The Problem of Boundary Control of the Thermal Process in a Rod

Abstract

This paper considers a rod with an insulated side surface, at the edges of which there is heat exchange with the external environment. It is assumed that the thermal process in the rod is controlled by the effect of the ambient temperature on the thermal state in the rod through its boundary temperatures. Using the technique of separation of variables and methods based on the theory of control of finite-dimensional systems, we propose a constructive approach to build the control function of the temperature conditions at the ends of the rod that change the temperature state distribution in the rod from a given initial state to a final state within a specified time interval. We have formulated the necessary and sufficient condition that the boundary control functions of the temperature modes of the rod must satisfy in order for the problem to be completely controllable under any allowable initial and final conditions. As an application of the proposed approach, we have built the temperature control functions at the ends of the rod for the first two harmonics.