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Reach-avoid problems involve driving a system to a set of desirableconfigurations while keeping it away from undesirable ones. Providingmathematical guarantees for such scenarios is challenging but have numerouspotential practical applications. Due to the challenges, analysis ofreach-avoid problems involves making trade-offs between generality of systemdynamics, generality of problem setups, optimality of solutions, andcomputational complexity. In this paper, we combine sum-of-squares optimizationand dynamic programming to address the reach-avoid problem, and provide aconservative solution that maintains reaching and avoidance guarantees. Ourmethod is applicable to polynomial system dynamics and to general problemsetups, and is more computationally scalable than previous related methods.Through a numerical example involving two single integrators, we validate ourproposed theory and compare our method to Hamilton-Jacobi reachability. Havingvalidated our theory, we demonstrate the computational scalability of ourmethod by computing the reach-avoid set of a system involving two kinematiccars.

Reach-Avoid Problems via Sum-of-Squares Optimization and Dynamic Programming