International Journal of Computer Mathematics Vol. 85, No. 11, November 2008, 1649–1671 Numerical solutions for constrained time-delayed optimal control problems M. El-Kady a * and Elsayed M.E. Elbarbary b a Department of Mathematics, Faculty of Science, SouthValley University, Aswan, Egypt; b Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt (Received 21 April 2006; revised version received 07 June 2007; accepted 24 June 2007 ) In this paper, time-delayed optimal control problems governed by delayed differential equation are solved. Two different techniques based on integration and differentiation matrices are considered.The time-delayed term of the problem has been approximated by Chebyshev interpolating polynomials. On this basis, the optimal control problem can be solved as a mathematical programming problem. The example illustrates the robustness, accuracy and efficiency of the proposed numerical techniques. Keywords: delayed differential equation; spectral methods; El-gendi method; differentiation matrix; optimal control 2000 AMS Subject Classification: 34H05; 34K35; 49J15 1. Introduction Many important engineering problems can be posed as time-delayed optimal control problems. Whereas, the classical control parameterization method is a flexible and efficient approach for a large class of optimal control problems, we find that the efficient and accurate solution techniques for delayed optimal control problems are not widely available. Many of the current techniques are difficult to use and require significant computational resources. Moreover, the motivation is to study an optimal control problem for a system presenting discontinuous case of time-delayed type, which appears in particular in the behaviour of thermostats. However, applications were given to an optimal control problem with a discrete version of the Preisach operator and for the continuous version case [1]. Wong et al. [16] presented the control parameterization enhancing transform (CPET) method to solve the time-delayed optimal control problems. Their model transformation is used to convert the time-delayed problem to an optimal control problem involving mixed boundary conditions, but without time-delayed arguments. The CPET is then used to solve this non-delayed problem. For a small delay, in comparison to the time horizon, this method has been creating a large number of new state variables, with attendant boundary value constraints [11, 16]. In [15], a transformation *Corresponding author. Email: mam_el_kady@yahoo.com ISSN 0020-7160 print/ISSN 1029-0265 online © 2008 Taylor & Francis DOI: 10.1080/00207160701542048 http://www.informaworld.com