Резюме. Study of nonlinear systems is important since most of the practical systems are nonlinear in nature. There are different notions of controllability like complete controllability, approximate controllability, exact controllability, partial exact controllability, null controllability, local controllability, etc. A current notion of controllability namely Trajectory controllability (T – controllability) has been introduced as a new direction in the field of control theory. We have discussed T- controllability of the first and second order nonlinear integro-differential system by using theory of monotone operators and tools of cosine operators. We also study an exact controllability of a nonlinear system without compactness of the semigroup or cosine operator. Results are obtained numerically also to justify the theory. Examples are given to illustrate the theory time to time.