Matlab program method of computing Carleman estimates and applications

TL;DR

This paper presents a Matlab-based method to compute Carleman estimates for a fourth-order PDE, enabling new controllability, observability, and inverse problem stability results.

Contribution

Introduction of a Matlab program method to compute Carleman estimates for a fourth-order PDE, leading to new applications in inverse problems and control theory.

Findings

Computed Carleman estimates with singular and regular weights.

Established controllability and observability results for 1-d fourth order equations.

Derived new stability results for inverse problems of fractional equations.

Abstract

In this paper, we introduce a Matlab program method to compute Carleman estimate for the fourth order partial differential operator $\gamma\partial_t+\partial_x^4\ (\gamma\in\mathbb{R})$. We obtain two kinds of Carleman estimates with different weight functions, i.e. singular weight function and regular weight function, respectively. Based on Carleman estimate with singular weight function, one can obtain the known controllability and observability results for the 1-d fourth order parabolic-type equation, while based on Carleman estimate with regular weight function, one can deduce not only the known result on conditional stability in the inverse problem of half-order fractional diffusion equation, but also a new result on conditional stability in the inverse problem of half-order fractional Schr\"odinger equation.