On the Game Riccati Equations Arising in $H_\infty$ Control Problems

• Authors: Pascal Gahinet
• Venue: SIAM Journal on Control and Optimization
• Year: 1994

In the state-space approach to $H_\infty$ optimal control, feasible closed-loop gains $\gamma$ are characterized via a pair of game Riccati equations depending on $\gamma$. This paper is concerned with the properties of these equations as $\gamma$ varies. The most general problem is considered $(D_{11} \neq 0)$ and the variations of the Riccati solutions are thoroughly analyzed. Insight is gained into the behavior near the optimum and into the dependence on $\gamma$ of the suboptimality conditions. In addition, concavity is established for a criterion that synthesizes the three conditions $X \geq 0, Y \geq 0$, and $\rho(XY) Most results presented here are extensions of earlier contributions. The main concern is to provide a complete and synthetic overview as well as results and formulas tailored to the development of numerically sound algorithms.