On the existence of periodic solutions to some nonlinear thermal control problems

Abstract

1 We consider the heat equation with a boundary condition involving a control function. The control function satisfies an ordinary differential equation with a right-hand side containing a nonlinear functional that provides the hysteresis phenomenon. The dependence of the functional on the “mean” temperature over the domain causes nonlocal effects. The problem under consideration occurs in the modeling of thermal control processes in chemical reactors and climate control systems. The solvability of the problem and the periodicity of its solutions are considered. 1. In chemical reactors and climate control systems, there arises the problem of temperature control inside a volume by means of some thermal elements on the boundary of the volume. We consider a mathematical model for such a thermal control process. In our model, the temperature distribution inside the domain obeys the heat equation, while the boundary condition involves a control function. The control function satisfies an ordinary differential equation whose right-hand side is given by a nonlinear functional depending on the “mean” temperature over the domain, which provides the so-called hysteresis phenomenon. The existence and uniqueness of solutions to twophase Stefan problems involving a boundary hysteresis control were studied in [1‐3]. In the present paper, we establish an existence and uniqueness result for the heat equation and investigate the periodicity of its solutions. We suggest the concepts of a strong periodic solution 1 The article was translated by the authors. and a mean-periodic solution and show that the existence of a mean-periodic solution implies the existence of a strong periodic solution with the same period. We also give an example in which a unique mean-periodic solution (hence, a unique strong periodic solution) exists.