  Irregular linear quadratic control (LQ, was called Singular LQ) has been a
long-standing problem since 1970s. This paper will show that an irregular LQ
control (deterministic) is solvable (for arbitrary initial value) if and only
if the LQ cost can be rewritten as a regular one by changing the terminal cost
$x'(T)Hx(T)$ to $x'(T)[H+P_1(T)]x(T)$, while the optimal controller can achieve
$P_1(T)x(T)=0$ at the same time. In other words, the irregular controller (if
exists) needs to do two things at the same time, one thing is to minimize the
cost and the other is to achieve the terminal constraint $P_1(T)x(T)=0$, which
clarifies the essential difference of irregular LQ from the standard LQ control
where the controller is to minimize the cost only.
  With this breakthrough, we further study the irregular LQ control for
stochastic systems with multiplicative noise. A sufficient solving condition
and the optimal controller is presented based on Riccati equations.
