The 2-block superoptimal AAK problem
This paper is a continuation of other works of some previous authors which solved the 1- and 2-block AKK problems. First, the authors outline the solution to the 2-block optimal AKK problem. Afterwards, the authors deal with the 2-block superoptimal AKK problem: Given a rational and antistable matrix function \(R(s)= [R_{11}(s), R_{12}(s)]\) and a nonnegative integer \(k\), find all superoptimal approximations \(Q(s)\), with no more than \(k\) poles in the right-half complex plane, that minimize the supremum, over the imaginary axis, of the singular values of the error function \(E(s)= [R_{11}(s), R_{12}(s)+ Q(s)]\), with respect to lexicographic ordering. The authors present a simple algorithm for the solution to both 1- and 2-block superoptimal AKK problems, namely, it is proved that, under certain conditions, it is possible to construct a sequence of 2-block functions \(R_1,\dots, R_t\), of the same form as \(R\) (but of progressively reduced dimensions and MacMillan degree) such that \(s_i(R,k)= s_1(R_i,k_i)\) and \({\mathcal S}_i(R,k)= {\mathcal T}_{i-1}[{\mathcal S}_1(R_i,k_i)]\), where \({\mathcal T}_1,{\mathcal T}_2,\dots\) is a certain sequence of operators and \(\{k_i\}\) is a nonincreasing integer sequence with \(k_0= k\). The notations used are defined by: \[ \begin{aligned} s_1(R,k) & :=\inf_{Q\in{\mathcal H}^{p\times m}_\infty(k)} s^\infty_1([R_{11}, R_{12}+ Q]),\;s^\infty_1(\cdot)= \|\cdot\|_\infty;\\ {\mathcal S}_1(R, K) & := \{Q\in{\mathcal H}^{p\times m}_\infty(k): s^\infty_1([R_{11}, R_{12}+ Q])= s_1(R,k)\};\\ s_i(R, k) & := \inf_{Q\in{\mathcal S}_{i-1}(R, k)} s^\infty_i([R_{11}, R_{12}+ Q]),\;i= 1,2,\dots;\\ {\mathcal S}_i(R, k) & := \{Q\in{\mathcal S}_{i-1}(R, k): s^\infty_i([R_{11}, R_{12}+ Q])= s_i(R, k)\},\;i= 1,2,\dots,\end{aligned} \] where \({\mathcal S}_0(R,k):= {\mathcal H}^{p\times m}_\infty(k)\) is the subset of \({\mathcal L}^{p\times m}_\infty\) consisting of the matrices with no more than \(k\) poles in \({\mathcal C}_+\). Thus, the solution to the \(i\)th-level superoptimal problem is reduced to the solution to \(i\) optimal AKK problems. Sufficient conditions are given for which the superoptimal approximation of \(R\) is unique. Also, the authors give an a priori upper bound on the second superoptimal level, which generalizes a previous result.
scientific article; zbMATH DE number 1234467
scientific article