Completion of the squares in the finite horizon $H^\infty$ control problem by measurement feedback

In this paper we study the finite horizon version of the standard $H^\infty$ control problem by measurement feedback. Given a finite-dimensional linear, time-invariant system, together with a positive real number $\gamma$, we obtain necessary and sufficient conditions for the existence of a possibly time-varying dynamic compensator such that the $L_2 [0,T]$-induced norm of the closed loop operator is smaller than $\gamma$. These conditions are expressed in terms of a pair of quadratic differential inequalities, generalizing the well-known Riccati differential equations that were introduced recently in the context of finite horizon $H^\infty$ control.