Decomposition and Parametrization of Semidefinite Solutions of the Continuous-Time Algebraic Riccati Equation

Abstract

Negative-semidefinite solutions of the $\operatorname{ARE}\mathcal{R}(X) = A^ * X + XA + XBB^ * X - C^ * C = 0$ are studied. With respect to an appropriate basis the ARE breaks up into a Lyapunov equation $A_0^ * X_0 + X_0 A_0 = 0$, where $A_0 $ has only purely imaginary eigenvalues, and an indecomposable Riccati equation $\mathcal{R}_r (X_r ) = A_r^ * X_r + X_r A_r + X_r B_r B_r^ * X_r - C_r^ * C_r = 0$ such that each solution $x \leq 0$ is of the form $X = {\operatorname {diag}}(X_0 ,X_r )$ . The focus is on the solutions $\mathcal{S} = \{ X| X . = {\operatorname {diag}}(0,X_r ),\mathcal{R}_r (X_r ) = 0,X_r \} \leq 0$. The set $\mathcal{S}$ has as an order-isomorphic image a well-defined set $\mathcal{N}$ of A-invariant subspaces. The characterization of $\mathcal{N}$ involves the stabilizable and the uncontrollable subspace of $(A,B,C)$.