Reachable Sets of Homogeneous Polynomial Dynamical Systems Using Exact Solutions

Reachability analysis is a powerful tool to analyze the behavior of dynamical systems. Typically, these tools are used to evaluate whether the dynamics of a system beginning from some initial set reaches some unsafe region of state space in a finite amount of time. To answer this question, these tools often construct over-approximations to the reachable sets of the dynamical systems, which can be overly conservative when applied to arbitrary systems. To address this challenge, this letter develops a novel technique for reachability analysis of Homogeneous Polynomial Dynamical Systems (HPDSs) by computing their exact solutions using tensor theory. In addition, this letter illustrates how to build tight over-approximations of the reachable set for HPDSs with constant control inputs. Simulation results highlight a significant improvement in the accuracy of reachable set estimates compared to established methods for HPDSs.