Stability of linear difference and differential inclusions
If \(S\) is an \(n\)-dimensional linear space and \(A\) the matrix of a linear transformation \({\mathcal A}: S\to S,\) then the space \(S\) can always be imbedded into an \(N\)-dimensional linear space \(E,\) \(N>n,\) such that there exists an extension \({\mathcal B}: E\to E\) of \({\mathcal A}\) with a nonnegative matrix \(B\) in some base. This property and some of its applications are the subject of this paper. In particular, the authors study the extensions \(B\) of the matrix \(A\) for which the spectral radius \(\rho(B)\) is close to \(\rho(A).\) The inclusion \(x^{k+1}\in {\mathbb A} x^{k}\) is also considered, where \({\mathbb A}\) is a compact set of matrices, and it is proved that this is asymptotically stable if and only if \({\mathbb A}\) can be extended to a set \({\mathbb B}\) of nonnegative matrices \(B\) with \(\| B\| _{1}\) or \(\| B\| _{\infty}<1.\) Similar results are derived for differential inclusions.
scientific article; zbMATH DE number 2176784