The difference and unity of irregular LQ control and standard LQ control and its solution

Abstract

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 quadratic cost functional can be rewritten as a regular one by changing the terminal cost x′(T)Hx(T) to x′(T)[H + P1(T)]x(T), while the optimal controller can achieve P1(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 P1(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.