Explicit Description of 2D Parametric Solution Sets

Consider a linear system A(p) · x = b(p), where the elements of the matrix and the right-hand side vector depend linearly on a m-tuple of parameters p = (p1, . . . , pm), the exact values of which are unknown but bounded within given intervals. Apart from quantifier elimination, the only known general way of describing the solution set {x ∈ Rn | ∃p ∈ [p],A(p)x = b(p)} is a lengthy and non-unique Fourier-Motzkin-type parameter elimination process that leads to a description of the solution set by exponentially many inequalities. In this work we modify the parameter elimination process in a way that has a significant impact on the representation of the inequalities describing the solution set and their number. An explicit minimal description of the solution set to 2D parametric linear systems is derived. It generalizes the Oettli-Prager theorem for non-parametric linear systems. The number of the inequalities describing the solution set grows linearly with the number of the parameters involved simultaneously in both equations of the system. The boundary of any 2D parametric solution set is described by polynomial equations of at most second degree. It is proven that when the general parameter elimination process is applied to two equations of a system in higher dimension, some inequalities become redundant.