On approximate controllability of impulsive functional evolution inclusion

The main aim of our presentation is to consider in a real Hilbert space H the following controllability problem (P )  ẏ(t) ∈ A(t)y(t) + F (t, yt) +Bu(t), for t ∈ J := [0, T ], T > 0, t 6= tk, k = 1, . . . , p, y(t) = φ(t), t ∈ I := [−τ, 0], τ > 0, y(tk ) = y(tk), y(t + k ) = y(tk) + Ik(ytk), k = 1, . . . , p, where F : J ×PC(I,H) ( H is a multivalued mapping with convex and compact values, which is an upper hemicontinuous (uhc) multifunction, yt(θ) := y(t + θ), for θ ∈ I, φ ∈ PC(I,H) is the initial trajectory, B is a bounded linear operator, and Ik : PC(I,H)→ H, k = 1, . . . , p, are given impulse (bounded) functions. Moreover, it is assumed that {A(t) : D(A) ⊂ H→ H}t∈J is a family of closed, densely defined bounded linear operators with a common domain, which generate a compact evolution system. It is a known fact that the systems of the form (P ) under the above-mentioned assumptions is never completely controllable (see [5, 6]). Therefore, we will give conditions for such a system to be approximately controllable [2]. We assume, for example, that the linear system associated to the problem (P ) is approximately controllable (cf. [4]). In order to show the existence of approximate controls we follow the papers [1, 3]. Sufficient conditions are formulated and proved using the resolvent of the grammian controllability operator and the fixed point theorem. Finally, we present an example to illustrate application of the proposed method.