This paper presents a comprehensive analysis of the infinite time least squares optimal control problem for linear systems, focusing on the algebraic Riccati equation (ARE) and its role in solving optimal control problems. The study addresses both cases where the terminal state is free and where it is constrained to be zero. The performance criterion is allowed to be fully quadratic in the control and state without requiring definiteness conditions typically assumed in the standard regulator problem. The paper derives frequency-domain and time-domain conditions for the existence of solutions to the ARE and provides a complete classification of all its solutions. It shows how the optimal control problems can be solved analytically via the ARE. The paper begins by discussing the importance of the infinite time least squares problem in modern systems theory, highlighting its applications in control theory, stability analysis, and other areas. It then introduces the quadratic performance criterion and the associated optimization problems, including the boundedness of the infima of the performance criterion. The paper presents several theorems that provide equivalent conditions for the boundedness of these infima, using both time-domain and frequency-domain conditions. The paper then focuses on the algebraic Riccati equation and its solutions. It establishes necessary and sufficient conditions for the existence of real symmetric solutions to the ARE and provides a classification of all solutions in terms of projection operators. The paper also discusses the relationship between the ARE and the stability of systems, showing how the ARE can be used to derive solutions to optimal control problems. The paper concludes with a classification of all solutions to the ARE, showing that they can be expressed as combinations of the maximum and minimum solutions. It also discusses the implications of these results for the stability of systems and the behavior of solutions under changes in the performance criterion and system parameters. The paper emphasizes the importance of the ARE in optimal control theory and its role in solving a wide range of control problems.This paper presents a comprehensive analysis of the infinite time least squares optimal control problem for linear systems, focusing on the algebraic Riccati equation (ARE) and its role in solving optimal control problems. The study addresses both cases where the terminal state is free and where it is constrained to be zero. The performance criterion is allowed to be fully quadratic in the control and state without requiring definiteness conditions typically assumed in the standard regulator problem. The paper derives frequency-domain and time-domain conditions for the existence of solutions to the ARE and provides a complete classification of all its solutions. It shows how the optimal control problems can be solved analytically via the ARE. The paper begins by discussing the importance of the infinite time least squares problem in modern systems theory, highlighting its applications in control theory, stability analysis, and other areas. It then introduces the quadratic performance criterion and the associated optimization problems, including the boundedness of the infima of the performance criterion. The paper presents several theorems that provide equivalent conditions for the boundedness of these infima, using both time-domain and frequency-domain conditions. The paper then focuses on the algebraic Riccati equation and its solutions. It establishes necessary and sufficient conditions for the existence of real symmetric solutions to the ARE and provides a classification of all solutions in terms of projection operators. The paper also discusses the relationship between the ARE and the stability of systems, showing how the ARE can be used to derive solutions to optimal control problems. The paper concludes with a classification of all solutions to the ARE, showing that they can be expressed as combinations of the maximum and minimum solutions. It also discusses the implications of these results for the stability of systems and the behavior of solutions under changes in the performance criterion and system parameters. The paper emphasizes the importance of the ARE in optimal control theory and its role in solving a wide range of control problems.