An adjoint control method for initial condition identification of the Abstract Cauchy problem

TL;DR

This paper introduces a novel adjoint control method for reconstructing initial conditions in abstract Cauchy problems within Hilbert spaces, effectively handling noisy data through theoretical analysis and numerical validation.

Contribution

It presents a new generic adjoint control approach for initial condition identification in abstract Cauchy problems, combining theoretical insights with practical numerical demonstrations.

Findings

Method accurately reconstructs initial conditions from noisy data.

Theoretical analysis confirms stability and convergence.

Numerical experiments validate effectiveness across examples.

Abstract

This paper develops and analyzes a generic method for reconstructing solutions to the abstract Cauchy problem in a general Hilbert space, from noisy measured data. The method is based on the relationship between a partial differential equation and its adjoint equation with control. We demonsrate the capability of the method through analysis and numerical experiments.