Advances in Difference Equations
Volume 2010 (2010), Article ID 279493, 9 pages
doi:10.1155/2010/279493
Research Article

Boundary Controllability of Nonlinear Fractional Integrodifferential Systems

Hamdy M. Ahmed

Higher Institute of Engineering, El-Shorouk Academy, P.O. 3 El-Shorouk City, Cairo, Egypt

Received 21 July 2009; Accepted 11 January 2010

Academic Editor: Toka Diagana

Copyright © 2010 Hamdy M. Ahmed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Sufficient conditions for boundary controllability of nonlinear fractional integrodifferential systems in Banach space are established. The results are obtained by using fixed point theorems. We also give an application for integropartial differential equations of fractional order.

1. Introduction

Let 𝐸 and 𝑈 be a pair of real Banach spaces with norms ‖ ⋅ ‖ and | ⋅ | , respectively. Let 𝜎 be a linear closed and densely defined operator with 𝐷 ( 𝜎 ) ⊆ 𝐸 and let 𝜏 ⊆ 𝑋 be a linear operator with 𝐷 ( 𝜎 ) and 𝑅 ( 𝜏 ) ⊆ 𝑋 , a Banach space. In this paper we study the boundary controllability of nonlinear fractional integrodifferential systems in the form 𝑑 𝛼 𝑥 ( 𝑡 ) 𝑑 𝑡 𝛼 = 𝜎 𝑥 ( 𝑡 ) + 𝑓 ( 𝑡 , 𝑥 ( 𝑡 ) ) + 𝑡 0 [ ] , 𝑔 ( 𝑡 , 𝑠 , 𝑥 ( 𝑠 ) ) 𝑑 𝑠 , 𝑡 ∈ 𝐽 = 0 , 𝑏 𝜏 𝑥 ( 𝑡 ) = 𝐵 1 𝑢 ( 𝑡 ) , 𝑥 ( 0 ) = 𝑥 0 , ( 1 . 1 ) where 0 < 𝛼 ≤ 1 and 𝐵 1 ∶ 𝑈 → 𝑋 is a linear continuous operator, and the control function 𝑢 is given in 𝐿 2 ( 𝐽 , 𝑈 ) , a Banach space of admissible control functions. The nonlinear operators 𝑓 ∶ 𝐽 × 𝐸 → 𝐸 and 𝑔 ∶ Δ × 𝐸 → 𝐸 are given and Δ ∶ ( 𝑡 , 𝑠 ) ; 0 ≤ 𝑠 ≤ 𝑡 ≤ 𝑏 .

Let 𝐴 ∶ 𝐸 → 𝐸 be the linear operator defined by 𝐷 ( 𝐴 ) = { 𝑥 ∈ 𝐷 ( 𝜎 ) ; 𝜏 𝑥 = 0 } , 𝐴 𝑥 = { 𝜎 𝑥 , f o r 𝑥 ∈ 𝐷 ( 𝐴 ) } . ( 1 . 2 ) The controllability of integrodifferential systems has been studied by many authors (see [1–6]). This work may be regarded as a direct attempt to generalize the work in [7, 8].

2. Main Result

Definition 2.1. System (1.1) is said to be controllable on the interval 𝐽 if for every 𝑥 0 , 𝑥 1 ∈ 𝐸 there exists a control 𝑢 ∈ 𝐿 2 ( 𝐽 , 𝑈 ) such that 𝑥 ( ⋅ ) of (1.1) satisfies 𝑥 ( 𝑏 ) = 𝑥 1 .

To establish the result, we need the following hypotheses. (H1) 𝐷 ( 𝜎 ) ⊂ 𝐷 ( 𝜏 ) and the restriction of 𝜏 to 𝐷 ( 𝜎 ) is continuous relative to the graph norm of 𝐷 ( 𝜎 ) .(H2)The operator 𝐴 is the infinitesimal generator of a compact semigroup 𝑇 ( 𝑡 ) and there exists a constant 𝑀 1 > 0 such that ‖ 𝑇 ( 𝑡 ) ‖ ≤ 𝑀 1 . (H3)There exists a linear continuous operator 𝐵 ∶ 𝑈 → 𝐸 such that 𝜎 𝐵 ∈ 𝐿 ( 𝑈 , 𝐸 ) , 𝜏 ( 𝐵 𝑢 ) = 𝐵 1 𝑢 , for all 𝑢 ∈ 𝑈 . Also 𝐵 𝑢 ( 𝑡 ) is continuously differentiable and ‖ 𝐵 𝑢 ‖ ≤ 𝐶 ‖ 𝐵 1 𝑢 ‖ for all 𝑢 ∈ 𝑈 , where C is a constant.(H4) For all 𝑡 ∈ ( 0 , 𝑏 ] and 𝑢 ∈ 𝑈 , 𝑇 ( 𝑡 ) 𝐵 𝑢 ∈ 𝐷 ( 𝐴 ) . Moreover, there exists a positive constant 𝐾 1 > 0 such that ‖ 𝐴 𝑇 ( 𝑡 ) ‖ ≤ 𝐾 1 . (H5)The nonlinear operators 𝑓 ( 𝑡 , 𝑥 ( 𝑡 ) ) and 𝑔 ( 𝑡 , 𝑠 , 𝑥 ( 𝑠 ) ) , for 𝑡 , 𝑠 ∈ 𝐽 , satisfy ‖ 𝑓 ( 𝑡 , 𝑥 ( 𝑡 ) ) ‖ ≤ 𝐿 1 , ‖ 𝑔 ( 𝑡 , 𝑠 , 𝑥 ( 𝑠 ) ) ‖ ≤ 𝐿 2 , ( 2 . 1 ) where 𝐿 1 ≥ 0 and 𝐿 2 ≥ 0 . (H6) The linear operator 𝑊 from 𝐿 2 ( 𝐽 , 𝑈 ) into 𝐸 defined by 𝑊 𝑢 = 𝛼 𝑏 0 ∞ 0 𝜃 ( 𝑡 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑏 − 𝑠 ) 𝛼 𝜃 ) 𝜎 − 𝐴 𝑇 ( ( 𝑏 − 𝑠 ) 𝛼 𝜃 ) 𝐵 𝑢 ( 𝑠 ) 𝑑 𝜃 𝑑 𝑠 , ( 2 . 2 ) where 𝜉 𝛼 ( 𝜃 ) is a probability density function defined on ( 0 , ∞ ) (see [9, 10]) and induces an invertible operator 𝑊 − 1 defined on 𝐿 2 ( 𝐽 , 𝑈 ) / k e r 𝑊 , and there exists a positive constant 𝑀 2 > 0 and 𝑀 3 > 0 such that ‖ 𝐵 ‖ ≤ 𝑀 2 and ‖ 𝑊 − 1 ‖ ≤ 𝑀 3 . Let 𝑥 ( 𝑡 ) be the solution of (1.1). Then we define a function 𝑧 ( 𝑡 ) = 𝑥 ( 𝑡 ) − 𝐵 𝑢 ( 𝑡 ) and from our assumption we see that 𝑧 ( 𝑡 ) ∈ 𝐷 ( 𝐴 ) . Hence (1.1) can be written in terms of 𝐴 and 𝐵 as 𝑑 𝛼 𝑥 ( 𝑡 ) 𝑑 𝑡 𝛼 = 𝐴 𝑧 ( 𝑡 ) + 𝜎 𝐵 𝑢 ( 𝑡 ) + 𝑓 ( 𝑡 , 𝑥 ( 𝑡 ) ) + 𝑡 0 𝑔 ( 𝑡 , 𝑠 , 𝑥 ( 𝑠 ) ) 𝑑 𝑠 , 𝑡 ∈ 𝐽 , 𝑥 ( 𝑡 ) = 𝑧 ( 𝑡 ) + 𝐵 𝑢 ( 𝑡 ) , 𝑥 ( 0 ) = 𝑥 0 . ( 2 . 3 ) If 𝑢 is continuously differentiable on [ 0 , 𝑏 ] , then 𝑧 can be defined as a mild solution to be the Cauchy problem 𝑑 𝛼 𝑧 ( 𝑡 ) 𝑑 𝑡 𝛼 𝑑 = 𝐴 𝑧 ( 𝑡 ) + 𝜎 𝐵 𝑢 ( 𝑡 ) − 𝐵 𝛼 𝑢 ( 𝑡 ) 𝑑 𝑡 𝛼 + 𝑓 ( 𝑡 , 𝑥 ( 𝑡 ) ) + 𝑡 0 𝑔 ( 𝑡 , 𝑠 , 𝑥 ( 𝑠 ) ) 𝑑 𝑠 , 𝑡 ∈ 𝐽 , 𝑧 ( 0 ) = 𝑥 0 − 𝐵 𝑢 ( 0 ) , ( 2 . 4 ) and the solution of (1.1) is given by 𝑥 ( 𝑡 ) = ∞ 0 𝜉 𝛼 ( 𝜃 ) 𝑇 ( 𝑡 𝛼 𝑥 𝜃 ) 0 − 𝐵 𝑢 ( 0 ) 𝑑 𝜃 + 𝐵 𝑢 ( 𝑡 ) + 𝛼 𝑡 0 ∞ 0 𝜃 ( 𝑡 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑡 − 𝑠 ) 𝛼 𝜃 ) 𝑓 ( 𝑠 , 𝑥 ( 𝑠 ) ) 𝑑 𝜃 𝑑 𝑠 + 𝛼 𝑡 0 ∞ 0 𝜃 ( 𝑡 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑡 − 𝑠 ) 𝛼 𝑑 𝜃 ) 𝜎 𝐵 𝑢 ( 𝑠 ) − 𝐵 𝛼 𝑢 ( 𝑠 ) 𝑑 𝑠 𝛼 𝑑 𝜃 𝑑 𝑠 + 𝛼 𝑡 0 ∞ 0 𝜃 ( 𝑡 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑡 − 𝑠 ) 𝛼 𝜃 ) 𝑠 0 𝑔 ( 𝑠 , 𝜏 , 𝑥 ( 𝜏 ) ) 𝑑 𝜏 𝑑 𝜃 𝑑 𝑠 ( 2 . 5 ) (see [11–13]).

Since the differentiability of the control 𝑢 represents an unrealistic and severe requirement, it is necessary of the solution for the general inputs 𝑢 ∈ 𝐿 2 ( 𝐽 , 𝑈 ) . Integrating (2.5) by parts, we get 𝑥 ( 𝑡 ) = ∞ 0 𝜉 𝛼 ( 𝜃 ) 𝑇 ( 𝑡 𝛼 𝜃 ) 𝑥 0 𝑑 𝜃 + 𝛼 𝑡 0 ∞ 0 𝜃 ( 𝑡 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑡 − 𝑠 ) 𝛼 𝜃 ) 𝜎 − 𝐴 𝑇 ( ( 𝑡 − 𝑠 ) 𝛼 𝜃 ) 𝐵 𝑢 ( 𝑠 ) 𝑑 𝜃 𝑑 𝑠 + 𝛼 𝑡 0 ∞ 0 𝜃 ( 𝑡 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑡 − 𝑠 ) 𝛼 𝜃 ) 𝑓 ( 𝑠 , 𝑥 ( 𝑠 ) ) 𝑑 𝜃 𝑑 𝑠 + 𝛼 𝑡 0 ∞ 0 𝜃 ( 𝑡 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑡 − 𝑠 ) 𝛼 𝜃 ) 𝑠 0 𝑔 ( 𝑠 , 𝜏 , 𝑥 ( 𝜏 ) ) 𝑑 𝜏 𝑑 𝜃 𝑑 𝑠 . ( 2 . 6 ) Thus (2.6) is well defined and it is called a mild solution of system (1.1).

Theorem 2.2. If hypotheses (H1)–(H6) are satisfied, then the boundary control fractional integrodifferential system (1.1) is controllable on 𝐽 .

Proof. Using assumption (H6), for an arbitrary function 𝑥 ( ⋅ ) define the control 𝑢 ( 𝑡 ) = 𝑊 − 1 𝑥 1 − ∞ 0 𝜉 𝛼 ( 𝜃 ) 𝑇 ( 𝑏 𝛼 𝜃 ) 𝑥 0 𝑑 𝜃 − 𝛼 𝑏 0 ∞ 0 𝜃 ( 𝑏 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑏 − 𝑠 ) 𝛼 𝜃 ) 𝑓 ( 𝑠 , 𝑥 ( 𝑠 ) ) 𝑑 𝜃 𝑑 𝑠 − 𝛼 𝑏 0 ∞ 0 𝜃 ( 𝑏 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑏 − 𝑠 ) 𝛼 𝜃 ) 𝑠 0 𝑔 ( 𝑠 , 𝜏 , 𝑥 ( 𝜏 ) ) 𝑑 𝜏 𝑑 𝜃 𝑑 𝑠 ( 𝑡 ) . ( 2 . 7 ) We shall now show that, when using this control, the operator defined by ( Φ 𝑥 ) ( 𝑡 ) = ∞ 0 𝜉 𝛼 ( 𝜃 ) 𝑇 ( 𝑡 𝛼 𝜃 ) 𝑥 0 𝑑 𝜃 + 𝛼 𝑡 0 ∞ 0 𝜃 ( 𝑡 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑡 − 𝑠 ) 𝛼 𝜃 ) 𝜎 − 𝐴 𝑇 ( ( 𝑡 − 𝑠 ) 𝛼 𝜃 ) 𝐵 𝑢 ( 𝑠 ) 𝑑 𝜃 𝑑 𝑠 + 𝛼 𝑡 0 ∞ 0 𝜃 ( 𝑡 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑡 − 𝑠 ) 𝛼 𝜃 ) 𝑓 ( 𝑠 , 𝑥 ( 𝑠 ) ) 𝑑 𝜃 𝑑 𝑠 + 𝛼 𝑡 0 ∞ 0 𝜃 ( 𝑡 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑡 − 𝑠 ) 𝛼 𝜃 ) 𝑠 0 𝑔 ( 𝑠 , 𝜏 , 𝑥 ( 𝜏 ) ) 𝑑 𝜏 𝑑 𝜃 𝑑 𝑠 ( 2 . 8 ) has a fixed point. This fixed point is then a solution of (1.1). Clearly, ( Φ 𝑥 ) ( 𝑏 ) = 𝑥 1 , which means that the control 𝑢 steers the nonlinear fractional integrodifferential system from the initial state 𝑥 0 to 𝑥 1 in time 𝑇 , provided we can obtain a fixed point of the nonlinear operator Φ .
Let 𝑌 = 𝐶 ( 𝐽 , 𝑋 ) and 𝑌 0 = { 𝑥 ∈ 𝑌 ∶ ‖ 𝑥 ( 𝑡 ) ‖ ≤ 𝑟 , f o r 𝑡 ∈ 𝐽 } , where the positive constant 𝑟 is given by 𝑟 = 𝑀 1 ‖ 𝑥 0 ‖ + 𝑏 𝛼 𝑀 1 ‖ 𝜎 ‖ + 𝐾 1 𝑀 2 𝑀 3 ‖ 𝑥 1 ‖ + 𝑀 1 ‖ 𝑥 0 ‖ + 𝑀 1 𝐿 1 𝑏 𝛼 + 𝑀 1 𝐿 2 𝑏 𝛼 + 1 + 𝑀 1 𝐿 1 𝑏 𝛼 + 𝑀 1 𝐿 2 𝑏 𝛼 + 1 . ( 2 . 9 ) Then 𝑌 0 is clearly a bounded, closed, and convex subset of 𝑌 . We define a mapping Φ ∶ 𝑌 → 𝑌 0 by ( Φ 𝑥 ) ( 𝑡 ) = ∞ 0 𝜉 𝛼 ( 𝜃 ) 𝑇 ( 𝑡 𝛼 𝜃 ) 𝑥 0 𝑑 𝜃 + 𝛼 𝑡 0 ∞ 0 𝜃 ( 𝑡 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑡 − 𝑠 ) 𝛼 𝜃 ) 𝜎 − 𝐴 𝑇 ( ( 𝑡 − 𝑠 ) 𝛼 𝜃 ) 𝐵 𝑊 − 1 × 𝑥 1 − ∞ 0 𝜉 𝛼 ( 𝜃 ) 𝑇 ( 𝑏 𝛼 𝜃 ) 𝑥 0 𝑑 𝜃 − 𝛼 𝑏 0 ∞ 0 𝜃 ( 𝑏 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑏 − 𝑠 ) 𝛼 𝜃 ) 𝑓 ( 𝑠 , 𝑥 ( 𝑠 ) ) 𝑑 𝜃 𝑑 𝑠 − 𝛼 𝑏 0 ∞ 0 𝜃 ( 𝑏 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑏 − 𝑠 ) 𝛼 𝜃 ) 𝑠 0 𝑔 ( 𝑠 , 𝜏 , 𝑥 ( 𝜏 ) ) 𝑑 𝜏 𝑑 𝜃 𝑑 𝑠 ( 𝑠 ) 𝑑 𝜃 𝑑 𝑠 + 𝛼 𝑡 0 ∞ 0 𝜃 ( 𝑡 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑡 − 𝑠 ) 𝛼 𝜃 ) 𝑓 ( 𝑠 , 𝑥 ( 𝑠 ) ) 𝑑 𝜃 𝑑 𝑠 + 𝛼 𝑡 0 ∞ 0 𝜃 ( 𝑡 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑡 − 𝑠 ) 𝛼 𝜃 ) 𝑠 0 𝑔 ( 𝑠 , 𝜏 , 𝑥 ( 𝜏 ) ) 𝑑 𝜏 𝑑 𝜃 𝑑 𝑠 . ( 2 . 1 0 ) Consider ≤ ‖ ‖ ‖ ‖ ( Φ 𝑥 ) ( 𝑡 ) ‖ ∞ 0 𝜉 𝛼 ( 𝜃 ) 𝑇 ( 𝑡 𝛼 𝜃 ) 𝑥 0 ‖ ‖ ‖ 𝑑 𝜃 + 𝛼 𝑡 0 ‖ ‖ ‖ ∞ 0 𝜃 ( 𝑡 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑡 − 𝑠 ) 𝛼 𝜃 ) 𝜎 − 𝐴 𝑇 ( ( 𝑡 − 𝑠 ) 𝛼 ‖ ‖ ‖ × ‖ ‖ 𝑊 𝜃 ) 𝑑 𝜃 ‖ 𝐵 ‖ − 1 ‖ ‖ ‖ ‖ 𝑥 1 ‖ ‖ − ‖ ‖ ‖ ∞ 0 𝜉 𝛼 ( 𝜃 ) 𝑇 ( 𝑏 𝛼 𝜃 ) 𝑥 0 ‖ ‖ ‖ 𝑑 𝜃 − 𝛼 𝑏 0 ‖ ‖ ‖ ∞ 0 𝜃 ( 𝑏 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑏 − 𝑠 ) 𝛼 ‖ ‖ ‖ × 𝜃 ) 𝑑 𝜃 ‖ 𝑓 ( 𝑠 , 𝑥 ( 𝑠 ) ) ‖ 𝑑 𝑠 − 𝛼 𝑏 0 ‖ ‖ ‖ ∞ 0 𝜃 ( 𝑏 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑏 − 𝑠 ) 𝛼 ‖ ‖ ‖ × ‖ ‖ ‖ 𝜃 ) 𝑑 𝜃 𝑠 0 𝑔 ‖ ‖ ‖ ( 𝑠 , 𝜏 , 𝑥 ( 𝜏 ) ) 𝑑 𝜏 𝑑 𝑠 ( 𝑠 ) 𝑑 𝑠 + 𝛼 𝑡 0 ‖ ‖ ‖ ∞ 0 𝜃 ( 𝑡 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑡 − 𝑠 ) 𝛼 𝜃 ‖ ‖ ‖ ) 𝑑 𝜃 ‖ 𝑓 ( 𝑠 , 𝑥 ( 𝑠 ) ) ‖ 𝑑 𝑠 + 𝛼 𝑡 0 ‖ ‖ ‖ ∞ 0 𝜃 ( 𝑡 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑡 − 𝑠 ) 𝛼 ‖ ‖ ‖ ‖ ‖ ‖ 𝜃 ) 𝑑 𝜃 𝑠 0 ‖ ‖ ‖ 𝑔 ( 𝑠 , 𝜏 , 𝑥 ( 𝜏 ) ) 𝑑 𝜏 𝑑 𝑠 ≤ 𝑀 1 ‖ ‖ 𝑥 0 ‖ ‖ + 𝑏 𝛼 𝑀 1 ‖ 𝜎 ‖ + 𝐾 1 𝑀 2 𝑀 3 ‖ ‖ 𝑥 1 ‖ ‖ + 𝑀 1 ‖ ‖ 𝑥 0 ‖ ‖ + 𝑀 1 𝐿 1 𝑏 𝛼 + 𝑀 1 𝐿 2 𝑏 𝛼 + 1 + 𝑀 1 𝐿 1 𝑏 𝛼 + 𝑀 1 𝐿 2 𝑏 𝛼 + 1 ≤ 𝑟 . ( 2 . 1 1 ) Since 𝑓 and 𝑔 are continuous and ‖ ( Φ 𝑥 ) ( 𝑡 ) ‖ ≤ 𝑟 , it follows that Φ is also continuous and maps 𝑌 0 into itself. Moreover, Φ maps 𝑌 0 into precompact subset of 𝑌 0 . To prove this, we first show that, for every fixed 𝑡 ∈ 𝐽 , the set 𝑌 0 ( 𝑡 ) = { ( Φ 𝑥 ) ( 𝑡 ) ∶ 𝑥 ∈ 𝑌 0 } is precompact in 𝑋 . This is clear for 𝑡 = 0 , since 𝑌 0 ( 0 ) = { 𝑥 0 } . Let 𝑡 > 0 be fixed and for 0 < 𝜖 < 𝑡 define Φ 𝜖 𝑥 ( 𝑡 ) = ∞ 0 𝜉 𝛼 ( 𝜃 ) 𝑇 ( 𝑡 𝛼 𝜃 ) 𝑥 0 𝑑 𝜃 + 𝛼 0 𝑡 − 𝜖 ∞ 0 𝜃 ( 𝑡 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑡 − 𝑠 ) 𝛼 𝜃 ) 𝜎 − 𝐴 𝑇 ( ( 𝑡 − 𝑠 ) 𝛼 𝜃 ) 𝐵 𝑊 − 1 × 𝑥 1 − ∞ 0 𝜉 𝛼 ( 𝜃 ) 𝑇 ( 𝑏 𝛼 𝜃 ) 𝑥 0 𝑑 𝜃 − 𝛼 𝑏 0 ∞ 0 𝜃 ( 𝑏 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑏 − 𝑠 ) 𝛼 𝜃 ) 𝑓 ( 𝑠 , 𝑥 ( 𝑠 ) ) 𝑑 𝜃 𝑑 𝑠 − 𝛼 𝑏 0 ∞ 0 𝜃 ( 𝑏 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑏 − 𝑠 ) 𝛼 𝜃 ) 𝑠 0 𝑔 ( 𝑠 , 𝜏 , 𝑥 ( 𝜏 ) ) 𝑑 𝜏 𝑑 𝜃 𝑑 𝑠 ( 𝑠 ) 𝑑 𝜃 𝑑 𝑠 + 𝛼 0 𝑡 − 𝜖 ∞ 0 𝜃 ( 𝑡 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑡 − 𝑠 ) 𝛼 𝜃 ) 𝑓 ( 𝑠 , 𝑥 ( 𝑠 ) ) 𝑑 𝜃 𝑑 𝑠 + 𝛼 0 𝑡 − 𝜖 ∞ 0 𝜃 ( 𝑡 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑡 − 𝑠 ) 𝛼 𝜃 ) 𝑠 0 𝑔 ( 𝑠 , 𝜏 , 𝑥 ( 𝜏 ) ) 𝑑 𝜏 𝑑 𝜃 𝑑 𝑠 . ( 2 . 1 2 )
Since 𝑇 ( 𝑡 ) is compact for every 𝑡 > 0 , the set 𝑌 𝜖 ( 𝑡 ) = { ( Φ 𝜖 𝑥 ) ( 𝑡 ) ∶ 𝑥 ∈ 𝑌 0 } is precompact in 𝑋 for every 𝜖 , 0 < 𝜖 < 𝑡 . Furthermore, for 𝑥 ∈ 𝑌 0 , we have ‖ Φ ( Φ 𝑥 ) ( 𝑡 ) − 𝜖 𝑥 ≤ ‖ ‖ ‖ 𝛼 ( 𝑡 ) ‖ 𝑡 𝑡 − 𝜖 ∞ 0 𝜃 ( 𝑡 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 𝑇 ( 𝜃 ) ( ( 𝑡 − 𝑠 ) 𝛼 𝜃 ) 𝜎 − 𝐴 𝑇 ( ( 𝑡 − 𝑠 ) 𝛼 𝜃 ) 𝐵 𝑊 − 1 × 𝑥 1 − ∞ 0 𝜉 𝛼 ( 𝜃 ) 𝑇 ( 𝑏 𝛼 𝜃 ) 𝑥 0 𝑑 𝜃 − 𝛼 𝑏 0 ∞ 0 𝜃 ( 𝑏 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑏 − 𝑠 ) 𝛼 𝜃 ) 𝑓 ( 𝑠 , 𝑥 ( 𝑠 ) ) 𝑑 𝜃 𝑑 𝑠 − 𝛼 𝑏 0 ∞ 0 𝜃 ( 𝑏 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑏 − 𝑠 ) 𝛼 𝜃 ) 𝑠 0 ( ‖ ‖ ‖ + ‖ ‖ ‖ 𝛼 𝑔 ( 𝑠 , 𝜏 , 𝑥 ( 𝜏 ) ) 𝑑 𝜏 𝑑 𝜃 𝑑 𝑠 𝑠 ) 𝑑 𝜃 𝑑 𝑠 𝑡 𝑡 − 𝜖 ∞ 0 𝜃 ( 𝑡 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑡 − 𝑠 ) 𝛼 ‖ ‖ ‖ + ‖ ‖ ‖ 𝛼 𝜃 ) 𝑓 ( 𝑠 , 𝑥 ( 𝑠 ) ) 𝑑 𝜃 𝑑 𝑠 𝑡 𝑡 − 𝜖 ∞ 0 𝜃 ( 𝑡 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑡 − 𝑠 ) 𝛼 𝜃 ) 𝑠 0 ‖ ‖ ‖ 𝑔 ( 𝑠 , 𝜏 , 𝑥 ( 𝜏 ) ) 𝑑 𝜏 𝑑 𝜃 𝑑 𝑠 ≤ 𝜖 𝛼 𝑀 1 ‖ 𝜎 ‖ + 𝐾 1 𝑀 2 𝑀 3 ‖ 𝑥 1 ‖ + 𝑀 1 ‖ 𝑥 0 ‖ + 𝑀 1 𝐿 1 𝑏 𝛼 + 𝑀 1 𝐿 2 𝑏 𝛼 + 1 + 𝑀 1 𝐿 1 𝜖 𝛼 + 𝑀 1 𝐿 2 𝜖 𝛼 𝑏 , ( 2 . 1 3 ) which implies that 𝑌 0 ( 𝑡 ) is totally bounded, that is, precompact in 𝑋 . We want to show that Φ ( 𝑌 0 ) = { Φ 𝑥 ∶ 𝑥 ∈ 𝑌 0 } is an equicontinuous family of functions. For that, let 𝑡 2 > 𝑡 1 > 0 . Then we have ‖ ‖ 𝑡 ( Φ 𝑥 ) 1 − 𝑡 ( Φ 𝑥 ) 2 ‖ ‖ ≤ ‖ ‖ ‖ 𝛼 𝑡 1 0 ∞ 0 𝜃 𝜉 𝛼 𝑡 ( 𝜃 ) 1 − 𝑠 𝛼 − 1 𝑇 𝑡 1 − 𝑠 𝛼 𝜃 𝑡 𝜎 − 𝐴 𝑇 1 − 𝑠 𝛼 𝜃 − 𝑡 2 − 𝑠 𝛼 − 1 𝑇 𝑡 2 − 𝑠 𝛼 𝜃 𝑡 𝜎 − 𝐴 𝑇 2 − 𝑠 𝛼 𝜃 × 𝐵 𝑊 − 1 𝑥 1 − ∞ 0 𝜉 𝛼 ( 𝜃 ) 𝑇 ( 𝑏 𝛼 𝜃 ) 𝑥 0 𝑑 𝜃 − 𝛼 𝑏 0 ∞ 0 𝜃 ( 𝑏 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑏 − 𝑠 ) 𝛼 𝜃 ) 𝑓 ( 𝑠 , 𝑥 ( 𝑠 ) ) 𝑑 𝜃 𝑑 𝑠 − 𝛼 𝑏 0 ∞ 0 𝜃 ( 𝑏 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑏 − 𝑠 ) 𝛼 𝜃 ) 𝑠 0 ( ‖ ‖ ‖ + ‖ ‖ ‖ 𝛼 𝑔 ( 𝑠 , 𝜏 , 𝑥 ( 𝜏 ) ) 𝑑 𝜏 𝑑 𝜃 𝑑 𝑠 𝑠 ) 𝑑 𝜃 𝑑 𝑠 𝑡 2 𝑡 1 ∞ 0 𝜃 𝜉 𝛼 ( 𝑡 𝜃 ) 2 − 𝑠 𝛼 − 1 𝑇 𝑡 2 − 𝑠 𝛼 𝜃 𝑡 𝜎 − 𝐴 𝑇 2 − 𝑠 𝛼 𝜃 𝐵 𝑊 − 1 × 𝑥 1 − ∞ 0 𝜉 𝛼 ( 𝜃 ) 𝑇 ( 𝑏 𝛼 𝜃 ) 𝑥 0 𝑑 𝜃 − 𝛼 𝑏 0 ∞ 0 𝜃 ( 𝑏 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑏 − 𝑠 ) 𝛼 𝜃 ) 𝑓 ( 𝑠 , 𝑥 ( 𝑠 ) ) 𝑑 𝜃 𝑑 𝑠 − 𝛼 𝑏 0 ∞ 0 𝜃 ( 𝑏 − 𝑠 ) 𝛼 − 1 𝜉 𝛼 ( 𝜃 ) 𝑇 ( ( 𝑏 − 𝑠 ) 𝛼 𝜃 ) 𝑠 0 ( ‖ ‖ ‖ + ‖ ‖ ‖ 𝛼 𝑔 ( 𝑠 , 𝜏 , 𝑥 ( 𝜏 ) ) 𝑑 𝜏 𝑑 𝜃 𝑑 𝑠 𝑠 ) 𝑑 𝜃 𝑑 𝑠 𝑡 1 0 ∞ 0 𝜃 𝜉 𝛼 𝑡 ( 𝜃 ) 1 − 𝑠 𝛼 − 1 𝑇 𝑡 1 − 𝑠 𝛼 𝜃 − 𝑡 2 − 𝑠 𝛼 − 1 𝑇 𝑡 2 − 𝑠 𝛼 𝜃 ‖ ‖ ‖ + ‖ ‖ ‖ 𝛼 𝑓 ( 𝑠 , 𝑥 ( 𝑠 ) ) 𝑑 𝜃 𝑑 𝑠 𝑡 0 ∞ 0 𝜃 𝜉 𝛼 𝑡 ( 𝜃 ) 1 − 𝑠 𝛼 − 1 𝑇 𝑡 1 − 𝑠 𝛼 𝜃 − 𝑡 2 − 𝑠 𝛼 − 1 𝑇 𝑡 2 − 𝑠 𝛼 𝜃 𝑠 0 ‖ ‖ ‖ + ‖ ‖ ‖ 𝛼 𝑔 ( 𝑠 , 𝜏 , 𝑥 ( 𝜏 ) ) 𝑑 𝜏 × 𝑑 𝜃 𝑑 𝑠 𝑡 2 𝑡 1 ∞ 0 𝜃 𝜉 𝛼 𝑡 ( 𝜃 ) 2 − 𝑠 𝛼 − 1 𝑇 𝑡 2 − 𝑠 𝛼 𝜃 ‖ ‖ ‖ + ‖ ‖ ‖ 𝛼 𝑓 ( 𝑠 , 𝑥 ( 𝑠 ) ) 𝑑 𝜃 𝑑 𝑠 𝑡 2 𝑡 1 ∞ 0 𝜃 𝜉 𝛼 𝑡 ( 𝜃 ) 2 − 𝑠 𝛼 − 1 𝑇 𝑡 2 − 𝑠 𝛼 𝜃 𝑠 0 ‖ ‖ ‖ . 𝑔 ( 𝑠 , 𝜏 , 𝑥 ( 𝜏 ) ) 𝑑 𝜏 𝑑 𝜃 𝑑 𝑠 ( 2 . 1 4 )
By using conditions (H2)–(H6), we get ‖ 𝑡 ( Φ 𝑥 ) 1 − 𝑡 ( Φ 𝑥 ) 2 ‖ ≤ 𝛼 𝑡 1 0 ‖ ‖ 𝑡 1 − 𝑠 𝛼 − 1 𝑇 𝑡 1 − 𝑠 𝛼 𝜃 𝑡 𝜎 − 𝐴 𝑇 1 − 𝑠 𝛼 𝜃 − 𝑡 2 − 𝑠 𝛼 − 1 𝑇 𝑡 2 − 𝑠 𝛼 𝜃 𝑡 𝜎 − 𝐴 𝑇 2 − 𝑠 𝛼 𝜃 ‖ ‖ × 𝑀 2 𝑀 3 ‖ 𝑥 1 ‖ + 𝑀 1 ‖ 𝑥 0 ‖ + 𝑀 1 𝐿 1 𝑏 𝛼 + 𝐿 2 𝑏 𝛼 + 1 + 𝑡 𝑑 𝑠 2 − 𝑡 1 𝛼 𝑀 1 𝑀 2 𝑀 3 ‖ 𝜎 ‖ + 𝐾 1 𝑀 2 𝑀 3 ‖ 𝑥 1 ‖ + 𝑀 1 ‖ 𝑥 0 ‖ + 𝑀 1 𝐿 1 𝑏 𝛼 + 𝐿 2 𝑏 𝛼 + 1 + 𝛼 𝑡 1 0 ‖ ‖ 𝑡 1 − 𝑠 𝛼 − 1 𝑇 𝑡 1 − 𝑠 𝛼 𝜃 − 𝑡 2 − 𝑠 𝛼 − 1 𝑇 𝑡 2 − 𝑠 𝛼 𝜃 ‖ ‖ × 𝐿 1 + 𝐿 2 𝑏 𝛼 𝑡 𝑑 𝑠 + 2 − 𝑡 1 𝛼 𝑀 1 𝐿 1 + 𝐿 2 𝑏 𝛼 . ( 2 . 1 5 ) The compactness of 𝑇 ( 𝑡 ) , 𝑡 > 0 , implies that 𝑇 ( 𝑡 ) is continuous in the uniform operator topology for 𝑡 > 0 . Thus, the right hand side of (2.15) tends to zero as 𝑡 2 → 𝑡 1 . So, Φ ( 𝑌 0 ) is an equicontinuous family of functions. Also, Φ ( 𝑌 0 ) is bounded in 𝑌 , and so by the Arzela- Ascoli theorem, Φ ( 𝑌 0 ) is precompact. Hence, from the Schauder fixed point in 𝑌 0 , any fixed point of Φ is a mild solution of (1.1) on 𝐽 satisfying ( Φ 𝑥 ) ( 𝑡 ) = 𝑥 ( 𝑡 ) ∈ 𝑋 . ( 2 . 1 6 ) Thus, system (1.1) is controllable on 𝐽 .

3. Application

Let Ω ⊂ 𝑅 𝑛 be bounded with smooth boundary Γ .

Consider the boundary control fractional integropartial differential system 𝜕 𝛼 𝑦 ( 𝑡 , 𝑥 ) 𝜕 𝑡 𝛼 − Δ 𝑦 ( 𝑡 , 𝑥 ) = 𝐹 ( 𝑡 , 𝑦 ( 𝑡 , 𝑥 ) ) + 𝑡 0 [ ] , 𝐺 ( 𝑡 , 𝑠 , 𝑦 ( 𝑠 , 𝑥 ) ) 𝑑 𝑠 , i n 𝑌 = ( 0 , 𝑏 ) × Ω , 𝑦 ( 𝑡 , 0 ) = 𝑢 ( 𝑡 , 0 ) o n Σ = ( 0 , 𝑏 ) × Γ , 𝑡 ∈ 0 , 𝑏 𝑦 ( 0 , 𝑥 ) = 𝑦 0 ( 𝑥 ) , f o r 𝑥 ∈ Ω . ( 3 . 1 ) The above problem can be formulated as a boundary control problem of the form of (1.1) by suitably taking the spaces 𝐸 , 𝑋 , 𝑈 and the operators 𝐵 1 , 𝜎 , and 𝜏 as follows.

Let 𝐸 = 𝐿 2 ( Ω ) , 𝑋 = 𝐻 − 1 / 2 ( Γ ) , 𝑈 = 𝐿 2 ( Γ ) , 𝐵 1 = 𝐼 , the identity operator and 𝐷 ( 𝜎 ) = { 𝑦 ∈ 𝐿 2 ( Ω ) ∶ Δ 𝑦 ∈ 𝐿 2 ( Ω ) } , 𝜎 = Δ . The operator 𝜏 is the trace operator such that 𝜏 𝑦 = 𝑦 | Γ is well defined and belongs to 𝐻 − 1 / 2 ( Γ ) for each 𝑦 ∈ 𝐷 ( 𝜎 ) and the operator 𝐴 is given by 𝐴 = Δ , 𝐷 ( 𝐴 ) = 𝐻 1 0 ( Ω ) ∪ 𝐻 2 ( Ω ) where 𝐻 𝑘 ( Ω ) , 𝐻 𝛽 ( Ω ) , and 𝐻 1 0 ( Ω ) are usual Sobolev spaces on Ω , Γ . We define the linear operator 𝐵 ∶ 𝐿 2 ( Γ ) → 𝐿 2 ( Ω ) by 𝐵 𝑢 = 𝑤 𝑢 where 𝑤 𝑢 is the unique solution to the Dirichlet boundary value problem Δ 𝑤 𝑢 𝑤 = 0 i n Ω , 𝑢 = 𝑢 i n Γ . ( 3 . 2 ) We also introduce the nonlinear operator defined by 𝑓 ( 𝑡 , 𝑥 ( 𝑡 ) ) = 𝐹 ( 𝑡 , 𝑦 ( 𝑡 , 𝑥 ) ) , 𝑔 ( 𝑡 , 𝑠 , 𝑥 ( 𝑠 ) ) = 𝐺 ( 𝑡 , 𝑠 , 𝑦 ( 𝑠 , 𝑥 ) ) . ( 3 . 3 ) Choose 𝑏 and other constants such that conditions (H1)–(H6) are satisfied. Consequently Theorem 2.2 can be applied for (3.1), so (3.1) is controllable on [ 0 , 𝑏 ] .

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