be treated in terms of its eigenvalues and its eigenelernents. These assumptions correspond to differential equations in domains bounded with respect to space variables but they are not true for domains unbounded with respect to space variables. In the present paper, we consider the following system
0w (x, t)
Ot
-A(D~)w(x,t)-B(D~)u(x,t),
. • R '~, t > h ,
(2)
where D~ = (-iO/Oxl,...,-iO/Ox.~), A(cr) and B(cr) are polynomial matrices (rn x rn), det B(cr) ~ 0 on IR'~, w : IR'~ × (h, + c o ) > C "~ is an unknown function, u : R 'r~ x (h, + c o )
> C "~ is a c o n t r o l , h > 0.
In Section 6 we investigate stabilizability of systems of the form (2) t h a t correspond to the telegraph equation, the wave equation, the heat equation, the SchrSdinger equation and another model equation. We use the following Sobolev spaces
c~ = {g • cq(Rr~) I Ilgll~ < + ~ } , < -- {g I [v~ • [0, h]g(.,~) • c~]
A[Voz
•
NS~(I~I _< q ~ D~g • C(R ~ x [0, h]))] A
Liiiglll8< ÷~]},
IIIgll18 = sup{llg(.,~)ll81 ~ • [0, h]}, cZ = {g I [v~ • [0, +oo)g(.,~) • c~] A [v~ • NS~(I~I _< ~ ~ Dgg • C(R ~ x [0, +oo)))] f[sup{llg(',t)ll~ It • [0, +oo)} < +oo]} where q • No = 1N n {0}, 7 • Z, (~ = ((~s,...,(~,~) • No is a multi-index and as + . . .
c4 - ~ =
I~1
=
+ ~,~, I" I is the Euclidean norm or R ~. w e also use the spaces C £ = N Of, U c~, c~_~ = qENo
U c~, ~ ~yER
=
N c~-~, s = qENo
N c%, and for P • ~ , denote qENo
by P(D.) the following operator P ( D . ) f = ~ - l ( p ~ f ) , f • S~ where ~ is the Fourier transform operator (such an operator P is called pseudodifferential).
L.V. Fardigola / Central European Journal of Mathematics 2 (2003) 141 156
143
In a d d i t i o n , we a s s u m e t h r o u g h o u t t h e p a p e r t h a t ~ _> 0 a n d h > 0 are fixed. Definition
1.1. S y s t e m (2) is said to be stabilizable in C4- ~ if t h e r e exists a m a t r i x
(rn x rn) P E k4 such t h a t for each p E N0 t h e r e exists q E N0 w i t h t h e p r o p e r t y t h a t for every s o l u t i o n of this s y s t e m w i t h t h e control
u(x,t) = P ( D . ) w ( x , t - h)
(3)
u n d e r t h e initial c o n d i t i o n w E eq we have w E C~r a n d
II (.,t)ll
,0
ast
,
(4)
Such a m a t r i x P is called a stabilizing m a t r i x for s y s t e m (2), a n d such a control u is called a stabilizing control for this s y s t e m . In t h e stabilizability p r o b l e m u n d e r c o n s i d e r a t i o n , a delay a p p e a r s in t h e control because, in fact, a w control c a n n o t be realized i n s t a n t l y ( w i t h o u t a delay). To investigate s y s t e m (2), we use t h e Fourier t r a n s f o r m m e t h o d t h a t was p r o p o s e d by I . C . P e t r o w s k y [14] to s t u d y t h e well-posedness p r o p e r t y of t h e C a u c h y p r o b l e m for e v o l u t i o n s y s t e m s o n a layer R'r~x [0, T]. L a t e r this m e t h o d was generalized by I . M . C e l f a n d a n d C . E . S h i l o v [7]. If we a p p l y t h e Fourier t r a n s f o r m to a control of t h e form (3) we o b t a i n (:Y'u(., t))(~) =
P(~)(:Yw(.,t-h))(~).
Hence for a fixed ~0 E R 'r~ we have a linear o p e r a t o r
d e t e r m i n e d by t h e m a t r i x P(cr0). T h a t is w h y we can c o n s t r u c t a stabilizing m a t r i x P(cr), t r e a t i n g er E R '~ as a p a r a m e t e r .
However, if we w a n t to i n t e r p r e t
$-l(P(er)$w(.,t)) in
s o m e way t h e n we have to a s s u m e t h a t P b e l o n g s to s o m e class of functions. It is n a t u r a l to a s s u m e t h a t P is such a f u n c t i o n t h a t
:Y 0, 7 • R, N{H} is the set of the real zeros of a polynomial H and d(n, M ) is the distance between a point n and a s e t M C R '~ (if N { d e t B} =(~ t h e n W ( F , 3 ' ) = ( ~ for all P > 0 and 3' • R). In Section 5 we prove the following Theorem
1.2. Assume that A and B satisfy the conditions Vn • W ( < , ~ l )
a0(n) < 0,
(5)
I det B(n)l 2 _> ~2(1 + Inl~)~ 2,
(6)
vn • R ~ 1 - hA0(n) _> ms(1 ÷ Inl~) ~3
(7)
vn • R'~\w(@I, ~ )
where ¢)s,¢)2 > 0, ¢)3 • (0,1], ~ s , ~ 2 , ~ 3 • Q and ~3 -< 0. Assume also that R > 0, (I)2(I)3R > e, r • Q, ~2 + ~3 + r _> 0 and B' is the adjoint matrix for B (if rn = 1 t h e n we set B ' = 1). T h e n system (2) is stabilizable in C4-°° and
P(n) -
1 e
det B(n)R(1 ÷ Inlb r B'(n)~ ~ u ) I det B(n)l~Rh(1 ÷ Inl~) r ÷ 1
(s)
is a stabilizing matrix for this system.
In Section 3 we prove that conditions (5), (6) are necessary for (9) and (7) is necessary for (10). Theorem
1.3. If system (2) satisfies the following two conditions Vn • R'"
[det B ( n ) = 0 W • R r~
:- A0(n) < 0],
1 i 0 ( n ) < )7
(9) (10)
then this system is stabilizable in C~ °°. Moreover, condition (9) is necessary for stabilizability of (2) in C4-°°, and in the case m = 1, condition (10) is also necessary for stabilizabilty of this system in C~ °°. In addition, in R e m a r k 5.4 we show t h a t for each rn E N there exist polynomial matrices A ( m x m) such t h a t condition (10) is necessary for the stabilizability of system (2) in C4-~. Finally, in R e m a r k 5.5 we show t h a t for the stabilizing control ~t corresponding to the matrix P constructed in Theorem 1.2 we have ~t E C~8 and II~t(.,t)ll _< #(t)IIIwlll q for some s E No where # E C[0, + c o ) , #(t) , 0 as t , +co. Note that, as a rule, a control of the form (3) with a di.ffer'er~tialoperator P(D,) destabilizes (2) and it can be unbounded. Note also that the problem of stabilizability by feedback control without delays (h = 0) was investigated in [5, 6] for equations and systems of the form (2).
L.V. Fardigola / Central European Journal of Mathematics 2 (2003) 141 156
2
The
sketch
of our
145
study
Consider system (2) with a control of the form (3):
- A (D.) ~ ( . , ~) - B ( D . ) P (D.) ~ ( . , ~ -
Ot
h),
. • E ~, t > h,
(11)
under the initial condition
~(.,t)
= ~o(.,t),
(12)
• • R r~, t • Eo, hi.
wherew•C~r,w °• eq,p,q•N0. Applying (formally) the Fourier transform (with respect to z) to problem (11), (12), we obtain
t>h,
- A(o->(o-, t ) - B(o-)P(o->(o-, t - h), at ~(o-,t) = ~°(o-,t), t • [0, hi,
(13)
(14)
where ~ ( . , t ) = Z ~ ( . , t ) , ~0 = Z~0. Assuming that V(cr, t) = e-tA(~)v(cr, t), V°(cr, t) = e-tA(~)vO(cr, t) we reduce problem (13), (14) to the following problem
av(o-,t) - B ( ~ ) P ( ~ ) v ( ~ , t - h), at v(o-,t) = v°(o-,t), t • [0, h].
t > h,
(15) (16)
Now we put
P(G) ~ /3(G)B-I(G)C hA(°-)
(17)
where /3 is a scalar function, /3/det B • ~V[. Substituting P in (15) we obtain
av(~,t)
de
+/3(~)v(~,t-
h) = 0,
(18)
t > h.
Set k(/3, t)
= 0 if t < 0, k(/3,0) = 1, k(/3,t) = Ji~)e~t/({ +/Je-h~)d{ if t > 0 where c is greater then the suprernurn of the real parts of the roots of the quasipolynornial + / J e -h~ (k(/3, t) does not depend on c). Here and henceforth throughout the paper we let J)~)f(A) dA = V.P. f+o~ f(c + i#)d#, V.P. means the principal value of the integral. According to [1, Theorem 6.2, 6,4] and Lernrna 4.4 we conclude that
v(~,t)-
k(9(~),t-h)v°(~,h)-9(~)
~ohk(9(o-),t-~--h)V°(o-,~-)a~ -,
is a unique solution of (18), (16), cr E R 'r~, and V(cr, Setting K(cr, t) = etA(~)k(/3(cr),t) we obtain that
_> h, (19)
.) E C[O, +co), V(cr, .) E US(h, +co).
h
~(~,t) - K ( ~ , t -
h ) ~ ° ( ~ , h ) - 9(~)
0•0
K(~,t-
7- h>°(~,~)&,
_> h,
(2o)
146
L.V. Fardigola / Central European Journal of Mathematics 2 (2003) 141 156
is a solution of (13), (14), (7 E IR'~, and v((7,.) E C[0, + o c ) , v((7,.) E Cl(h, +(x~). One can see t h a t to study the solution (20) of problem (13), (14) we should investigate some properties of the roots of the quasipolynomial ~ ( / 3 , ~ ) = ~ + ~e -h~ where /3 depends on (7. It is not very easy. But if we consider an arbitrary matrix P E 3V[ t h e n we should investigate properties of the roots of det H(%, (7) where H(%, (7) = I% - A((7) + B((7)P((7)e -ha. It is a more complicated problem. Note t h a t if P has the form (17) t h e n H(%,(7) = ~ ( / ~ ( ( 7 ) , I ~ - A((7)) and detH(%,(7) = 0 e=~ [~{0 ~ C ~ 0 ~ C ~(/~((7),{0) = 0 A det(I%0 - A((7)) = 0 A ~ = ~0 + {0]. Thus in this case the zeros of H are d e t e r m i n e d by the zeros of ~ and the s p e c t r u m of A. Finally, applying the inverse Fourier transform with respect to (7 to the solution v of problem (13), (14) we obtain a solution of problem (11), (12) and study its properties. Thus to investigate the stabilizability problem under consideration we (1) analyze conditions (5) (7) of T h e o r e m 1.2 and conditions (9) (10) of T h e o r e m 1.3 (Section 3); (2) study the zero dispositions ofl}((/~, ~) and properties of k(/~,t) and K((7, t) (Section 4); (3) build and study a stabilizing matrix P of the form (17) (Section 5); (4) apply obtained results to the telegraph equation, the wave equation, the heat equation, the SchrSdinger equation and another model equation (Section 6).
3
An analysis of the conditions of Theorems
1.2 a n d 1.3
In this section we prove that conditions (5), (6) are necessary for (9) and (7) is necessary for (10). L e m m a 3.1. Suppose t h a t condition (9) holds for a polynomial matrix (rn x rn) A. T h e n there exist (I)1, F > 0, ~ 1 , 7 E Q such t h a t (5) is true. Moreover, there exist F > 0, 7 E Q such that
A0((7) < - r ( 1 + 1(71~)~,
(7 ~ W ( ~ ,
~),
(21)
P r o o f . We can represent the set W((I)I,~I) in the form W ( d p l , ~ l ) = {(7 E IR'r~ 13rl E R'r~Kdet B(~) = 0 A I~ -- (71 < r ( 1 + 1(712)~1]}. Let ~(r) = inf{d > 0 13(7 E R'~3~I E R'~[d = 1(7 - ~]1 a X0((T) _> 0 a det B(~]) -- 0 a 1(71 -- ~]}. From (90 it follows t h a t L,(r) > 0 (r _> 0). It is easy to see that for every r0 > 0 there exists C(r0) > 0 such t h a t L,(r) _> C(r0), r E [0, r0]. Using the Tarski Seidenberg t h e o r e m [17] and its corollaries [8, A p p e n d i x A] we obtain t h a t u(r) = + o c as r ) + o c or u(r) = Nr2~1(1 + o(1)) as r ) + o c where N > 0, ~1 E Q. Therefore L,(r) _> 2(I)1 (1 + r2) vl, r _> 0, where (I)~ > 0, ~ E Q. Hence (5) holds. Applying the Tarski Seidenberg t h e o r e m [17] and its corollaries [8, A p p e n d i x A]
to ~(r) - supTA0((7) I (7 ~ w ( ~ , ~ ) a 1(71 -- ~} we conclude that ~(~) _< r ( l + ~ F , r _> 0, where F > 0, ~ E Q. We use the same reasoning for obtaining this estimate as we did for obtaining the analogous estimate for L,(r). Thus (21) is true as was to be proved. Lemma
3.2. Let (I)l,P > 0, ~1,~ E Q be constants such t h a t (21) holds. T h e n there
L.V. Fardigola / Central European Journal of Mathematics 2 (2003) 141 156
147
exist (I)2 > 0 a n d ~2 • Q such t h a t (6) is true. P r o o f . From [9, L e m m a 2], we get
I det B( )I _> ~ (1 +
1 12)
(diet, N { d e t B}]) 9 ,
cr • R '~,
(22)
where 13 > 0, (~ • Q , / 3 • Q, m o r e o v e r , / 3 > 0 if N { d e t B} ¢ (~ and /3 = 0 otherwise. Hence (6) holds. The l e m m a is proved. Lemma true.
a . a . Let (10) hold. T h e n there exist (ha • (0, 1] and ~3 • Q such t h a t (7) is
P r o o f . Taking into account (10) a n d applying the Tarski Seidenberg t h e o r e m [17] a n d its corollaries [8, A p p e n d i x A] to #(r) = inf{1 - hA1 I 3A2 • R 3or • R ' ~ [ d e t ( A ( c r ) (A1 + iA2)I) = 0 A Icrl = r]} we conclude t h a t (7) holds. We use the same reasoning for obtaining this e s t i m a t e as we did for obtaining the analogous e s t i m a t e (5). The l e m m a is proved.
4
Properties of the resolving function
K(t, ~)
In this section we study the complex roots of the q u a s i p o l y n o m i a l fie(/3, ~) and e s t i m a t e the functions k(/3,t) a n d K(cr, t). D e n o t e A(/3) = sup { ~ Lemma
[ ffC(/3,~) = 0}.
4.1. For all/3 • C we have A(/3) _> - 1 l b . Moreover, A(/3) = - 1 / h iff/3 = 1/(eh).
P r o o f . Let h = 1. Let us prove t h a t for all /3 • C \ { 1 / e } there exists { • C such t h a t :}f(/3,{) = 0 and R{ > - 1 . Suppose /3 = be~ where b _> 0, ~ • I - r e , r c). Assume F = F1 [..J F2 [..JF3 [..J F4 where F1 = { - 1 t•
I-1,N]},
Ca =
i(t-
~) I t • I-re, re]}, F2 = {t + i ( - r c + ~ ) l =
I-1,N]},
N > 0. Assume also t h a t D is the d o m a i n b o u n d e d by the curve F. Let N > 0 be sufficiently large. W h e n { goes a r o u n d the curve F, the a r g u m e n t i n c r e m e n t of J£(/3, {) is equal to 2re if 0 _< be < 1 and it is equal to 4re otherwise. F r o m the a r g u m e n t principle we conclude t h a t there exists at least one zero of ~(/3, ~) in D. If fl = 1/e t h e n ~ = - 1 is a zero of J£(/3, {). Thus for h = 1 we have A(/3) _> - 1 . Moreover, A(/3) = - 1 iff/3 = 1/e. Let 0 < h ¢ 1. Set ~ = ~/h. T h e n A(/3) 2 - 1 / h . Moreover, A(/3) = - 1 / h iff /3 = 1/(eh). The l e m m a is proved. Lemma
4.2. If 0 _ - x a n d x _> - 1 t h e n x _> x0. It follows t h a t hA(/~) -- x0 _< xl where xl is the greatest root of the e q u a t i o n ~h(x 2 + (2 - e)x + 1) = - x . It is easy to see t h a t xl _< ( - 1 + V/e(1 - e~h))/(e~h). The l e m m a is proved. Lemma
4.3. Let 0 _0, -
(23)
where M~ > 0, w > 0. P r o o f . Since ]~(~1 + i~2,/~)] 2 is a real analytic function of (~1,~2,/~) on R a t h e n from [11, Section 17] we conclude t h a t ]~(~1 + i~2,/~)] _> f~(d[(~l,~2,/~),N]) ~, I~11 _< l / h ,
1~21 _< 2/h, 0 ~ I~(~) - x(~)l" ,
(240
We have k(~,t) = Ji~(~)) et~/~ d~ - ~ Ji~(~)) e(t-h)~/(~J~(~'~))d~' t > 0. Taking into account the inequality 1c(/3) + ip[ 2 / h , and (24) we get
Ik(9,t)l _< e ~(9/' { 2 ~ + ~
2
e(~)2 + #2 +
-2/h ~le(~)l le(~) - ~(~)1 ~
Thus (23) holds for I = 0. Since (0/0/3) ~ ( 1 / ~ ( / 3 , ~ ) ) = (--1)~l!e-~h~/(~(~,~)) ~+1 t h e n
Olfl
< - Kle~(~)t 21+1
(~(9)2 + #2)(~+1//2 +
d#
-2/~ ~ 1~(9)- a(9)l ~
, t > 0,
where Kl > 0, I > 0. It follows from here t h a t (23) is true for I > 0. The l e m m a is proved. Lemma (5)
4.4. Let ~1, ~2 > 0, ~3 • (0, 1], 7)1, 7)2, 7)3 • Q, 7)3 _< 0 be constants such t h a t
(7) hold.
1 ]det B(cr)]2Rh (1 + 1~12)r where R > 0, ~ 2 ~ 3 R > e, Let ~(cr) = eh ]det B(~)12Rh (1 + 1~12)r + 1
r • Q, 7)2 + 7)3 + r _> 0, and let K(cr, t) = etA(~)k(/3(cr), t). T h e n for each multi-index (~
IID~K(~,~)II ~ ~,~, (1 Jr 1~12)~'~' ~-~ 0, nl~ I E R. At first we consider (or, t) E (R'r~\W((I)I,~I)) x [0,+oc). W i t h regard to L e m m a s 3.1 3.3 we get V / 1 - e£(cr)h < (1 + 1~12)~/(O2Rh). Taking into account L e m m a s 4.2, 3.3 and setting c(/3(cr)) = ( - 1 + eV/1 - e/3(cr)h)/(e/3(cr)h 2) we obtain that hA(/3(cr)) < According to L e m m a 4.2 we have Ic(/~(cr))- a(~(~))l _> ( ~ - v~)~/1- e/~(cr)h/h >_ ~(1 + I~lb-b÷'r/~ where b -- deg(det B), K > 0. Now assume that (or, t) ~ W ( ( I ) ~ , ~ ) x [0,+oc). Set c(cr) = L' (1 +l~lb ~" where L ' = min{1/h,r/2}, l'= m i n { 0 , 7 } . Hence A0(cr) + c(cr) < -12(1 + I~1~/~/2. According to L e m m a 4.2 we have
I~(~(~)) - a(~(~))l _> r (1 ÷ I~1~)~/2.
All this implies that for each multi-index p we have I D ' ~ ( ~ ) I _< 7~,,, (1 + I~lb r'~', cr ~ R '~, where 7~1~1, rl~ I ~ R. W i t h regard to L e m m a 4.3 we have t h a t for each multiindex ff an estimate of the form (26) is valid. Using [7, Chapter 1, §6] we obtain I I D g ( ~ ) l l _< ~ , , , ( 1 ÷ [~[)"~'~'e~a°(~), ~ ~ N ~, where Adl~ I > 0, m M e R. Therefore (25) is true. The l e m m a is proved.
we
Taking into account the estimate for get
D~e tA(~) obtained in the proof of this l e m m a
L e m m a 4.5. Let (I)1, (I)2 > 0, (I)3 E (0, 1], ~1~2, ~3 E Q, ~3 -< 0 be constants such that (5) (8) hold. T h e n for P defined by (8) we have P E ~V[.
5
C o n d i t i o n s for stabilizability
S t a t e m e n t 5.1. Assume t h a t for system (2) conditions (9) and (10) hold. T h e n there exists a matrix ( m x m) P E ~ such that for any p E N0 there exist q E N0 and a continuous function p(t) on [0, ÷ o c ) , p(t) > 0 as t > ÷ o c , such t h a t for each solution w of system (2) with control (3) under the initial condition w E e q we have w E C~ and
vt > h
II~(,t)l15 s ~(t)II1~111~.
(27)
P r o o f . It follows from L e m m a s 3.1 3.3 t h a t conditions (5) (7) hold. Let P be a matrix of the form (8). T h e n B(cr)P(cr) - /3(cr)I for /3 defined in L e m m a 4.4. From L e m m a s 4.4, 4.5 we have that P E ~ and estimate (25) is true. Let g' be the dual space for g. Let p E N0 be fixed. Assume that q _> p + ~+,r~+l + 7, w0 e e~ where ~+,r~+l is the constant from estimate (25). For system (12) consider a problem under initial condition (12). W i t h regard to L e m m a 4.4 we conclude that v defined by (20) is a solution of (13), (14) in g' and v(cr,.) E C [ 0 , + o c ) , v(cr,.) E Cl(h, ÷oc). Hence w(.,t) = ~ - l v ( . , t ) is a solution of (11), (12) in g'. Now we prove t h a t w E C~ and (27) is true. Henceforth t h r o u g h o u t the proof we assume t h a t x E R '~, cr E R '~, t _> h. Let e(x) be an infinite differentiable function on IR'~, let s u p p e C { x E R ' ~ I I x I _< 1}, and let E e ( x - k ) - 1. Denote w~(x,t)° IEZ ~
150
L.V. Fardigola / Central European Journal of Mathematics 2 (2003) 141 156
~(.>o(.
+ k,t), %(.,t) o = ~Yw°(.,t). W i t h regard to (20) we have
vk(cr, t ) = K ( c r , t-h)v°(cr, h)-B(cr)P(cr)
/0
K(cr, t - r - h ) v ° ( c r ,
r) dr(g'), t > h (28)
is a solution of (13), (14) w i t h v ° = v ° therefore w k ( x , t ) - (~Y-lvk(.,t))(x) is a solution of (11), (12) in $' with w0 __ wk0 where k • Z '~. Obviously, IIwO(.,T)II q 0 does not d e p e n d on r • [0, h] and k • g 'r~. T h e n we have craD~ (cr~v°(cr, t)) _< c II1~°111~(1 + Ikl)~ where C > 0, 191 + I~1 -< q, I~1 -- ~ + 7 + 1. W i t h regard to (28) a n d L e m m a 4.4 this gives D~ (cr~vk(cr, t)) 0,
I~1 -- deg B +ec~+.~+s +rt + 1,191 _< p. Applying the inverse Fourier t r a n s f o r m with respect to or, we get D~wk(x,t) 0 for all matrices (m x m) P • 3V[ t h e n this system is not stabilizable in C4- ~ Proof.
Let P
E 3V[ and Ap(cr0) _> 0 for some cr0 E R 'r~. Let det{%0I - A(cr0) +
B(cro)P(cro)e -ha°} = 0 and A0(cr0) = ~A0 _> 0 for A0 ¢ C. Also let v0, I~01 -- 1, be a vector such t h a t ( k 0 I - A(cr0) + B(Cro)P(cro)e-ha°)Vo = 0. Consider system (2) with the control u ( . , t ) = P ( D x ) w ( . , t h), i.e. the system of the form (11), under the initial condition w(x,t) = exp{tA0 + i(x, Cro)}Vo, x • R '~, t • [0, hi, where (.,.) is the scalar p r o d u c t corresponding to the E u c l i d e a n n o r m in R '~. It is easy to see t h a t
w(x,t) - exp{tA0 + i(x, ~0)}v0, x • R '~, t • [0, + c o l , is a solution of this problem. Since
I w ( z , t ) l - e x p { t ~ a 0 } and ~a0 _> 0 t h e n
lira
t----+--co
IIw(.,t)ll ° > 0, i.e., condition (4) is not
satisfied. Therefore system (2) is not stabilizable in C4- ~ . The s t a t e m e n t is proved.
L.V. Fardigola / Central European Journal of Mathematics 2 (2003) 141 156
151
From this statement and Lernrna 4.1 we obtain C o r o l l a r y 5.3. If the system (2) is stabilizable in C~- ~ t h e n (9) holds. Moreover if, in addition, rn = 1 then (10) also holds. W i t h regard to Statement 5.1 and Corollary 5.3 we conclude that Theorem 1.3 is true. R e m a r k 5.4. For all rn E N, it is possible to construct a polynomial matrix A (rn x rn) such that for some or0 E R '~ it has the form aI where a >_ 1/h. Let P E 3V[ and let #j, j = 1,m, be the eigenvalues of the matrix B(cr0)P(cr0). T h e n d e t { / ~ 0 I - A(cr0) + TD+ B(cr0)P(cr0)} = H / = I ( ~ - a + #/e-h~). W i t h regard to L e m m a 4.1 we obtain that Av(cr0) > 0. Taking into account Statement 5.2 we conclude that condition (10) is necessary for stabilizability of the system (2) with this matrix A in C~-°+. R e m a r k 5.5. Let conditions (9), (10) be valid for system (2). The stabilizing matrix P E 3V[ that has been found in the proof of Statement 5.1 has the form (8). It can be shown t h a t if u(x,t) = P ( D , ) w ( x , t h) t h e n u(x,t) = W ( x , h , t ) where W ( x , r , t ) is a solution of the Cauchy problem
OW(z'T't)-A(Dx)W(z,T,t),
z E R '~, ~ ~ [0, hi,
[Idet B (D~)I 2 2~h (1 + ID~12)'r + 1] B' (D~)W(x, 0, t) =detB(Dx)R(l+lDxl2)'rw(z,t-h), z E R '~,
(30)
(31)
and t _> h is a parameter. From [4, Corollary 2] we obtain that for each s E N0 there exist p E N0 and C > 0 such t h a t for w(.,t) E C~ (t >_ h) the function W(z,T,t) is the unique solution of the problem c I I w ( . , t - h)ll~, ~ ~ [0, h], t _> have t h a t for each s E N0 there #(t) > 0 as t > + o c , such
s (30), (31) for t >_ h, W(.,T,t) E C 7s and IIW(',~, ~ )11~ _< h. Taking into account L e m m a 4.5 and Statement 5.1 we exist q E N0 and a continuous function #(t) on [0, + o c ) , that for each solution w of system (2) with control (3)
under the initial condition w E e q we have ~t E Cv8 and
6 6.1
II~(,t)ll; _< #(t)II1~111~, t _> 0.
Applications The telegraph equation and the wave equation
Consider the system
( cOwl
- ~
+ b~ (D~) ~ + b~ (D~) ~ z E R 'r~, t > h ,
( 0t
- 2 k - ~ , ~ + z X ~ + b ~ (D~) ~ + b~ (D~) ~ OZ
(32)
152
L.V. Fardigola / Central European Journal of Mathematics 2 (2003) 141 156
where B
{bij }i,j=l 2 is a p o l y n o m i a l matrix, k • R. This system corresponds to the telegraph equation. It is easy to see t h a t =
I~1 _> Ikl
-Io-I~ 2k
k + @k~-I~1 ~, I~1 ~ Ikl
Let us consider three cases: i) k < 0, ii) k = 0, iii) k > 0. i) Let k < 0. T h e n A0(0r) < 0 if 0r ¢ 0 and A0(0) -- 0. Taking into account T h e o r e m 1.3 we o b t a i n Statement
6.1. Let k < 0. T h e n system (32) is stabilizable in C4- ~ iff det B(0) ¢ 0.
Let us find a stabilizing control of the form (8) for this system, given t h a t det B(0) ¢ 0. P u t ~2 = d(0, N { d e t B } ) / 2 , N" = {0r E R '~ II det B(~)I _< ~ } , ,(~) -- sup{~ ~ a; I I~1-r}, r _> 0. Using the same reasoning as for obtaining e s t i m a t e (5) in Section 3 we conclude t h a t , _< ( P s ( I + r 2 ) ~*, r _> 0, where (Ps > 0, ~ • Q. Hence W((Ps, ~ ) n N" n N { d e t B}. Therefore estimates (5), (6) hold w i t h these (Ps, (P2, ~ a n d ~2 = 0. Obviously, e s t i m a t e (7) is true for ~3 = 1, ~3 = 0. P u t R = 2 e / ~ 2 , r = 0. Erom T h e o r e m 1.2 we conclude that
P(~)-
2detB(~) B,(~) [ ~ Idet g(~)122& + ~2
s i n ( h v / , c r , 2 - k 2) (
(32) in the case k < 0. ii) Let k = 0. Then system (32) corresponds to
/
-k
1
is a stabilizing m a t r i x for
th~ ~
~v~teo~.
We have A0(~) = 0,
0r E R a. F r o m T h e o r e m 1.3 we o b t a i n Statement
6.2. Let k = 0. T h e n system (32) is stabilizable in C~- - o o iff V0r E R 'r~
det B(0r) ¢ 0.
(33)
Let us find a stabilizing control of the form (8) for this system, given t h a t (33) holds. W i t h regard to (22) we conclude t h a t there exist ~2 > 0, ~2 E Q such t h a t [det B(cr)[ 2 _> (P2(1 + [cr[2)~2, cr E R 'r~. Hence estimates (5), (6) hold w i t h these (P2, ~2 and a r b i t r a r y ~1 > 0, ~1 E Q. Clearly, e s t i m a t e (7) is true for ~3 = 1, ~3 = 0. P u t R = 2 e / ~ 2 , r = 0. From T h e o r e m 1.2 we conclude t h a t
P(~)-
2detg(~) g'(~) ( cos(hl~l) (sin(hi-I))/1"1 "~ I Idet g(~)122& ÷ ~2 ~,-I~1 sin (hl~l) cos (hl~l) )
is a stabilizing m a t r i x for (32) in the case k = 0.
L.V. Fardigola / Central European Journal of Mathematics 2 (2003) 141 156
153
iii) Let k > 0. T h e n k _< A0(cr) _< 2k, cr E R'% A p p l y i n g T h e o r e m 1.3 a n d R e m a r k 5.4 we o b t a i n Statement
6.3. Let k > 0. If h < 1 / ( 2 k ) a n d (33) holds t h e n s y s t e m (32) is stabilizable
in C4- ~ . If (33) is not t r u e t h e n this s y s t e m is not stabilizable in C4- ~ . Let us find a stabilizing control of t h e f o r m (8) for this s y s t e m , given t h a t h < 1 / ( 2 k ) a n d (33) holds. W i t h r e g a r d to (22) we c o n c l u d e t h a t t h e r e exist dp2 > 0, ~2 E Q such
that I det B(~)I 2 > ¢)2(1 + 1~12)~2, ~ ~ R ~. Hence e s t i m a t e s (5), (6) hold w i t h these do2, ~2 a n d a r b i t r a r y dol > 0, ~1 E Q. Obviously, e s t i m a t e (7) is t r u e for dp3 = 1 -
2kh,
~3 = 0. P u t R = 2 e / ( e 2 ( 1 - 2kh)), r = 0. F r o m T h e o r e m 1.2 we c o n c l u d e t h a t
P(~)-
2det
B(cr)
g,(cr)[ekhsin(hv/lcrl2-k2) ( -k 1 I
I det g(~)122eh + ~2(1 - 2kh)
+1o_12
it _1o_12/)
is a stabilizing m a t r i x for (32) in t h e case k > 0.
6.2
The
Consider
heat equation
the heat equation Ow -Ot
A w + B ( D . ) u,
. ~ R r', t > h,
where B is a p o l y n o m i a l . It is easy to see t h a t A(cr) = A0(cr) =
(34)
-I~12 on
R r'. A c c o r d i n g
to T h e o r e m 1.3 we o b t a i n Statement
6.4. E q u a t i o n (34) is stabilizable in C4- ~ iff B ( 0 ) ¢ 0.
Let us find a stabilizing control of t h e f o r m (8) for this s y s t e m , given t h a t B ( 0 ) • 0. P u t do2 =
d(O,N{detB})/2.
Using t h e s a m e r e a s o n i n g as in e x a m p l e 6.1 (k > 0) we
c o n c l u d e t h a t e s t i m a t e s (5)
(7) hold w i t h s o m e dpl, ~1, this dp2, ~2 = 0, dp3 = 1, ~3 = 0. 2B(cr)e-hl~l 2 P u t R = 2e/alP2, r = 0. F r o m T h e o r e m 1.2 we c o n c l u d e t h a t P(cr) = is
IB(~)122eh + +2
a stabilizing m a t r i x for (34).
6.3
The
Consider
SchrSdinger
equation
the SchrSdinger equation Ow
Ot
= i A ~ + B (D~)~,
• ~ R ~, ~ > h,
(3~)
154
L.V. Fardigola / Central European Journal of Mathematics 2 (2003) 141 156
where B is a polynomial. It is easy to see t h a t A ( a ) = - i l a l 2, A0(a) = 0 on IR'~. According to T h e o r e m 1.3 we o b t a i n Statement
6.5. E q u a t i o n (35) is stabilizable in C4- ~ iff g a • IR'~ B ( a ) ¢ 0.
Let us find a stabilizing control of the form (8) for this system, given t h a t B ( a ) a • R '~. W i t h regard to (22) we conclude t h a t there exists dP2 > 0, ~2 • Q such IB(a)l 2 _> dP2(1 + la12) ~ , a • R '~. Hence estimates (5), (6) hold with these dp2, ~2 arbitrary (I)x > 0, ~x • Q. Evidently, e s t i m a t e (7) is true for (I)3 = 1, ~3 = 0. R = 2e/alP2, r = 0. From T h e o r e m 1.2 we conclude t h a t P ( a ) =
# 0, that and Put
2B(a)e-ihl~l 2 is a IB(a)122eh + +2
stabilizing m a t r i x for (35).
6.4
A Model e q u a t i o n
Consider the e q u a t i o n
Ow
O2w
O2w
a-7 - a.-7 + o.-7 Clearly, A ( a ) -
10(a) -
O2w
O2u
2-57xo.2 + ~ 1-
(
