Linear systems with two-point boundary Lyapunov and Riccati equations

Abstract

This paper extends some well-known system theories for algebraic Lyapunov and Riccati equations. These extended results deal with the existence and uniqueness properties of the solutions to matrix differential equations with two-point boundary conditions and are shown to include conventional results as special cases. Necessary and sufficient conditions are derived under which linear systems are stabilizable with periodic feedback gains derived from the two-point boundary matrix differential equations. An easy iterative method for solving the two-point boundary differential Riccati equation is given with an initial guess which is obtained from the intervalwise receding horizon control. The results in this paper are related to periodic feedback gain controls and also to the quadratic cost problem with a discrete state penalty.