Analysis of Regular Riccati Equations of Continuous-Time Mixed H₂/H_∞ Control

This paper investigates the continuous-time mixed H<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf>/H<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> control problem for linear systems under positive semi-definite (singular) control weighting matrices. While traditional methods typically assume strictly positive-definite weights, the semi-definite scenario often leads to non-uniqueness and other technical challenges. We address these by developing a two-step optimization framework. First, we solve an H<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf>-type minimax problem—yielding a partially undetermined control law. Next, we remove the resulting freedom through an H<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf>-optimal design, where a differential Riccati equation ensures that a unique standard solution emerges. Under suitable regularity conditions, our approach provides an explicit closed-form control law that simultaneously achieves robustness and performance objectives.