Page 199 - Mathematical Techniques of Fractional Order Systems

188 Mathematical Techniques of Fractional Order Systems

0
x 0 5φAB;x ð0Þ5x 1 ð7:2Þ
α
c
where J: 5 ½0; bŠ, D is the Caputo fractional derivative of order 1 , α , 2,

A generates an t α21 ; 1 -regularized family S α ðtÞ. Here xðUÞ takes values in
ΓðαÞ
the separable Hilbert space H with inner product hU; Ui and norm OUO H . Let
H
K be another separable Hilbert space with inner product hU; Ui K and norm
OUO K . Suppose that W is a given K-valued Wiener process with a finite trace
nuclear covariance operator Q $ 0 defined on the filtered complete probabil-
ity space ðΩ; F; fF t g t $ 0 ; PÞ. The control function uðUÞ is given in L 2 ðJ; UÞ of
admissible control functions, where L 2 ðJ; UÞ is the Hilbert space of all
F t -adapted, square integrable process, U is a Hilbert space. B is a bounded
linear operator from U into H. The history x t :ð2N; 0Š-H; x t ðθÞ 5 xðt 1 θÞ
for t $ 0 belongs to some abstract space B defined axiomatically. Here
F:J 3 B 3 H-PðHÞ; g:J 3 J 3 B-H, and σ:J 3 J 3 B-LðK; HÞ are the
appropriate functions, where LðK; HÞ denotes the space of all bounded linear
operators from K into H.

7.2.1.1 Preliminaries
Let ðΩ; F; Ffg t $ 0 ; PÞ be a filtered complete probability space satisfying the
usual conditions, which means that the filtration fF t g t $ 0 is a right continu-
ous increasing family and F 0 contains all P-null sets. Suppose fWðtÞ: t $ 0
f
be Q- Wiener process defined on ðΩ; F; F t g t $ 0 ; PÞ with a finite trace
nuclear covariance operator Q $ 0, such that TrðQÞ , N. We assume that
there exists a complete orthonormal basis fe k g k $ 1 in K, a bounded sequence
of nonnegative real numbers λ k such that Qe k 5 λ k e k ; k 5 1; 2; ? and a
sequence of independent Brownian motions fβ g such that
k k $ 1
N
ffiffiffiffiffi
X p
hwðtÞ; ei 5 λ k he k ; ei β ðtÞ; eAK; t $ 0:
K k
K
k51
We employ the axiomatic definition of the phase space B introduced by
Hale and Kato (1978). The axioms of the space B are established for F 0
measurable functions from ð2N; 0Š into H, endowed with a seminorm OUO B .
We assume that B satisfies the following axioms
(ai) If x:ð2N; aÞ-H; a . 0 is continuous on ½0;aÞ and x 0 in B, then for
every tA½0;aÞ the following conditions hold:
(i) x t is in B;
^
(ii) OxðtÞO # LOx t O B ;
^
^
^
(iii) Ox t O B # KðtÞsupfOxðsÞO:0 # s # tg 1 NðtÞOx 0 O B , where L . 0isa
^
^
^
^
constant; K; N:½0; NÞ-½0; NÞ, K is continuous, N is locally
^
^
^
bounded and L; K; N are independent of xðUÞ.
(aii) For the function xðUÞ in (ai), x t is a B-valued function.
(aiii) The space B is complete.