On the instant-controllability of a second order multidimensional differential equation subjected to damping term and impulses.

In this paper the existence of admissible trajectory control-pairs for an impulsive problem driven by a multidimensional differential equation with a nonlinear Balakrishnan-Taylor type damping term, is investigated. This purpose is achieved rewriting the impulsive problem in an abstract form governed by a semilinear second order differential inclusion in which the nonlinear term also depends on the first derivative. The method used leads to a preliminary study of the existence of mild solutions for a non-impulsive multivalued problem on a closed and bounded interval, stating two new results in non reflexive Banach spaces. Then, the mild solution in [0,∞)for the impulsive abstract multivalued problem is obtained glueing the solutions defined on the bounded intervals. This approach allows to not require the continuity on the impulsive functions. Applying the abstract impulsive multivalued results we achieve the desired existence of admissible trajectory control-pairs for the impulsive phenomena described by the multidimensional differential equation. The paper concludes with the study of an instant-controllability relatively to a suitable functional for the impulsive problem in exam. The established results improve recent theorems present in the literature and obtained in reflexive Banach spaces and assuming the continuity on the impulsive functions.