Monotonicity and convexity of H∞ Riccati solutions in general case

State space formulas for H^∞ optimal control problem involve two H^∞ Riccati equations, whose solutions can be used to construct an optimal or suboptimal H^∞ controller. This paper studies the existence of the solutions to the two H^∞ Riccati equations in Glover-Doyle´s formulation which is the most general one yet been considered, and shows that the solutions are nonincreasing convex functions in the domain of interest. The monotonicity and convexity of those H^∞ Riccati solutions guarantee that the spectral radius of the product of those two Riccati solutions is also a nonincreasing convex function of γ in the domain of interest. According to these properties, a quadratically convergent algorithm is developed to compute the optimal H^∞.