The generalised continuous algebraic Riccati equation and impulse-free continuous-time LQ optimal control

The purpose of this paper is to investigate the role that the continuous-time generalised Riccati equation plays within the context of singular linear-quadratic optimal control. This equation has been defined following the analogy with the discrete-time generalised Riccati equation, but, differently from the discrete case, to date the importance of this equation in the context of optimal control is yet to be understood. This note addresses this point. We show in particular that when the continuous-time generalised Riccati equation admits a symmetric solution, the corresponding linear-quadratic (LQ) problem admits an impulse-free optimal control.