• Previous Article
    Nonlinear optimization to management problems of end-of-life vehicles with environmental protection awareness and damaged/aging degrees
  • JIMO Home
  • This Issue
  • Next Article
    A low-dimensional SDP relaxation based spatial branch and bound method for nonconvex quadratic programs
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  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 & 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. LiJ. 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. PitchaiJ. J. ChenS. BirlaD. 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. WeiH. 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. LiJ. 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. PitchaiJ. J. ChenS. BirlaD. 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. WeiH. 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
[1]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

[2]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019

[3]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046

[4]

Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325

[5]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[6]

Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347

[7]

Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213

[8]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110

[9]

Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032

[10]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

[11]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[12]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326

[13]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[14]

Dorothee Knees, Chiara Zanini. Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 121-149. doi: 10.3934/dcdss.2020332

[15]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

[16]

José Madrid, João P. G. Ramos. On optimal autocorrelation inequalities on the real line. Communications on Pure & Applied Analysis, 2021, 20 (1) : 369-388. doi: 10.3934/cpaa.2020271

[17]

Sergio Conti, Georg Dolzmann. Optimal laminates in single-slip elastoplasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 1-16. doi: 10.3934/dcdss.2020302

[18]

Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020

[19]

Tommi Brander, Joonas Ilmavirta, Petteri Piiroinen, Teemu Tyni. Optimal recovery of a radiating source with multiple frequencies along one line. Inverse Problems & Imaging, 2020, 14 (6) : 967-983. doi: 10.3934/ipi.2020044

[20]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (134)
  • HTML views (587)
  • Cited by (1)

Other articles
by authors

[Back to Top]