Upper bounds for mixed H2/H∞ control

We consider the mixed H²/H^∞ control problem of choosing a controller to minimize the H² norm of a given closed-loop map, subject to the H^∞ norm of another closed-loop map being less than a prescribed value γ. Let d₂ and γ₂ denote the H² and H^∞ norms for the pure H²-optimal solution (without any H^∞ constraint), and let d_c and γ_c<γ denote the H² and H^∞ norms for any solution that yields an H^∞ norm strictly less than γ (such as, say, the central solution). Then if d_m denotes the optimal H² norm that can be achieved in the mixed H²/H^∞ control problem, we show that d_m²-d₂ ²/d_c²-d₂² ⩽(γ₂-γ/γ₂-γ _c)²<(γ₂²-γ ²/γ₂²-γ_c² )²<1