A Sum-of-Squares-Based Procedure to Approximate the Pontryagin Difference of Semialgebraic Sets

The P-difference between two sets $\mathcal{A}$ and $\mathcal{B}$ is the set of all points, $\mathcal{C}$, such that the addition of $\mathcal{B}$ to any of the points in $\mathcal{C}$ is contained in $\mathcal{A}$. Such a set difference plays an important role in robust model predictive control and in set-theoretic control. In the paper we demonstrate that an inner approximation of the P-difference between two semialgebraic sets can be computed using the Sums of Squares Programming, and we illustrate the procedure using several computational examples.