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September  2020, 16(5): 2103-2116. doi: 10.3934/jimo.2019045

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

 1 School of Mathematics and Statistics, Guizhou University, Guiyang, Guizhou 550025, China 2 Department of Mathematics, Guizhou Education University, Guiyang, Guizhou 550018, China 3 Department of Mathematics, Guizhou Minzu University, Guiyang, Guizhou 550025, China

* Corresponding author: Wei Wei

Received  October 2018 Published  September 2020 Early access  May 2019

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.

Citation: Yumei Liao, Wei Wei, Xianbing Luo. Existence of solution of a microwave heating model and associated optimal frequency control problems. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2103-2116. doi: 10.3934/jimo.2019045
##### References:
 [1] V. Barbu, Aanalysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Boston, 1993. [2] L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, Rhode Island, 1998. doi: 10.1090/gsm/019. [3] H. O. Fattorini, Infinite Dimensional Linear Control System: The Time Optimal and Norm Optimal Problem, North-Holland Mathematics Studies, Elsevier, 2005. [4] D. Kleis and E. W. Sachs, Optimal Control of the Sterilization of Prepackaged Food, SIAM J.Optim., 10 (2000), 1180-1195.  doi: 10.1137/S1052623497331208. [5] J. C. Kuang, General Inequality (Fourth Eedition), Shandong Science and Technology Press, Shandong, 2010. [6] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, AMS Trans., 23, Providence., R.I, 1968. [7] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Ⅰ. Abstract Parabolic Systems, in: Encyclopedia of Mathematics and its Applications, vol. 74, Cambridge University Press, Cambridge, 2000. [8] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Ⅱ. Abstract Hyperbolic-like Systems Over a Finite Time Horizon, Encyclopedia of Mathematics and its Applications, vol. 75, Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511574801.002. [9] X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4. [10] B. Li, J. Tang and H. M. Yin, Optimal control microwave sterilization in food processing, Int. J. Appl. Math., 10 (2002), 13-31. [11] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971. [12] A. C. Metaxas, Foundations of Electroeat, A Unified Aproach, John Wiley and Sons, New York, 1996. [13] A. C. Metaxas and R. J. Meredith, Industrial Microwave Heating in I.E.E Power Engineering Series Vol.4, Per Peregrimus Ltd., London, 1983. [14] K. Pitchai, J. J. Chen, S. Birla, D. Jones and J. Subbiah, Modeling microwave heating of frozen mashed potato in a domestic oven incorporating electromagnetic frequency spectrum, Journal of Food Engineering, 173 (2016), 124-131.  doi: 10.1016/j.jfoodeng.2015.11.002. [15] Z. Tang, T. Hong, Y. H. Liao and etc, Frequency-selected Method to Improve Microwave Heating Performance, Applied Thermal Engineering, 131 (2018), 642-648. doi: 10.1016/j.applthermaleng.2017.12.008. [16] F. Troltzsch, Optimal Control of Partial Differential Equations, Theory, Methods and Applications, Graduate Studies in Mathematics. Vol.112, AMS, Providence, Rhode Island, 2010. doi: 10.1090/gsm/112. [17] W. Wei, H. M. Yin and J. Tang, An Optimal Control Problem for Microwave Heating, Nonlinear Analysis, 75 (2012), 2024-2036.  doi: 10.1016/j.na.2011.10.003. [18] H. M. Yin and W. Wei, A nonlinear optimal control problem arising from a sterilization process for packaged foods, Applied Mathematics and Optimization, 77 (2018), 499-513.  doi: 10.1007/s00245-016-9382-0. [19] H. M. Yin, Regularity of solutions of maxwell's equations in quasi-stationary electromagnetic field and applications, Partial Differential Equations, 22 (1997), 1029-1053.  doi: 10.1080/03605309708821294. [20] H. M. Yin, Regularity of weak solutions of maxwell's equations and applications to microwave heating, J.Differential Equations, 200 (2004), 137-161.  doi: 10.1016/j.jde.2004.01.010. [21] H. M. Yin and W. Wei, Regularity of weak solution for a coupled system arising from a microwave heating model, European Journal of Applied Mathematics, 25 (2014), 117-131.  doi: 10.1017/S0956792513000326. [22] E. Zeidler, Nonlinear Functional and Its Applications Ⅱ, Springer, New York, 1990.  doi: 10.1007/978-1-4612-0985-0.

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##### References:
 [1] V. Barbu, Aanalysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Boston, 1993. [2] L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, Rhode Island, 1998. doi: 10.1090/gsm/019. [3] H. O. Fattorini, Infinite Dimensional Linear Control System: The Time Optimal and Norm Optimal Problem, North-Holland Mathematics Studies, Elsevier, 2005. [4] D. Kleis and E. W. Sachs, Optimal Control of the Sterilization of Prepackaged Food, SIAM J.Optim., 10 (2000), 1180-1195.  doi: 10.1137/S1052623497331208. [5] J. C. Kuang, General Inequality (Fourth Eedition), Shandong Science and Technology Press, Shandong, 2010. [6] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, AMS Trans., 23, Providence., R.I, 1968. [7] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Ⅰ. Abstract Parabolic Systems, in: Encyclopedia of Mathematics and its Applications, vol. 74, Cambridge University Press, Cambridge, 2000. [8] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Ⅱ. Abstract Hyperbolic-like Systems Over a Finite Time Horizon, Encyclopedia of Mathematics and its Applications, vol. 75, Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511574801.002. [9] X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4. [10] B. Li, J. Tang and H. M. Yin, Optimal control microwave sterilization in food processing, Int. J. Appl. Math., 10 (2002), 13-31. [11] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971. [12] A. C. Metaxas, Foundations of Electroeat, A Unified Aproach, John Wiley and Sons, New York, 1996. [13] A. C. Metaxas and R. J. Meredith, Industrial Microwave Heating in I.E.E Power Engineering Series Vol.4, Per Peregrimus Ltd., London, 1983. [14] K. Pitchai, J. J. Chen, S. Birla, D. Jones and J. Subbiah, Modeling microwave heating of frozen mashed potato in a domestic oven incorporating electromagnetic frequency spectrum, Journal of Food Engineering, 173 (2016), 124-131.  doi: 10.1016/j.jfoodeng.2015.11.002. [15] Z. Tang, T. Hong, Y. H. Liao and etc, Frequency-selected Method to Improve Microwave Heating Performance, Applied Thermal Engineering, 131 (2018), 642-648. doi: 10.1016/j.applthermaleng.2017.12.008. [16] F. Troltzsch, Optimal Control of Partial Differential Equations, Theory, Methods and Applications, Graduate Studies in Mathematics. Vol.112, AMS, Providence, Rhode Island, 2010. doi: 10.1090/gsm/112. [17] W. Wei, H. M. Yin and J. Tang, An Optimal Control Problem for Microwave Heating, Nonlinear Analysis, 75 (2012), 2024-2036.  doi: 10.1016/j.na.2011.10.003. [18] H. M. Yin and W. Wei, A nonlinear optimal control problem arising from a sterilization process for packaged foods, Applied Mathematics and Optimization, 77 (2018), 499-513.  doi: 10.1007/s00245-016-9382-0. [19] H. M. Yin, Regularity of solutions of maxwell's equations in quasi-stationary electromagnetic field and applications, Partial Differential Equations, 22 (1997), 1029-1053.  doi: 10.1080/03605309708821294. [20] H. M. Yin, Regularity of weak solutions of maxwell's equations and applications to microwave heating, J.Differential Equations, 200 (2004), 137-161.  doi: 10.1016/j.jde.2004.01.010. [21] H. M. Yin and W. Wei, Regularity of weak solution for a coupled system arising from a microwave heating model, European Journal of Applied Mathematics, 25 (2014), 117-131.  doi: 10.1017/S0956792513000326. [22] E. Zeidler, Nonlinear Functional and Its Applications Ⅱ, Springer, New York, 1990.  doi: 10.1007/978-1-4612-0985-0.
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