Optimal Cooling Control of a Sphere

In a cooling process,the rate of surface temperature drop will affect the magnitude of temperature gradient in the body. The sooner the surface temperature is lowered,the larger the temperature gradient is,and vice-versa.The objective of this thesis is to apply Optimal Control theory to a simplified heat conduction system to obtain an optimal way of cooling in a sense of archiving a fast temperature drop while not causing large temperature gradient in the solid body.We first define an functional constructed by temperature and temperature gradient as an object to be optimized. Since the temperature and temperature gradient must conform to the law of heat conduction, a Lagrangeultiplier is introduced and the object function is augmented with the multiplier.In order to derive the necessary conditions for optimization, method of Variation is applied to the object function, and a system of PDEs(composed of an Euler-Lagrange equation and heat conduction equation) and relevant boundary conditions are obtained. The equations are then solved numerically for the required optimal temperature drop on the body surface.After comparing with other pre-specified processes,it is confirmed that the resulting cooling process is indeeded an optimal one as expected.