Semi-global stabilization of linear time-delay systems with control energy constraint

This note is concerned with the problem of semi-globally stabilizing a linear system with an input delay and a constraint on the energy of its input. Under the condition of null controllability with vanishing energy, the parametric Lyapunov equation based L2L2 low gain feedback is adopted to solve the problem. The proposed approach is applied to the linearized model of the relative motion in the orbit plane of a spacecraft with respect to another spacecraft in a circular orbit around the Earth to validate its effectiveness.