ON THE CONTROL PROBLEM FOR A ONE-DIMENSIONAL PSEUDO-PARABOLIC EQUATION WITH A NONLOCAL BOUNDARY CONDITION

Abstract

This paper studies a control problem for a one-dimensional pseudo-parabolic equation with a nonlocal boundary condition of the Bitsadze-Samarskii type. The control function appears as a distributed source term on the right-hand side of the equation, and the ob jective is to steer the system from the zero initial state so that the weighted average temperature follows a prescribed trajectory. The equation contains a third-order mixed derivative term that accounts for memory or viscoelastic effects i n h eat conduction, which is particularly relevant for materials with internal structure such as polymers, biological tissues, and granular media. Using spectral decomposition and a biorthogonal basis, the problem is reduced to a Volterra integral equation of the first kind. The kernel of this integral equation is proved to be p ositive and continuous. Under suitable assumptions on the weight function and the spatial distribution of the control, the existence of an admissible control is established for a class of desired trajectories with sufficiently small norm. The Laplace transform method is employed to solve the integral equation and to obtain bounds on the control function. The results quantify the influence o f t he n onlocal b oundary c ondition a nd the memory parameter on the controllability of the system.