Exponential stability for systems of delay differential equations with block matrices

We obtain efficient exponential stability tests for a system ẋ(t)=A(t)x(h(t)), where A is a block matrix and h(t) is a delay function, in terms of norms and matrix measures of blocks. Compared to the analysis of the whole matrix A, handling blocks can be more manageable even for the system ẋ(t)=A(t)x(t) without delay. In a presented example, the criterion applied to A fails, while considering appropriate blocks leads to exponential stability. Another example illustrates efficiency even in the case of a non-delay system.