On a functional equation arising in the stability theory of difference-differential equations
Abstract : The functional differential equation Q'(t) = AQ(t) + B((Q(tau - t)) to the T-th power), - infinity t infinity, where A,B are n x n constant matrices, tau or = 0, Q(t) is a differentiable n x n matrix and (Q(t)) to the T-th power is its transpose, is studied. Existence, uniqueness and an algebraic representation of its solutions is given. This equation, of considerable interest in its own right, naturally arises in the construction of Liapunov functionals of difference differential equations of the type dx(t)/dt = Cx(t) + Dx(t-tau), where C,D are constant n x n matrices. The role played by the matrix Q(t) is analogous to the one played by a positive definite matrix in the construction of Liapunov functions for ordinary differential equations. In this paper, we show that, in spite of the functional nature of this equation, the linear vector space of its solutions is n squared; moreover, ,we give a complete algebraic characterization of its solutions and indicate computationally simple methods for obtaining these solutions, which we illustrate through an example. Finally, we briefly indicate how to obtain solutions for the nonhomogeneous problem, through the usual variation of constants method.