Page 597 - Mathematical Techniques of Fractional Order Systems

Control and Synchronization Chapter | 19 567

Mittag Leffler stability and extension of Lyapunov direct method for frac-
tional order nonlinear systems is given by Li et al. (2010).Mittag Leffler
stability is given by following theorem.

Theorem 1: (Li et al., 2010) The solution of following fractional order non-
autonomous system
α
D xtðÞ 5 fðt; xÞ ð19:8Þ
t
is said to be Mittag Leffler stable if
α b
:xðtÞ: # fm½xðt 0 ފE α ð2λðt2t 0 Þ Þg ð19:9Þ
where t 0 is the initial time, αA 0; 1ð Þ, λ . 0,b . 0,m 0ðÞ 5 0,m xðÞ $ 0 and
n
mxðÞ is locally lipschitz on xABAR with Lipschitz constant m 0 .
In the above theorem, a function E α has been used. This function is called
Mittag Leffler function (Mathai and Haubold, 2008)and hassignificant
importance in fractional calculus. It was proposed by Agarwal and Humbert in
1953. Mittag Leffler function in two parameter form can be defined as:
N k
X z
E α;β zðÞ 5
ð
Γαk 1 βÞ
k50
where, zAC;α; βAC and R αðÞ . 0
Also for α 5 β 5 1, the function reduces to conventional exponential
function, E 1;1 zðÞ 5 e z
For β 5 1, Mittag Leffler function can be defined in one parameter form
as follows:
N k
X z
E α;1 zðÞ 5 E α ðzÞ
Γðαk 1 1Þ
k50
A number of researchers have tried to extend Lyapunov stability criterion
to fractional order systems. The following theorem can be stated for the same.
Theorem 2: (Li et al., 2010) Let x 5 0 be an equilibrium point for the non-
autonomous fractional order system (19.8). Assume that there exists a
Lyapunov function Vt; xtðÞÞ and class K functions γ ði 5 1; 2; 3Þ satisfying
ð
i
γ :x: # Vt; xtðÞð Þ # γ :x: ð19:10Þ
2
1
α
D Vt; xtðÞð Þ #2 γ :x: ð19:11Þ
t 3
where t $ 0, αA 0; 1Þ. Then x 5 0 is Mittag Leffler stable,
ð
asymptotically.
To use the above theorems in our problem, the following lemma and the-
orem are described here.