Approximate controllability of a second-order neutral stochastic differential equation with state-dependent delay

摘要

In this paper, the existence and uniqueness of mild solution is initially obtained by useof measure of noncompactness and simple growth conditions. Then the conditions for approximatecontrollability are investigated for the distributed second-order neutral stochastic differential systemwith respect to the approximate controllability of the corresponding linear system in a Hilbertspace. We construct controllability operators by using simple and fundamental assumptions onthe system components. We use Lemma 3, which implies the approximate controllability of theassociated linear system. Lemma 3 is also described as a geometrical relation between the rangeof the operator B and the subspaces N?i, i = 1; 2; 3, associated with sine and cosine operatorsin L2([0; a];X) and L2([0; a];LQ). Eventually, we show that the reachable set of the stochasticcontrol system lies in the reachable set of its associated linear control system. An example isprovided to illustrate the presented theory.