Weak observability conditions for exponential and prescribed-time stabilization of non-autonomous bilinear systems

This paper explores the stabilization of abstract non-autonomous bilinear systems, focusing on two distinct forms of stabilization of the closed-loop system; namely the exponential stabilization which consists of converging exponentially fast towards an equilibrium point as time progresses, and prescribed time stabilization, which ensures the convergence within a specific time. We construct a feedback control by computing the time derivative of the system’s “energy”. Based on the theories of nonlinear maximal monotone operators and fixed point theory, we analyze the well-posedness of the system in closed-loop. Then, using a specific weak observability inequality, we prove the uniform exponential stability of the system in closed loop. Subsequently, We utilize the acquired findings to establish prescribed time stability. Finally, applications to some partial differential equations are given.