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Controllability of Neutral Differential Equation with Impulses on Time Scales Muslim Malik1

· Vipin Kumar1

© Foundation for Scientific Research and Technological Innovation 2019

Abstract In this article, we establish the controllability results for time-varying neutral differential equation with impulses on time scales. We also examine the exact controllability results for the integro and corresponding nonlocal problem. Banach fixed point theorem is used to establish the controllability results. In the end, an example is given to illustrate the application of these results. Keywords Controllability · Neutral differential equation · Impulsive condition · Time scales Mathematics Subject Classification: 93B05 · 34K40 · 34K45 · 34N05

Introduction There are many physical problems that are discriminated by sudden changes in their states, such sudden changes are known as impulsive effects in the problem. The theory of impulsive differential equation has many applications in engineering and science. Impulsive problems are arise in many consequential areas of research such as population dynamics, ecology, biotechnology and so forth. For the references please see the monograph given by Benchohra et al. [1], Bainov and Simeonov [2], Lakshmikantham et al. [3]. In 1960, Kalman [4,5] was the first person who introduced the concept of controllability which becomes the backbone of modern control theory. Roughly speaking, controllability means the steering of a dynamical control system from an initial state to the desired final state. These results have been studied in depth in both the continuous and discrete cases, please see (see [6–8] and references therein). In 1988, Hilger [9] introduced the theory of time scales in his PhD thesis. The study of dynamic systems on time scales encapsulates both the continuous as well as discrete analysis of the system. Since its inception, the study of dynamical systems on time scales has gained a great deal of international attention and

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Muslim Malik [email protected] Vipin Kumar [email protected]

1

School of Basic Sciences, Indian Institute of Technology Mandi, Mandi, India

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Differential Equations and Dynamical Systems

many researchers have since found the applications of time scales in control system [10], population dynamics [11] and as well as in economics [12,13]. For the recent work on the controllability for the dynamical systems on time scale, please see [14–18]. In particular, Davis et al. [14] considered the following linear equation on time scales T x  (t) = A(t)x(t) + B(t)u(t), t ∈ T, y(t) = C(t)x(t) + D(t)u(t), x(t0 ) = x0 ∈ Rn , t0 ∈ T, where x(t) ∈ Rn is a state variable. A(t), B(t), C(t) and D(t) are the rd-continuous matrices of suitable order defined on T. u(t) ∈ Rm be the control input. They give the controllability and observability results using the Gramian matrix. Lupulescu et al. [16] have considered the following control systems with impulses on an unbounded time scale T ∞ y  = A(t)y + B(t)u, t ∈ T0 \ {ti }i=1 ,

y(ti+ ) = (1 + ci )y(ti