2020 年 2 月
ISSN 0258-2724 DOI：10.35741/issn.0258-2724.55.1.4
Research article Mathematics
O
BSERVABILITY OF
F
RACTIONAL
-O
RDER
I
MPULSIVE
C
ONTROL
I
NTEGRO
-D
IFFERENTIAL
S
YSTEM WITH
N
ONLOCAL
I
NITIAL
C
ONDITION
具有非局部初始条件的分数阶脉冲控制积分-微分系统的可观测
性
Sameer Qasim Hasan
Department of Mathematics, University of Mustansiriyah P.O. Box: 14022, Palestine St., Baghdad, Iraq
Abstract
The article describes a new concept for initial and exactly observability of nonlocal fractional-order impulsive control integro-differential system. This is based on the concepts of the abstract Cauchy problem, which depended on some necessary and sufficient conditions. These conditions established on the semigroup theory of bounded operators as a dynamical operator system, which generated by bounded linear operators. Moreover, invertible operators play a primary role, and we presented a necessary condition for some nonlinear multi variables functions. Thus, all these operators were treated in nonlinear functional analysis to guaranty the initial observable and exactly observability. Therefore, from the mild solution of the system and exactly homogenous part, we proved the equivalent concepts between the initial observability and exactly the observability. Thus, our approach in this article is to prove the uniqueness of initial nonlocal values with admissible control, which belongs to the second-order Lebesgue integrable. The interest of observability results in this article lies by proving a unique fixed point, which is nonlocal initial values that are described in the proposal formula by using Banach’s fixed point theory. The processing observability for complexly systems (such as this system) with all components and properties was established and can be used for many control system applications.
Keywords: Observability, Integro-Differential Equation, Nonlocal Initial Condition, Fractional Order
摘要 本文介绍了一种新的概念，用于非局部分数阶脉冲控制积分-微分系统的初始和可观测性。
这基于抽象柯西问题的概念，该概念取决于一些必要条件和充分条件。这些条件基于有界线性算 子生成的作为动态算子系统的有界算子的半群论建立。此外，可逆算符起主要作用，我们为某些 非线性多变量函数提供了必要条件。因此，所有这些算子都在非线性泛函分析中进行了处理，以 保证初始的可观察性和精确可观察性。因此，从系统的温和解和正好均匀的部分，我们证明了初
始可观性和正可观性之间的等效概念。因此，本文中的方法是证明可控制的初始非局部值的唯一 性，该值属于二阶勒贝格可积。本文对可观察性结果的兴趣在于证明一个唯一的不动点，该不动 点是使用巴纳赫的不动点理论在提案公式中描述的非局部初始值。建立了具有所有组件和属性的 复杂系统（例如该系统）的处理可观察性，并可将其用于许多控制系统应用程序。 关键词: 可观性，积分微分方程，非局部初始条件，分数阶
I.
I
NTRODUCTIONThe observability theory for linear and linear impulsive systems is available for many years, even in infinite-dimensional and piecewise continuous space [3], [5]. The observability of fractional-order impulsive control integro-differential system (with the nonlocal initial condition) is limited and depended on classes of nonlinearity part and the choice of the appropriate method. Many methods investigated for observability approach, such as a fixed point theorem, vector field and algebras, perturbation method, as well as various Banach’s fixed point theorems(such as Darboux and Schauder's fixed point) [6], [8], [9], [11].
Consider the following impulsive fractional integro-differential equations [10]: (1) (2) (0) + g( ) = , (3) y(t) = C (t). where , 0
< , is the Caputo’s fractional derivative. Assume a bounded operator A(t): (
Banach space , ,
, 0 , 0
= ,
, ,
denote the left and the right limit of at , respectively, g: PC([0, T]; is a given function. Also, y(.) is referred to as the output, which is belonged to Banach space Y.C:
is a bounded linear operator.
We aim to study and present the observability of impulsive fractional integro-differential equations with the nonlocal initial condition (1-3),
as well as the approach of Cauchy problem with necessary and sufficient conditions that guarantee the system initial observable.
II.
P
RELIMINARIESBy considering mathematical relations of our initial observable, the following definitions and results have essential roles.
A. Definition 2.1 [7]
The Riemann-Liouville fractional integral of order for a function f is
(4) B. Definition 2.2 [2]
The Caputo fractional derivative of order n-1 < , n for a function f is defined as:
(5) where f(t) is an absolutely continuous derivative up to n-1q. If 0< then
. C. Lemma 2.3 [7]
Let and f be a suitable function. Then, we have
(6) D. Definition 2.4 [10]
The function is a mild
solution for problems (1)-(3), which is equivalent to:
If satisfies the Integral (7). E. Theorem 2.5 [10]
1. Assume that the (H1)-(H7) are satisfied,
then, the fractional impulsive integro-differential systems (1)-(3) have a unique solution on [0, T].
Let with
(H1): where is a
bounded linear operator and
(H2): is continuous and there
exist constant > 0 and such that , and
(H3): A continuous function
with fixed constants and
satisfy , and
(H4): and
for each and k = 1, …, m, (H5): , for (H6): , where K = (H7): G + m + < 1.
2. Assume that hypotheses (H1)-(H5) are
satisfied; moreover, suppose that.
(H8): The bounded linear operator
W: , defined by Wu
= has an inverse
operator , which takes values in
, , and , (H9): , where . (H10): G + m , where .
Then, Equations (1)-(3) are controllable on [0, T].
3. The Observability of fractional impulsive integro-differential system in [10]:
We have the following system:
C is a bounded linear operator defined from X into Y. And consider the homogenous part of (7), the system
(8) and
(9) Let :Y), now assume the operator H:
as:
(10) Based on the results in [4], we have the
following remarks: F. Remark 2.6
1. The system in (8) is initial observable if kernel = {0}.
2. The system in (8) is continuously
initially observable if .
3. If the system in (8) is initially observable then the map H is injective but not surjective.
4. when the system in (8) is continuous, initially, observable implies that : Y exists and bounded that there exists > 0.
Such that for all
Based on the results in [1], we have the following lemma:
G. Lemma 2.7
The system in (8) is continuously initially observable on [0, T] if and only if the system is exactly controllable on [0, T].
H. Concluding Remark 2.8
From Theorem 2.5(2), we have that the mild solution in (7) is to the 1-3 systems
is exactly controllable on [0, T], thus, the special case is also exactly controllable on [0. T]. Then, by Remark 2.6, the system defined in (1-3) is continuously initially observable on [0, T]. Thus, Remark 2.6(4) is satisfied.
1. The system in (8) is continuously initially observable; then, the initial state of the system (8) can be obtained as follows
(11) From (11), the Equation (8) becomes
(12) In the following formulation, we generalized
the concluding Remark 2.8.
III.
T
HEP
ROBLEMF
ORMULATION The integro-fractional differential impulsive and nonlocal conditions ofintegro-fractional differential
control impulsive and nonlocal conditions in (1-3). Now substitutes (7) into
(13) For u(.) and computing the
finite-time observer, our aim is to construct the initial state implicitly as a function x(.) for an arbitrary
control function u(.) ; the nonlocal initial state of the system (13) can be obtained by:
(14) Now, since H is an invertible operator, from
Equation (14), we get:
From the Equations (10) and (15), we get:
(16) By substituting Equation (16) into (7), we
have that:
(17) A. Remark 3.1
The equation in (17) is a finite-time observer and guarantee a fixed point for the mild solution x(.) for the all control functions
u(.) .
IV.
M
AINR
ESULTSTo study the observability of mild solution (7) for the integro-fractional deferential impulsive control system (1-3), the previous results and hypotheses (H1-H5), as well as the following
hypotheses, are adopted in the main result: (G1) Let C: X is a bounded Linear
operator, there exists L > 0 such that .
(G2) If T where is the set of positive
numbers. (G3)(i) , , . (G3)(ii) T < min , . A. Lemma 4.1
Consider the integro-fractional deferential impulsive control system (1-3). Let where a complex Banach
space with the norm
and assume M= be is a nonempty subset of Z,
which is dependent on the control function u(.) . Then, M is a closed subset of Z.
Proof: Let the sequence such that
as where is a sequence of a continuous function at t , k = 1, 2, …, m and left continuous at t . Right limit x( exists dependent on the control function u(.)
which is pointwise convergent to ; to prove , we need the following: prove that and show that for all control functions u(.) .
Now, to prove that since
pointwise convergent, we have that the sequence is uniformly convergence to m, we have that
. To prove that , since the sequence is uniformly convergent to and , where Z is a complex Banach space;
then, , as
for all 0 , we have that
. Therefore, M is a closed subset of Z.
B. Lemma 4.2
Assume the hypotheses (H1), (H2), (H3), and
(H5) hold, and the nonlinear
observation which is defined in (13); also, are nonlinear observations defined as follows:
If , , Then, 1. 2. Proof: , , .
From condition (G1), we get:
, from condition (H2)
From condition (H5)
, from condition (H3),
Now, , where
For , we have that:
; from condition (G1), we get:
; from conditions (H1-H5), as proved
previously, we have that
;
hence, , where
. For t , we need to prove the Lipschitz
; from condition (H3), we get that:
LG + ; hence, ; then, we obtain , for t . For , we need to
prove the Lipschitz prosperity as follows
From conditions (G1), (H1), (H2), and (H5), we
get that:
LG +
; from condition (H4), we get that:
LG +
; hence,
;
then, we obtain
.
For .
C. Theorem 4.3
Assuming that the hypotheses (H1-H5) are
satisfied with the condition (G3[i] and [ii]). Then,
the integro-fractional differential impulsive with
nonlocal conditions has a unique fixed point x(.) for all control functions u(.) .
Proof: Define the nonlinear map as
follows:
by:
(18) for all control functions u(.) .
We aim to prove that . Thus, the fixed point theorem is our interested to get a fixed point as follows:
Step 1: is a closed subset
of Z .
Step 2: for all control functions u(.) .
Step 3: is a contraction on for an arbitrary control function u(.) . By using Lemma 4.1.
Then, Step 1 is satisfied. For proving Step 2, let ; to show that, we need the following, which is in proof of Lemma 4.1:
1. for all control functions u(.) .
2. , for all control functions u(.) . From the map in (18), it is clear that (1) is satisfied. To prove (2), we have that:
(19) From (19) and boundedness of in
From condition (G3)(i), we get:
, for and
.
From condition (G3)(ii), hence,
. Therefore, is contraction.
Thus, , and we had C ;
hence, C
