Stability with respect to a subspace of solutions of linear systems with variable delay (Q1022197)

From the text: Consider the system \[ \begin{cases} \dot{\mathbf x}(t)-{\mathbf{Ax}}(t-r(t))={\mathbf f}(t), &t\geq\tau;\\ {\mathbf x}(\xi)={\mathbf 0}, &\xi<\tau, \end{cases} \] where \(\tau\in\mathbb R_+\), \({\mathbf A}\in\mathbb C^{n\times n}\), the function \(r:\mathbb R_+\to\mathbb R_+\) is measurable, the vector-function \({\mathbf f}\) is summable on each finite segment \([\tau,\beta]\). We reduce the problem about the properties of projections of solutions of linear systems with a constant matrix and a variable delay onto an arbitrarily defined subspace (we call it initial) to the classic problem about the properties of the norm of solutions. We constructively complement the basis of the initial subspace up to a basis of a subspace such that one can consider the behavior of projections of solutions onto it regardless of the behavior of projections of solutions onto its complement. Asymptotic properties of projections of solutions of the system onto the initial and constructed subspaces are equivalent.