An extension of the algebraic Riccati equation for the stationary control problem without stabilizability condition

Existence of stationary (constant gain) optimal control is established and some optimality conditions are derived for the linear-quadratic control problem under the following condition weaker than stabilizability: there exists a control strategy which makes the sequence of cost bounded. For example, a system with an unassignable eigenvalue on the unit circle satisfies this condition. Although the associated algebraic Riccati equation usually has no solution for the extended class, the existence of the stationary control which minimizes the average cost (or the average expected cost) per unit time is proved. A complete optimality condition is given by part of the algebraic Riccati equation. Weakened conditions of detectability are also introduced.