A new approach for determining multi-objective optimal control of semilinear parabolic problems
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A new approach for determining multi-objective optimal control of semilinear parabolic problems H. Alimorad1 Received: 24 April 2018 / Revised: 22 January 2019 / Accepted: 28 January 2019 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019
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. Keywords Multi-objective optimal control problem · Pareto solution · Evolutionary algorithm · Radon measure Mathematics Subject Classification 90C29 · 49M27
1 Introduction In real applications, optimization problems are often described by introducing several objective functions conflicting with each other. This leads to multi-objective or multicriterial optimization problems; see, e.g., Ehrgott (2005). In the area of control engineering, multiobjective optimization has been discussed by control engineers [see, e.g., Gambier and Bareddin (2007)]. These objectives often involve conflict situations of many criteria, such as control energy, tracking performance and robustness. A suitable introduction on the concepts of MOOCP may be found in Gambier and Jipp (2011). Also, one may find an overview on multi-objective optimization applications in control engineering in Liu et al. (2003). Over the years, some indirect and direct approaches have been presented to extract analytical and approximate Pareto solutions of MOOCP ’s Yalcin Kaya and Maurer (2014) and El-Kady et al. (2003). But, these approaches are facing some difficulties. For instance, convexity of
Communicated by Maria do Rosário de Pinho.
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H. Alimorad [email protected] Department of Mathematics, Jahrom University, P.O. Box: 74135-111, Jahrom, Iran
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the objectives is a basic requirement which limits the scope of applications of such methods Maity and Maiti (2005). Rubio Rubio (1986) applied the embedding method for solving a control system governed by an elliptic equation to find the global control for the described system. The history of these ideas can be find for instance in Rubio (1990). Based on these papers, here we modify this method. In this manner, we present the problem in a variational form; next, it is transferred into a new theoretical measure problem in which one unknown positive Radon measure in a space of measures is sought. Then, a two-stage approximation is used to convert the optimal control problem to a finite dimensional NLP. The solution of this NLP is used to construct an approximate solution to the original mul
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