Optimal bounds for the boundary control cost of one-dimensional fractional Schr\"odinger and heat equations

TL;DR

This paper establishes precise bounds on the boundary control costs for one-dimensional fractional Schrödinger and heat equations, utilizing complex analysis and the moment method to analyze control costs.

Contribution

It provides the first sharp bounds for boundary control costs of fractional Schrödinger and heat equations, combining complex analysis and the moment method.

Findings

Derived sharp bounds for control costs

Analyzed lower bounds via singular boundary control problems

Estimated Fourier transforms for compactly supported functions

Abstract

We derive sharp bounds for the boundary control cost of the one-dimensional fractional Schr\"odinger and heat equations. The analysis of the lower bound is based on the study of the control cost of a related singular boundary control problem in finite time, using tools from complex analysis. The analysis of the upper bound relies on the moment method, involving estimates of the Fourier transform of a class of compactly supported functions.