Computation of Reachable Sets Based on Hamilton-Jacobi-Bellman Equation with Running Cost Function (2107.11941v3)

Abstract: A novel method for computing reachable sets is proposed in this paper. In the proposed method, a Hamilton-Jacobi-BeLLMan equation with running cost functionis numerically solved and the reachable sets of different time horizons are characterized by a family of non-zero level sets of the solution of the Hamilton-Jacobi-BeLLMan equation. In addition to the classical reachable set, by setting different running cost functions and terminal conditionsof the Hamilton-Jacobi-BeLLMan equation, the proposed method allows to compute more generalized reachable sets, which are referred to as cost-limited reachable sets. In order to overcome the difficulty of solving the Hamilton-Jacobi-BeLLMan equation caused by the discontinuity of the solution, a method based on recursion and grid interpolation is employed. At the end of this paper, some examples are taken to illustrate the validity and generality of the proposed method.