Structure of stability sets and asymptotic stability sets of families of linear differential systems with parameter multiplying the derivative: I (Q610330)

Consider the one-parameter family of \(n\)-dimensional linear differential systems \[ {\frac{dx}{dt}=A(t,\mu )x,~\;\;x=(x_{1},\dots,x_{n})^{T}\in \mathbb R^{n},~\;\;t\geq 0,\;\;(\ast )} \] whose matrix \(A(.,.):[0,\infty )\rightarrow\mathrm{End}\,\mathbb R^{n}~\)\ \ is piecewise continuous with respect to \(t\geq 0\) \ for each fixed \ \(\mu \in\mathbb R\) \ and ensures that the solutions of system (\(*\)) continuously depend on \(\mu\) . The stability (respectively, asymptotic stability) set \(S_{A}\) (respectively, \(S_{aA}\)) of such a family is defined as the set of all values of the parameter for which the corresponding systems in the family are stable (respectively, asymptotically stable). The structure of \(S_{A}\) and \(S_{aA}\) is completely described for (\(*\)) and for the subclass that consists of all families (\(*\)) whose coefficient matrix \(A(.,.)\) is bounded for each fixed \ \(\mu \in\mathbb R\).