Optimal Actuator Design across Ranks and Control Horizons

We study how actuator rank and control horizon jointly determine worst-case control performance for finite-dimensional linear systems under a fixed total actuator-gain budget. The design variable is the input covariance $X=B{B}^{\mathrm{\top }}$. Scalar actuators give rank-one matrices $X=b{b}^{\mathrm{\top }}$; dropping the rank constraint yields their convex hull and a semidefinite benchmark. We identify regimes in which low-rank designs necessarily fall short and others in which they attain the benchmark. At small time, the relaxed maximizer is unique and full rank. If $A$ is cyclic, then, for every $k