Abstract
In this paper we study an optimization problem arising from a sterilization process for packaged foods by using a microwave heating method. The goal of the optimal control is to find the optimal frequency function such that the temperature profile at the final stage has a relative uniform distribution in the food product. The underlying state variables are electric, magnetic fields and temperature which satisfy the coupled nonlinear Maxwell’s system and a nonlinear heat equation. The control variable for the system is chosen to be the electric frequency function. We show that there exists an optimal frequency which minimizes the cost functional. Moreover, an optimality condition is also derived.
Similar content being viewed by others
References
Alberti, G.S., Capdeboscq, Y.: Elliptic regularity theory applied to time harmonic anisotropic Maxwell’s equations with less than Lipschitz complex coefficients. SIAM J. Math. Anal. 46, 998–1016 (2014)
Barbu, V.: Analysis and Control of Nonlinear Infinite Dimensional System. Academic Press, New York (1993)
Eller, M.M., Masters, J.E.: Exact boundary controllability of electromagnetic fields in a general region. Appl. Math. Optim. 45, 99–123 (2002)
Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence (1998)
Fu, X., Yong, J., Zhang, X.: Exact controllability for multidimensional semilinear hyperbolic equations. SIAM J. Control Optim. 46(5), 1578–1614 (2007)
Hazard, C., Lenoir, M.: On the solution of time-Harmonic scattering problems for Maxwell’s equations. SIAM J. Math. Anal. 27, 1597–1630 (1996)
Lagnese, J.E.: Exact boundary controllability of Maxwell’s equations in a general region. SIAM J. Control Optim. 27, 374–388 (1989)
Li, X., Yong, J.: Optimal Control Theory for Infinite Dimensional Equations. Birhauser, Boston (1995)
Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)
Kleis, D., Sachs, E.W.: Optimal control of the sterization of prepacaged food. SIAM J. Optim. 10, 1180–1195 (2000)
Krigman, S.S., Wayne, C.E.: Boundary controllability of Maxwell’s equations with nonzero conductivity inside a cube. I. Spectral controllability (English summary). J. Math. Anal. Appl. 329(2), 1375–1396 (2007)
Lasiecka, I., Triggiani, R.: 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)
Lasiecka, I., Triggiani, R.: Control theory for partial differential equations: continuous and approximation theories. II. Abstract hyperbolic-like systems over a finite time horizon. In: Encyclopedia of Mathematics and its Applications, vol. 75. Cambridge University Press, Cambridge (2000)
Li, B., Yin, H.M., Tang, J.: Optimal control of microwave sterilization in food processing. Int. J. Appl. Math. 10, 13–31 (2002)
Metaxas, A.C.: Foundations of Electroheat, A Unified Approach. Wiley, New York (1996)
Metaxas, A.C., Meredith, R.J.: Industrial Microwave Heating. I.E.E. Power Engineering Series, vol. 4. Per Peregrimus Ltd., London (1983)
Müller, C.: Foundations of the Mathemtical Theory of Electromagnetic Waves. Springer, NewYork (1969)
Resurreccion, F.P., Luan, D., Tang, J., Liu, F., Tang, Z., Pedrow, P.D.: Effect of changes in microwave frequency on heating patterns of foods in a microwave assisted thermal sterilization system. J. Food Eng. 150, 99–105 (2015)
Raymond, J.P., Zidani, H.: Hamiltonian Pontryagin’s principles for control problems governed by semilinear parabolic equations. Appl. Math. Optim. 39, 143–177 (1999)
Tang, J., Liu, F., Pathak, S., Eves, G.: Apparatus and method for heating objectes with microwaves, US Patent No. 7,119,313, approved 10 Oct 2006
Sun, T., Tang, J., Powers, J.: Antioxidant activity and quality of asparagus affected by microwave-circulated water combination and conventional sterilization. Food Chem. 100, 813–819 (2007)
Troltzsch, F.: Optimal Control of Partial Differential Equations, Theory, Methods and Applications, Graduate Studies in Mathematics, vol. 112. AMS, Providence, Rhode Island (2010)
Wei, W., Yin, H.M., Tang, J.: An Optimal control problem for microwave heating. Nonlinear Anal. Theory Methods Appl. 75, 2024–2036 (2012)
Weck, N.: Exact boundary controlability of a Maxwell’s problem. SIAM J. Control Optim. 38, 736–750 (2000)
Tsering-Xiao, B., Xiang, W.: Regularity of solutions to time-harmonic Maxwell’s system with various lower than Lipschitz coefficients. arXiv:1603.01922, March 7, 2016
Yin, H.M.: Regularity of weak solution to Maxwell’s equations and applications to microwave heating. J. Differ. Equ. 200, 137–161 (2004)
Yin, H.M., Wei, W.: Regularity of weak solution for a coupled system arising from a microwave heating model. Eur. J. Appl. Math. 25(1), 117–131 (2014)
Zeidler, E.: Nonlinear Functional and Its Applications II. Springer, NewYork (1990)
Acknowledgments
Many thanks to the anonymous referees for their comments and suggestions. The main result of this paper was reported by the first author at SIAM Conference on Control and its Applications from July 8-10, 2015, at Paris, France. The work of the second author is supported by a Chinese Natural Science Research Grant No. 11261011.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yin, HM., Wei, W. A Nonlinear Optimal Control Problem Arising from a Sterilization Process for Packaged Foods. Appl Math Optim 77, 499–513 (2018). https://doi.org/10.1007/s00245-016-9382-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00245-016-9382-0