Least squares stationary optimal control and the algebraic Riccati equation

TLDR

Optimal control of linear systems with quadratic performance criteria over an infinite horizon is studied. The study considers both free and zero terminal state cases, allowing a fully quadratic integrand without the usual definiteness assumptions. Frequency‑ and time‑domain solvability conditions are derived, the algebraic Riccati equation is fully classified, and the optimal control problems are solved analytically using it.

Abstract

The optimal control of linear systems with respect to quadratic performance criteria over an infinite time interval is treated. Both the case in which the terminal state is free and that in which the terminal state is constrained to be zero are treated. The integrand of the performance criterion is allowed to be fully quadratic in the control and the state without necessarily satisfying the definiteness conditions which are usually assumed in the standard regulator problem. Frequency-domain and time-domain conditions for the existence of solutions are derived. The algebraic Riccati equation is then examined, and a complete classification of all its solutions is presented. It is finally shown how the optimal control problems introduced in the beginning of the paper may be solved analytically via the algebraic Riccati equation.