On the Nehari problem for a certain class of \(L^{\infty}\)-functions appearing in control theory
The authors obtain an explicit computable solution for finding \({\bar \rho}:= \inf_{q\in H^{\infty}(D)}\| W-mq\|\) where \(W\in H^{\infty}(D)\) is rational and \(m\in H^{\infty}(D)\) is an arbitrary inner function. The formula involves calculating where the function of \(\rho\) given by \(\nu(\rho)=\) trace of \((I=T^*T^{*n})P_{\rho}^{- 1}\) blows up; here T is the compression of the shift operator on \(H^ 2(D)\) (multiplication by z) to the orthogonal complement of its invariant subspace \(mH^ 2(D)\) and \(P_{\rho}= q(T)(I- \rho^{- 2}W(T)W(T)^*)q(T)^*\) where \(W=p/q\). In practice this in turn comes down to finding the roots of a transcendental equation. In the last section an explicit example is worked out for illustration of the approach. The problem is motivated by the \(H^{\infty}\)-optimization problem from control theory.
\(H^{\infty }\)-optimization problem from control theory
scientific article; zbMATH DE number 4034463
scientific article