On the four block problem and simultaneous \(H^{\infty}\) and \(L^ 2\) suboptimization (Q2367619)

Inspired by \textit{V. Kaftal}, \textit{D. Larson} and \textit{G. Weiss} [J. Funct. Anal. 107, No. 2, 387-401 (1992), \textit{C. Foias} and \textit{A. E. Frazho} [SIAM J. Math. Anal. 23, No. 4, 984-994 (1992; Zbl 0783.47013)] used the Commutant Lifting Theorem to obtain simultaneously (with respect to \(L^ \infty\) and \(L^ 2\) norms) a suboptimal solution to the two- sided Nehari optimization problem. In this paper, we apply the version of the Commutant Lifting Theorem derived there for this sort of problem to extend the result to four block problems arising in robust control. In particular we prove that for \(A\), \(B\), \(C\), \(D\) matrix-valued \(L^ \infty\)-functions there exists a matrix-valued function \(Q_{\text{opt}}\in H^ \infty\) such that \[ \left\|\left[{A- Q_{\text{opt}}\atop C}{B\atop D}\right]\right\|_ \infty\leq \sqrt 2\inf_{Q\in H^ \infty} \left\|\left[{A-Q\atop C}{B\atop D}\right]\right\|_ \infty \] and \[ \left\|\left[{A- Q_{{opt}}\atop C}{B\atop D}\right]\right\|_ 2\leq \sqrt 2\inf_{Q\in H^ \infty}\left\|\left[{A- Q\atop C}{B\atop D}\right]\right\|_ 2. \] {}.