Nonlinear optimal control problem in a two-point boundary regime for a pseudoparabolic equation with Samarskii–Ionkin type conditions

This paper is devoted to study a optimal movable point control problem for a pseudoparabolic equation with nonlinear control function in a two-point nonlinear boundary condition. The equation is studied with Samarskii–Ionkin type boundary conditions on spatial variable $x$. Spectral problem is studied and eigenvalues, eigenfunctions and optimality conditions are found. Loaded nonlinear functional equations are obtained with respect to control function. We prove the existence and uniqueness of the control function by the method of compressing mapping. The state function is determined. Convergence of the Fourier series for the state function is proved.