A Novel Unified Framework for Solving Reachability, Viability and Invariance Problems

TL;DR

This paper introduces a recursive and interpolation-based method for solving reachability, viability, and invariance problems that overcomes limitations of the level set method, especially for complex nonlinear systems.

Contribution

It proposes a novel approach that avoids solving PDEs directly, enhancing applicability to non-affine nonlinear systems and irregular target sets.

Findings

The method accurately characterizes reachable and invariant sets.

It demonstrates better generality and ease of handling complex systems.

Examples confirm the method's effectiveness and accuracy.

Abstract

The level set method is a widely used tool for solving reachability and invariance problems. However, some shortcomings, such as the difficulties of handling dissipation function and constructing terminal conditions for solving the Hamilton-Jacobi partial differential equation, limit the application of the level set method in some problems with non-affine nonlinear systems and 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.