A characterization of backward reachable sets for nonlinear dynamical systems via the pseudospectral legendre method

Abstract

In this paper, we provide a new approach to analyze backward reachable sets for nonlinear dynamical systems, which are formulated as constrained optimal control problems. This class of problems is related to characterization of the level set of the value function by solving the associated Hamilton-Jacobi-Bellman (HJB) equation, a first-order nonlinear PDE. However, in many cases, solving HJB equations numerically requires high performance computation. Instead of solving HJB equations, we use the pseudospectral Legendre method to reformulate backward reachability problems as nonlinear programs (NLPs). The backward reachability problem is discretized by state and control variables parameterized at Legendre-Gauss-Lobatto collocation points using the pseudospectral method. We convert several reachability examples to the corresponding NLPs, and characterize explicit backward reachable sets using the gradient projection method.