Stability and the Infinite-Time Quadratic Cost Problem for Linear Hereditary Differential Systems

Abstract

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.