Approximate Controllability of Second-Order Semilinear Control System

This paper deals with the approximate controllability of second-order semilinear control system in Hilbert spaces under the assumption that the corresponding linear system is approximately controllable. The control function for this system is suitably constructed, and the sufficient conditions for the approximate controllability of the proposed problem in Hilbert spaces are established. The results are obtained when the nonlinearity satisfying the monotone condition and integral contractor condition. Finally, an example is provided to illustrate the application of the obtained results.