- Email: [email protected]
Stochastic B-evolutions on Hilbert spaces
Nonlinear
Analysis,
Theory, Methods
Pergamon PII: SO362-546X(%)00197-6
& Applications, Vol. 30, No. 1. pp. l!XJ-XM.1997 Proc. 2nd World Congress of Nonlinear Analysts Q 1997 Elswier Science Ltd Printed in Great Britain. Ail tights reserved 0362-546X197 $17.00 + 0.00
STOCHASTIC B- EVOLUTIONS ON HILBERT SPACES N.U.AHMED Department of Mathematics Department of Electrical Engineering University of Ottawa Key words and phrases: B-evolutions, Stochastic systems, Dynamic tions, Semigroups, Hilbert spaces, Heat equations, Boundary noise.
boundary
condi-
INTRODUCTION. For the first timein the literature we consider stochastic evolution equations governed by B-evolutions involving two different Hilbert spaces. This allows dynamic boundary conditions along with noisy boundary data. Our results are illustrated by two practical examples. Detailed proof of the results presented here will appear in [13]. The paper theory, is organized as follows. Section 1 gives motivation , section 2 Basic B-evolution Section 3 Deterministic Semilinear Systems, Section 4 Stochastic Semilinear Systems and Section 5 Examples. 1. MOTIVATION For motivation let us consider the following (deterministic) heat transfer equation with dynamic boundary condition. Let 0 c R”, (n = 1,2,3) be an open bounded domain with smooth boundary which consists of two parts dR = rc U rr. The material (e.g.fluid) in the interior of the domain receives heat energy through the boundary l?r from an external source distributed on the exterior of the boundary layer l?r. Taking into account the dynamics of heat source the problem can be modeled as follows:
Here T denotes the space-time k : fi H [0, 00) and is given by
temperature
distribution
in the interior
of the domain,
w = w(t, [) E R3 denotes the transport velocity of the material and f represents the internal heat source possibly nonlinear. The function g represents a nonlinear heat transfer characteristic of the boundary , u = u(t,<) denotes the (control) temperature of the external source on the part rr and D, denotes the outward normal derivative. Here 199
200
Second World
Congress
we have dynamic boundary condition. differential equation given by:
of Nonlinear
Analysts
More generally
we consider
the stochastic
partial
t>o,EER (l-2)
where R is an open bounded connected subset of R”. The processes Nd and Nb are suitable random fields distributed in the interior and the boundary of the set R respectively. The operators L, M, and 7 are linear partial differential operators and Fi, Gi, i = 1,2, are nonlinear as described below:
Fi(t,t,$) Gi(t,C,4)
= fi(t,~,~,D~,..,Dm-l~),i = gi(t,t,A
Q/A
= 172
..$,“%W
= 172.
Using appropriate Sobolev spaces and the Hilbert spaces Lz(s2) and Lz(dR), the system (1.2) can be realized as an abstract Stochastic differential equation on two Hilbert spaces XandY: d(Bz(t)) = Az(t)dt + F(t, Bz)dt + u(t, Bx)dW, t 2 0 (1.3) s - tliy+ Bz(t) = y. + With reference to system (1.2), here B, A, F and ~7 are appropriate realizations of the pair of operators {J, T} ,{L, M}, {FI, Gi} and {Fz, Gz} respectively where J denotes the identity operator in X.
2. BASIC
B-EVOLUTION
THEORY
In this section for the convenience of the reader we present some facts from the theory of the so called B-evolution. Let X and Y be two (real or complex) Banach spaces and B a linear operator with domain D(B) c X and range R(B) c Y.
Definition a B-evolution
2.1 A family
if
S(t)(Y)
of bounded linear operators for all t > 0 and
{S(t),
t > 0) defined on Y is culled
C D(B)
S(t + s) = S(s)BS(t),
for all t, s > 0.
(2.1)
Associated with any B-evolution S(t), t > 0, there is a semigroup of bounded linear operators {R(t), t > 0) in Y given by R(t) = BS(t), t > 0.
(2.2)
Second World
The B-evolution
property
Congress
of Nonlinear
can be also expressed
qt
+ s) = S(t)R(s)
201
Analysts
in terms of the semigroup
= S(s)R(t),
as follows:
t, s > 0.
It is clear from this that R(t + s) = R(t)R(s), t,s > 0. The B-evolution S(t) is said to be strongly continuous if {R(t), t > 0) is a Cc semigroup in Y. Since in such cases, for w > 0, ePwtS(t) is also a B-evolution, without loss of generality we may assume that R(t) is uniformly bounded, that is, 11R(t)
Ij.cc~jI
Definition 2.2 A B-evolution N > 0 such that
M,M
> 0, for all t 2 0.
S(t) is said to be uniformly
bounded if there exists a number
II s(t) I~L(Y,x) 0, and it is said to be X-compatible S(t)Bx For x E D(B)
if -
x
as t -+ 0 for each x E D(B).
and T > 0, define the operator A, as A,x = (l/r)(BS(r)B
Definition
2.3The
infinitesimal D(A)
generator = {x E D(B)
Ax E Zim+,oA,.x,
- B)z.
A of a B-evolution : Zim,,oA,x for
S(t)
is given by
exists}
x E D(A).
It is clear from the definition of the infinitesimal generator A that D(A) c D(B). A fundamental result due to Sauer [l] that will be useful in the sequel is quoted here for easy reference. Lemma 2.4 (1 Sauer, Theorem 2.1, p289).Let S(t) be a strongly continuous B-evolution. Then (a): for x E D(A), S(t)Bx E D(A) for t > 0, and AS(t)Bx
= BS(t)Ax
(b): if Ay is the infinitesimal D(Ay) and for such x
generator
= (d/dt)BS(t)Bx. of R(t)
then x E D(A)
if
and only if Bx E
Ax = AyBx. (c): B(D(A)) is dense in Y. (d): for y E Y, the mapping t -
S(t)y
In view of this result we can now consider (d/dt)(Bx(t)) s - tliy+ Bx(t) +
is right continuous. the Cauchy problem = Ax(t), = y.
t > 0, (2.3)
Second World
202
It follows from the above result problem (2.3) is given by
Congress
of Nonlinear
Analysts
that for each y E B(D(A)) x(t)
the solution
of the Cauchy
= S(t)y, t > 0,
and that t --+ k(t), t > 0, is once differentiable and z satisfies equation (2.3). This is the classical solution. Note that since B need not be bounded or closed, equation (2.3b) says nothing about the Zimt,ox(t). The Lemma formly (a): (b):
following result due to Sauer is also useful in the sequel. 2.5 (l,Sauer, Theorem 2.3, p290) Let S(t), t > 0, be a strongly continuous bounded B-evolution. Then S(t)y, t > 0, is strongly continuous with values in X. for each y E Y, t there exists an operator C E J~(Y, X) such that CY = tlir,rjr+S(t)y
+
for y E Y;
uni-
and S(t)y = CR(t)y, t > 0.
(c): C restricted to the range of the operator
B, R(B), is the right inverse of B.
As in the case of semigroup theory dealing with standard Cauchy problems f Ax, x(0) = <, we can introduce the notion of mild solutions as follows:
=
Definition 2.6 (Mild Sol.) For each y E Y, an element x E C((O,T],X), T < 00, given by x(t) E S(t)y, t > 0, is called a mild solution of the Cauchy problem (2.8). In other words mild solutions are defined for arbitrary y E Y though S(t)y does not satisfy the equation (2.3). However since, by Lemma 2.4(c), B(D(A)) is dense in Y, mild solutions are the limits of classical solutions. Hence the mild solution makes good sense. Now consider the nonhomogeneous Cauchy problem: (d/dt)(Bx(t))
= Ax(t)
Bx(t)It=o
+ f(t), t > 0,
= y.
(24
For this problem we can also define the mild solution as x E C(1, X) given by t x(t)
=
S(t)y
S(t - s)f (s)ds, t > 0.
+
(2.5).
s 0
This expression is well defined for each y E Y and f E Ll(I, Y). However in this general situation, z need not satisfy equation (2.4). Again as for the standard Cauchy problems [ see lo], we can also introduce the notion of strong solutions for the problem (2.4). Definition 2.7(Strong Sol.) A n element x E C(I, X) is said to be a strong solution of the Cauchy problem (2.4) if (i): t Bx(t), t > 0, is differentiable a.e on I E (0, T] a.e (ii): fort > 0,x(t) E D(A) (iii): (d/dt)Bx(t) = Ax(t) + f(t), for a.e t > 0.
Second World
Congress
of Nonlinear
Analysts
203
Proposition 2.8(Existence) Let S(t), t > 0, denote the B-evolution corresponding to the generating pair < A, B > and let x, given by (2.5), be the mild solution of the Cauchy problem (2.4) and z(t) E s,“S(t - s)f(s)ds. If Y E B(WA)), f E b(I,J’-), z(t) E D(A), for a.e t > 0, and B is closed then x is a strong solution. Remark If the operator I3 has a bounded inverse, that is, B-l E L(Y, X), the closedness assumption is superfluous. For more comments on the assumptions see the second remark following Theorem 4.5. Proposition 2.9 Sufficient conditions guaranteeing z(t) E D(A), t > 0, are (i) A is closed (ii) f(t) E B(D(A)) for all t E [0, T] and f E C((O,T], Y) fl Ll(I, Y) 3. SEMILINEAR (A) : Semilinear:
DETERMINISTIC
Nonholomorphic
PROBLEMS
case
First we consider the time invariant semilinear evolution equation of the form: (d/dt)Bx Bx]t,o
= Ax + F(Bx),
(3.1)
= y.
The time varying case can be treated in a similar We assume here that the operator B : D(B) c use the same symbol B to denote its closure if it vector space Xg 3 {x E X :I[ 2 /Ix< oo, 11Bx IIy< induced by the graph norm II x llxB=lI
t > 0
way. X +-+Y is either closed or closeable. We is not already closed. We introduce the oo} and furnish this with the topology
x Ilx + II Bx IIY for x E XB.
Since B is closed, Xg is a Banach space with respect to this norm topology. Define the operator FB = FOB as the composition of the operators F and B as they appear in equation (3.1). Using the B-evolution S(t), t > 0, and the variation of constants formula we can write this equation as an integral equation: x(t) = S(t)y +
ts 0
S(t - s)F&x(s))ds,
t E (0, T], .
(3.2)
For I = (O,T], let C(I,Xg) d enote the space of bounded continuous functions defined on I and taking values from the Banach space Xg. Furnished with the sup norm topology this is a Banach space. Definition 3.1(Mild.Sol.) An element x E C(I,Xg) is said to be a mild solution of equation (3.1) 2f 2‘t asa solution of the integral equation (3.2). In the following theorem we present an existence result. Theorem 3.2(Existence) Let S(t), t > 0, denote the B-evolution with the generating par < A, B > and suppose the nonlinear map FB : XB H Y and satisfy the following properties: there exists a K > 0 and, for each ball D, c XB of radius r around the origin, there exists a constant L, > 0, such that
II F&t)
11~5 K(l+
II FB(<)
- F~(rl)
II E Ilx& IIY I L
E XB,
II t - rl llxe >t> 71 E Dr.
204
Second World Congress
Then, for each y E Y, the evolution of definition (3.1). (B):
Semilinear:
Holomorphic
equation
of Nonlinear
Analysts
(3.1) has a unique mild solution
in the sense
case
In the preceding section we considered general B-evolution but with the assumption that B is either closed or at least closeable. In the case of Holomorphic B-evolution it is not required to make this assumption; it s&ices if the pair < A, B > is closed which is a weaker condition. The Laplace transform of a strongly continuous uniformly bounded B-evolution S(t), t > 0, given by
00 pry
=
J
eAxtS(t)ydt
0
is well defined for all ReX > 0 and y E Y. The B-evolution S(t), t > 0, is said to be of type L if &(X)y E D(B) f or all y E Y and all X such that ReX > 0. In this case Bh?(X)Y
emXtBS(t)ydt
=
E
Jrn
eeXtR(t)ydt.
0
If, in addition, R(t), t 2 0, is a holomorphic semigroup, then S(t), t > 0, is said to be a holomorphic B-evolution of type L. The generating pair of a B-evolution of type L is characterized as follows. Lemma 3.3 ( Sauer 1, Theorem 5.1, p 296) The pair < Ao, BO >, where A0 and Bo are suitable restrictions of A and B respectively, is the generating pair of a B-evolution S(t), t > 0, of type L if and only if: (a): BO has a bounded inverse on its range R(Bo) c Y, (b): AoBg’ generates a uniformly bounded Co-semigroup R(t), t 2 0, in Y, (c): The bounded linear operator C which is the strong limit of C, = (Bo - (l/n)Ao)-1 is invertible on E-t R(Bo) where E 5 &,oR(t)(Y). In case R(t), t > 0, is holomorphic or BO is closeable the last condition is superfluous. In this case the pair < Ao, Bo > coincides with the pair < A, B > . In case R(t), t 2 0, is a holomorphic semi group in Y the pair < A, B > is called the generating pair of a holomorphic B-evolution of type L. In this case we can write
equation (d/dt)z
(3.1) as = AB-‘z
+ F(z),
t > 0
z(0) = y. and call s(t) E Cz(t), t > 0, as the generalized solution of equation (3.1) where mild solution of equation (3.5) and C E L(Y, X) is the operator given by Lemma
z is the 2.5.
Thus in the case of holomorphic B-evolution the problem (3.1) is related to the classical problem (3.5). In case F G 0, the solution is given by s(t) = Cz(t) = B-lz(t) since z(t) = R(t)y E R(B) and C/R(B) = B-‘.
Second World
Congress
of Nonlinear
205
Analysts
In the nonlinear case, that is F # 0, r(t) ma y not be in the range of B and hence the definition of generalized solution z(t) = CZ(~) makes good sense. We have the following result. Theorem 3.4(Existence) Suppose the morphic B evolution S(t), t > 0, of type Lipschitx and has at mast linear growth unique generalized solution x E C( (0, T],
pair < A, B > is the generating pair of a holoL. The nonlinear operator F : Y I-+ Y is locally in Y. Then for every y E Y, equation (3.1) has a X).
Remark Since C restricted to R(B) is the right inverse of B, i.f.z(t) then x, as dejined above, is the genuine mild solution of equation (3.1). 4. SEMILINEAR
= Ax(t)dt
B&o
c
Y a.e
STOCHASTIC
In this section we consider stochastic systems of the form (1.3). we may assume that both F and g are time invariant giving d(Bx(t))
E R(B)
+ F(Bx)dt
+ o(Bx)dW,
Again for simplicity
t 2 0 (4-l)
= y.
Let (C, F, Ft tc F, P) denote a complete probability space with an increasing family completed subsigma algebras .F* c F. Let H be a separable Hilbert space and {W(t), t 0}, a Brownian motion with values in H having mean zero and covariance operator Q Fa that E{W(t),h)P’(t),g)l = t(Qh,g) f or every h, g E H. Assumed throughout: a{bv(t) - W(s), t 2 s > 0). (A):
Semilinear:
Nonholomorphic
of > so I
case
Here also we shall assume that B is either closed or is closeable with its closed extension again denoted by B. Again we introduce the vector space Xg and furnish it with the scalar product and associated norm defined by (t, ox,
= (
+ CR,
BOY
7t, c E xl3
II c-cIlxB = (II x II; + II Bz Il$)1’2,x E XB. Let Mz(I, X,) denote the equivalence classes of XB-valued stochastic I} which are 3t-adapted and have finite second moments, that is,
II x IlA42=gp The space Mz furnished
with
II x(t) llsJ’2
the above norm topology
processes
{x(t),
t
E
< 00. is a Banach space.
Definition 4.1(Mild sol.) An element x E M~(I,XB) i a as a solution of the stochastc integral equation (4.1) f ‘t
is said to be a mild solution equation
of
t x(t) =S(t)y
+ t
+
s0
s0
S(t - s)F(Bx(s))ds
S(t - s)o(Bx(s))dW(s),
(4.2) t E I.
206
Second World
Again we introduce
Congress
the composition FB(X)
of Nonlinear
Analysts
operators = (FOB)(X)
and
OB(X) = (aoB)(x). Theorem 4.2(Existence)Suppose the following assumptions hold: (AI): the pair < A, B > is the generating pair of a uniformly bounded B-evolution s(t), t > 0. (A2): the maps FB : Xg I-+ Y and ag : Xg H C(H, Y) are continuous and there exists a constant K > 0 such that for all 5, C E’XB
{II FB(C)Ilk II dC) II;(H,Y$ I K2U+ II C II!& {II FE&) - FB(C)Ilk II ~(6) - cm(C)ll&,~$ I K2(ll t - c llL>. (As): the covariance operator Q E L(H) is positive nuclear with eigen vectors {ei} corresponding to the eigen values {Xi} so that Qei = Xiei and {ei} is a complete orthonormal basis of H. Then, for every y E &I(&,, Y; P), the evolution equation (4.1) has a unique mild solution x E Mz(I,
XB).
In the following corollary we prove temporal regularity of the solution process 2. Corollary 4.3 Under the assumptions of Theorem 4.2, the mild solution of the evolution equation (4.1) belongs to C(I, &(3, X,)). Remark. In case of cylindrical Brownian motion, Q = IH, the identity operator in H. In this situation the conclusions of Theorem 4.2 and its Corollary 4.3 remain valid provided the assumptions on the operator og are replaced by 11 ‘-(c)
&(H,Y)-
11 “B(J)
-
“B(C)
< K2(1+ II c II%,>, IIi,(H,Y)~ K”(ll 6 -I ll%,h
where Ca(H, Y) denotes the space of Hilbert-Schmidt operators from H to Y. The proof is identical. (B): Semilinear:
Holomorphic
case
Here we consider the Stochastic evolution equation (4.1) without sumption on B. We replace the definition (4.1) by the following.
the closeability as-
Definition 4.4 An element 2 E Ma(I,X), given by x E Cz,is said to be a generalized as a solution of the stochastc integral equation solution of equation (4.1) if z E Ma(I,Y) t z(t) = R(t)y + R(t - s)F(z(s))ds s0 (4.15) t + R(t - s)o(z(s))dW(s), t E I. s0
Second World
Congress
of Nonlinear
207
Analysts
Theorem 4.5 Suppose the following assumptions hold: (Al): the pair < A, B > is the generating pair of a uniformly bounded holomorphic B-evolution S(t), t > 0, of type L. are continuous and there exists a (A2): the maps F : Y HY ando:Y HL(H,Y) constant K > 0 such that for all <, 5 E Y
{II F(C) Ilk II 40 II&f,Y)I I K2U+ II I lIc4> {II WI - F(C) I& II 45) - 4C) lI&,Y)) I K2(ll E- 5 II”,>. (A3): same as assumption (A3) of Theorem 4.2. Then for every Y E L2(.7Q,, Y; P) the evolution equation solution x E A&(I, X). Remark. Using similar x E C(I, Lz(F’, x>>.
(4.1)
has a unique generalized
procedure as given in the proof of Corollary
4.3 we can verify that
Remark. Before considering an example we make some remark on the basic assumptions on the pair of operators {A, B}. In Brill [8], it is assumed that B is dominant, {A, B} closed, and B is bijective with compact inverse; in Favini [9], A is dominant and {A, B} closed; in Sauer [1,2] and Van Dalsen [3,4,5], the pair < A, B > is closed. The last assumption is the (weakest) most general one so far. In dealing with some of the semilinear problems considered here, we assumed, as in Sauer and Van Dalsen, that the pair < A, I3 > is closed and also used the additional assumption that B is closeable. We do not know at this time how to avoid this assumption for the stochastic problems. 5. EXAMPLES (A)
Example
(Heat
Transfer)
(Al) Deterministic Case: Considering the heat transfer problem (l.l), trace operator yi, i = 0,l. as -yiq5 = +lri and the formal differential operators L$ = div(k
define the
V4) + v.Vq5
Mq5 = -PD&. We t&e X E L2(s2) and Y = La(a)
x Lz(I’,)
with
the norm topology
II Y lb- (II 91ll&-2) + II Y2 ll;2(rI))1’2,Y The operators
= {Yl,YZI
A and B are defined as follows: D(A) 4 D(B)
~(4
E H2(s2)
={W,
M4),
~(4
E H1(fl)
B# ={A
n4),
: -/e+ = 0) 4 E D(A); : ,ya+ = 0)
4 E D(B).
and
on Y given by E y.
208
Defining
Second World
x(t)(.)
Congress
of Nonlinear
E T(t, .) and, for a fixed control
and yc 3 {To, Ti}, the heat transfer equation evolution in the two Hilbert spaces {X, Y} :
(d/dt)Ba: Bzlt,o
Analysts
21E Lz(l, Lz(I’i)),
(1.1) can be written
= Az + F(Bz), = yo.
as an abstract
B-
t > 0, (5.1)
Following standard procedure one can verify that A and B satisfy the following properties: (i) R(B), range of B, is dense in Y (ii) A is closed (iii) B is injective and has a bounded inverse on R(B) C Y. Thus AB-1 : D(AB-‘) E B(D(A)) c Y ++ Y is a closed densely defined linear operator. Following similar procedure as in [Sauer 1, Van Dalsen 5] one can show that for each ti E R(B) II (AB - A)B-'$IJY~ (A - w> (I 1LI/Y, for X E R, X > w,
II (XB - AW1$ II> l1m-VII ti II for X E C, where w( > 0) is dependent on the L, bound of 2, and the material constant K. In fact w = ((II w llL,)2/4K). F-l-om these estimates it follows that there exists a constant it4 2 1, and 0 < 6 < (7r/2) such that for X in the sectorial domain
Cw,a E {A E C, ReX > w, -(7r/2 + S) < argX < (7r/2 + 6) the operator (XB - A)B-1
has a bounded inverse satisfying
II NAB - 4-l
IIL(Y)<
M/(ReX
-w), ReX > w.
(5.2)
Hence AB-l generates an analytic or holomorphic semi group see [ 1; 10 Theorem 3.2.7, ~821 . Thus by Lemma 3.3, the pair < A, B > is the generating pair of a holomorphic B-evolution of type L. As for the nonlinear terms it suffices to assumethat f (respectively g for fixed u) is a Caratheodory function on I x 0 x R (respectively I x Ii x R) and that they are locally Lipschitz having at most linear growth. Under these assumptions the operator F as defined above satisfies the hypotheses of Theorem 3.4. Thus, for ye E Y, equation (5.1) has a unique generalized solution and hence the heat transfer equation (1.1) has a unique solution. (A2) Stochastic Case: Considering again the heat transfer equation (1.1) subject to random disturbances as indicated in model (1.2), equation (5.1) becomes
dBz = Azdt + F(t, Bz)dt + (~(Bz)diV, t 2 0, B&o = yo.
(5.3)
209
Second World Congress of Nonlinear Analysts
The stochastic term in equation (5.3) is used to model the combined effect of turbulence in the flow field in the interior of the domain St and random fluctuation of the heat source on the boundary I’r. In this case we choose H = Y and the dispersion operator u as
4YP where fi : Lz(s1) ++ L{&(a)) K > 0 so that II 4~)
= U2(Yl~h,92(Y2)W
and gz kc(~)5
II 4~) - 49
: Jh(rl)
H
452(h))
and there exists a constant
K2(l+ II Y Il$>,;v E Y llq~,S
K2 II Y - 5 Ilk
Y,V E Y.
Under these assumptions Theorem 4.5 applies and hence equation (5.3) has a unique generalized solution.
REFERENCES 1. SAUER N.,Linear evolution equations in two Banach spaces, Proceedings of the Royal Society of Edinburgh,SlA, (19$2), 287-303. 2. SAUER N., Dynamical Processes Associated with Dynamic Boundary Conditions for partial Differential Equations, Proc. Intern. Conf.on Theory and Applications of Differential Equations , Ohio University, 2, (1988),374-378. 3. VAN DALSEN M., Die Teorie van Nie-stasionere Evolusies geassosieermet Dinamies Gekoppelde Randwaardeprobleme., Doctoral Thesis, Pretoria University, 1978. 4. VAN DALSEN M., Evolution Problems involving non-stationary Operators between two Banach Spaces I. Existence and Uniqueness Theorems, Quaestiones Mathematicae 8(2), (1985) 97-129. 5. VAN DALSEN M., Semilinear Evolution Equations and Fractional Powers of a Closed Pair of Operators, Proc.of the Royal Sot. of Edinburgh, 105 A,(1987), 101-115. 6. AHMED N.U., Stochastic Initial-Boundary Value Problems for a Class of Second Order Evolution Equations, Proc. Intern. Conf.on Theory and Applications of Differential Equations , Ohio University, 1,(1988), 13-19 7. AHMED N.U., Relaxed Controls for Stochastc Boundary Value Problems in Infinite Dimension, Proc.IFIP WG 7.2 Intern. Conf., Irse,(1990),1-10. 8. BRILL H., A semilinear evolution equation in a Banach space, J. Differential Equations, 24(1977), 412-425. 9. FAVINI A., Laplace Transform Method for a class of Degenerate Evolution Problems, Rend. Mat. 12, (1979), 511-536. 10. AHMED N.U., Semigroup Theory with Applications to Systems and Control, Pitman Res.Notes in Math. Ser. 246., Longman Scientific and Technical and John Wiley, London, New York, 1991. 11. LI P., LIM S.S., VUKOVICH G. & AHMED N.U., Stability and Robustness Analysis of Boundary Control of Flexible Space Structure, Int. J. Systems Sci. , 26,10,(1995), 1759-1776. 12. DA PRATO G. & ZABCZYK J., Stochastic Equations In Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press(1992). 13. AHMED N.U. & SEBTI K., Stochastic Systems Governed by B-Evolutions on Hilbert Spaces, Proc. Royal Sot. Edinburgh, 1997( to appear)
Analysis,
Theory, Methods
Pergamon PII: SO362-546X(%)00197-6
& Applications, Vol. 30, No. 1. pp. l!XJ-XM.1997 Proc. 2nd World Congress of Nonlinear Analysts Q 1997 Elswier Science Ltd Printed in Great Britain. Ail tights reserved 0362-546X197 $17.00 + 0.00
STOCHASTIC B- EVOLUTIONS ON HILBERT SPACES N.U.AHMED Department of Mathematics Department of Electrical Engineering University of Ottawa Key words and phrases: B-evolutions, Stochastic systems, Dynamic tions, Semigroups, Hilbert spaces, Heat equations, Boundary noise.
boundary
condi-
INTRODUCTION. For the first timein the literature we consider stochastic evolution equations governed by B-evolutions involving two different Hilbert spaces. This allows dynamic boundary conditions along with noisy boundary data. Our results are illustrated by two practical examples. Detailed proof of the results presented here will appear in [13]. The paper theory, is organized as follows. Section 1 gives motivation , section 2 Basic B-evolution Section 3 Deterministic Semilinear Systems, Section 4 Stochastic Semilinear Systems and Section 5 Examples. 1. MOTIVATION For motivation let us consider the following (deterministic) heat transfer equation with dynamic boundary condition. Let 0 c R”, (n = 1,2,3) be an open bounded domain with smooth boundary which consists of two parts dR = rc U rr. The material (e.g.fluid) in the interior of the domain receives heat energy through the boundary l?r from an external source distributed on the exterior of the boundary layer l?r. Taking into account the dynamics of heat source the problem can be modeled as follows:
Here T denotes the space-time k : fi H [0, 00) and is given by
temperature
distribution
in the interior
of the domain,
w = w(t, [) E R3 denotes the transport velocity of the material and f represents the internal heat source possibly nonlinear. The function g represents a nonlinear heat transfer characteristic of the boundary , u = u(t,<) denotes the (control) temperature of the external source on the part rr and D, denotes the outward normal derivative. Here 199
200
Second World
Congress
we have dynamic boundary condition. differential equation given by:
of Nonlinear
Analysts
More generally
we consider
the stochastic
partial
t>o,EER (l-2)
where R is an open bounded connected subset of R”. The processes Nd and Nb are suitable random fields distributed in the interior and the boundary of the set R respectively. The operators L, M, and 7 are linear partial differential operators and Fi, Gi, i = 1,2, are nonlinear as described below:
Fi(t,t,$) Gi(t,C,4)
= fi(t,~,~,D~,..,Dm-l~),i = gi(t,t,A
Q/A
= 172
..$,“%W
= 172.
Using appropriate Sobolev spaces and the Hilbert spaces Lz(s2) and Lz(dR), the system (1.2) can be realized as an abstract Stochastic differential equation on two Hilbert spaces XandY: d(Bz(t)) = Az(t)dt + F(t, Bz)dt + u(t, Bx)dW, t 2 0 (1.3) s - tliy+ Bz(t) = y. + With reference to system (1.2), here B, A, F and ~7 are appropriate realizations of the pair of operators {J, T} ,{L, M}, {FI, Gi} and {Fz, Gz} respectively where J denotes the identity operator in X.
2. BASIC
B-EVOLUTION
THEORY
In this section for the convenience of the reader we present some facts from the theory of the so called B-evolution. Let X and Y be two (real or complex) Banach spaces and B a linear operator with domain D(B) c X and range R(B) c Y.
Definition a B-evolution
2.1 A family
if
S(t)(Y)
of bounded linear operators for all t > 0 and
{S(t),
t > 0) defined on Y is culled
C D(B)
S(t + s) = S(s)BS(t),
for all t, s > 0.
(2.1)
Associated with any B-evolution S(t), t > 0, there is a semigroup of bounded linear operators {R(t), t > 0) in Y given by R(t) = BS(t), t > 0.
(2.2)
Second World
The B-evolution
property
Congress
of Nonlinear
can be also expressed
qt
+ s) = S(t)R(s)
201
Analysts
in terms of the semigroup
= S(s)R(t),
as follows:
t, s > 0.
It is clear from this that R(t + s) = R(t)R(s), t,s > 0. The B-evolution S(t) is said to be strongly continuous if {R(t), t > 0) is a Cc semigroup in Y. Since in such cases, for w > 0, ePwtS(t) is also a B-evolution, without loss of generality we may assume that R(t) is uniformly bounded, that is, 11R(t)
Ij.cc~jI
Definition 2.2 A B-evolution N > 0 such that
M,M
> 0, for all t 2 0.
S(t) is said to be uniformly
bounded if there exists a number
II s(t) I~L(Y,x)
if -
x
as t -+ 0 for each x E D(B).
and T > 0, define the operator A, as A,x = (l/r)(BS(r)B
Definition
2.3The
infinitesimal D(A)
generator = {x E D(B)
Ax E Zim+,oA,.x,
- B)z.
A of a B-evolution : Zim,,oA,x for
S(t)
is given by
exists}
x E D(A).
It is clear from the definition of the infinitesimal generator A that D(A) c D(B). A fundamental result due to Sauer [l] that will be useful in the sequel is quoted here for easy reference. Lemma 2.4 (1 Sauer, Theorem 2.1, p289).Let S(t) be a strongly continuous B-evolution. Then (a): for x E D(A), S(t)Bx E D(A) for t > 0, and AS(t)Bx
= BS(t)Ax
(b): if Ay is the infinitesimal D(Ay) and for such x
generator
= (d/dt)BS(t)Bx. of R(t)
then x E D(A)
if
and only if Bx E
Ax = AyBx. (c): B(D(A)) is dense in Y. (d): for y E Y, the mapping t -
S(t)y
In view of this result we can now consider (d/dt)(Bx(t)) s - tliy+ Bx(t) +
is right continuous. the Cauchy problem = Ax(t), = y.
t > 0, (2.3)
Second World
202
It follows from the above result problem (2.3) is given by
Congress
of Nonlinear
Analysts
that for each y E B(D(A)) x(t)
the solution
of the Cauchy
= S(t)y, t > 0,
and that t --+ k(t), t > 0, is once differentiable and z satisfies equation (2.3). This is the classical solution. Note that since B need not be bounded or closed, equation (2.3b) says nothing about the Zimt,ox(t). The Lemma formly (a): (b):
following result due to Sauer is also useful in the sequel. 2.5 (l,Sauer, Theorem 2.3, p290) Let S(t), t > 0, be a strongly continuous bounded B-evolution. Then S(t)y, t > 0, is strongly continuous with values in X. for each y E Y, t there exists an operator C E J~(Y, X) such that CY = tlir,rjr+S(t)y
+
for y E Y;
uni-
and S(t)y = CR(t)y, t > 0.
(c): C restricted to the range of the operator
B, R(B), is the right inverse of B.
As in the case of semigroup theory dealing with standard Cauchy problems f Ax, x(0) = <, we can introduce the notion of mild solutions as follows:
=
Definition 2.6 (Mild Sol.) For each y E Y, an element x E C((O,T],X), T < 00, given by x(t) E S(t)y, t > 0, is called a mild solution of the Cauchy problem (2.8). In other words mild solutions are defined for arbitrary y E Y though S(t)y does not satisfy the equation (2.3). However since, by Lemma 2.4(c), B(D(A)) is dense in Y, mild solutions are the limits of classical solutions. Hence the mild solution makes good sense. Now consider the nonhomogeneous Cauchy problem: (d/dt)(Bx(t))
= Ax(t)
Bx(t)It=o
+ f(t), t > 0,
= y.
(24
For this problem we can also define the mild solution as x E C(1, X) given by t x(t)
=
S(t)y
S(t - s)f (s)ds, t > 0.
+
(2.5).
s 0
This expression is well defined for each y E Y and f E Ll(I, Y). However in this general situation, z need not satisfy equation (2.4). Again as for the standard Cauchy problems [ see lo], we can also introduce the notion of strong solutions for the problem (2.4). Definition 2.7(Strong Sol.) A n element x E C(I, X) is said to be a strong solution of the Cauchy problem (2.4) if (i): t Bx(t), t > 0, is differentiable a.e on I E (0, T] a.e (ii): fort > 0,x(t) E D(A) (iii): (d/dt)Bx(t) = Ax(t) + f(t), for a.e t > 0.
Second World
Congress
of Nonlinear
Analysts
203
Proposition 2.8(Existence) Let S(t), t > 0, denote the B-evolution corresponding to the generating pair < A, B > and let x, given by (2.5), be the mild solution of the Cauchy problem (2.4) and z(t) E s,“S(t - s)f(s)ds. If Y E B(WA)), f E b(I,J’-), z(t) E D(A), for a.e t > 0, and B is closed then x is a strong solution. Remark If the operator I3 has a bounded inverse, that is, B-l E L(Y, X), the closedness assumption is superfluous. For more comments on the assumptions see the second remark following Theorem 4.5. Proposition 2.9 Sufficient conditions guaranteeing z(t) E D(A), t > 0, are (i) A is closed (ii) f(t) E B(D(A)) for all t E [0, T] and f E C((O,T], Y) fl Ll(I, Y) 3. SEMILINEAR (A) : Semilinear:
DETERMINISTIC
Nonholomorphic
PROBLEMS
case
First we consider the time invariant semilinear evolution equation of the form: (d/dt)Bx Bx]t,o
= Ax + F(Bx),
(3.1)
= y.
The time varying case can be treated in a similar We assume here that the operator B : D(B) c use the same symbol B to denote its closure if it vector space Xg 3 {x E X :I[ 2 /Ix< oo, 11Bx IIy< induced by the graph norm II x llxB=lI
t > 0
way. X +-+Y is either closed or closeable. We is not already closed. We introduce the oo} and furnish this with the topology
x Ilx + II Bx IIY for x E XB.
Since B is closed, Xg is a Banach space with respect to this norm topology. Define the operator FB = FOB as the composition of the operators F and B as they appear in equation (3.1). Using the B-evolution S(t), t > 0, and the variation of constants formula we can write this equation as an integral equation: x(t) = S(t)y +
ts 0
S(t - s)F&x(s))ds,
t E (0, T], .
(3.2)
For I = (O,T], let C(I,Xg) d enote the space of bounded continuous functions defined on I and taking values from the Banach space Xg. Furnished with the sup norm topology this is a Banach space. Definition 3.1(Mild.Sol.) An element x E C(I,Xg) is said to be a mild solution of equation (3.1) 2f 2‘t asa solution of the integral equation (3.2). In the following theorem we present an existence result. Theorem 3.2(Existence) Let S(t), t > 0, denote the B-evolution with the generating par < A, B > and suppose the nonlinear map FB : XB H Y and satisfy the following properties: there exists a K > 0 and, for each ball D, c XB of radius r around the origin, there exists a constant L, > 0, such that
II F&t)
11~5 K(l+
II FB(<)
- F~(rl)
II E Ilx& IIY I L
E XB,
II t - rl llxe >t> 71 E Dr.
204
Second World Congress
Then, for each y E Y, the evolution of definition (3.1). (B):
Semilinear:
Holomorphic
equation
of Nonlinear
Analysts
(3.1) has a unique mild solution
in the sense
case
In the preceding section we considered general B-evolution but with the assumption that B is either closed or at least closeable. In the case of Holomorphic B-evolution it is not required to make this assumption; it s&ices if the pair < A, B > is closed which is a weaker condition. The Laplace transform of a strongly continuous uniformly bounded B-evolution S(t), t > 0, given by
00 pry
=
J
eAxtS(t)ydt
0
is well defined for all ReX > 0 and y E Y. The B-evolution S(t), t > 0, is said to be of type L if &(X)y E D(B) f or all y E Y and all X such that ReX > 0. In this case Bh?(X)Y
emXtBS(t)ydt
=
E
Jrn
eeXtR(t)ydt.
0
If, in addition, R(t), t 2 0, is a holomorphic semigroup, then S(t), t > 0, is said to be a holomorphic B-evolution of type L. The generating pair of a B-evolution of type L is characterized as follows. Lemma 3.3 ( Sauer 1, Theorem 5.1, p 296) The pair < Ao, BO >, where A0 and Bo are suitable restrictions of A and B respectively, is the generating pair of a B-evolution S(t), t > 0, of type L if and only if: (a): BO has a bounded inverse on its range R(Bo) c Y, (b): AoBg’ generates a uniformly bounded Co-semigroup R(t), t 2 0, in Y, (c): The bounded linear operator C which is the strong limit of C, = (Bo - (l/n)Ao)-1 is invertible on E-t R(Bo) where E 5 &,oR(t)(Y). In case R(t), t > 0, is holomorphic or BO is closeable the last condition is superfluous. In this case the pair < Ao, Bo > coincides with the pair < A, B > . In case R(t), t 2 0, is a holomorphic semi group in Y the pair < A, B > is called the generating pair of a holomorphic B-evolution of type L. In this case we can write
equation (d/dt)z
(3.1) as = AB-‘z
+ F(z),
t > 0
z(0) = y. and call s(t) E Cz(t), t > 0, as the generalized solution of equation (3.1) where mild solution of equation (3.5) and C E L(Y, X) is the operator given by Lemma
z is the 2.5.
Thus in the case of holomorphic B-evolution the problem (3.1) is related to the classical problem (3.5). In case F G 0, the solution is given by s(t) = Cz(t) = B-lz(t) since z(t) = R(t)y E R(B) and C/R(B) = B-‘.
Second World
Congress
of Nonlinear
205
Analysts
In the nonlinear case, that is F # 0, r(t) ma y not be in the range of B and hence the definition of generalized solution z(t) = CZ(~) makes good sense. We have the following result. Theorem 3.4(Existence) Suppose the morphic B evolution S(t), t > 0, of type Lipschitx and has at mast linear growth unique generalized solution x E C( (0, T],
pair < A, B > is the generating pair of a holoL. The nonlinear operator F : Y I-+ Y is locally in Y. Then for every y E Y, equation (3.1) has a X).
Remark Since C restricted to R(B) is the right inverse of B, i.f.z(t) then x, as dejined above, is the genuine mild solution of equation (3.1). 4. SEMILINEAR
= Ax(t)dt
B&o
c
Y a.e
STOCHASTIC
In this section we consider stochastic systems of the form (1.3). we may assume that both F and g are time invariant giving d(Bx(t))
E R(B)
+ F(Bx)dt
+ o(Bx)dW,
Again for simplicity
t 2 0 (4-l)
= y.
Let (C, F, Ft tc F, P) denote a complete probability space with an increasing family completed subsigma algebras .F* c F. Let H be a separable Hilbert space and {W(t), t 0}, a Brownian motion with values in H having mean zero and covariance operator Q Fa that E{W(t),h)P’(t),g)l = t(Qh,g) f or every h, g E H. Assumed throughout: a{bv(t) - W(s), t 2 s > 0). (A):
Semilinear:
Nonholomorphic
of > so I
case
Here also we shall assume that B is either closed or is closeable with its closed extension again denoted by B. Again we introduce the vector space Xg and furnish it with the scalar product and associated norm defined by (t, ox,
= (
+ CR,
BOY
7t, c E xl3
II c-cIlxB = (II x II; + II Bz Il$)1’2,x E XB. Let Mz(I, X,) denote the equivalence classes of XB-valued stochastic I} which are 3t-adapted and have finite second moments, that is,
II x IlA42=gp The space Mz furnished
with
II x(t) llsJ’2
the above norm topology
processes
{x(t),
t
E
< 00. is a Banach space.
Definition 4.1(Mild sol.) An element x E M~(I,XB) i a as a solution of the stochastc integral equation (4.1) f ‘t
is said to be a mild solution equation
of
t x(t) =S(t)y
+ t
+
s0
s0
S(t - s)F(Bx(s))ds
S(t - s)o(Bx(s))dW(s),
(4.2) t E I.
206
Second World
Again we introduce
Congress
the composition FB(X)
of Nonlinear
Analysts
operators = (FOB)(X)
and
OB(X) = (aoB)(x). Theorem 4.2(Existence)Suppose the following assumptions hold: (AI): the pair < A, B > is the generating pair of a uniformly bounded B-evolution s(t), t > 0. (A2): the maps FB : Xg I-+ Y and ag : Xg H C(H, Y) are continuous and there exists a constant K > 0 such that for all 5, C E’XB
{II FB(C)Ilk II dC) II;(H,Y$ I K2U+ II C II!& {II FE&) - FB(C)Ilk II ~(6) - cm(C)ll&,~$ I K2(ll t - c llL>. (As): the covariance operator Q E L(H) is positive nuclear with eigen vectors {ei} corresponding to the eigen values {Xi} so that Qei = Xiei and {ei} is a complete orthonormal basis of H. Then, for every y E &I(&,, Y; P), the evolution equation (4.1) has a unique mild solution x E Mz(I,
XB).
In the following corollary we prove temporal regularity of the solution process 2. Corollary 4.3 Under the assumptions of Theorem 4.2, the mild solution of the evolution equation (4.1) belongs to C(I, &(3, X,)). Remark. In case of cylindrical Brownian motion, Q = IH, the identity operator in H. In this situation the conclusions of Theorem 4.2 and its Corollary 4.3 remain valid provided the assumptions on the operator og are replaced by 11 ‘-(c)
&(H,Y)-
11 “B(J)
-
“B(C)
< K2(1+ II c II%,>, IIi,(H,Y)~ K”(ll 6 -I ll%,h
where Ca(H, Y) denotes the space of Hilbert-Schmidt operators from H to Y. The proof is identical. (B): Semilinear:
Holomorphic
case
Here we consider the Stochastic evolution equation (4.1) without sumption on B. We replace the definition (4.1) by the following.
the closeability as-
Definition 4.4 An element 2 E Ma(I,X), given by x E Cz,is said to be a generalized as a solution of the stochastc integral equation solution of equation (4.1) if z E Ma(I,Y) t z(t) = R(t)y + R(t - s)F(z(s))ds s0 (4.15) t + R(t - s)o(z(s))dW(s), t E I. s0
Second World
Congress
of Nonlinear
207
Analysts
Theorem 4.5 Suppose the following assumptions hold: (Al): the pair < A, B > is the generating pair of a uniformly bounded holomorphic B-evolution S(t), t > 0, of type L. are continuous and there exists a (A2): the maps F : Y HY ando:Y HL(H,Y) constant K > 0 such that for all <, 5 E Y
{II F(C) Ilk II 40 II&f,Y)I I K2U+ II I lIc4> {II WI - F(C) I& II 45) - 4C) lI&,Y)) I K2(ll E- 5 II”,>. (A3): same as assumption (A3) of Theorem 4.2. Then for every Y E L2(.7Q,, Y; P) the evolution equation solution x E A&(I, X). Remark. Using similar x E C(I, Lz(F’, x>>.
(4.1)
has a unique generalized
procedure as given in the proof of Corollary
4.3 we can verify that
Remark. Before considering an example we make some remark on the basic assumptions on the pair of operators {A, B}. In Brill [8], it is assumed that B is dominant, {A, B} closed, and B is bijective with compact inverse; in Favini [9], A is dominant and {A, B} closed; in Sauer [1,2] and Van Dalsen [3,4,5], the pair < A, B > is closed. The last assumption is the (weakest) most general one so far. In dealing with some of the semilinear problems considered here, we assumed, as in Sauer and Van Dalsen, that the pair < A, I3 > is closed and also used the additional assumption that B is closeable. We do not know at this time how to avoid this assumption for the stochastic problems. 5. EXAMPLES (A)
Example
(Heat
Transfer)
(Al) Deterministic Case: Considering the heat transfer problem (l.l), trace operator yi, i = 0,l. as -yiq5 = +lri and the formal differential operators L$ = div(k
define the
V4) + v.Vq5
Mq5 = -PD&. We t&e X E L2(s2) and Y = La(a)
x Lz(I’,)
with
the norm topology
II Y lb- (II 91ll&-2) + II Y2 ll;2(rI))1’2,Y The operators
= {Yl,YZI
A and B are defined as follows: D(A) 4 D(B)
~(4
E H2(s2)
={W,
M4),
~(4
E H1(fl)
B# ={A
n4),
: -/e+ = 0) 4 E D(A); : ,ya+ = 0)
4 E D(B).
and
on Y given by E y.
208
Defining
Second World
x(t)(.)
Congress
of Nonlinear
E T(t, .) and, for a fixed control
and yc 3 {To, Ti}, the heat transfer equation evolution in the two Hilbert spaces {X, Y} :
(d/dt)Ba: Bzlt,o
Analysts
21E Lz(l, Lz(I’i)),
(1.1) can be written
= Az + F(Bz), = yo.
as an abstract
B-
t > 0, (5.1)
Following standard procedure one can verify that A and B satisfy the following properties: (i) R(B), range of B, is dense in Y (ii) A is closed (iii) B is injective and has a bounded inverse on R(B) C Y. Thus AB-1 : D(AB-‘) E B(D(A)) c Y ++ Y is a closed densely defined linear operator. Following similar procedure as in [Sauer 1, Van Dalsen 5] one can show that for each ti E R(B) II (AB - A)B-'$IJY~ (A - w> (I 1LI/Y, for X E R, X > w,
II (XB - AW1$ II> l1m-VII ti II for X E C, where w( > 0) is dependent on the L, bound of 2, and the material constant K. In fact w = ((II w llL,)2/4K). F-l-om these estimates it follows that there exists a constant it4 2 1, and 0 < 6 < (7r/2) such that for X in the sectorial domain
Cw,a E {A E C, ReX > w, -(7r/2 + S) < argX < (7r/2 + 6) the operator (XB - A)B-1
has a bounded inverse satisfying
II NAB - 4-l
IIL(Y)<
M/(ReX
-w), ReX > w.
(5.2)
Hence AB-l generates an analytic or holomorphic semi group see [ 1; 10 Theorem 3.2.7, ~821 . Thus by Lemma 3.3, the pair < A, B > is the generating pair of a holomorphic B-evolution of type L. As for the nonlinear terms it suffices to assumethat f (respectively g for fixed u) is a Caratheodory function on I x 0 x R (respectively I x Ii x R) and that they are locally Lipschitz having at most linear growth. Under these assumptions the operator F as defined above satisfies the hypotheses of Theorem 3.4. Thus, for ye E Y, equation (5.1) has a unique generalized solution and hence the heat transfer equation (1.1) has a unique solution. (A2) Stochastic Case: Considering again the heat transfer equation (1.1) subject to random disturbances as indicated in model (1.2), equation (5.1) becomes
dBz = Azdt + F(t, Bz)dt + (~(Bz)diV, t 2 0, B&o = yo.
(5.3)
209
Second World Congress of Nonlinear Analysts
The stochastic term in equation (5.3) is used to model the combined effect of turbulence in the flow field in the interior of the domain St and random fluctuation of the heat source on the boundary I’r. In this case we choose H = Y and the dispersion operator u as
4YP where fi : Lz(s1) ++ L{&(a)) K > 0 so that II 4~)
= U2(Yl~h,92(Y2)W
and gz kc(~)5
II 4~) - 49
: Jh(rl)
H
452(h))
and there exists a constant
K2(l+ II Y Il$>,;v E Y llq~,S
K2 II Y - 5 Ilk
Y,V E Y.
Under these assumptions Theorem 4.5 applies and hence equation (5.3) has a unique generalized solution.
REFERENCES 1. SAUER N.,Linear evolution equations in two Banach spaces, Proceedings of the Royal Society of Edinburgh,SlA, (19$2), 287-303. 2. SAUER N., Dynamical Processes Associated with Dynamic Boundary Conditions for partial Differential Equations, Proc. Intern. Conf.on Theory and Applications of Differential Equations , Ohio University, 2, (1988),374-378. 3. VAN DALSEN M., Die Teorie van Nie-stasionere Evolusies geassosieermet Dinamies Gekoppelde Randwaardeprobleme., Doctoral Thesis, Pretoria University, 1978. 4. VAN DALSEN M., Evolution Problems involving non-stationary Operators between two Banach Spaces I. Existence and Uniqueness Theorems, Quaestiones Mathematicae 8(2), (1985) 97-129. 5. VAN DALSEN M., Semilinear Evolution Equations and Fractional Powers of a Closed Pair of Operators, Proc.of the Royal Sot. of Edinburgh, 105 A,(1987), 101-115. 6. AHMED N.U., Stochastic Initial-Boundary Value Problems for a Class of Second Order Evolution Equations, Proc. Intern. Conf.on Theory and Applications of Differential Equations , Ohio University, 1,(1988), 13-19 7. AHMED N.U., Relaxed Controls for Stochastc Boundary Value Problems in Infinite Dimension, Proc.IFIP WG 7.2 Intern. Conf., Irse,(1990),1-10. 8. BRILL H., A semilinear evolution equation in a Banach space, J. Differential Equations, 24(1977), 412-425. 9. FAVINI A., Laplace Transform Method for a class of Degenerate Evolution Problems, Rend. Mat. 12, (1979), 511-536. 10. AHMED N.U., Semigroup Theory with Applications to Systems and Control, Pitman Res.Notes in Math. Ser. 246., Longman Scientific and Technical and John Wiley, London, New York, 1991. 11. LI P., LIM S.S., VUKOVICH G. & AHMED N.U., Stability and Robustness Analysis of Boundary Control of Flexible Space Structure, Int. J. Systems Sci. , 26,10,(1995), 1759-1776. 12. DA PRATO G. & ZABCZYK J., Stochastic Equations In Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press(1992). 13. AHMED N.U. & SEBTI K., Stochastic Systems Governed by B-Evolutions on Hilbert Spaces, Proc. Royal Sot. Edinburgh, 1997( to appear)