Reach-Avoid Problems via Sum-of-Squares Optimization and Dynamic Programming
This work addresses the challenge of providing mathematical guarantees for reach-avoid problems, which is important for safety-critical applications in control systems, but it is incremental as it builds on existing optimization and dynamic programming techniques.
The paper tackles the reach-avoid problem by combining sum-of-squares optimization and dynamic programming to provide a conservative solution with guarantees for polynomial system dynamics, achieving computational scalability compared to prior methods like Hamilton-Jacobi reachability, as validated in numerical examples including two single integrators and two kinematic cars.
Reach-avoid problems involve driving a system to a set of desirable configurations while keeping it away from undesirable ones. Providing mathematical guarantees for such scenarios is challenging but have numerous potential practical applications. Due to the challenges, analysis of reach-avoid problems involves making trade-offs between generality of system dynamics, generality of problem setups, optimality of solutions, and computational complexity. In this paper, we combine sum-of-squares optimization and dynamic programming to address the reach-avoid problem, and provide a conservative solution that maintains reaching and avoidance guarantees. Our method is applicable to polynomial system dynamics and to general problem setups, and is more computationally scalable than previous related methods. Through a numerical example involving two single integrators, we validate our proposed theory and compare our method to Hamilton-Jacobi reachability. Having validated our theory, we demonstrate the computational scalability of our method by computing the reach-avoid set of a system involving two kinematic cars.