Stabilization with decay estimate of infinite dimensional second order systems by bilinear control damping

This paper deals with stabilization and energy decay rates of second-order bilinear systems in a Hilbert space. Namely, we examine a class of bounded feedback control to investigate weak, strong, and exponential stabilization. Firstly, we discuss the well-posedness of mild solutions of the considered systems using the cosine family. Secondly, we provide some sufficient conditions to ensure feedback stabilization for the bilinear systems which are presented in terms of observation estimates. Moreover, we derive an explicit decay estimate of the energy in the case of strong stabilization. Finally, we offer examples with numerical simulations to demonstrate the applicability of our theoretical results.