  We study in this paper the linear quadratic optimal control (linear quadratic
regulation, LQR for short) for discrete-time complex-valued linear systems,
which have shown to have several potential applications in control theory.
Firstly, an iterative algorithm was proposed to solve the discrete-time
bimatrix Riccati equation associated with the LQR problem. It is shown that the
proposed algorithm converges to the unique positive definite solution
(bimatrix) to the bimatrix Riccati equation with appropriate initial
conditions. With the help of this iterative algorithm, LQR problem for the
antilinear system, which is a special case of complex-valued linear system, was
carefully examined and three different Riccati equations based approaches were
provided, namely, bimatrix Riccati equation, anti-Riccati equation and normal
Riccati equation. The established approach is then used to solve the LQR
problem for discrete-time time-delay system with one step state delay and a
numerical example was used to illustrate the effectiveness of the proposed
methods.
