Controllability Results for Non Densely Defined Impulsive Fractional Differential Equations in Abstract Space
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Controllability Results for Non Densely Defined Impulsive Fractional Differential Equations in Abstract Space Ashish Kumar1 · Dwijendra N. Pandey1 © Foundation for Scientific Research and Technological Innovation 2019
Abstract In this paper, we study controllability results for non-densely defined impulsive fractional differential equation by applying the concepts of semigroup theory, fractional calculus, and Banach Fixed Point Theorem. An example is also discussed to illustrate the obtained results. Keywords Fractional differential equation · Controllability · Impulsive conditions · Non dense domain Mathematics Subject Classification 34K05 · 93B05 · 34K30
Introduction Fractional calculus is the study of integrals and derivatives of arbitrary orders(both real and complex). Although it was introduced to the end of the seventeenth century, the main contributions have been made during the last few decades due to its applications in many areas of science and engineering such as, in visco-elasticity and damping, electromagnetism, diffusion and wave propagation, signal processing, robotics, traffic systems, genetic algorithms, control systems as well as in economy, finance, biological population models, optics and signals processing. There are several definitions of fractional derivatives such as Hadamard derivative, Grünwald–Letnikov derivative, Riemann–Liouville fractional derivative, Caputo fractional derivative etc. For the basics of fractional calculus, one can refer the monographs [15,26] and for recent developments in this field, we can make references to the papers [1,9,38] and references cited therein. The concept of controllability plays a vital role in control theory and engineering. There are two basic concepts of controllability namely, exact and approximate controllability which are equivalent in finite dimensional systems and different in the case of infinite dimensional systems. Klamka [16] discussed the controllability of linear systems in finite-dimensional
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Ashish Kumar [email protected] Dwijendra N. Pandey [email protected]
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Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India
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Differential Equations and Dynamical Systems
spaces and controllability of fractional evolution dynamical systems in finite dimensional spaces is well established in [4,37]. Malik and Abbas [24] proved the controllability of nonautonomous nonlinear differential system with non-instantaneous impulses via Rothe’s fixed point theorem. Controllability of infinite dimensional systems has been extensively studied due to its various applications. Several authors studied the approximate controllability of semi-linear control systems, see [8,22,25,28,34] and references therein. The concept of approximate controllability of the several types of nonlinear systems under different conditions is further discussed in [5,6,17,21,29,31] and references therein. Prato [7] was among the first to introduce the concept of differential equations with the operator having non-dense domain. Liu an
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