The paper by J. C. Willems addresses the optimal control of linear systems over infinite time intervals, focusing on quadratic performance criteria. It considers both cases where the terminal state is free and where it is constrained to be zero. The integrand of the performance criterion can be fully quadratic in the control and state, without necessarily satisfying the definiteness conditions typically assumed in standard regulator problems. The paper derives frequency-domain and time-domain conditions for the existence of solutions and examines the algebraic Riccati equation (ARE). It presents a complete classification of all solutions to the ARE and demonstrates how the optimal control problems can be solved analytically via the ARE. The paper also discusses the boundedness of the infima of the performance criterion under different terminal state conditions and provides necessary and sufficient conditions for the existence of real symmetric solutions to the ARE. The results are applied to various control problems, including those with conflicting objectives, network synthesis, stability theory, and second variations. The paper concludes with a discussion on the implications of the findings and their applications in different areas of control theory.The paper by J. C. Willems addresses the optimal control of linear systems over infinite time intervals, focusing on quadratic performance criteria. It considers both cases where the terminal state is free and where it is constrained to be zero. The integrand of the performance criterion can be fully quadratic in the control and state, without necessarily satisfying the definiteness conditions typically assumed in standard regulator problems. The paper derives frequency-domain and time-domain conditions for the existence of solutions and examines the algebraic Riccati equation (ARE). It presents a complete classification of all solutions to the ARE and demonstrates how the optimal control problems can be solved analytically via the ARE. The paper also discusses the boundedness of the infima of the performance criterion under different terminal state conditions and provides necessary and sufficient conditions for the existence of real symmetric solutions to the ARE. The results are applied to various control problems, including those with conflicting objectives, network synthesis, stability theory, and second variations. The paper concludes with a discussion on the implications of the findings and their applications in different areas of control theory.