Efficient Computation of State-Constrained Reachability Problems Using Hopf-Lax Formulae

• 成果类型: Article
• 署名作者: Lee, Donggun; Tomlin, Claire J.
• 署名单位: University of California System; University of California Berkeley; University of California System; University of California Berkeley
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2023.3241180
• 发表日期: 2023
• 页码: 6481-6495
• 关键词: Nonlinear control systems; optimal control; scalability

摘要

This article considers two state-constrained reachability problems: computing 1) control-invariant and 2) reach-avoid sets, both under state constraints. Prior research has developed Hamilton-Jacobi (HJ) partial differential equations (PDEs) that characterize the optimal cost functions of these two problems. Unfortunately, solving the HJ PDEs by grid-based methods, such as level-set methods, suffers from exponentially growing computational complexity in the system state dimension. In order to alleviate this computational issue, this article proposes a Hopf-Lax formula for each reachability problem's HJ PDE. The advantage of the Hopf-Lax formulae is that they have more favorable convexity conditions than the corresponding problems. Thus, direct methods may be used to solve Hopf-Lax formulae and thus efficiently compute the optimal solution of the reachability problems under specified conditions. This article provides an example for each reachability problem and demonstrates the performance of the proposed Hopf-Lax method.