The Null Boundary Controllability for a Fourth-Order Parabolic Equation with Samarskii–Ionkin-Type Boundary Conditions

This paper presents a moment method approach to solve the null boundary controllability problem for a fourth-order parabolic equation subject to Samarskii–Ionkin-type boundary conditions. The problem is solved in two stages. First, we demonstrate that the eigenfunctions of the system, which are not self-adjoint under these boundary conditions, form a Riesz basis in L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2$$\end{document} space. Using Fourier’s method, we construct a biorthonormal system of functions to express the series solution. In the second stage, we use these spectral results to show that the system is null boundary controllable for a specific class of initial data. Our approach extends the existing literature on null boundary controllability of parabolic equations and provides new insights into the properties of systems subject to Samarskii–Ionkin-type boundary conditions.