Lower Bounds in Multi-Objective Hz / H , Problems

In this paper we discuss a multi-objective control problem that involves several Hz-norm and H,norm constraints. In an accompanying paper we show how to efficiently compute a sequence of upper bounds that converges monotonically from above to the optimal value of such problems. If more than one H, constraint is involved, it is so far unknown and revealed in this paper how to compute as well a sequence of lower bounds that converges to the optimum. This complements the tools available for multi-objective LMI based control by a stopping criterion. Notation. 12nxrn denotes the space of sequences X , E U? ' " satisfying IlXll$ := trace(X;X,) < 0. The space of all bounded linear operators IT into 1; is denoted as & ( 1 ~ , 1 ~ ) and equipped with the induced norm (I.I(. Hzx" or H;' " denote the normed algebra or inner product space (with the inner product written as (.,.)) of CnXm-valued mappings f which are analytic in D := { z E C : Iz( < 1) and which S U ~ ~ ~ ~ < ~ s-, trace(f(re")*f(re") d r < co respectively. Obviously, H Z m c Hrxm. RHkXrn denotes the subalgebra of all real-rational proper functions in H,. For ease of notation we will often refrain from explicitly mentioning the dimension n x rn. satisfy Ilfllg := suP,Ep Ilf(4Il < or rlfrl; :=