Parameter Identification and Control in Heat Transfer Processes

The article is devoted to the study of some mathematical models describing heat transfer processes. We examine an inverse problem of recovering a control parameter providing a prescribed temperature distribution at a given point of the spatial domain. The parameter is a lower order coefficient depending on time in a parabolic equation. This nonlinear problem is reduced to an operator equation whose solvability is established with the help of a priori estimates and the fixed point theorem. Existence and uniqueness theorems of solutions to this problem are stated and proved. Stability estimates are exposed. The main result is the global (in time) existence of solutions under some natural conditions of the data. The proofs rely on the maximum principle. The main functional spaces used are the Sobolev spaces.

Keywords

heat transfer; distributed control; mathematical model; parabolic equation; inverse problem; boundary value problem.

References

1. Alifanov O.M. Inverse Heat Transfer Problems. Berlin, Heidelberg, Springer-Verlag, 1994. DOI:10.1007/978-3-642-76436-3
2. Ozisik M.N., Orlando H.R.B. Inverse Heat Transfer. N.-Y., Taylor & Francis, 2000.
3. Dehghan M. Numerical Computation of a Control Function in a Partial Differential Equation. Applied Mathematics and Computation, 2004, vol. 147, no. 2, pp. 397-408. DOI: 10.1016/S0096-3003(02)00733-6
4. Dehghan M., Shakeri F. Method of Lines Solutions of the Parabolic Inverse Problem with an Overspecification at a Point. Numerical Algorithms, 2009, vol. 50, no. 4, pp. 417-437. DOI: 10.1007/s11075-008-9234-3
5. Dehghan M. Parameter Determination in a Partial Differential Equation from the Overspecified Data. Mathematical and Computer Modelling, 2005, vol. 41, no. 2-3, pp. 196-213. DOI: 10.1016/j.mcm.2004.07.010
6. Iskenderova A.D., Akhundov A.Ya. Inverse Problem for a Linear System of Parabolic Equations. Doklady Mathematics, 2009, vol. 79, no. 1, pp. 73-75. DOI:10.1134/S1064562409010219
7. Kuliev M.A. A Multidimensional Inverse Problem for a Parabolic Equation in a Bounded Domain. Nonlinear Boundary Value Problems, 2004, vol. 14, pp. 138-145.
8. Prilepko A.I., Orlovsky D.G., Vasin I.A. Methods for Solving Inverse Problems in Mathematical Physics. N.-Y., Marcel Dekker, 1999.
9. Pyatkov S.G., Samkov M.L. On Some Classes of Coefficient Inverse Problems for Parabolic Systems of Equations. Siberian Advances in Mathematics, 2012, vol. 22, no. 4, pp. 287-302. DOI:10.3103/S1055134412040050
10. Cannon J.R., Yin H.-M. A Class of Non-Linear Non-Classical Parabolic Equations. Journal of Differential Equations, 1989, vol. 79, issue 2, pp. 266-288. DOI:10.1016/0022-0396(89)90103-4
11. Shidfar A. An Inverse Heat Conduction Problem. Southeast Asian Bulletin of Mathematics, 2003, vol. 26, no. 3, pp. 503-507. DOI:10.1007/s10012-002-0503-0
12. Ivanchov N.I., Pabyrivska N.V. On Determination of Two Time-Dependent Coefficients in a Parabolic Equation. Siberian Mathematical Journal, 2002, vol. 43, no. 2, pp. 323-329. DOI:10.1023/A:1014749222472
13. Ivanchov M. Inverse Problems for Equations of Parabolic Type. Lviv, WNTL Publishers, 2003.
14. Cannon J.R. An Inverse Problem of Finding a Parameter in a Semi-linear Heat Equation. Journal of Mathematical Analysis and Applications, 1990, vol. 145, issue 2, pp. 470-484. DOI:10.1016/0022-247X(90)90414-B
15. Kozhanov A.I. Parabolic Equations with an Unknown Time-Dependent Coefficient. Computational Mathematics and Mathematical Physics, 2005, vol. 45, no. 12, pp. 2085-2101.
16. Ismailov M.I., Kanca F. Inverse Problem of Finding the Time-Dependent Coefficient of Heat Equation from Integral Overdetermination Condition Data. Inverse Problems In Science and Engineering, 2012, vol. 20, issue 2, pp. 463-476. DOI:10.1080/17415977.2011.629093
17. Hussein M.S., Lesnic D. Simultaneous Determination of Time-Dependent Coefficients and Heat Source. International Journal for Computational Methods in Engineering Science and Mechanics, 2016, vol. 17, issue 5-6, pp. 401-411. DOI:10.1080/15502287.2016.1231241
18. Kamynin V.L. Unique Solvability of the Inverse Problem of Determination of the Leading Coefficient in a Parabolic Equation. Differential Equations, 2011, vol. 47, no. 1, pp. 91-101. DOI:10.1134/S0012266111010101
19. Triebel H. Interpolation Theory. Function Spaces. Differential Operators. Berlin, VEB Deutscher verlag der wissenschaften, 1978. DOI:10.1002/zamm.19790591227
20. Denk R., Hieber M., Pr'uss J. Optimal L_{p}-L_{q}-estimates for Parabolic Boundary Value Problems with Inhomogeneous Data. Mathematische zeitschrift, 2007, vol. 257, issue 1, pp. 193-224. DOI:10.1007/s00209-007-0120-9
21. Ladyzhenskaya O.A., Solonnikov V.A., Ural'tseva N.N. Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, 1968.
22. Amann H. Nonautonomous Parabolic Equations Involving Measures. Journal of Mathematical Sciences, 2005, vol. 130, no. 4, pp. 4780-4802. DOI:10.1007/s10958-005-0376-8
23. Amann H. Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary-Value Problems. Function Spaces, Differential Operators and Nonlinear Analysis, Stuttgart, Teubner, 1993, vol. 133, pp. 9-126.
24. Grisvard P. Equations differentielles abstraites. Annales scientifiques de l'Ecole Normale Superieure, 1969, vol. 2, issue 2, pp. 311-395.
25. Ladyzhenskaya O.A., Ural'tseva N.N. Linear and Quasilinear Elliptic Equations. N.-Y., London, Academic Press, 1968.
26. Pyatkov S.G., Tsybikov B.N. On Some Classes of Inverse Problems for Parabolic and Elliptic Equations. Journal of Evolution Equations, 2011, vol. 11, no. 1, pp. 155-186. DOI:10.1007/s00028-010-0087-6
27. Lieberman G.M. Second Order Parabolic Differential Equations. Singapore, World Scientific Publishing, 1998.
28. Triebel H. Theory of Function Spaces. Basel, Boston, Stuttgart, Birkhauser verlag, 1983. DOI: 10.1007/978-3-0346-0416-1