IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN:2319-765X. Volume 10, Issue 3 Ver. IV (May-Jun. 2014), PP 10-15 www.iosrjournals.org www.iosrjournals.org 10 | Page On Construction of a Control Operator Applied In ECGM Algorithm 1 F. M. Aderibigbe, 2 K. J. Adebayo and 3 B. Ojo 1,2,3 Department of Mathematical Sciences, Ekiti State University, Ado Ekiti, Nigeria. Abstract: In this paper, we constructed a control operator, G, which enables an Extended Conjugate Gradient Method (ECGM) to be employed in solving for the optimal control and trajectories of continuous time linear regulator problems. Similar operators constructed in the past by various authors have limited application. This call for the construction of the control operator that is aimed at taking care of any of the Mayer’s, Lagrange’s and Bolza’s cost form of linear regulator problems. The authors of this paper desire that, with the construction of the operator, one will circumvent the difficulties undergone using the classical methods and its application will further improve the result of the Extended Conjugate Gradient Method in solving this class of optimal control problem. Keywords: Continuous Linear Regulator Problem, Control Operator, Extended Conjugate Gradient Method and Optimal Control. I. Introduction Consider the linear regulator problem of the Bolza type as in [6] and [1]: Problem (P1): ( )( ) ∫ { } 1.1 subject to the differential state equation ̇ 1.2 where H and are real symmetric positive semi-definite matries. is a real symmetric positive definite matrix, the initial time, and the final time, are specified. is an n-dimensional state vector, is the m-dimensional plant control input vector. are not constrained by any boundaries. are specified constants which are not necessarily positive.For H = 0, (1.1) is called a Lagrange problem, but if Q(t) = R(t) = 0, it is called a Mayer problem. The form (1.1) may be rewritten as: ∫ { ( )( )} ∫ { } 1.3 ∫ { ̇} ∫ { } ∫ { ̇ } 1.4 As customary with penalty function techniques, constrained problem equations (1.2) and (1.4) may be put into the following equivalent form: ∫ { ̇ ‖̇ ‖ } 1.5 where is the penalty parameter and ‖̇ ‖ is the penalty term. Let us denote by the product space [ ] [ ] 1.6 is the product space of Sobolev space [ ] of absolutely continuous function such that, both ̇ are square integrable over the finite interval [ and the Hilbert space [ of equivalence classes of real valued functions on [ with norm defined by: ‖ ‖ [ (∫ || ) [ . 1.7 Then, the inner product on is given by [ ] [ ] 1.8 Suppose denotes the ordered pair ( ) [ ] [ ], 1.9 then, we seek to determine the operator G on such that