Analyticity of Exponential Dirichlet Series and Applications to the Approximate Controllability of Parabolic Equations

TL;DR

This paper explores the analyticity of exponential Dirichlet series, determines their power series coefficients, and applies these findings to analyze the approximate controllability of linear parabolic equations using the moment method.

Contribution

It provides new insights into the analyticity and coefficient determination of exponential Dirichlet series and applies these results to control theory of parabolic PDEs.

Findings

Explicit coefficients for the power series of exponential Dirichlet series

Estimates for the remainder in the series expansion

Application to approximate controllability of parabolic equations

Abstract

In this paper, we investigate the analyticity of a class of exponential Dirichlet series. We then explicitly determine the coefficients of their power series decomposition and provide an estimate for the remainder. As an application, we study the approximate controllability property of linear parabolic equations with locally distributed or lumped controls by employing the moment method, which relies on the exponential Dirichlet series associated with the spectrum of the system's operator.