This paper is concerned with the finite-horizon version of the H8 problem with measurement feedback. Given a finite-dimensional linear, time-varying system, together with a positive real number ¿, we obtain necessary and sufficient conditions for the existence of a possibly time-varying dynamic compensator such that the 2([0, t1])-induced norm of the closed-loop operator is smaller than ¿. These conditions are expressed in terms of a pair of quadratic differential inequalities, generalizing the well-known Riccati differential equations introduced recently in the context of finite-horizon H8 control.
Stoorvogel, A. A., & Trentelman, H. L. (1993). The finite-horizon singular $H_\infty$ control problem with dynamic measurement feedback. Linear Algebra and Its Applications, 187, 113-161. https://doi.org/10.1016/0024-3795(93)90132-8