Page 204 - Mathematical Techniques of Fractional Order Systems

Controllability of Single-valued Chapter | 7 193

The approximate controllability of the considered system (7.1) (7.2) is
studied under the following hypotheses

(H 1 ) A generates an t α21 ; 1 -regularized family S α ðtÞ such that there
ΓðαÞ
exist ω . 0 and M . 0 such that
ωt
OS α ðtÞO # Me ; ’tAJ:
(H 2 ) For each ðt; sÞAJ 3 J, the function gðt; s; UÞ:B-H is continuous and
for each φAB the function gðU; U; φÞ:J 3 J-H is strongly measurable.
(H 3 ) The multivalued map F:J 3 B 3 H-P bd;cl;cv ðHÞ satisfies the follow-
ing conditions
(i) for each tAJ, the map Fðt; U; UÞ:B 3 H-P bd;cl;cv ðHÞ is u.s.c.
(ii) for each ðφ; yÞAB 3 H, the map FðU; φ; yÞ:J-P bd;cl;cv ðHÞ is measur-
able and the set

ð t
1
N F;φ 5 fAL ðJ; HÞ: fðtÞAFt; φ; gðt; s; φÞds for a:etAJ
0
is nonempty.
(H 4 ) For each positive number r . 0, there exists a positive function μðrÞ
depending on r such that

^
2
sup OFðt; φ; LφÞO # μðrÞ; lim μðrÞ 5 δ , N
2
OφO # r r-N r
B
^
and there exists constant d . 0 such that
EOFðt; φ; LφÞO
^ 2
^
0 # lim sup 2 # d:
OφO -N tAJ OφO
2
B B
(H 5 ) The function σ:J 3 J 3 B-LðK; HÞ satisfies the following
(i) for each ðt; sÞAJ 3 J, the function σðt; s; UÞ:B-LðK; HÞ is continuous
and for each φAB, the function σðU; U; φÞ:J 3 J-LðK; HÞ is strongly
measurable.
1
(ii) there is a positive integrable function mAL ð½0;bŠÞ and a continuous
nondecreasing function Λ σ : ½0;NÞ-ð0;NÞ such that for every
ðt; s; xÞAJ 3 J 3 B, we have
ð t Λ σ ð4rÞ
2
2
EOσðt; s; xÞO ds # mðtÞΛ σ ðOxO Þ; lim 5 Δ , N:
Q B
0 r-N r
(H 6 ) The linear system corresponding to the fractional order system
(7.1) (7.2) is approximately controllable.
(H 7 ) The functions Fðt; ψ; yÞ:J 3 B 3 H-P bd;cl;cv ðHÞ; gðt; s; ψÞ:J 3 J 3
B-H; and σðt; s; ψÞ:J 3 J 3 B-LðK; HÞ are uniformly bounded for all
tAJ; ψAB and yAH.