A new approach on the approximate controllability of fractional differential evolution equations of order 1 < r < 2 in Hilbert spaces
Introduction
Models with real systems of fractional power should be good enough, which could compare with the model of standard integer order. The derivative of the model sum in the real system could be in an exact manner when it comes to fractional order. This system could apply to abundant models in wave propagation, signal processing, robotics, etc. It defines more about the theory and details of applications; also, the readers can review the books [10], [11], [23], [52], [53] and the research articles related to the method of fractional differential systems [1], [3], [7], [8], [12], [18], [19], [22], [26], [27], [28], [29], [38], [40], [43], [46], [47], [48], [49], [50], [51]. As well known, mathematical control theory has many fundamental perceptions, mainly controllability is one among them. Roughly speaking, controllability has the meaning that one can govern the state of the vigorous system suitable by using control actual in the framework. Discussion about theory and applications related to controllability, one can verify the research articles [7], [9], [12], [13], [14], [15], [26], [27], [28], [29], [35], [36], [37], [38], [39], [40], [41], [47], [48], [49].
Recently, in Shu and Xu [30], Shu and Wang [31], Shu et al. [32], Wang and Shu [42], Shu et al. investigated the existence of mild solutions and controllability for fractional differential systems of order 1 < α < 2 by applying the concepts related to sectorial operators via suitable fixed point theorems. In [20], [21], [25], Muthukumar et al. proved the existence of solutions and approximate controllability of fractional impulsive stochastic differential systems of order 1 < q < 2 with infinite delay and Poisson jumps by using the ideas about the resolvent operators and fixed point technique. In [29], Sakthivel et al. studied the approximate controllability of fractional dynamical systems of order 1 < q < 2 by applying the concepts related to sectorial operators and Krasnoselskii’s fixed point theorem. In [24], Qin et al. established the approximate controllability and optimal controls of fractional dynamical systems of order 1 < q < 2 by using the concepts related to sectorial operators and Krasnoselskii’s fixed point theorem. In [44], Yan et al. investigated the optimal controls for fractional stochastic functional differential system of order α ∈ (1, 2] by using the α - order cosine family and Krasnoselskii–Schaefer-type fixed point theorem.
Very recently, in He et al. [8], Zhou and He [49], Zhou et al. proved the existence and controllability of fractional differential evolution equations with order 1 < r < 2 by using the cosine and sine function of operators. Nonetheless, upmost definitely, the investigation of approximate controllability of fractional differential evolution equations of order 1 < r < 2 by utilizing the concepts on cosine and sine function of operators discussed in the article has not been addressed, and this gives the inspiration for the current manuscript.
Consider the fractional differential evolution equations of order 1 < r < 2 has the following formwhere denotes the Caputo fractional derivative of order 1 < r < 2, the state variable z( · ) takes values in the Hilbert space X, A is the infinitesimal generator of a strongly continuous cosine family {N(t)}t ≥ 0 of uniformly bounded linear operators in X; and the control function x( · ) is given in is a Hilbert space. Further, is a bounded linear operator from into X, and g: V × X → X is a given appropriate function satisfying some assumptions. z0, z1 are elements of space X and c is a finite positive number. The purpose of this paper is to show the approximate controllability of fractional differential evolution equations of order 1 < r < 2 has the form (1.1) under simple and fundamental assumptions on the system operators, in particular that the corresponding linear system is approximate controllable.
We now subdivide our article into the following Sections. Section 2, we introduce a few important facts and definitions associated with our study that is employed, which utilizes throughout the discussion of this article. Section 3 is reserved for discussion about approximate controllability of (1.1). Section 4, we continue our discussion to the approximate controllability of (1.1) with nonlocal conditions. Lastly, in Section 5, we present theoretical and practical applications to support the validity of the study.
Section snippets
Preliminaries
We present essential facts, ideas and lemmas desired to organize the main results of our paper. We consider X is a Hilbert space along with ‖ · ‖. Assume and C(V, X) be a Banach space of continuous functions from V → X with here z ∈ C(V, X). Assume E(X) the space of all bounded linear operator from X to X with where Q ∈ E(X) and z ∈ X. The domain and range of an operator A are defined by D(A) and respectively, if A: X → X is a linear
Approximate controllability
In this section, we formulate and prove conditions for the approximate controllability of fractional evolution control systems (1.1). Before stating and proving the main results, we impose the following hypotheses on data of the problem:
- (H1)
The operator {M(t)} is compact for t ≥ 0.
- (H2)
For all t ∈ [0, c], g(t, · ): X → X is continuous and for all z ∈ C([0, c], X), g( · , z): [0, c] → X is strongly measurable.
- (H3)
There exists a constant v ∈ [0, b] and such that |g(t, z)| ≤ p(t) for all z ∈ X and
Nonlocal conditions
‘Nonlocal conditions’ concept has been presented by Byszewski for the extension of problems based on classical conditions. When comparing nonlocal initial conditions with the classical initial condition, which is more accurate to depict the nature marvels, since more information is considered, along these lines lessening the negative impacts initiated by a potential incorrect single estimation taken toward the beginning time. Very useful discussion about differential systems including nonlocal
Application - I
This section mainly focusing the uses of our theoretical outcomes. Assume be an open C2 bounded domain and . Consider the following fractional systemwhere is the Caputo fractional partial derivative of order 1 < r < 2, η > 0. Assume and A be the Laplace operator along Dirichlet boundary conditions given by andIt is clear that we have
Conclusion
Our manuscript is mainly focused on approximate controllability for fractional differential evolution equations of order 1 < r < 2. By using the results on fractional calculus, cosine and sine functions of operators, and Schauder’s fixed point theorem, a new set of sufficient conditions are formulated which guarantees the approximate controllability of fractional differential evolution systems. Then, we developed our conclusions to the ideas of nonlocal conditions. Lastly, we presented
CRediT authorship contribution statement
M. Mohan Raja: Conceptualization, Writing - original draft. V. Vijayakumar: Conceptualization, Writing - original draft, Formal analysis, Software, Methodology. R. Udhayakumar: Conceptualization, Writing - original draft, Formal analysis. Yong Zhou: Conceptualization, Writing - original draft, Formal analysis, Methodology.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The work was supported by the Fundo para o Desenvolvimento das Ciências e da Tecnologia of Macau (Grant no. 0074/2019/A2).
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