- Open Access
Stability of a nonlinear non-autonomous fractional order systems with different delays and non-local conditions
© El-Sayed and Gaafar; licensee Springer. 2011
- Received: 1 March 2011
- Accepted: 27 October 2011
- Published: 27 October 2011
In this paper, we establish sufficient conditions for the existence of a unique solution for a class of nonlinear non-autonomous system of Riemann-Liouville fractional differential systems with different constant delays and non-local condition is. The stability of the solution will be proved. As an application, we also give some examples to demonstrate our results.
- Riemann-Liouville derivatives
- nonlocal non-autonomous system
- time-delay system
- stability analysis
where D α denotes the Riemann-Liouville fractional derivative of order α ∈ (0, 1), x(t) = (x 1(t), x 2(t), ..., x n (t))', where ' denote the transpose of the matrix, and f i , g i : [0, T] × R n → R are continuous functions, Φ(t) = (ϕ i (t)) n × 1are given matrix and O is the zero matrix, r j ≥ 0, j = 1, 2, ..., n, are constant delays.
Recently, much attention has been paid to the existence of solution for fractional differential equations because they have applications in various fields of science and engineering. We can describe many physical and chemical processes, biological systems, etc., by fractional differential equations (see [1–9] and references therein).
In this work, we discuss the existence, uniqueness and uniform of the solution of stability non-local problem (1)-(3). Furthermore, as an application, we give some examples to demonstrate our results.
For the earlier work we mention: De la Sen  investigated the non-negative solution and the stability and asymptotic properties of the solution of fractional differential dynamic systems involving delayed dynamics with point delays.
by the method of steps, where A, B, C are bounded linear operators defined on a Banach space X.
Sabatier et al.  delt with Linear Matrix Inequality (LMI) stability conditions for fractional order systems, under commensurate order hypothesis.
by the method of steps, where A 0, A 1 are constant matrices and studied the finite time stability for it.
Let L 1[a, b] be the space of Lebesgue integrable functions on the interval [a, b], 0 ≤ a < b < ∞ with the norm .
where Γ (.) is the gamma function.
The following theorem on the properties of fractional order integration and differentiation can be easily proved.
, and if f(t) ∈ L 1 then .
, n = 1,2,3,... uniformly.
, α ∈ (0,1), f (t) is absolutely continuous.
, α ∈ (0,1), f (t) is absolutely continuous.
Let X = (C n (I), || . ||1), where C n (I) is the class of all continuous column n-vectors function. For x ∈ C n [0, T], the norm is defined by , where N > 0.
and , .
Then there exists a unique solution × ∈ X of the problem (1)-(3).
Now choose N large enough such that , so the map F : X → X is a contraction and hence, there exists a unique column vector x ∈ X which is the solution of the integral equation (4).
Now we complete the proof by proving the equivalence between the integral equation (4) and the non-local problem (1)-(3). Indeed:
which completes the proof of the equivalence between (4) and (1).
Then from (4),. Also from (2), we have .
In this section we study the stability of the solution of the non-local problem (1)-(3)
Definition 5 The solution of the non-autonomous linear system (1) is stable if for any ε > 0, there exists δ > 0 such that for any two solutions x(t) = (x 1(t), x 2(t), ..., x n (t))' and with the initial conditions (2)-(3) and respectively, one has , then for all t ≥ 0.
Theorem 3 The solution of the problem (1)-(3) is uniformly stable.
Therefore, for δ > 0 s.t., we can find s.t. which proves that the solution x(t) is uniformly stable.
where A(t) = (a ij (t)) n×n and are given continuous matrix, then the problem has a unique uniformly stable solution x ∈ X on (-∞, T], T < ∞
where B(t) = (b ij (t)) n×n , and are given continuous matrices, then the problem has a unique uniformly stable solution x ∈ X on (-∞, T], T < ∞
where A(t) = (a ij (t)) n×n B(t) = (b ij (t)) n×n , and H(t) = (h i (t)) n×1are given continuous matrices, then the problem has a unique uniformly stable solution x ∈ X on (-∞, T], T < ∞.
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