Abstract
In this paper, two approaches based on evolutionary algorithms are applied to solve a multi-objective optimal control problem governed by semilinear parabolic partial differential equations. In this approach, first, we change the problem into a measure-theoretical one, replace this with an equivalent infinite dimensional multi-objective nonlinear programming problem and apply approximating schemes. Finally, non-dominated sorting genetic algorithm and multi-objective particle swarm optimization are employed to obtain Pareto optimal solutions of the problem. Numerical examples are presented to show the efficiency of the given approach.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alimorad H, Fakharzadeh JA (2017) A theoretical measure technique for determining 3D symmetric nearly optimal shapes with a given center of mass. Comput Math Math Phys 57(7):12251240
Borzabadi AH, Hasanabadi M, Sadjadi N (2016) Approximate Pareto optimal solutions of multi-objective optimal control problems by evolutionary algorithms. Control Optim Appl Math 1(1):1–19
Ehrgott M (2005) Multicriteria optimization. Springer, Berlin
Eichfelder G (2009) An adaptive scalarization method in multi-objective optimization. SIAM J Optim 19(4):1694–1718
El-Kady MM, Salim MS, El-Sagheer AM (2003) Numerical treatment of multi-objective optimal control problems. Automatica 39:4755
Fakharzadeh JA, Rubio JE (1999) Shape and measure. J Math Control Inf 16:207–220
Fakharzadeh JA, Alimorad H, Rafiei Z (2013) Using linearization and penalty approach to solve optimal shape design problem with an obstacle. J Math Comput Sci 7:43–53
Gambier A, Bareddin E (2007) Multi-objective optimal control: an overview. In: IEEE conference on control applications, CCA, Singapore, pp 170–175
Gambier A, Jipp M (2011) Multi-objective optimal control: an introduction. In: Proceedings of the 8th Asian control conference (ASCC11). Kaohsiung, Taiwan, pp 15–18
Introduction to Galerkin Methods (2016) http://fischerp.cs.illinois.edu/tam470/refs/galerkin2.pdf. Accessed October 2018
Iapichino SL, Volkwein ST (2015) Reduced-order multi-objective optimal control of semilinear parabolic problems. Konstanzer Online-Pablikations Syst 347:1–8
Kumar V, Minz S (2014) Multi-objective particle swarm optimization: an introduction. Smart Comput Rev 4(5):335–353
Liu GP, Yang JB, Whidborne JF (2003) Multi-objective optimization and control. Research Studies Press Ltd., Exeter
Maity K, Maiti M (2005) Numerical approach of multi-objective optimal control problem in imprecise environment. Fuzzy Optim Decision Mak 4:313–330
Munch A (2009) Optimal internal dissipation of damped wave equation using a topological approach. Int J Appl Math Comput Sci 19:15–37
Rosenbloom PC (1956) Qudques classes de problems exteremaux. Bull Soc Math France 80:183–216
Rubio JE (1986) Control and optimization: the linear treatment of nonlinear problems. Manchester university Press, Manchester
Rubio JE (1990) Modern trends in calculus of variations and optimal control theory. In: Geise R (ed) Vortagsauszuge, Mathematiker kongress, 155–163, Berlin Mathematische Gesellschaft
Rudin W (1983) Real and complex analysis, 2nd edn. Tata McGraw-Hill Publishing Co Ltd., New Dehli
Yalcin Kaya C, Maurer H (2014) A numerical method for nonconvex multi-objective optimal control problems. Comput Optim Appl 57(3):685–702
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Maria do Rosário de Pinho.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Alimorad, H. A new approach for determining multi-objective optimal control of semilinear parabolic problems. Comp. Appl. Math. 38, 27 (2019). https://doi.org/10.1007/s40314-019-0809-5
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-019-0809-5