Explicit Characterization of a Class of Parametric Solution Sets

Consider a linear system A(p) · x = b(p), where the elements of the ma-trix and the right-hand side vector depend affine-linearly on a m-tuple of pa-rameters p = (p1,..., pm) varying within given intervals. It is a fundamen-tal problem how to describe the parametric solution set Σ (A(p), b(p), [p]):= {x ∈ Rn | ∃p ∈ [p], A(p)x = b(p)}. So far, the solution set description can be obtained by a lengthy (and not unique) parameter elimination process. In this paper we introduce a new classification of the parameters with respect to the way they participate in the equations of the system and give numerical charac-terization for each class of parameters. For a class of parametric linear systems, where each uncertain parameter occurs in only one equation of the system and does not matter how many times within that equation, an explicit character-ization of the parametric solution set is derived which generalizes the famous Oettly-Prager theorem for non-parametric linear systems. Key words: linear systems, solution set, interval parameter