Boundary control of temperature distribution in a spherical shell with spatially varying parameters

This paper presents a solution to the control (stabilization) problem of temperature distribution in spherical shells with spatially varying properties. The desired temperature distribution satisfies the steady-state heat conduction equation. For the spherical shell under consideration, it is assumed that material properties such as thermal conductivity, density, and specific heat capacity may vary in radial, polar, and azimuthal directions of the spherical shell; the governing heat conduction equation of the shell is a second-order partial differential equation. Using Lyapunov's theorem, it is shown how to obtain boundary heat flux required for producing a desired steady-state distribution of the temperature. Finally, numerical simulation is provided to verify the effectiveness of the proposed method such that by applying the boundary transient heat flux, in-domain distributed temperature converges to its desired steady-state temperature.