Distributed Space-Based Telescopes (DST), particularly those employing aperture synthesis interferometry, are essential for next-generation exoplanet detection and high-resolution astrophysical observation. Such systems offer significant advantages compared to monolithic telescopes, including enhanced angular resolution, reconfigurable baselines, and improved fault tolerance. A typical DST formation generally consists of one beam combiner satellite and multiple collectors, demanding strict formation geometry to establish precise optical paths.  During multi-target observation campaigns, the formation plane must be reconfigured to realign with the new target region. This involves the simultaneous maneuvering of all collectors from their initial positions to the designated terminal states, with the requirement of maintaining terminal position errors within a specified tolerance. Compared to chemical propulsion, electric propulsion features smaller thrust magnitude and higher specific impulse, making it more suitable for the precise control requirement of telescope formations. Moreover, previous studies have indicated that large-amplitude Halo orbits around the Sun-Earth L2 point are ideal candidates for distributed space telescopes dedicated to exoplanet detection, owing to their stable thermal and illumination conditions. Consequently, this paper investigates the fuel-optimal low-thrust reconfiguration problem for DST formations deployed in Halo orbits within the Sun-Earth Circular Restricted Three-Body Problem (CR3BP).
For simplicity, the low-thrust continues trajectory optimization for the individual satellite transfer scenario is considered, with the initial state and target state given by the scientific requirements. Methods for such problem are generally categorized into direct methods, indirect methods. Direct methods transform the continuous-time optimal control problem into a nonlinear programming problem via direct transcription. They usually suffer from a heavy computational burden and are hard for online use. Indirect methods derive the first-order necessary conditions for optimality based on the Pontryagin's Maximum Principle (PMP), seeking solutions by solving the two-point boundary value problem. However, the fuel-optimal control using indirect methods is generally bang-bang control, whose solution is difficult to obtain due to the small convergence radius and the sensitivity of the initial values of costates. The Linear Quadratic Regulator (LQR) yields an energy-optimal control law in state-feedback form by solving the Riccati equation, which has a relatively large convergence radius with the continuous control and is computationally efficient. Therefore, this paper focus on the low-thrust fuel-optimal strategy through homotopy continuation from the LQR solution. First, the CR3BP model and the linearized dynamical model for relative motion near a Halo orbit are presented. Second, the time-optimal problem is formulated and solved using PMP with the maximal thrust magnitude constraint, which determines the minimum transfer time. Then, a finite-time LQR is solved to obtain the energy-optimal control inputs. Finally, the homotopic approach is adopted, where the fuel-optimal problem is connected with the energy-optimal problem through a homotopy parameter. Numerical simulations are performed to validate the effectiveness of the proposed strategy and the terminal accuracy in the nonlinear dynamical model is tested using the obtained fuel-optimal control. This paper extends the low-thrust homotopic approach commonly used in long-duration transfer trajectory design to solve the fuel-optimal reconfiguration problem for close formations near Halo orbits. It is expected that this strategy serves as an effective reference for the reconfiguration problem of distributed space telescopes, with the potential to extend the operational lifetime via fuel-optimal maneuvering.