×

zbMATH — the first resource for mathematics

Analysis of a multistate control problem related to food technology. (English) Zbl 1147.49003
Summary: This paper is concerned with an optimal control problem related to the determination of an optimal profile for the steam temperature into the autoclave along the processing of canned foods. The problem studies a system coupling the evolution Navier-Stokes equations with the heat transfer equation by natural convection (the so-called Boussinesq equations), and with the microorganisms removal equation. The essential difficulties in the study of this multistate control problem arise from the lack of uniqueness for the solution of the state system. Here we obtain – after a careful analysis of the mathematical formulation of the problem – the uniqueness of part of the state, and the existence of optimal solutions.

MSC:
49J20 Existence theories for optimal control problems involving partial differential equations
35B37 PDE in connection with control problems (MSC2000)
49N90 Applications of optimal control and differential games
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abergel, F.; Temam, R., Optimality conditions for some nonqualified problems of distributed control, SIAM J. control optim., 27, 1-12, (1989) · Zbl 0681.49025
[2] Alvarez-Vázquez, L.J.; Marta, M.; Martínez, A., Sterilization of canned viscous foods: an optimal control approach, Math. models methods appl. sci., 14, 355-374, (2004) · Zbl 1069.49028
[3] Alvarez-Vázquez, L.J.; Martínez, A., Modelling and control of natural convection in canned foods, IMA J. appl. math., 63, 247-265, (1999) · Zbl 0936.93039
[4] Blanchard, D.; Murat, F., Renormalised solutions of nonlinear parabolic problems with \(L^1\) data: existence and uniqueness, Proc. roy. soc. Edinburgh sect. A, 127, 1131-1152, (1997) · Zbl 0895.35050
[5] Boccardo, L.; Murat, F.; Puel, J., Existence results for some quasilinear parabolic equations, Nonlinear anal. theory methods appl., 13, 373-392, (1989) · Zbl 0705.35066
[6] Brezis, H., Analyse fonctionelle, (1993), Masson Paris
[7] Casas, E., The navier – stokes equations coupled with the heat equation: analysis and control, Control cybernet., 23, 605-620, (1994) · Zbl 0901.49003
[8] A.K. Datta, Numerical modeling of natural convection and conduction heat transfer in canned foods with application to on-line process control, PhD thesis, University of Florida, 1985
[9] Engelman, M.; Sani, R.L., Finite element simulation of an in-package pasteurization process, Numer. heat transfer, 6, 41-61, (1983)
[10] Fattorini, H.O.; Sritharan, S.S., Optimal control problems with state constraints in fluid mechanics and combustion, Appl. math. optim., 38, 159-192, (1998) · Zbl 1068.49501
[11] Kavian, O., Introduction à la théorie des points critiques et applications aux problèmes elliptiques, (1993), Springer-Verlag Paris · Zbl 0797.58005
[12] Kumar, A.; Bhattacharya, M.; Blaylock, J., Numerical simulation of natural convection heating of canned thick viscous liquid food products, J. food sci., 55, 1403-1411, (1990)
[13] Ladyzhenskaya, O.A., The mathematical theory of viscous incompressible flow, (1969), Gordon and Breach New York · Zbl 0184.52603
[14] Li, S., Optimal control of Boussinesq equations with state constraints, Nonlinear anal., 60, 1485-1508, (2005) · Zbl 1100.49026
[15] P.M. Stevens, Lethality calculations including effects of product movement, for convection heating and broken-heating foods in still-cook retorts, PhD thesis, University of Massachusetts, 1972
[16] Temam, R., Navier – stokes equations, (1979), North-Holland Amsterdam · Zbl 0454.35073
[17] Trenchea, C., Periodic optimal control of the Boussinesq equation, Nonlinear anal., 53, 81-96, (2003) · Zbl 1039.49026
[18] Wang, G., Optimal controls of 3-dimensional navier – stokes equations with state constraints, SIAM J. control optim., 41, 583-606, (2002) · Zbl 1022.93026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.