On a functional equation arising in the stability theory of difference-differential equations

The functional differential equation \[ Q ′ ( t ) = A Q ( t ) + B Q T ( τ − t ) , − ∞ > t > ∞ Q’\left ( t \right ) = AQ\left ( t \right ) + B{Q^T}\left ( {\tau - t} \right ), - \infty > t > \infty \] , where A A , B B are n × n n \times n constant matrices, τ ≥ 0 \tau \ge 0 , Q ( t ) Q\left ( t \right ) is a differentiable n × n n \times n matrix and Q T ( t ) {Q^T}\left ( t \right ) is its transpose, is studied. Existence, uniqueness and an algebraic representation of its solutions are given.