State Feedback Stabilization with Guaranteed Transient Bounds

We analyze under which conditions linear time-invariant state space systems can be stabilized by static linear state feedback such that prescribed transient bounds hold pointwise in time for the trajectories of the closed loop system. We introduce the concepts of (M,β)-stability and quadratic (M,β)-stability which guarantee a transient bound M along with a decay rate β < 0. Necessary and sufficient conditions for quadratic (M,β)-stabilizability are derived which can be expressed in terms of linear matrix inequalities. A full characterization of general (M,β)-stabilizability is still missing. However, we obtain necessary and sufficient criteria under which a given system can be transformed by linear state feedback into a closed loop system generating a strict contraction semigroup with respect to the spectral norm.