Tracking Trojan asteroids in periodic and quasi-periodic orbits around the Jupiter Lagrange points using LDV techniques

Linear dynamically varying (LDV) control is a technique for getting a natural (nonlinear, possibly chaotic) trajectory and a perturbed trajectory asymptotically synchronized given an initial condition offset. Probably the best illustrative example, which motivates this paper, is tracking the natural (possibly quasi-periodic) motion of a Trojan asteroid near the L4 point of Jupiter with a spacecraft that follows a trajectory perturbed by the non conservative propulsion forces. The tracking error is linearized around the natural dynamics of the Trojan body, leading to an LDV model of the tracking error. This in turn leads to a dynamically varying controller, itself given as the solution to a Partial Differential Riccati Equation, solved via the method of characteristics. It is shown that this technique allows for accurate tracking of the complicated dynamics around the L4 point.