Existence of solution of a microwave heating model and associated optimal frequency control problems

Abstract

Microwave heating has been widely used in various fields during recent years. However, it also has a common problem of uneven heating. In this paper, optimal frequency control problem for microwave heating process is considered. The cost function is defined such that the temperature profile at the final stage has a relative uniform distribution in the field. The controlled system is a coupled by Maxwell equations with nonlinear heating equation. The existence of a weak solution for coupled system is proved. The weak continuity of the solution operator is also shown. Moreover, the existence of a global minimizer of the optimal frequency control problems is proved.

The Regularity of Solution for Weakly Coupled System Derived by Microwave Heating Model

“…We have derived weak coupling mathematical model of microwave heating system with frequency variation in [13]. In paper [13], the existence of global weak solutions is established by assuming suitable conditions.…”

“…We have derived weak coupling mathematical model of microwave heating system with frequency variation in [13]. In paper [13], the existence of global weak solutions is established by assuming suitable conditions. By applying Lax-Milgram theorem and monotone operator theory, the existence and uniqueness of solutions for weakly coupled systems are proved in [13].…”

“…In paper [13], the existence of global weak solutions is established by assuming suitable conditions. By applying Lax-Milgram theorem and monotone operator theory, the existence and uniqueness of solutions for weakly coupled systems are proved in [13]. However, the regularity of the weak solution has not been studied for the coupled system in [13].…”

“…By applying Lax-Milgram theorem and monotone operator theory, the existence and uniqueness of solutions for weakly coupled systems are proved in [13]. However, the regularity of the weak solution has not been studied for the coupled system in [13]. Particularly, we consider the uniformity and boundedness of temperature.…”

“…We study that the temperature is Hölder continuous by appropriate assumptions.The regularity results provide the basis about the derivation of optimal conditions for uniform microwave heating This paper is organized as follows. In Section 2, for completeness we introduce the mathematical model of microwave heating with frequency variable from [13]. In Section 3, we impose some basic assumptions which ensure the well-posedness of the underlying system.…”