Approximate controllability for finite delay nonlocal neutral integro-differential equations using resolvent operator th
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Approximate controllability for finite delay nonlocal neutral integro-differential equations using resolvent operator theory KAMAL JEET Indian Institute of Technology Roorkee, Roorkee 247 667, India E-mail: [email protected]
MS received 29 November 2018; revised 9 February 2020; accepted 11 March 2020 Abstract. In this paper, our purpose is to study the approximate controllability of abstract nonlocal neutral integro-differential equations with finite delay in a Hilbert space using the resolvent operator theory. We derive a variation of parameters formula for representing a solution of the given neutral integro-differential system in the form of resolvent operators and then define a mild solution of the system. We also study the existence of a mild solution of the system with the help of resolvent operator theory. The fractional power theory, α-norm, resolvent operator theory, semigroup theory and Krasnoselskii’s fixed point theorem are used to prove the approximate controllability of the system. Finally, we illustrate our results with the help of an example. Keywords. Approximate controllability; semigroup; resolvent operator; α-norm; finite delay; Krasnoselskii’s fixed point theorem. Mathematics Subject Classification.
93B05, 34K30, 34G20.
1. Introduction In the few past years, the neutral integro-differential equations have been extensively studied by various authors (see [1,5,13,15,16,25] and references therein). The integrodifferential equations of neutral type are used to describe many physical phenomena arising in fluid dynamics, electronics, chemical kinetics and so on. Grimmer et al. [9] used the resolvent operator theory to establish the existence results for the integro-differential equations
y (t) = −Ay(t) + y(0) = y0 ∈ Y,
t 0
γ (t − s)y(s)ds + f (t), t ∈ [0, ∞),
(1.1)
where f : R+ → Y, Y a Banach space. The resolvent operator solves the differential equations (1.1) in weak as well as strict sense. In [10,11], Grimmer represented the solutions of integro-differential equations in the form of resolvent operators. Santos et al. [13,25] investigated the existence and regularity of a mild solution for an abstract integro-differential equation of neutral type using resolvent operators. We refer the readers to the papers © Indian Academy of Sciences 0123456789().: V,-vol
62
Page 2 of 19
Proc. Indian Acad. Sci. (Math. Sci.)
(2020) 130:62
[1,5,14,16] and the books [12,23] for the study of abstract integro-differential equations via analytic resolvent operators. In a mathematical control problem, we obtain an appropriate control function such that we can drive the state of a dynamical system to the desired final state. The controllability problem of various nonlinear systems is an important and interesting subject for many researchers. That is why many authors have analyzed the controllability of various nonlinear systems during the last decade, see for instance [8,17–21,24,26–29,31] and references therein. The approximate controllability enables us to steer the system to an arbitrarily sm
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