Control Synthesis for Multiple Reach-Avoid Tasks via Hamilton-Jacobi Reachability Analysis

We investigate the control synthesis problem for continuous-time time-varying nonlinear systems with disturbance under a class of multiple reach-avoid (MRA) tasks. Specifically, the MRA task requires the system to reach a series of target regions in a specified order while satisfying state constraints between each pair of target arrivals. This problem is more challenging than standard reach-avoid tasks, as it requires considering the feasibility of future reach-avoid tasks during the planning process. To solve this problem, we define a series of value functions by solving a cascade of time-varying reach-avoid problems characterized by Hamilton-Jacobi variational inequalities. We prove that the super-level set of the final value function computed is exactly the feasible set of the MRA task. Additionally, we demonstrate that the control law can be effectively synthesized by ensuring the non-negativeness of the value functions over time. We also show that the Linear temporal logic task control synthesis problems can be converted to a collection of MRA task control synthesis problems by properly defining each target and state constraint set of MRA tasks. The effectiveness of the proposed approach is illustrated through four case studies on robot planning problems under time-varying nonlinear systems with disturbance.