Optimal timing of control law updates for unstable systems with continuous control

The optimal control of a linear system is studied relative to a periodic unstable trajectory using continuous control. Gaussian state uncertainties are included, which induces a statistical cost of controlling the state over a long period of time. The length of time between control law updates directly impacts this statistical cost. When uncertainties are present in a hyperbolically unstable system, the time between control updates can take an optimal value. We apply these ideas to study the statistical cost of controlling a spacecraft in the vicinity of a relative equilibrium point and a Halo orbit in the Hill three-body problem.