On the stability of a time-dependent source identification problem for a Schrödinger-type involutory differential equation

Abstract

A study is presented of a time-dependent source identification problem for a Schrödinger-type involutory differential equation. The problem is formulated in an abstract Hilbert space with a self-adjoint positive definite operator. The unique solvability of the abstract problem is established, and stability estimates for its solution are obtained. These results are further applied to four specific time-dependent source identification problems: A one-dimensional problem with nonlocal conditions, a one-dimensional problem with spatial involution, and multi-dimensional problems with Dirichlet and Neumann boundary conditions. This demonstrates the importance of the presented operator approach for studying various classes of problems for Schrödinger-type involutory differential equations with unknown time-dependent source terms.