New results on exact controllability of a class of fractional neutral integro-differential systems with state-dependent delay in Banach spaces

https://doi.org/10.1016/j.jfranklin.2018.12.001Get rights and content

Abstract

This article deals with the use of Krasnoselskii’s fixed point theorem and Leray–Schauder alternative theorem combined with resolvent operator and some analytical methods to investigate the exact controllability and continuous dependence of a class of neutral fractional integro-differential systems with state-dependent delay in Banach spaces. An application to exemplify the concept is provided at the end.

Introduction

The Fractional calculus is a generalization of classical calculus concerned with integrals and derivatives of non-integer order. It has become a powerful tool in modeling several complex phenomena in numerous seemingly diverse and widespread fields of science and engineering such as electromagnetics, viscoelasticity, fluid mechanics, electrochemistry, biological population models, optics and signals processing. Fractional differential equations (FDEs) are recognized as an indispensable tool to compile the dynamical behavior of real-life phenomena in a precise manner and can be described very successfully by models using mathematical tools from the fractional calculus. Case in point, the nonlinear oscillation of earthquake can be effectively displayed with the fractional derivative. This includes fluid flow, rheology, dynamical processes in self-similar and porous structures, dielectric polarization, electrode-electrolyte polarization, electromagnetic wave, electrical networks, traffic model with fractional derivative, control theory of dynamical systems and so on. Many problems in engineering systems can be resolved by incorporating fractional calculus.

Controllability plays a significant role in the evolution of modern mathematical control theory. This is a qualitative property of dynamical control systems and is of appropriate significance in control theory. Many fundamental problems of control theory such as pole-assignment, stabilizability and optimal control may be solved under the presumption that the system is controllable. Controllability is generally referred that, it is possible to steer dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. It has many significant applications not only in control theory and systems theory, but also in such fields as industrial and chemical process control, reactor control, control of electric bulk power systems, aerospace engineering and recently in quantum systems theory. To have an effective illustration one can refer to [1], [5], [7], [9], [20], [21], [23], [25], [30], [31], [32], [33], [35], [36], [39], [49], [50].

Controllability is also strongly related to the theory of realization and so-called minimal realization and canonical forms for linear time-invariant control systems such as the Kalman canonical form, the Jordan canonical form and the Luenberger canonical form. It should be mentioned that, for many dynamical systems there exists a formal duality between the concepts of controllability and observability. Moreover, it is strongly connected with the minimum energy control problem for many classes of linear finite dimensional, infinite dimensional dynamical systems, and delayed systems both deterministic and stochastic.

The study of delay differential equations(DDEs) was originally motivated by problems in feedback control theory [24] and have become popular in biological models, such as those involving population dynamics, epidemiology and physiology. It may be a special kind of functional ordinary differential equations and their evolution includes the past values of the state variable. The dynamical structure exhibited by DDEs is richer than that of ordinary differential equations. It is worth mentioning that many practical delay systems can be modeled as differential systems of neutral type, the differential expression of which depends on both present state and past state. Some concrete instances in engineering involve lossless circuits and transmission lines, population dynamics, heat exchanges, and so on. Recently, several number of research theories have dealt with fractional-order problems with state-dependent delay(SDD) and infinite delay [3], [11], [16], [19], [34], [37], [38], [40], [41], [42], [44], [45], [46] and the sources therein.

The most inviting aspects in terms of existence and controllability, as well as the qualitative and quantitative of FDEs can be referred in [12], [13], [18], [19], [26], [29], [43]. Santos et al. [4], [14], [15] discussed the existence of solutions for a fractional integro-differential equation with an unbounded delay in Banach spaces. Vijayakumar et al. [42] analyzed the approximate controllability for a class of fractional neutral integro-differential inclusions with SDD using Dhage fixed point theorem which plays an important role. In addition, Kailasavalli et al. [22] acknowledged the existence and controllability of fractional neutral integro-differential systems with SDD with Banach contraction and resolvent operator technique as the main reference. Dabas and Chauhan [10] studied the existence, uniqueness and continuous dependence of mild solution for an impulsive neutral fractional order differential equation with infinite delay.

Recently, Ma and Liu [27] interpreted the exact controllability and continuous dependence of fractional neutral integro-differential equations with state-dependent delay in Banach spaces. Also Yan [45] discussed the approximate controllability of neutral integro-differential delay systems with inclusion type in Hilbert space by using the fixed point theorem of discontinuous multi-valued operators supported by Dhage fixed point technique with the αresolvent operator. Additionally, Yan and Jia [46] explained the approximate controllability of partial fractional neutral stochastic functional integro-differential inclusions with state-dependent delay.

A factual report, at this position, is to admit the controllability and continuous dependence of mild solutions for fractional neutral integro-differential systems with SDD in Bh space adages have not yet been completely examined. The above verdict is in fact driven us to explore the exact controllability of the structure (1.1) and (1.2) in Banach spaces. As a contrast to the current, we include the integral term in the non-linear terms G and F which act an appropriate notion of mild solution of the system (1.1) and (1.2). Instead of Dhage’s fixed point theorem pointed in [45] and [46], by employing Krasnoselskii’s fixed point theorem and Leray–Schauder alternative theorem combined with the resolvent operator, contraction principle respectively, we have presented the exact controllability for the fractional-order control system with state-dependent delay. Besides, there is a focus on the continuous dependence of the mild solution as well while many results of approximate controllability problems.

This manuscript emphasizes the exact controllability and continuous dependence of a class of neutral fractional integro-differential systems with state-dependent delay of the modelCDtα[x(t)+g(t,xρ(t,xt))]=Ax(t)+Bu(t)+0tH(ts)x(s)ds+G(t,xρ(t,xt),0tk1(t,s,xρ(s,xs))ds)+F(t,xρ(t,xt),0tk2(t,s,xρ(s,xs))ds),tI=[0,T],x(t)=ϕ(t)Bh,x(0)=0,t(,0],where α ∈ (1, 2) and the unknown x( · ) takes the values in a Banach space with norm ‖ · ‖, CDtαf(t) represents Caupto derivative and defined by Dtαf(t):=0tg(nα)dndsnf(s)ds, where n is the smallest integer greater than or equal to α and gβ(t):=t(β1)Γ(β),t>0,β>0. A is the infinitesimal generator of semigroup (T(t))t0.B is a bounded linear operator on a Banach space, u(·)L2[I,U], a Banach space of admissible control functions and Bh is a phase space endowed with a seminorm .Bh. For almost any continuous function x characterized on (,T] and for almost any t ≥ 0, we designate by xt the part of Bh characterized by xt(θ)=x(t+θ) for θ ≤ 0. Now xt( · ) speaks to the historical drop of the state from every θ(,0], likely the current time t. Further G,F:I×Bh×XX,ki:D×BhX,i=1,2;D=(t,s)I×I:0stT,ρ:I×Bh(,T] are apposite functions.

This manuscript is structured as follows. Section 2, we have listed some basic definitions and introductory facts that would be employed in this work to attain our primary results. In Section 3, the controllability issues of the system are analyzed using Krasnoselskii’s fixed point theorem and some additional techniques. The continuous dependence with regard to initial values and control is addressed in Section 4. As the final part, in Section 5, the applications are offered to intensify the abstract techniques.

Section snippets

Preliminaries

In this part, we provide some definitions, lemmas and preparatory facts from functional analysis, solution operator and fractional calculus theory which are employed in the course of this work. Throughout this paper, A and H for t ≥ 0, are closed linear operators defined on D(A), which is dense in X. Let 0 ∈ ρ(A), the resolvent set of A and define fraction power Aξ,  0 < ξ < 1, as a closed linear operator on D(Aξ). We can define the inverse fraction power Aξ for any ξ > 0 by Aξ:=1Γ(ξ)0tξ1T(

Fractional control systems

In this segment, we formulate and prove conditions for exact controllability for the system (1.1) and (1.2). We have illustrated that under certain assumptions, the exact controllability of Eqs. (1.1) and (1.2) is implied by the exact controllability of the corresponding linear systems. Initially, we present the mild solution of the system (1.1) and (1.2).

Definition 3.1

A function x[,T]X is called a mild solution of the system (1.1)(1.2) on [0, T] if x0=ϕ,xρ(s,xs)Bh, for every s ∈ [0, T], the

Continuous dependence

Below, we discuss the continuous dependence of mild solution for (1.1)-(1.2).

Theorem 4.1

Suppose ϕj(0)Bh,ujL2(I,U),j=1,2 and the hypotheses (H0), (H1), (H2) and (H4) are satisfied and the system (2.11)and (2.12) is exactly controllable. Moreover for every (t,φ)I×Bh such that ρ(t, φ) ≤ t,xj(t)={ϕj(t),t(,0],Rα(t)(ϕj(0)+g(0,ϕj(0)))g(t,xρ(t,xtj)j)0tASα(ts)g(s,xρ(s,xsj)j)ds0tH(sτ)0sSα(ts)g(τ,xρ(τ,xτj)j)dτds+0tSα(ts)G(s,xρ(s,xsj)j,0sk1(s,τ,xρ(τ,xτj)j)dτ)ds+0tSα(ts)F(s,xρ(s,xsj)j,0sk2(s,τ,xρ(τ

Application I

To exemplify our theoretical results, we consider the following modelDtα[x(t,ξ)+te2(st)a2x(sρ1(s)ρ2(x(s)),ξ)ds]=2ξ2x(t,ξ)+0t(ts)δ¯eγ(ts)2ξ2x(s,ξ)ds+μ(t,ξ)+te2(st)4x(sρ1(s)ρ2(x(s)),ξ)ds+0tG(st)se2(ηs)b2x(ηρ1(η)ρ2(x(η)),ξ)dηds+te2(st)c2x(sρ1(s)ρ2(x(s)),ξ)ds+0tH(st)se2(ηs)d2x(ηρ1(η)ρ2(x(η)),ξ)dηds,x(t,0)=0=x(t,π),t[0,T]x(t,ξ)=ϕ(0,ξ),t0,0ξπ,where Dtα is Caputo’s fractional derivative of order α(1,2),δ¯,γ and a2, b2, c2, d2 are positive numbers. In

Conclusions

This manuscript illustrates that the exact controllability and continuous dependence of fractional neutral integro-differential system with state-dependent delay in a Banach space by using the fractional calculus, resolvent operator, Krasnoselskii’s fixed point theorem and Leray–Schauder alternative theorem. Our theorem guarantees the effectiveness of exact controllability which is the results of the system concerned.

Acknowledgments

The authors are highly grateful to Editor/Associate Editor of the journal for their comments. Also the authors are thankful to the referees for their valuable suggestions. The first author wishes to thank the President Dr. M. Aruchami; Secretary and Director Dr. C. A. Vasuki, Kongunadu Arts and Science College (Autonomous), Coimbatore-641 029, Tamilnadu, India, for their constant encouragement and support for this research work. The research of Nieto has been partially supported by the AEI of

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