A Lyapunov Inequality Characterization of and a Riccati Inequality Approach to $L_{\infty}$ and $L_{2}$ Low Gain Feedback

This paper is concerned with a Lyapunov inequality characterization of the eigenstructure assignment-based low gain feedback laws. With this characterization and our earlier characterizations of other low gain feedback design approaches, all existing low gain feedback designs are unified under this Lyapunov inequality framework, which in turn implies that all of these low gain feedback laws are both $L_{\infty}$ and $L_{2}$ low gain feedback. This Lyapunov inequality characterization also leads to a quadratic Lyapunov function for the closed-loop system, which is expected to play an important role in solving other control problems. This characterization also motivates a new Riccati inequality-based low gain feedback design, which not only possesses the appealing features of the existing low gain designs but also is computationally easy to carry out.