In the present paper we shall see that philosophizing on the specific nature of Linear-Quadratic optimal Control Problems (LQCPs) yields several a priori statements that are valid for the entire set of these problems. For instance, the real symmetric matrix that represents the optimal cost for a particular LQCP necessarily is a rank minimizing solution of the dissipation inequality (DI). Since, in case of a positive definite input weighting matrix, the set of these solutions of the DI is equivalent to the set of real symmetric solutions of the algebraic Riccati equation (ARE), our result thus covers both the regular and the singular case. In addition, we will provide a characterization of the afore-mentioned set of solutions of the DI.
Next, a serious attempt is made at reducing general (indefinite) LQCPs to nonnegative definite LQCPs. Moreover, a distributional framework for singular LQCPs is proposed.
Keywords: System theory, optimal control, stability, dissipation inequality, matrix algebra, numerical methods.