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

The purpose of this paper is to investigate the role that the so-called constrained generalized Riccati equation plays within the context of continuous-time singular linear–quadratic (LQ) optimal control. This equation has been defined following the analogy with the discrete-time setting. However, while in the discrete-time case the connections between this equation and the linear–quadratic optimal control problem has been thoroughly investigated, to date very little is known on these connections in the continuous-time setting. This note addresses this point. We show, in particular, that when the continuous-time constrained generalized Riccati equation admits a solution, the corresponding linear–quadratic problem admits an impulse-free optimal control. We also address the corresponding infinite-horizon LQ problem for which we establish a similar result under the additional constraint that there exists a control input for which the cost index is finite.