EVOLUTION EQUATIONS AND doi:10.3934/eect.2020103 CONTROL THEORY APPROXIMATE CONTROLLABILITY OF SECOND ORDER IMPULSIVE SYSTEMS WITH STATE-DEPENDENT DELAY IN BANACH SPACES Soniya Singh and Sumit Arora Department of Applied Science and Engineering Indian Institute of Technology Roorkee Roorkee, Uttarakhand 247667, India Manil T. Mohan Department of Mathematics Indian Institute of Technology Roorkee Roorkee, Uttarakhand 247667, India Jaydev Dabas ∗ Department of Applied Science and Engineering Indian Institute of Technology Roorkee Roorkee, Uttarakhand 247667, India (Communicated by George Avalos) Abstract. In this paper, we consider the second order semilinear impulsive differential equations with state-dependent delay. First, we consider a linear second order system and establish the approximate controllability result by using a feedback control. Then, we obtain sufficient conditions for the approx- imate controllability of the considered system in a separable, reflexive Banach space via properties of the resolvent operator and Schauder’s fixed point theo- rem. Finally, we apply our results to investigate the approximate controllability of the impulsive wave equation with state-dependent delay. 1. Introduction. Second order differential equations emerge in many areas of sci- ence and engineering. One aspect of studying second order systems is through an equivalent formulation of first order equations, but this transformation may lack important information about the original evolution systems. Therefore, it is more advantageous to study a second order system directly. For the basic theory of second order differential equations, the interested readers are referred to see [12, 21, 49], etc. We observe that the strongly continuous cosine family is an important tool in studying the second order evolution equations. The theory of strongly continu- ous cosine family was introduced by Fattorini in [19, 20]. Later, Travis and Webb have made essential additions to the theory of strongly continuous cosine family (cf. [49, 50]). There are several realistic evolutionary processes subject to abrupt changes in state occur at certain negligible time instant. These processes are mathematically 2020 Mathematics Subject Classification. 34K06, 34A12, 37L05, 93B05. Key words and phrases. Impulsive functional differential equations, approximate controllabil- ity, Schauder’s fixed point theorem, cosine family, resolvent operator. * Corresponding author: Jaydev Dabas. 1