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246 Mathematical Techniques of Fractional Order Systems
By (8.5), Φ is a contraction and thus, by Lemma 8.5, Φ has a fixed point
which is the solution to (8.1) on J:
Theorem 8.4: (Complete controllability) Assume that the hypotheses (H1)
(H5) are satisfied, and the corresponding linear stochastic system is
completely controllable on J: Then, the system (8.1) is completely
controllable.
Proof: By Theorems 8.2 and 8.3 the operator Φ has a fixed point. So, the
control
q2p ÞW 21 q2p
u x ðtÞ 5 B E q2p;q ðA ðT2tÞ Ex 1 2 E q2p ðAT Þx 0
1 AT q2p E q2p;q2p11 ðAT q2p Þx 0 2 TE q2p;2 ðAT q2p Þx 0
0
ð T ð s
2 ðT2sÞ q21 E q2p;q ðAðT2sÞ q2p Þ vðsÞ 1 Gðθ; xðθÞÞdwðθÞ ds :
0 0
transfers the system (8.1) from x 0 to x 1 at time T: By Definition 8.6, the sys-
tem is completely controllable on J: &
8.4 DISCUSSION
In this chapter, the nonlinear fractional higher order stochastic integrodiffer-
ential inclusion has been considered to investigate the complete controlla-
bility in finite dimensional space. Controllability Grammian martrix has been
formulated using fractional calculus and the Mittag Leffler matrix function
and utilized to derive the sufficient conditions in stochastic settings to guar-
antee that the system (8.1) is completely controllable. The main advantage of
the proposed technique relies on some hypotheses and fixed point theorems,
namely the Bohnenblust Karlin fixed point theorem for the convex case and
the Covitz Nadler for the nonconvex case for establishing the proposed
results. The result can be extended to the infinite dimensional Hilbert space
for approximate controllability results.
Further, impulsive fractional differential equations and inclusions have
become important in recent years as mathematical models of many phenom-
ena in both physical and social sciences. Upon making some appropriate
assumptions on system functions, by adapting the techniques and ideas estab-
lished in this chapter with suitable modifications, one can easily prove that
complete controllability results of impulsive fractional higher order stochas-
tic integrodifferential inclusions are driven by fractional Brownian motion in
finite dimensional space.