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Approximate controllability of time-dependent impulsive semilinear retarded differential equations with infinite delay and nonlocal conditions

Abstract:

Here we study the approximate controllability of time dependent impulsive retarded semilinear differential equations with infinite delay and nonlocal conditions, where some ideas are taking from a previous works for this kind of systems with impulses, nonlocal conditions and finite delay, this is done using a techniques evading fixed point theorems used by A.E. Bashirov et al. In this case we have to impose some conditions on the nonlinear term depending of the last time impulse t<inf>p</inf>, so that we can prove the approximate controllability of this system by living the impulses behind on a fixed solution curve in a small interval of time, and from this position, we are able to reach a ball of center the final state and radius ɛ > 0 small enough, at time τ, by assuming that the associated linear control system is exactly controllable on any interval [t<inf>0</inf>, τ], 0 < t<inf>0</inf> < τ.