Page 213 - Mathematical Techniques of Fractional Order Systems

202 Mathematical Techniques of Fractional Order Systems

7.2.2 Solvability and Optimal Control Results for FSDEs
This subsection investigates the solvability and optimal controls for FSDEs
of order 1 , α , 2 in Hilbert space by using ða; kÞ-regularized families of
bounded linear operators. Sufficient conditions are formulated to prove that
the system has a unique mild solution by using the classical Banach contrac-
tion mapping principle. Then, the existence of optimal control for the corre-
sponding Lagrange optimal control problem is investigated. Consider the
following form of FSDE
ð t
c α c α21
D xðtÞ5AxðtÞ1 D BðtÞuðtÞ1fðt;xðtÞÞ1 σðs;xðsÞÞdWðsÞ ; tAJ ð7:9Þ
t t
0
xð0Þ5x 0 ; x ð0Þ50 ð7:10Þ
0
α21
where 1 , α , 2, A generates an t ; 1 -regularized family S α ðtÞ, u is a
ΓðαÞ
given control function, it takes values from another separable reflexive
Hilbert space U, B is a linear operator from U into H. Here f:J 3 H-H and
σ:J 3 H-LðK; HÞ are the appropriate functions.

7.2.2.1 Preliminaries
The collection of all strongly-measurable, p-integrable H-valued random
variables, denoted by L p ðΩ; F; P; HÞ L p ðΩ; HÞ is a Banach space equipped
p 1
5 ðEOxðU; wÞO Þ . Let CðJ; L p ðΩ; HÞÞ be the Banach space
p
H
with norm OxðUÞO L p
of all continuous maps from J to L p ðΩ; HÞ satisfying the condition
p
sup EOxðtÞO , N:
tAJ
1
0
Let L 5 L 2 ðQ 2K; HÞ be the space of all Hilbert-Schmidt operators from
2
1
Q 2K to H with inner product hψ; πi 0 5 trðψQπ Þ.
L
2
Let C be the closed subspace of all continuous process x that belong to
the space CðJ; L p ðΩ; HÞÞ consisting of F t -adapted measurable processes such
that the F 0 -adapted processes xð0Þ with a seminorm OUO C is defined by
1
p p
OxO C 5 sup tAJ OxðtÞO L p : It is easy to verify that C furnished with the norm
topology as defined above is a Banach space.
Consider the following form of FSDEs

ð t
c α c α21
D xðtÞ 5 AxðtÞ 1 D BðtÞuðtÞ 1 f ðt; xðtÞÞ 1 σðs; xðsÞÞdWðsÞ
t t
0
by taking the Laplace transform on both sides of the above equation, one
can get
α
^
λ ^ xðλÞ 2 λ α21 xð0Þ 2 λ α22 0 1 h ^ uðλÞ 1 fðλÞ 1 ^σðλÞ i ð7:11Þ
x ð0Þ 5 A^ xðλÞ 1
λ 12α