Computation of Reachable Sets Based on Hamilton-Jacobi-Bellman Equation
with Running Cost Function
AbstractA novel method for computing reachable sets is proposed in this paper.
In the proposed method, a Hamilton-Jacobi-Bellman equation with running
cost function is 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 conditions of 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.