Communications in Nonlinear Science and Numerical Simulation
A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces
Research highlights
► Fractional delay nonlinear integrodifferential controlled system via analytic semigroup is studied. ► Existence result for α-mild solutions is shown. Existence result for Lagrange type optimal controls problem is presented.
Introduction
Fractional differential equations have been proved to be valuable tools in the modeling of many phenomena in various fields of engineering, physics and economics. We can find numerous applications in viscoelasticity, electrochemistry, control and electromagnetic. There has been a significant development in fractional differential equations. One can see the monographs of Kilbas et al. [17], Miller and Ross [16], Podlubny [20], Lakshmikantham et al. [18], the survey of Agarwal et al. [1], [2].
Recently, there are some papers focused on fractional delay differential equations or inclusions in Banach spaces [3], [6], [7], [9], [12], [13], [19], [22], [28], [29], [30]. To study the theory of abstract fractional differential equations involving the Caputo derivative in Banach spaces, the nature of the difficulty is how to introduce a concept of a mild solution. A pioneering work has been reported by El-Borai [10], [11]. Hernández et al. [14] pointed that some recent works [2], [8], [13], [22] of abstract fractional differential equations in Banach spaces were incorrect and used another approach to treat abstract equations with fractional derivatives based on the well developed theory of resolvent operators for integral equations. Moreover, Wang and Zhou [25], Zhou and Jiao [31], [32] also introduced a suitable definition of the mild solutions based on the well known theory of Laplace transform and probability density functions.
To our knowledge, fractional delay integrodifferential equations has not been extensively studied. Especially, the results dealing with the optimal control problems of fractional differential equations are relatively scarce [25]. Here, motivated by [13], [24], [25], [26], [31], [32], we will consider the following fractional delay nonlinear integrodifferential controlled systemwhere denotes the Caputo fractional derivative of order q, −A : D(A) → X is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators {S(t), t ⩾ 0} on a separable reflexive Banach space X, f is X-value function and g is Xα-value function where Xα = D(Aα) is a Banach space with the norm ∥x∥α = ∥Aαx∥ for x ∈ Xα. u takes value from another separable reflexive Banach space Y, B is a linear operator from Y into X. xt : [−r, 0] → Xα, t ⩾ 0, which is defined by setting xt = {x(t + s)∣s ∈ [−r, 0]}, represents the history of the state from time t − r up to the present time t.
To study fractional delay integrodifferential equations, Hu et al. introduced the so called mild solutions (Definition 3, [13]) which was inspired by Jaradat et al. [15]. In fact, this definition is not suitable for these settings although it has been utilized by several authors. Here, we firstly introduce a new concept of a mild solution (Definition 2.6) for system (1) based on our early works [25], [31], [32]. Secondly, we prove the existence and uniqueness of mild solutions for system (1). We also give the continuous dependence result of mild solutions. Thirdly, we consider a Lagrange problem (P) and an existence result of optimal controls for system (1) is presented.
The rest of this paper is organized as follows. In Section 2, some notations and preparation results are given. In Section 3, the existence, uniqueness and continuous dependence results of mild solutions for system (1) are given. In Section 4, the Lagrange problem (P) of system (1) is formulated and existence result of optimal controls are presented. At last, an example is given to demonstrate the applicability of our result.
Section snippets
Preliminaries
Throughout this paper, X and Y are two separable reflexive Banach spaces, Lb(X, Y) denote the space of bounded linear operators from X to Y. −A : D(A) → X is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators {S(t), t ⩾ 0}. This means that there exists M > 1 such that ∥S(t)∥ ⩽ M. In the usual way, we introduce the fractional power operator Aα(0 < α < 1) (see [21]) having dense domain D(Aα) which is endowed with graph norm ∥·∥α to be a fractional power space Xα. Then, Xβ ↪ X
Existence of α-mild solutions
We make the following assumptions.
- [HF]:
f: J × C−r,0,α × Xα → X satisfies:
(i) f is measurable.
(ii) For arbitrary ξ1, ξ2 ∈ C−r,0,α, η1, η2 ∈ Xα satisfying ∥ξ1∥−r,0,α, ∥ξ2∥−r,0,α, ∥η1∥α, ∥η2∥α ⩽ ρ, there exists a constant Lf(ρ) > 0 such that
(iii) There exists a constant Mf > 0 such that
(iv) φ(0) ∈ Xβ(α < β < 1).
- [HG]:
Let D = {(t, s) ∈ J × J∣0 ⩽ s ⩽ t}. g : D × C−r,0,α → Xα satisfies:
(i) g is continuous.
(ii) For each (t, s) ∈ D
Existence of optimal controls
In the following, we consider the optimal controls problem of system (1).
We consider the Lagrange problem (P):
Find a control u0 ∈ Uad such thatwherewhere is the integrable cost function and xu denotes the α-mild solution of system (1), corresponding to the control u ∈ Uad. {u, xu} is called an admissible state control pair, or simply admissible pair.
For the existence of solution for problem (P), we shall introduce the
An example
As an application we consider the following problem:with the cost functionwhere Ω ⊂ RN is a bounded domain, ∂Ω ∈ CN, Δ is the Laplace operator, , h ∈ L1([−r, T + r], R) and is continuous.
Define for x ∈ D(A). Then A can generate an
Acknowledgements
This work is supported by Tianyuan Special Funds of the National Natural Science Foundation of China (11026102), National Natural Science Foundation of China (10971173, 10961009), National Natural Science Foundation of Guizhou Province (2010, No.2142).
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