A Novel Unified Framework for Solving Reachability and Invariance Problems

Abstract

The level set method is a widely used tool for solving reachability and invariance problems. However, some shortcomings, such as large amount of storage space consumption and the difficulty of constructing terminal conditions for solving the Hamilton-Jacobi partial differential equation, limit the application of the level set method in some problems, especially those with irregular target sets. This paper proposes a method that can effectively avoid the above tricky issues and thus has better generality. In the proposed method, the reachable or invariant sets with different time horizons are characterized by some non-zero sublevel sets of a value function. This value function is not obtained by solving a viscosity solution of the partial differential equation but by recursion and interpolation approximation. At the end of this paper, some examples are taken to illustrate the accuracy and generality of the proposed method.