Representation of Solutions to a Two-Sided Matrix Delay Differential Equations

In this paper, we investigate the representation of solution for the following linear matrix delayed differential equation Y˙(t)=A1Y(t)+Y(t)A2+BY(t−τ)+Y(t−τ)C+F(t),t∈[0,∞) where t is an independent variable, Y(t) is an n×n unknown variable matrix, τ>0 is a delay, A1,A2,B, and C are given n×n constant matrices, and F(t) is a given n×n variable matrix. Without requiring any commutativity condition between the coefficient matrices and Φ and F, we establish a formula for the initial problem Y(t)=Φ(t), t∈[−τ,0], where Φ(t) is an n×n variable matrix. The proof uses the vectorization technique and the method of steps. Our result settles Open Problems 1 and 2 posed by Diblík.