On stability of linear periodic differential difference equations

Abstract

The paper considers the equation

∞                        

∑a_j(t)x' (t - h_j) + ∫^∞_o m(t,s)x' (t-s) ds

j=o

∞ 

+∑b_j(t)x (t-h_j) + ∫^∞_o n(t,s)x(t - s) ds = f(t)

j=o

where the operator-valued bounded functions a_j and b_j are 2π-periodic, and the operator-valued kernels m and n are 2π-periodic with respect to the first argument. The connection between the input-output stability of the equation and the invertibility of a family of operators acting on the space of periodic functions is investigated.

Keywords: Stability, differential difference equation, functional differential equation, delay, causal invertibility, differential operator