Feedback stabilization of a class of multivalued nonlinear distributed parameter systems

Mmlrneor Printed

Analysis, Theory. in Great Britain.

Methods

& A~~lrcotrons,

Vol.

19,

No.

10,

pp.

911-921.

0362-546X/92 55.00+ .OO C’ 1992 Pergamon Presr Ltd

1992

W. L. CHAN,~ M. Q. XIAO and Y. ZHAO$ t The Chinese University

of Hong Kong, Shatin, New Territories, $ Zhongshan University, Guangzhou, China

(Received

1 March

1991; received

in revised form

Key words and phrases:

Multivalued

1 November nonlinear

Kong and

receivedfor publication

1991,

system,

Hong

dissipative

subset,

14 February

1992)

stabilization.

I. INTRODUCTION RECENTLY, the theory of multivalued nonlinear control systems has been developing fast, because it is related to many practical problems with free boundary and moving boundary conditions of partial differential equations. In this paper we consider the feedback stabilization problem for a multivalued nonlinear distributed parameter system as follows

x(t) E Ax(t)

+ V(t)Bx(t) (1.1)

x(4?) = x0 where A is a closed and dissipative subset of H x H, H is a real Hilbert space, B is a (possibly nonlinear) operator from H into H and V(t) is a real valued control. In 1978 Jurdjevic and Quinn [l] and Ball and Stemrod [2] showed that if H = R” and A is the infinitesimal generator of a linear semigroup eAt of isometries, the condition for all t E R + - y = 0

(eA’y, B(eA’y)>Rn = 0 is sufficient In 1979 dissipative We will dissipative

for the stabilization of (1.1). this result was generalized to the infinite-dimensional case with A being a linear operator under the assumption that B is sequentially continuous from H to H [3]. generalize this result to the case with A being a multivalued nonlinear closed operator under the assumption that B is FrCchet differentiable in H. 2. EXISTENCE

Consider

(1.2)

the abstract

AND

PROPERTIES

OF SOLUTIONS

system as follows x(t) E Ax(t)

+ Ax(t),

0 (2.1)

x(0) = xg where

f: X

x

A is a given closed [0, w) - X is a given

t Research $ Research

supported supported

dissipative set in XX X, X is a reflexive Banach space, function, and x0 E X is the initial value. In what follows, we

by UPGC Direct Grant, CUHK. in part by the Foundation of Zhongshan 911

University

Advance

Research

Centre.

912

W. L. CnAN et al.

assume the readers are familiar with the notions and basic properties of dissipative subsets of XX X, the duality map of the spaces Wksp(T, X), ACksp(T, X), the resolvent map .Zx(*) and the Yosida approximation A,(.) (cf. [4, p. 731). THEOREM 2.1. Let X* be uniformly convex and C a closed convex cone of X. Assume (1) R(Z - AA) 3 C, JxC c C, where Jx = (I - IA)-’ v ,I E (0,d);

that:

(2) 11 JAX, - Jxxzii5 ah - x,il, x1, x2 E X, cy E (0, 1); (3) D(A) n C f 0; (4) f: X x [0, 03) -+ X is continuous in t and locally Lipschitz continuous in x E X uniformly with respect to t on bounded intervals andf(t, x) E C, f(x( s), a) E W”‘(Z; X) with x E C(Z, X) for any bounded and closed interval Z c [0, co); (5) x,~D(A)nc# 0. Then, the system (2.1) has a unique solution x(t) defined on the maximal interval [0, t,,,), moreover, if x E C(0, &l,,; X), x E L”(O, t,,, ; X), x(t) E D(A) n C for a.e. t E [0, t,,,); t max < co, we have x(t) + 13 as t + t,,, . In order to prove theorem

2.1 we need the following

lemmas

[4].

LEMMA 2.1. Let X be a Banach space and x E Lp(O, T; X), 1 5 p I 00. Then conditions are equivalent: (1) x E WkTp(O, T; X); (2) there is xi E ACk*p(O, T; X) such that x(t) = xl(t) a.e. on [0, T]. LEMMA 2.2. Let X be a Banach space with the Radon-Nikodym Property X-valued absolutely continuous function in (0, T) is a.e. differentiable

(RNP) and

the following

[5]. Then every

‘I

x(t) = x(0) +

OrtsT.

X(s) ds / .0

LEMMA2.3. Let x(t) be an X-valued function on an interval of real numbers. Suppose that x(t) has a weak derivative i(s) E X at t = s. Assume also that t + Ilx(t)/ is differentiable at t = s. Then

Ilx(s)II$ IlX(~)II= Whf), where F is the duality Remark.

This lemma

for every f E F(x(s))

mapping. is very useful

in an evolution

equation

and is originally

due to T. Kato.

LEMMA 2.4. If the dual space X* of a real Banach space X is uniformly convex, then the duality mapping F: X -+ X* is uniformly continuous on every bounded subset of X. Remark. reflexive

The hypothesis that X* is uniformly (Milman’s theorem).

convex,

implies

that X* and hence X are also

913

Multivalued nonlinear systems

Proof of theorem 2.1. For A > 0 we define A, = and consider

the approximate

equation

P(J,

- I)

as follows

-);hW= A,x,(O + f(x,W, 0

(2.2)

x(to) = x0 which is equivalent

to

x,(t) = exp(-A-‘(t

- to))xo

-1

! w-A-‘0

A-’

+

- Wkf~xk(4,

4 + Jxxk(41 ds

to

=:

(QxdO.

(2.3)

Let X, = C(0, T; X), Co = {x E X, :x(t) E C, t E [to, t,]j. Since C is a closed convex cone we can see that Co is a closed subset in X, and R(Q) c Co by (1) and (5) where the length d(t,, Ilxoll) of [to, t,] is defined as follows

llxoll)= minil,tW(Wo)9to+ lVWo)+ Wo))-‘J

&to,

(2.4)

where K(fo) = N(t,)

2llxoll+ 1,

= max(Ilf(0,

L(K(t,),

t)ll, 0 5 t 5 to + 11,

to + 1) is the local Lipschitz

Let t, = to + &to, Ilxoll), it follows

constant

off.

from (2) that

llQ(xd- QWdllx,= (1 - exp(-f/~))b + ~P)llxx- %llx, where j3 = L(K(t,),

to + 1). Consequently,

(2.5)

for

0 < A < 6 = min(d,

(1 - ol)/p)

(2.3) has a unique solution xx E C(t,, t, ; X) by the Picard theorem and xx(t) E C which satisfies (2.2). The xx(t) is uniformly bounded for t E [to, tl], A E (0,6) in view of the following estimate:

1’

w-~-‘(t - ~))[II~f~xx(~), s) - .fUkdll + ~ll.f(O, 41 + 441 ds LflJ 5 Ilxoll+ (1 - ewW’(t - to)NWUWo),to+ lYWo)+ Wt,) + cWt,)l

liQxxllx, 5

Ilxoll

+

5 Ilxoll+ (uK(to),to+ l)Hfo) + Wo)V(fo3x0) + aHto)

I llx,ll + + + dqt,) 5 According to lemmas everywhere differentiable

2K(f,).

2.1 and 2.2, we can see that f(xx(t), by (2) and (4). Hence, we have IkU)II

;

Il%(f)II = (d(%,(t))/dt,

t), A,x,(t)

K%(t)))

and X,(t) are almost

(2.6)

914

W. L. CHAN ef al.

by lemma

2.3. Note that since A, is dissipative, ((A,xx(t

Setting

we have

+ h) - A,x,(t))/h,

F(x,(t

+ h) - x,(t))//?)

I 0.

h --, 0, one has (d(Axxx(t))/dt,

W,(t)))

5 0

so that Il%(0ll$

Il%(0]i = (d(A,x,(t) 5 (d&,(t),

+ f(xx(0, Wdt,

t))/dt,

K%(f)))

U%(t)))

by (2.2) and (2.6), i.e. dll%(t)ll/dt

5 llf(x,W,

t)ll.

Hence

IlK(Olt- b,(~o)ll5 I

and

Ik(f)II

5

!”IIf(

s)llds

fo

IIA~xoli+ IIf(xo~fo)II+ I

1’ IIfWQ, s)/lds to

“I

5 M%l + Iho [Ill + I /l.h,W, 4 ll ds IIf”

(2.7)

where IAX,~ = inf(l]yll: y E Ax,]. Because xx(r) is uniformly bounded for t E [to, t,] and L E (0,6), it follows from (2.7) and (4) that xx(t) is also uniformly bounded for t E [to, tl] and I. E (0,6). Moreover, from this fact and (2.2) we can deduce that Il~~~x~(r)]i is bounded for t E [0, T] and ,I E (0, 8). On the other hand,

uniformly

by lemma

(2.8)

2.3 and (2.2) we have

; $ Ilxh,,w- xx,,pl12

= (A,,, XA,W - ~x,,,xx,,,W+ .&,,W~ f) - .f(X,,,,(O~ 0, W,J) 5

(Ax,,%,,(f) - ~,,,XA,,,W~ w,,,w

-

- xx,,,(O)>

XA,,,UN - WAnXx,,,W - Jh,,,XA,,,W))

+ Pll%,,(O- .%,,,(f)l12

(2.9)

for 0 < A,,, /I,,, < 8 since Ax,,x,,,(t) E A(J,,,x,,,(t)) and A is dissipative with the hypotheses the theorem taken into account. Applying Gronwall’s lemma to (2.9) we obtain

of

Ilxx,,U)- k,,,(f)II ” I

- xx,,,(d) 5 2 ewW(f%), to+ 1)) lb,,,.%,,(d- 4,,,x,,,,(GlllIK%,~(~) II 10 _ W,,,x,,,W - Jx,,,xx,,,W)ll ds.

(2.10)

Multivalued

Noting

nonlinear

915

systems

that

a%%(~)ll + 0

IIJx%(G - Xx(011 =

(2.11)

asA+

and by (2.8), we have R%,(s)

- XXJS)) - WX,X,“(G

- Jx,x&))

for s E [to, tl] as A,, , &,, + 0 in view of lemma 2.4. It follows from (2.10) to (2.12) that (xx,] is a Cauchy x E C(0, T; X) such that in C(0, T; X),

xX, + x

(2.12)

+ 0

sequence.

Thus

xx,(t) + x(t)

there

exists

an

(2.13)

uniformly for t E [to, tJ in X. It follows from (2.13) and lemma 2.2 that x(t) is almost everywhere differentiable in (to, tJ and x(t) E C for t E [to, tJ because C is closed. In what follows, we want to prove that x(t) is the solution of (2.1). Let t* be a differentiable point of x(t) and KY)

(2.14)

XX”= x - &J.

E A,

It is clear that (2.15)

J = Ax,.fx,.

According to (2.13) we can see that there is a subsequence of {A,,), still denoted by {A,), such that xk,(t*) 3 x(t*) as A,, --t 0 in X. Hence by (2.1), (2.13) and (2.15) one obtains

[AA,%, It follows

from (2.13),

- AAn-%”

(2.15) and lemma F(x&*)

7 Let*> - r - _Mt*), t*)l.

2.4 that

- Xx,) + F(x(t*)

- x)

as A, -+ 0

and (x(t*)

- _F- f(x(t*),

t*), F(x(t*)

= x’:‘Foa,,xx,(t*) because Ax, is dissipative. On the other hand, for small enough

- 2))

- Ax,.q,,

F(x(t*)

- fh”)> 5 0

(2.16)

h (to I h I t*) we have

x(t* - h) = x(t*) - hx(t*) + g(h)

(2.17)

where lim Ilg(h)ll/h

(2.18)

= 0.

h-0

Note that R(I - &4) 1 C, f(x(t*),

t*) E C and x(t*) E C. There exists (xh, yh) E A such that

x(t* - h) + hf(x(t*), t*) = x, - hy,,. According

to (2.17) and (2.19), taking 0 5 Wx(t*)

X = x, , J = yh in (2.16),

- y, - f(x(t*),

we have

t*)), F(x(t*) - xh))

= (x(t*) - x, + g(h),F(x(t*)

- xh)).

(2.19)

916

W. L. CHAN

Consequently,

we can deduce

that Ilx(t*) - &II/h

It follows

from (2.17),

et al.

Ilscw~ + 0.

=

(2.20)

(2.19) and (2.20) that xfi + x(r*),

Y/l + (x(t*) - f(x(t*),

t*))

as h -+ 0 and thus we have x(t*) - f(x(t*),

t*) E AX(t*)

because A is closed. So x(t) satisfies (2.1) and x(t) E D f’ C a.e. t E [to, tJ by the selection of t*. In addition, it is easy to check that X(e) E L”(t,, t, ; X). From what we have just proved we can see that if x(t) is a solution of (2.1) on the interval [to, tJ it can be extended to the interval [tl , t, + S,] with some 6, > 0 by defining x(t) = w(t) on [ti , t, + S,] where w(t) is the solution of the equation as follows w(t) E Aw(t)

+ f(w(0,

0

(2.21)

w(f,) = x(t,) where 6, depends only on Ilx(tl)il and jlN(t,)ll. Let [0, t,,,) be the maximal interval for the solutions of (2.1) to exist. If t,,, must have lim Ilx(t)ll = co. f - frnax

< co then we

Otherwise, there is a sequence t, + t,,, such that Ilx(t,)ll _-E c f or all n. This would imply, by what we have just proved, that for each t,, near enough to t,,,, x(t) depends on [0, t,] extending to [0, t,, + 61 where 6 > 0 is independent of tn. Hence x(t) can be extended beyond t maxcontradicting the definition of t,,, and the proof is complete. 2.2. Let H be and the hypotheses of 0 E D(A), AO; (2) for /I > - AA)-’ is in H; (3) 5 0 for x E H and for any bounded absolutely

2.1 hold. Assume

THEOREM

g(t): 10, m) + H, then (a) the solution x(t) of (2.1) is defined (b) (x(t): t E R+J is precompact in H.

f(g(*),

continuous

that:

function

*) E L’(R+, H);

on [0, 03);

Proof. From the system (2.1) it is easy to deduce that Ilx(t)II 5 /lx,ll by (1) and (3) since A is dissipative. This means that x(t) is uniformly bounded for any t L 0. Applying theorem 2.1 we obtain (a). Let + H, for any absolutely continuous L: = (f:HxR+

x(t): [O, ~0) -+ H, f(x(-), W: = (f:Hx

.) E L’(R +, H)J

R+ + H, for any absolutely

x(0: 10, ~0) + H, f(x(*),

.) E H’(R

continuous +, H)l

Multivalued

nonlinear

917

systems

and X = (D(A) n C) x L. We define II(xcl~f)llx = Ilxoll + im I/f(x(h s)ll d5

,O

where x(t) is the solution (x(t)&)

of (2.1) with initial

value x0. Define

S(t): X -+ X mapping

(x0 ,f)

to

where ./X*(r), r) = f(*(t

TER+.

+ r), t + r),

It is not difficult to see that S(t) is a semigroup on X with the fixed point generator of S(t), then D(B) = (D(A) tl C) x W and

(0,O). Let B be the

Nxo f) = ‘,‘,gNNXO f) - (x0 f)vt 9

3

= (d+x(O)/dt, (Ax,

Since A is dissipative,

9

d+f(x(r),

+ f(xo,

r)/dt)

01, d+.fM%

WW.

for x,, y, E D(A) f~ C we have

tlx(0 - y(t)11 5 Ilxo - yell + ia IIfW),

(Ox)m 3 L qaE!a 103 pue ‘Ox qzc?a .1o3 las lue!.~~e~u! Lldtuauou e s! (Ox)m ‘H uo dno@uras e saur3ap (Ox :1)x = Ox(l)s ‘a~ow~aqyn;I ‘(v)Q 3 Ox qzea 103 8 uo (Ox !j)x uo!lnIos anb!un 1? sassassod (I’Z) uaqL ‘Z’Z uraloaqi 30 sasaqlodr(q aql ICjs!it?s (1-Z) walsk la? ‘Z-E IWXO~H~ ‘H UI las lu~y~u!

lul?yAu!

Lldurauou

e s! (x)m uaql lzeduro3ald

pue ‘ +x 3 I IIE .IOJ 3 5 S(J)S 3! lueyeAu!

@A!!+

s! (x)+Q 31 ‘[Z] 1’~ JWXOEIH~

‘+tl 3 J IF Jo3 3 = 20)s aq 01 p!es sy H 30 3 lasqns

3! v

*(CO+ z4 sr? L + x(~I)~ leql q3ns 00 +u~e~~~1a~uanbaseE:~3~}=(x)m icq (Lldwa Qq!ssod)

x 30 las lyurq-c~ aql pup

+831 X(J)S n = (x)+e

dnol8yas

e awauaiJ

Lq x q8nolql l!qJo aizysod aql augap ff 3 x log ‘ff uo (1)s QwauaFi) v pue azeds iJaqI!H 1ea.I B aq H la7 .I ‘E uo!gy!J‘aa

(Icauquou SNOI.Ln-IOS

30

ZIOIAVHEII8

31LOLdllVASV

‘E

.[9] H u! 1DeduroDaId ospz SI t+ 8 3 1 :(1&l

ssaualalduro2

aql Ag .lzeduro2

‘7 x 0

s! [_(vy

x~I(s‘Ox)II + XII(S‘“x)allu

u

W)a)

3 (S‘W

~03 leql apnI3uoD aA4 uraJoaq1 lzeduroDa.Id s! (+x 3 1 :(1)x) layi OS

- 1) asnwaq

5 II(J)X + (I)XoVY- II = II(M,v-Y

- z)II

leql (SZ’Z) Pus (EZ’Z) moJ3 sA’oIIo.4 11 ‘II(l)xov~~ = 110 ‘(1)x).4-)/ + I/(1 ‘(l)W)I

(SZ’Z) 1P (lP/(L II

+ 1 ‘(1 + W)s,P)

-

O ’ + 110‘WlJII m,i /I

llWxovll

=

- llWx,v~)

7 XII(S‘“x)wsall OS

816

Multivalued

nonlinear

919

systems

Let y E 0(x,), then there exists a sequence (t,), t, -+ ~0, such that S(t,)x, By (3.2) we have m f”+f lim (f(S(s)x,, s), S(s)x,) ds = lim (f(WS(t,)x,, 9, WS(t,W n-00 s 0 n-m i f” for each t E R +. On the other hand, d, %W(f,)xd

lim U(S(s)S(~,)xO, n-m

= (.fCWY,

for each s E [0, t]. Hence by the dominated convergence (f(S(t)y, t), S(t)y) = 0 for all t E R+ as required. 4. THE

STABILIZATION

+ y as n + ~0.

d.s = 0

4, Wx0)

theorem

and

f being continuous

PROBLEM

Definition 4.1. System (1.1) is called stabilizable if there exists a continuous feedback control I/: H -+ R such that (1.1) with V(t) = V(x(t)) satisfies the following properties: (1) for each x0 there exists a unique solution x(t; x,,) of (1.1) defined for all t E R ’ ; (2) 0 is a stable equilibrium of (1.1); (3) x(t; x0) + 0 as t - co for all x0 E D(A). A natural approach to the stabilization trajectories of (1.1) and obtaining

problem

is to formally

differentiate

x(t) along

the

dllx(t)l12/dt E 2(A(x(t)), x(t)> + 2Vt)(x(t), B(x(t))). An obvious

choice of a feedback

control

(though

V(x) = -(x, since this control

yields the “dissipating

energy

not the only one) is B(x)), inequality”

dllx(t)IIZ/dt5 -Xx(t), BC4t)))fWO). For this choice of V(x) the feedback

control

system becomes

x(t) E A(x) - (x(0, B(x(0))B(x(0) and we have the following

THEOREM 4.1.Let H be a real Hilbert space and A be a dissipative C is a closed convex cone of H and (1) for A E (0, a), .I, = (I - ,IA)-’ is compact in H; (2) for x1, x2 E H, IIJ~x, - JAx2115 4x, v a (0, 11, (3) E D(A), E AO; (4) R(Z IA) > and A closed; (5) is a differentiable in H some constant L(c) > 0 such that IlBx - BYI] E H with

for

absolutely X” E D(A)

continuous

n

(4.1)

theorem.

[0,

-+ H;

~11

set in H x H. Assume

that

E (0,4; c 2 0 there

is a constant

920

W. L. Cmiv et al.

Let and

v(t) = -(x(t),

Rx(t)).

If G(QY, R(S(0Y)) then,

(1.1) is strongly

Proof.

for all t E R + =Y = 0,

= 0

(4.2)

stabilizable.

Set f(x)

= -(x,

Bx)B(x).

(4.3)

It is easy to verify that the hypotheses of theorem 2.2 hold. x(t; x0) = T(t)x, defined on R+. Let y E o(x,J, we have

$ ILl(t

= 0

vteR+.

It follows that [u(t)1 I const. v t E R+, that means u(t) = 0 V t E R+. Hence, system (5.3) is stabilizable in H = H’(Q) x L2(sZ). So systems (5.1) and (5.2) are stabilizable in H’(a). REFERENCES 1. JURDIEVIC V. & QUINN J., Controllability and stability, J. d$f. Eqns 28, 281-289 (1978). 2. BALL J. M. & SLEMR~D M., Feedback stabilization of distributed semilinear control systems, Appl. Math. Optim. 5, 169-179 (1979). 3. DAFERMOS C. M. & SLEMROD M., Asymptotic behavior of nonlinear contractions semigroups, _Kfunct. Analysis 13, 97-106 (1973). 4. BARBU V., Nonlinear Semigroups and Differential Equations in Banach Space. Noordhoff Leyden-Ed, Academiei, Bucuresti (1976). 5. DIESTEZ-VHL. Vector measures, Math. Surveys, Vol. 15. American Mathematical Society, Rhode Island (1977). 6. TAYLOR A. E. & LAY P. C., Introduction to Functional Analysis. Wiley, New York (1980). I. HARAUX A., Nonlinear evolution equations-global behavior of solutions, Lecture Notes in Mathematics, Vol. 841. Springer, Berlin (1981). 8. ZHAO Yr, A class of nonlinear distributed parameter control system with closed dissipative operator, Fourth IFAC Symp. on Control of Distributed Parameter Systems. Los Angeles, U.S.A. Preprint (1986). 9. PAZY A., Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983). 10. SLEMRODM., Feedback stabilization of a linear control system in a Hilbert space with an apriori bounded control, in Mathematics of Control Signals and Systems, Vol. 2, pp. 265-285 (1989).