Inner estimation of the parametric tolerable solution set

We consider a linear algebraic system A(p)x=b(q)A(p)x=b(q), where the elements of the matrix and the right-hand side vector are linear functions of uncertain parameters varying within given intervals. The linear tolerance problem for the so-called parametric tolerable solution set Σtol(A(p),b(q),[p],[q])={x∈Rn∣(∀p∈[p])(∃q∈[q])(A(p)x=b(q))}Σtol(A(p),b(q),[p],[q])={x∈Rn∣(∀p∈[p])(∃q∈[q])(A(p)x=b(q))} requires an inner estimation of this solution set, that is an interval vector [y][y], such that [y]⊆Σtol(A(p),b(q),[p],[q])[y]⊆Σtol(A(p),b(q),[p],[q]). In this paper we consider the first methods for finding inner estimation of the parametric tolerable solution set, namely, we propose parametric generalization of the so-called centered approach and of the vertex approach. The results obtained by the two approaches are compared on some numerical examples. The advantages of the parametric approach are demonstrated on problems with independent nonparametric entries and in controllability analysis of linear dynamical systems involving interval uncertainties.