Understanding how Nehari's problem connects with robust stabiliziation and Nevanlinna-Pick

Question

I'm reading Young's "An Introduction to Hilbert space". In chapter 15 he writes about robust stabilization in control theory and ends with that this boils down to an interpolation problem called the Nevanlinna-Pick problem. In the next chapter he states what he calls Nehari's problem which is approximating a function in $L^{\infty}$ with a function in $H^{\infty}$ and refers back to the chapter on robust stabilization as an application of this. Here the leap of faith becomes to wide for me. Is the Nevanlinna-Pick problem a special case of Nehari's problem?

Answer 1

1. Nevanlinna-Pick interpolation (as well as Caratheodory-Fejer interpolation) is a special case of Nehari's problem. Indeed, NP interpolation is to find a function $f$ from the unit ball in $H^\infty$ that interpolates the given values $f(\zeta_i)=\omega_i$ in the unit disc. If $L(z)$ is the Lagrange interpolation polynomial with $L(\zeta_i)=\omega_i$ then $L-f$ is an analytical function with zeros at $\zeta_i$, i.e. $$ L(z)-f(z)=B(z)h(z),\qquad B(z)=\prod_i\frac{z-\zeta_i}{1-\bar\zeta_i z},\quad h\in H^\infty. $$ Now to find an interpolant $f$ such that $\|f\|_\infty\le 1$ is equivalent to finding an approximant $h$ such that $\|L-Bh\|_\infty\le 1$ or (since $|B(z)|=1$ on the unit circle) $$ \left\|\frac{L}{B}-h\right\|_\infty\le 1. $$ It is often desirable to make $f$ as small as possible, which corresponds to minimization of the $H^\infty$ norm in Nehari's problem.
2. I guess it was just an example with Nevanlinna-Pick interpolation to illustrate how $H^\infty$ optimization turns out to be important in robust control. In most robust control literature, the problem states directly as Nehari's problem, see, for example, B.Francis, A Course in $H^\infty$ Control Theory.