Linear Quadratic Control when Riccati Equation is Irregular

The optimal linear quadratic controller is usually designed based on a Riccati equation. However, when the Riccati is irregular, the problem becomes much more difficult since it is not clear what tools should be applied instead to design the controller. This paper is concerned with the linear quadratic control problem governed by continuous-time system. We show that the solvability of the open-loop control can be fully depicted by a Gramian matrix and a specified matrix. The controller is given via the Gramian matrix and a standard Riccati equation associated with a subsystem. The key to solve the problem is to convert the open-loop solvability into the controllability of a differential equation based on the maximum principle and the solution of a forward and backward differential equation. It is noted that the derived results can be applied to solve the closed-loop control and the stochastic linear quadratic control.