A Consecutive Quasilinearization Method for the Optimal Boundary Control of Semilinear Parabolic Equations

Abstract

Optimal boundary control of semilinear parabolic equations requires efficient solution methods in applications. Solution methods bypass the nonlinearity in different approaches. One approach can be quasilinearization (QL) but its applicability is locally in time. Nonetheless, consecutive applications of it can form a new method which is applicable globally in time. Dividing the control problem equivalently into many finite consecutive control subproblems they can be solved consecutively by a QL method. The proposed QL method for each subproblem constructs an infinite sequence of linear-quadratic optimal boundary control problems. These problems have solutions which converge to any optimal solutions of the subproblem. This implies the uniqueness of optimal solution to the subproblem. Merging solutions to the subproblems the solution of original control problem is obtained and its uniqueness is concluded. This uniqueness result is new. The proposed consecutive quasilinearization method is numerically stable with convergence order at least linear. Its consecutive feature prevents large scale computations and increases machine applicability. Its applicability for globalization of locally convergent methods makes it attractive for designing fast hybrid solution methods with global convergence.

Cite this paper

Nayyeri, M. and Kamyad, A. (2014) A Consecutive Quasilinearization Method for the Optimal Boundary Control of Semilinear Parabolic Equations.*Applied Mathematics*, **5**, 691-706. doi: 10.4236/am.2014.54067.

Nayyeri, M. and Kamyad, A. (2014) A Consecutive Quasilinearization Method for the Optimal Boundary Control of Semilinear Parabolic Equations.

References

[1] Hinze, M., Pinnau, R., Ulbrich, M. and Ulbrich, S. (2009) Optimization with PDE Constraints. Springer, Berlin.

[2] Trotzsch, F. (1999) On the Lagrange-Newton-SQP Method for the Optimal Control of Semilinear Parabolic Equations. SIAM Journal on Control and Optimization, 38, 294-312.

http://dx.doi.org/10.1137/S0363012998341423

[3] Bellman, R. and Kalaba, R. (1965) Quasilinearization and Nonlinear Boundary-Value Problems. American Elsevier Publishing Company, New York.

[4] Bellman, R. (1955) Functional Equations in the Theory of Dynamic Programming. V. Positivity and Quasi-Linearity. Proceedings of the National Academy of Sciences of the United States of America, 41, 743-746.

http://dx.doi.org/10.1073/pnas.41.10.743

[5] Buica, A. and Precup, R. (2002) Abstract Generalized Quasilinearization Method for Coincidences. Nonlinear Studies, 9, 371-386.

[6] Lakshmikantham, V. and Vatsala, A.S. (1995) Generalized Quasilinearization for Nonlinear Problems. Kluwer Academic Publishers, Dordrecht.

[7] Carl, S. and Lakshmikantham, V. (2002) Generalized Quasilinearization and Semilinear Parabolic Problems. Nonlinear Analysis, 48, 947-960. http://dx.doi.org/10.1016/S0362-546X(00)00225-X

[8] Beckenbach, E.F. and Bellman, R. (1961) Inequalities. Springer-Verlag, Berlin.

[9] Kalaba, R. (1959) On Nonlinear Differential Equations, the Maximum Operation, and Monotone Convergence. Journal of Mathematics and Mechanics, 8, 519-574.

[10] Raymond, J.P. and Zidani, H. (1999) Hamiltonian Pontryagin’s Principles for Control Problems Governed by Semilinear Parabolic Equations. Applied Mathematics and Optimization, 39, 143-177.

http://dx.doi.org/10.1007/s002459900102

[11] Showalter, R.E. (1997) Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations. American Mathematical Society, Providence.

[12] Evans, L.C. (1998) Partial Differential Equations. American Mathematical Society, Providence.

[13] Ladyzenskaja, O.A., Solonnikov, V.A. and Ural’ceva, N.N. (1968) Linear and Quasi-Linear Equations of Parabolic Type. American Mathematical Society, Providence.

[14] Zeidler, E. (1990) Nonlinear Functional Analysis and Its Applications Vol II/A: Linear Monotone Operators. Springer, Berlin.

[15] Showalter, R.E. (1994) Hilbert Space Methods for Partial Differential Equations. Electronic Journal of Differential Equations, Monograph 01, Dover Publications, Mineola.

[16] Ulbrich, M. (2011) Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces. SIAM. http://dx.doi.org/10.1137/1.9781611970692

[17] Neittaanmaki, P. and Tiba, D. (1994) Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms, and Applications. Marcel Dekker Inc., New York.