This paper studies the infinite-time quadratic cost control problem for a general class of linear autonomous hereditary differential systems. It uses an approach which clarifies the system-theoretic relationship between stabilizability, stability and existence of a solution of an associated operator equation of Riccati type. For this purpose the stability problem is studied and an operator equation of the Lyapunov type is derived. In both cases we obtain equations which characterize the kernels of the Lyapunov and the Riccati equations.

It is shown that all solutions of a class of first order autonomous functional differential equations are oscillatory if the fundamental solution has a first zero $T_1 $. If the fundamental solution does not have any zeros, we show that every solution with positive initial data is nonoscillatory. Upper and lower bounds are given for the distance between successive zeros in terms of $T_1 $ and the delay a.