Under-Approximating Reach Sets for Polynomial Continuous Systems

In this paper we suggest a method based on convex programming for computing semi-algebraic under-approximations of reach sets for polynomial continuous systems with initial sets being the zero sub-level set of a polynomial function. It is well-known that the reachable set can be formulated as the zero sub-level set of a value function to a Hamilton-Jacobi partial differential equation (HJE), and our approach in this paper consequently focuses on searching for approximate analytical polynomial solutions to associated HJEs, of which the zero sub-level sets converge to the exact reachable set from inside in measure, without discretizing the state space. Such approximate solutions can be computed via a classical hierarchy of convex programs consisting of linear matrix inequalities, which are constructed by sum-of-squares decomposition techniques. In contrast to traditional numerical methods approximately solving HJEs, such as level-set methods, our method reduces HJE solving to convex optimization, avoiding the complexity associated to gridding the state space. Compared to existing approaches computing under-approximations, the approach described in this paper is structurally simpler as the under-approximations are the outcome of a single semi-definite program. Furthermore, an over-approximation of the reach set, shedding light on the quality of the constructed under-approximation, can be constructed via solving the same semi-definite program. Several illustrative examples and comparisons with existing methods demonstrate the merits of our approach.