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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

Yumei Liao 1,2, , Wei Wei 3,, and  Xianbing Luo 1,

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.

Keywords: Optimal frequency control, microwave heating, existence of weak solutions, existence of optimal control, coupled system.
Mathematics Subject Classification: Primary: 35Q60, 35K55, 49J20; Secondary: 35J70.
Citation: Yumei Liao, Wei Wei, Xianbing Luo. Existence of solution of a microwave heating model and associated optimal frequency control problems. Journal of Industrial & 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.   Google Scholar
[2]

L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, Rhode Island, 1998. doi: 10.1090/gsm/019.  Google Scholar

[3]

H. O. Fattorini, Infinite Dimensional Linear Control System: The Time Optimal and Norm Optimal Problem, North-Holland Mathematics Studies, Elsevier, 2005.  Google Scholar

[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.  Google Scholar

[5] J. C. Kuang, General Inequality (Fourth Eedition), Shandong Science and Technology Press, Shandong, 2010.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar

[10]

B. Li, J. Tang and H. M. Yin, Optimal control microwave sterilization in food processing, Int. J. Appl. Math., 10 (2002), 13-31.   Google Scholar

[11] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971.   Google Scholar
[12] A. C. Metaxas, Foundations of Electroeat, A Unified Aproach, John Wiley and Sons, New York, 1996.   Google Scholar
[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.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22] E. Zeidler, Nonlinear Functional and Its Applications Ⅱ, Springer, New York, 1990.  doi: 10.1007/978-1-4612-0985-0.  Google Scholar

show all references

References:
[1] V. Barbu, Aanalysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Boston, 1993.   Google Scholar
[2]

L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, Rhode Island, 1998. doi: 10.1090/gsm/019.  Google Scholar

[3]

H. O. Fattorini, Infinite Dimensional Linear Control System: The Time Optimal and Norm Optimal Problem, North-Holland Mathematics Studies, Elsevier, 2005.  Google Scholar

[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.  Google Scholar

[5] J. C. Kuang, General Inequality (Fourth Eedition), Shandong Science and Technology Press, Shandong, 2010.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar

[10]

B. Li, J. Tang and H. M. Yin, Optimal control microwave sterilization in food processing, Int. J. Appl. Math., 10 (2002), 13-31.   Google Scholar

[11] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971.   Google Scholar
[12] A. C. Metaxas, Foundations of Electroeat, A Unified Aproach, John Wiley and Sons, New York, 1996.   Google Scholar
[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.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22] E. Zeidler, Nonlinear Functional and Its Applications Ⅱ, Springer, New York, 1990.  doi: 10.1007/978-1-4612-0985-0.  Google Scholar
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