Exponential Stabilization of Unstable Bilinear Systems in Finite- and Infinite-Dimensional Spaces

This article discusses the uniform exponential stabilization of bilinear systems with multiplicative control inputs. We first consider the case of finite-dimensional bilinear systems, and we present stabilization results concerning the case of open-loop stable systems using a bang-bang feedback control. For open-loop unstable systems, a normalized quadratic time-varying feedback control is proposed. These results are then applied to infinite-dimensional case. Necessary and sufficient conditions are formulated in terms of unique continuation properties and integral estimates involving the control operator and the state of the uncontrolled system. Applications to finite-dimensional systems and coupled partial differential equations (PDE) are further provided.