  As it is popular known, Riccati equation is the key basic tool for optimal
control in the modern control theory. The solvability conditions of optimal
control, stabilization conditions and controller design are all based on the
Riccati equation. However, these results highly depends on a strictly
assumption that the Riccati equation is regular. If the Riccati equation is
irregular, the controller could not be derived from the equilibrium condition.
This paper is concerned with the general stochastic linear quadratic (LQ) with
irregular Riccati equation. Different from the classical control theory for
regular LQ problems, a new approach of `multi-layer optimization' is proposed.
With the approach, we show that different controller entries of irregular-LQ
controller need to be derived from different equilibrium conditions and a
specified terminal constraint condition in different layers, which is much
different from the classical regular LQ control where all the controller
entries can be obtained from equilibrium condition in one layer based on
regular Riccati equation. The presented results clarify the differences of
open-loop control from closed-loop control in the aspects of solvability and
controller design and also explores in essentially the differences of regular
control from irregular control. Several examples are presented to show the
effectiveness of the proposed approach.
