Automatica 39 (2003) 47–55 www.elsevier.com/locate/automatica Brief Paper Numerical treatment of multiobjective optimal control problems  M.M. El-Kady a ; *; 2 , M.S. Salim b , A.M. El-Sagheer a a Faculty of Science, South Valley University, Aswan, Egypt b Faculty of Science, El-Azhar University, Assiut, Egypt Received 23 December 2000; received in revised form 29 May 2002; accepted 3 June 2002 Abstract We present a numerical procedure for solving optimal control problems with both linear terminal constraints and multiple criteria. Using a Chebyshev spectral procedure, the problem reduces to a constrained optimization problem which can be solved using hybrid penalty partial quadratic interpolation (HPPQI) technique. The proposed procedure compares quite favorably with other methods on a sample of well-known examples. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Parametric optimal control; Multiobjective constraints 1. Introduction Constrained optimization problems are of widespread in- terest because of their connection with optimal control and practical problems. Optimal control problem with linear ter- minal constraints have been treated in many approaches: Mehra and Davis (1972) use nite-dierence approxima- tions that yield a static optimization problem in a large num- ber of variables. Sirisena and Tan (1974) provide an algorithm for solving optimal control problems with terminal state variable con- straints in which a piecewise polynomial parameterization of the control variable is employed to reduce the original problem to static optimization problems. Sirisena and Chou (1976) present a modication of the previous approach by parameterizing the system state vari- ables rather than the control variables. This modication yields a static optimization problem with linear constraints whenever the terminal constraints are linear in the state vari- ables irrespective of whether or not the dynamic system it- self is linear. The static optimization problem is solved using  This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor J.B. Lasserre under the direction of Editor Alain Hansie. ∗ Corresponding author. KSA, P.O. Box 1144, Tabuk, Saudi Arabia. E-mail address: m el kady@hotmail.com (M.M. El-Kady). 1 Present address: Department of Computer Science, Tabuk Teachers College, P.O. Box 1144, Tabuk, Saudi Arabia. the quadratically convergent, Goldfarb–Lapidus algorithm (1968). In recent years, the analysis of several conicting objec- tives has become an important issue. Such multiobjective optimization problems reect the complexity of decision making in the real world. During the past two decades, both theory and method- ology for multiobjective optimization have grown by leaps and bounds. Results concerning a unied treatment of both theory and methodology are discussed by Chankong and Hamies (1983). See also Salukvadze’s book (1979) devoted to vector-valued optimization problems in control theory. Multiobjective optimization problem Duan (1996) opti- mizes a set of objectives with respect to the free controller parameters. Here the diculty is that there is no unique “op- timal ” solution but rather a set of “noninferior” solutions. Fleming (1985) presented a modication for his approach, and he showed that, instead of representing conicting per- formance measures in a scalar objective function, these mea- sures are treated individually in an attempt to optimize the objectives. Throughout this paper, a comparison between Fleming’s technique and our proposed technique for solving the Salukvadze problem will be given. The proposed algorithm describes an alternative technique based on the expansion of the control variables in Cheby- shev series with unknown coecients. In the system dynam- ics, the state variables can be obtained by transforming the boundary value problem for ordinary dierential equations 0005-1098/03/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII:S0005-1098(02)00109-7