Approximate Controllability of Impulsive Differential Equations with Nonlocal Conditions ∗

It is well known that the issue of controllability plays an important role in control theory and engineering (see [1]-[5]) because they have close connections to pole assignment, structural decomposition, quadratic optimal control, observer design etc. In recent years, the problem of controllability for various kinds of differential and impulsive differential systems has been extensively studied by many authors (see [6]-[13], [4]) using different approaches. Recently, Chang et al. [14] studied the controllability of impulsive neutral functional differential systems with infinite delay in Banach spaces by using Dhage’s fixed point theorem. More recently, Chang et al. [15] proved the existence of solutions for non-densely defined neutral impulsive differential inclusions with nonlocal conditions by using the Leray-Schauder theorem of the alternative for kakutani maps. In all these works ([6], [7], [9], [4], [13], [14]) contain the assumption of compactness of the semigroup, as well as the supposition of the controllability of corresponding linear system, i.e., the invertibility of the linear controllability operator W. But it is known (see [16], [17]) that in infinite-dimensional case these hypotheses are in contradiction to each other. Actually, in this situation the controllability may be provided only on the subspace RangeW under additional assumption that it is invariant w.r.t. the semigroup, i.e., that the linear system can be steered to this subspace. Therefore, it is important to study the weaker concept of controllability, namely approximate controllability for differential equations. Motivated by the above approach, the goal of the present paper is to study the approximate controllability for the following impulsive differential equations with nonlocal conditions ⎨⎩ x ′(t) = Ax(t) + f(t, x(t)) +Bu(t), t ∈ J := [0, b], t ∕= ti, x(0) = x0 − g(x), △x(ti) = Ii(x(ti)), i = 1, 2, ⋅ ⋅ ⋅ , p, 0 < t1 < t2 < ⋅ ⋅ ⋅ < tp < b, (1)