  Input-affine dynamical systems often arise in control and modeling scenarios,
such as the data-driven case when state-derivative observations are recorded
under bounded noise. Common tasks in system analysis and control include
optimal control, peak estimation, reachable set estimation, and maximum control
invariant set estimation. Existing work poses these types of problems as
infinite-dimensional linear programs in auxiliary functions with sum-of-squares
tightenings. The bottleneck in most of these programs is the Lie derivative
nonnegativity constraint posed over the time-state-control set. Decomposition
techniques to improve tractability by eliminating the control variables include
vertex decompositions (switching), or facial decompositions in the case where
the polytopic set is a scaled box. This work extends the box-facial
decomposition technique to allow for a robust-counterpart decomposition of
semidefinite representable sets (e.g. polytopes, ellipsoids, and projections of
spectahedra). These robust counterparts are proven to be equivalent to the
original Lie constraint under mild compactness and regularity constraints.
Efficacy is demonstrated under peak/distance/reachable set data-driven analysis
problems and Region of Attraction maximizing control.
