Int. Journal of Math. Analysis, Vol. 4, 2010, no. 23, 1105 - 1116
Controllability of Fractional Semilinear Mixed Volterra-Fredholm Integrodifferential Equations with Nonlocal Conditions Mohammed Matar Mathematics Department, Al-Azhar University-Gaza Jamal Ebed El-Naser St., Gaza Strip, PA mohammed
[email protected] Abstract In this article, the controllability problem of fractional semilinear mixed VolterraFredholm integrodifferential equations in Banach space together with nonlocal conditions is investigated. An example is given to illustrate the theory.
Mathematics Subject Classification: 34A34, 34G20, 26A33 Keywords: Controllability, Volterra-Fredholm integrodifferential equation, fractional calculus
1
Introduction
Fractional differential equations are emerged as a new branch of applied mathematics by which many physical and engineering approaches can be modelled [5]. The fact that fractional differential equations are considered as alternative models to nonlinear differential equations which induced extensive researches in various fields including the theoretical part [8],[9],[13]. The existence of fractional semilinear integrodifferential equations are one of the theoretical fields that investigated by many authors [11]. On the other hand, the controllability is one concept of control dynamic systems that some classes of such systems can be represented by nonlinear differential equations [2], [3], [6]. So it is naturally to extend this concept of such systems to be represented by fractional differential equations [1], [7], [10], [14]. Recently, controllability of fractional nonlinear differential systems in infinite dimensional spaces has been investigated by many authors [4], [15]. Motivated by these works, we show that a particular class of fractional integrodifferential systems in Banach spaces is controllable provided that some conditions have to be satisfied. The paper is organized as follows. In section 2, some definitions, lemmas, and assumptions are introduced to be used in the sequel. Section 3 and 4 will involve the main
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results and proofs of controllability problems of the given system with classical initial and nonlocal conditions respectively. Finally, in section 5, an example is introduced to illustrate the theory.
2
Preliminaries
We need some basic definitions and properties of fractional calculus which will be used in this paper. Definition 2.1 A real function f (t), t ≥ t0 is said to be in the space Cμ , μ ∈ R, if there exists a real number p > μ, such that f (t) = (t − t0 )p f1 (t), where f1 ∈ C[t0 , ∞), and it is said to be in the space Cμn if and only if f (n) ∈ Cμ , n ∈ N. Definition 2.2 A function f ∈ Cμ , μ ≥ −1 is said to be fractional integrable of order α > 0 if 1 I α f (t) = (I α f )(t) = Γ (α)
t (t − s)α−1 f (s)ds < ∞, t0
where t0 ≥ 0; and if α = 0, then I 0 f (t) = f (t). Moreover, I α (Cμ ) denotes the space of all fractional integrable functions of order α. Next, we introduce the Caputo fractional derivative. Definition 2.3 The fractional derivative in the Caputo sense is defined as dα f (t) = I 1−α dtα
df (t) dt
1 = Γ (1 − α)
t (t − s)
−α
df (s) ds ds
t0
for 0 < α ≤ 1, t0 ≥ 0, f ∈ C−1 . The properties of the above operators and the general theory of fractional differential equations can be found in [12] and [13]. Consider the following fractional semilinear integrodifferential system ⎧ ⎨ ⎩
dα x(t) dtα
t T = Ax(t) + Bu(t) + f (t, x(t), k(t, s, x(s))ds, h(t, s, x(s))ds),
x(t0 ) = x0 ∈ X.
t0
t0
(1)
where t ∈ J = [t0 , T ], t0 ≥ 0, 0 < α < 1, x ∈ Y = C(J, X) is a continuous function on J with values in the Banach space X and xY = maxt∈J x(t)X . The control function
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u ∈ L2 (J, U), a Banach space of admissible control functions with U as a Banach space. B : U → X is a bounded linear operator, and the nonlinear functions f : J × X × X × X → X, k : D × X → X, and h : D0 × X → X are continuous. Here D = α {(t, s) ∈ R2 : t0 ≤ s ≤ t ≤ T } , and D0 = J × J. The operator dtd α denotes the Caputo fractional derivative of order α. For brevity let us take T t k(t, s, x(s))ds, and Hx(t) = h(t, s, x(s))ds. Kx(t) = t0
t0
The common norm · is used throughout the sequel. Definition 2.4 ([4]) A continuous solution x(t) of the integral equation 1 x(t) = S(t − t0 )x0 + Γ(α)
t (t − s)α−1 S(t − s) (Bu(s) + f (s, x(s), Kx(s), Hx(s))) ds t0
(2) is called a mild solution of equation (1), where S(·) is a strongly continuous semigroup of bounded linear operator generated by A. If α → 1, the mild solution (2) becomes the standard mild solution given by t S(t − s) (Bu(s) + f (s, x(s), Kx(s), Hx(s))) ds
x(t) = S(t − t0 )x0 + t0
which is the mild solution of the system ⎧ t T ⎨ dx(t) = Ax(t) + Bu(t) + f (t, x(t), k(t, s, x(s))ds, h(t, s, x(s))ds), dt t0 t0 ⎩ x(t0 ) = x0 .
(3)
The non-fractional system (3) was consider by [6] in the inclusion form. Analogous to the conventional controllability concept, the controllability of fractional dynamical system (1) is defined as in [7]. Definition 2.5 The fractional system (1) is said to be controllable on the interval J if for every x0 , x1 ∈ X, there exists a control u ∈ L2 (J, U) such that the solution x(·) of (1) satisfies x(T ) = x1 . To proceed, we need the following assumptions: (A1) S(·) is a C0 −semigroup generated by the operator A on X which satisfies M = maxt∈J S(t) .
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(A2) There exist positive continuous real-valued functions f1 , k1 , and h1 defined respectively on J, D, and D0 , such that ⎧ ⎨ f (t, x1 , y1 , z1 ) − f (t, x2 , y2, z2 ) ≤ f1 (t)(x1 − x2 + y1 − y2 + z1 − z2 ) k(t, s, x1 ) − k(t, s, x2 ) ≤ k1 (t, s) x1 − x2 ⎩ h(t, s, x1 ) − h(t, s, x2 ) ≤ h1 (t, s) x1 − x2 for all x1 , y1, z1 , x2 , y2 , z2 ∈ Y. Moreover, assume that k2 (t) = sups∈J {k1 (t, s), k(t, s, 0)} and h2 (t) = sups∈J {h1 (t, s), h(t, s, 0)} for any t ∈ J. (A3) The real-valued functions g(t) = f1 (t)(1 + (t − t0 )k2 (t) + (T − t0 )h2 (t)) and b(t) = (t−t0 )f1 (t)k2 (t) + (T −t0 )f1 (t)h2 (t)+ f (t, 0, 0, 0) are fractional integrable of order α for any t ∈ J. Moreover, The linear operators F , and G are defined on I α (Cμ ) such that ⎧ T ⎪ ⎪ 1 ⎪ F (g) = (T − s)α−1 g(s)ds ⎨ Γ(α) ⎪ ⎪ ⎪ ⎩ G(b) =
t0
1 Γ(α)
T
(T − s)α−1 b(s)ds
t0
are also fractional integrable of order α. (A4) The linear operator W : L2 (J, U) → X defined by 1 Wu = Γ(α)
T (T − s)α−1 S(T − s)Bu(s)ds t0
defined on L2 (J, U)/ ker W , and there exists a induces an invertible operator W −1 positive constant K such that B W ≤ K. Remark 2.6 In view of (A2), one can get the following estimates Kx(t) ≤ (t − t0 )k2 (t) (x + 1) Hx(t) ≤ (T − t0 )h2 (t) (x + 1) and
Kx1 (t) − Kx2 (t) ≤ (t − t0 )k2 (t) x1 − x2 Hx1 (t) − Hx2 (t) ≤ (T − t0 )h2 (t) x1 − x2
for any t ∈ J. In this section, we prove the main results for controllability of the system (1).
Controllability of fractional Volterra-Fredholm systems
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Theorem 2.7 If the hypotheses (A1)-(A4) are satisfied, and if q −1 MI α [MKF (g)(t) + g(t)] < 1, 0 < q < 1, f orany t ∈ J, then the fractional integrodifferential system (1) is controllable on J. Proof. Let Br = {x ∈ Y : x ≤ r} be a closed subspace of Y, where r is any fixed finite real number which satisfies the inequality MK α α (1 − q)r ≥ M x0 + (x1 + M x0 ) (T − t0 ) + MI (MKG (b) (t) + b(t)) . Γ(α + 1) Define the control u ∈ L2 (J, U) by ⎡
−1 ⎣x1 − S(T − t0 )x0 − u(t) = W
1 Γ(α)
T
⎤ (T − s)α−1 S(T − s)f (s, x(s), Kx(s), Hx(s))ds⎦ (t).
t0
We shall show that when using the above control, the operator Ψ : Y → Y defined by t 1
−1 [x1 − S(T − t0 )x0 Ψx(t) = T (t − t0 )x0 + (t − s)α−1 S(t − s)B W Γ(α) t0 ⎤ T 1 − (T − r)α−1 S(T − r)f (r, x(r), Kx(r), Hx(r))dr ⎦ (s)ds Γ(α) t0 t 1 (t − s)α−1 S(t − s)f (s, x(s), Kx(s), Hx(s))ds. + Γ(α) t0 has a Banach fixed point which is a solution of the system (1). It is clear that if Ψ has a fixed point x(·), then Ψx(T ) = x1 . Hence the system (1) is controllable on J. It remains to prove the existence of such fixed point. Firstly, we show that the operator Ψ maps Br into itself. For this, by using assumptions (A1), (A2), and triangle inequality, we have t 1
−1 [x1 − S(T − t0 )x0 (t − s)α−1 S(t − s)B W Ψx(t) ≤ M x0 + Γ(α) t0 ⎤ T 1 α−1 ⎦ + − (T − r) S(T − r)f (r, x(r), Kx(r), Hx(r))dr (s)ds Γ(α) t0 t 1 α−1 (t − s) S(t − s)f (s, x(s), Kx(s), Hx(s))ds + Γ(α) t0
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MK ≤ M x0 + Γ(α)
t (t − s)α−1 (x1 + M x0 ) ds t0
2
+
M K (Γ(α))2
t
⎛ T α−1 ⎝ (t − s) (T − r)α−1 f1 (r)(x(r) + Kx(r) + Hx(r))
t0
t0
M + f (r, 0, 0, 0) dr) (s)ds + Γ(α)
t (t − s)α−1 × t0
[f1 (s) (x(s) + Kx(s) + Hx(s)) + f (s, 0, 0, 0)] ds, by Remark (2.6), we have MK (x1 + M x0 ) (t − t0 )α Γ(α + 1) ⎛ T t 2 M K α−1 ⎝ + (t − s) (T − r)α−1 f1 (r)× 2 (Γ(α))
Ψx(t) ≤ M x0 +
t0
t0
(x(r) + k2 (r) (x + 1) (r − t0 ) + h2 (r)(T − t0 )(x + 1)) + f (r, 0, 0, 0) dr) (s)ds t M + (t − s)α−1 f1 (s)(x(s) + k2 (s) (x + 1) (s − t0 ) + h2 (s)(T − t0 )(x + 1)) Γ(α) t0
+ f (s, 0, 0, 0)) ds therefore MK (x1 + M x0 ) (t − t0 )α Γ(α + 1)
Ψx(t) ≤ M x0 + M 2K (Γ(α))2
+ t
⎛
(t − s)α−1 ⎝
t0
+
T
⎞ (T − r)α−1 f1 (r)(1 + (r − t0 )k2 (r) + (T − t0 )h2 (r)) x(r) dr ⎠ (s)ds
t0
M 2K (Γ(α))2
t (t − s)α−1 × t0
⎛ T ⎞ ⎝ (T − r)α−1 (f1 (r)k2 (r)(r − t0 ) + f1 (r)h2 (r)(T − t0 ) + f (r, 0, 0, 0)) dr ⎠ (s)ds t0
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Controllability of fractional Volterra-Fredholm systems
M + Γ(α)
t (t − s)α−1 f1 (s)(1 + (s − t0 )k2 (s) + (T − t0 )h2 (s)) x(s) ds t0
M + Γ(α)
t (t − s)α−1 (f1 (s)k2 (s)(s − t0 ) + f1 (s)h2 (s)(T − t0 ) + f (s, 0, 0, 0)) ds t0
using the hypothesis (A3), we get MK (x1 + M x0 ) (T − t0 )α Ψx(t) ≤ M x0 + Γ(α + 1) t t 2 M 2K M K + (t − s)α−1 G(b)(s)ds x (t − s)α−1 F (g) (s)ds + Γ(α) Γ(α) t0
+
M x Γ(α)
t0
t (t − s)α−1 g(s)ds + t0
M Γ(α)
t (t − s)α−1 b(s)ds t0
MK (x1 + M x0 ) (T − t0 )α Γ(α + 1) + MI α (MKG (b) (t) + b(t)) + MI α (MKF (g) (t) + g(t)) x . = M x0 +
Hence, if x ∈ Br then Ψx(t) ≤ (1 − q)r + qr = r, which proves that Ψ(Br ) ⊂ Br . Next, we prove that Ψ is a contraction mapping. For this, pick any x1 , x2 ∈ Br , we have Ψx1 (t) − Ψx2 (t) ⎛ t T 1 α−1 −1 ⎝ 1 ˜ (t − s) S(t − s)B W (T − r)α−1 S(T − r) × = Γ(α) Γ(α) t0
t0
[f (r, x2 (r), Kx2 (r), Hx2 (r)) − f (r, x1 (r), Kx1 (r), Hx1 (r))] dr) (s)ds t 1 α−1 + (t − s) S(t − s) [f (s, x1 (s), Kx1 (s), Hx1 (s)) − f (s, x2 (s), Kx2 (s), Hx2 (s))] ds Γ(α) t0 ⎛ t T 1 M 2K (t − s)α−1 ⎝ (T − r)α−1 × ≤ Γ(α) Γ(α) t0
t0
f (r, x2 (r), Kx2 (r), Hx2 (r)) − f (r, x1 (r), Kx1 (r), Hx1(r)) dr) (s)ds t M (t − s)α−1 f (s, x1 (s), Kx1 (s), Hx1 (s)) − f (s, x2 (s), Kx2 (s), Hx2(s)) ds + Γ(α) t0
using (A2) we have Ψx1 (t) − Ψx2 (t)
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M K ≤ Γ(α)
t
⎛ 1 (t − s)α−1 ⎝ Γ(α)
t0
T (T − r)α−1 f1 (r)× t0
[x2 (r) − x1 (r) + Kx2 (r) − Kx1 (r) + Hx2 (r) − Hx1 (r)] dr) (s)ds t M (t − s)α−1 f1 (s) [x2 (s) − x1 (s) + Kx2 (s) − Kx1 (s) + Hx2 (s) − Hx1 (s)] ds + Γ(α) t0 ⎛ t T 2 M K 1 ≤ (t − s)α−1 ⎝ (T − r)α−1 f1 (r)× Γ(α) Γ(α) t0 t0 ⎡ ⎤ ⎞ r T ⎣x2 (r) − x1 (r) + k1 (r, θ) x1 (θ) − x2 (θ) dθ + h1 (r, θ) x1 (θ) − x2 (θ) dθ⎦ dr ⎠ (s)ds t0
t0
t M + (t − s)α−1 f1 (s) × Γ(α) t0 ⎤ ⎡ s s ⎣x2 (s) − x1 (s) + k1 (s, r) x1 (r) − x2 (r) dr + h1 (s, r) x1 (r) − x2 (r) dr ⎦ ds t0
t0
and by (A3) we have Ψx1 (t) − Ψx2 (t) M 2K x2 − x1 × ≤ Γ(α) ⎡ ⎤ ⎞ ⎛ t T r T 1 (t − s)α−1 ⎝ (T − r)α−1 f1 (r) ⎣1 + k1 (r, θ)dθ + h1 (r, θ)dθ⎦ dr ⎠ (s)ds Γ(α) t0 t0 t0 t0 ⎡ ⎤ t s T M + x2 − x1 (t − s)α−1 f1 (s) ⎣1 + k1 (s, r)dr + h1 (s, r)dr ⎦ ds Γ(α) t0
t0
t0
2
M K x2 − x1 × Γ(α) ⎞ ⎛ t T 1 (t − s)α−1 ⎝ (T − r)α−1 f1 (r) [1 + (r − t0 )k2 (r) + (T − t0 )h2 (r)] dr ⎠ (s)ds Γ(α) ≤
t0
+
t0
M x2 − x1 Γ(α)
t (t − s)α−1 f1 (s) [1 + (s − t0 )k2 (s) + (T − t0 )h2 (s)] ds t0
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M 2K ≤ x2 − x1 Γ(α)
t (t − s)
α−1
M F (g)(s)ds + x2 − x1 Γ(α)
t0
t (t − s)α−1 g(s)ds t0
= MI (g(t) + MKF (g)(t)) x2 − x1 ≤ q x2 − x1 . α
Hence, Ψ has a unique fixed point x = Ψ(x) ∈ Br , which is a solution of (2), and hence is a mild solution of (1).
3
Nonlocal problem
We discuss in this section, the controllability problem of the fractional integrodifferential system (1) with a nonlocal condition of the form x(t0 ) + g(x) = x0 ,
(4)
where g : Y → Y is a given function that satisfies the following condition (A5) g is a continuous function and there exists a positive constant G such that g(x) − g(y) ≤ G x − y , for x, y ∈ Y. Theorem 3.1 If the hypotheses (A1)-(A5) are satisfied, and if MK −1 α α (T − t0 ) < 1, 0 < q < 1, q M I [MKF (g) (t) + g(t)] + G 1 + Γ(α + 1) then the fractional integrodifferential system (1) with nonlocal condition (4) is controllable on J. Proof. Let Br = {x ∈ Y : x ≤ r} be a closed subspace of Y, where r is any fixed finite real number which satisfies the inequality MK (x1 + M x0 + M g(0)) (T − t0 )α r ≥ (1 − q) −1 M (x0 + g(0)) + Γ(α + 1) +MI α (MKG (b) (t) + b(t))) . Define the control u ∈ L2 (J, U) by ˜ −1 [x1 − S(T − t0 ) (x0 − g(x)) u(t) = W ⎤ T 1 (T − s)α−1 S(T − s)f (s, x(s), Kx(s), Hx(s))ds⎦ (t). − Γ(α) t0
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Applying the same technique as in proof of Theorem 2.7, on the operator Φ : Y → Y defined by Φx(t) 1 = T (t − t0 ) (x0 − g(x)) + Γ(α)
T
˜ −1 [x1 − S(T − t0 ) (x0 − g(x)) (t − s)α−1 S(t − s)B W
t0
−
1 Γ(α)
T
⎤
(T − r)α−1 S(T − r)f (r, x(r), Kx(r), Hx(r))dr ⎦ (s)ds
t0
1 + Γ(α)
t (t − s)α−1 S(t − s)f (s, x(s), Kx(s), Hx(s))ds. t0
to show that, this operator has a fixed point x = Φ(x) ∈ Br . This fixed point is then a solution of the system (1) and (4). On the other hand, by using this fixed point of the operator Φ, it is clear that Φx(T ) = x1 . Hence the system (1-4) is controllable on J. The rest of the details of the proof is similar to Theorem 2.7 and hence it is omitted.
4
Example
Consider the following nonlinear partial integrodifferential system ⎧ t T ⎪ ∂α ∂2 ⎪ ω(t, y) = ω(t, y) + μ(t, y) + f (t, ω(t, y), k(t, s, ω(s, y))ds, h(t, s, ω(s, y))ds), ⎪ α 2 ⎪ ∂y ⎨ ∂t t0 t0 ω(t, 0) = ω(t, π) = 0, t ∈ J = [0, π], ⎪ m ⎪ ⎪ ⎪ ⎩ ω(0, y) + ck φ(tk , y) = ω0 (y), φ ∈ Br , 0 ≤ y ≤ π, k=1
(5) where 0 < α < 1, ck > 0, and μ : J × [0, π] → [0, π] is continuous. Let X = U = L2 [0, π], Y = C(J, L2 [0, π]), Br = {x ∈ Y : x ≤ r} for some r. Put x(t) = ω(t, ·), (Bu)(t)(·) = m μ(t, ·), and g(φ(t, ·)) = ck φ(tk , y). Let A : D(A) ⊂ X → X defined by Ax = x , where k=1
D(A) = x ∈ X : x, x absolutely continuous, x ∈ X, x(0) = x(π) = 0 . Therefore Ax =
∞ n=1
n2 (x, xn )xn , x ∈ D(A),
Controllability of fractional Volterra-Fredholm systems
1115
where xn (s) = π2 sin (ns), n = 1, 2, 3, ...is the orthogonal set of eigenfunctions of A. It can be easily shown that A is the infinitesimal generator of an analytic semigroup S(t), t ≥ 0 in X and is given by S(t)x =
∞
exp(−n2 t)(x, xn )xn , x ∈ X,
n=1
and S(t) satisfies (A1). With the above choices, we see that the system (5) is the abstract formulation of (1) and (4). Assume that the operator W defined by 1 (W μ) (y) = Γ(α) n=1 ∞
T
(T − s)α−1 exp(−n2 (T − s))(μ(t, y), xn )xn ds
0
˜ −1 in L2 (J, U)/ ker W. Moreover, if the functions f , has a bounded invertible operator W k, and h satisfy the hypotheses (A2) and (A3), then we deduce that the system (5) is controllable on J.
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