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

The heat equation is considered with a boundary condition involving a control function that satisfies ordinary differential equation with a right hand side containing a nonlinear functional that provides the hysteresis phenomenon. 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 that report that in chemical reactors and climate control systems, there arises a problem of temperature control inside a volume by means of some thermal elements on the boundary of the volume. A mathematical model is considered for such a thermal control process and here the temperature distribution inside the domain obeys the heat equation, while the boundary condition involves a control function. The boundary condition contains a real-valued control function that regulates the temperature on the boundary.