Stabilization in $H^\infty_{\mathbb{R}}(\mathbb{D})$

TL;DR

This paper proves a stabilization theorem in the real Hardy space $H^ ext{infty}_ ext{R}( ext{D})$, establishing conditions under which certain bounded functions can be combined to produce an identity, with implications for control theory.

Contribution

The paper introduces a new stabilization result in $H^ ext{infty}_ ext{R}( ext{D})$, extending classical results to real Hardy spaces with specific boundary conditions.

Findings

Existence of stabilizing functions $g_1, g_2$ with controlled norms.

Construction of solutions under positivity and boundary conditions.

Bounded inverse of $g_1$ established within the Hardy space.

Abstract

In this paper we prove the following theorem: Suppose that $f_1,f_2\in H^\infty_\R(\D)$, with $\norm{f_1}_\infty,\norm{f_2}_{\infty}\leq 1$, with $$ \inf_{z\in\D}(\abs{f_1(z)}+\abs{f_2(z)})=\delta>0. $$ Assume for some $\epsilon>0$ and small, $f_1$ is positive on the set of $x\in(-1,1)$ where $\abs{f_2(x)}<\epsilon$ for some $\epsilon>0$ sufficiently small. Then there exists $g_1, g_1^{-1}, g_2\in H^\infty_\R(\D)$ with $$ \norm{g_1}_\infty,\norm{g_2}_\infty,\norm{g_1^{-1}}_\infty\leq C(\delta,\epsilon) $$ and $$ f_1(z)g_1(z)+f_2(z)g_2(z)=1\quad\forall z\in\D. $$