IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN:2319-765X. Volume 9, Issue 5 (Jan. 2014), PP 20-24 www.iosrjournals.org www.iosrjournals.org 20 | Page On Construction of A Control Operator Introduced To ECGM Algorithm for Solving Discrete-Time Linear Quadratic Regulator Control Systems with Delay-I K. J. ADEBAYO and F. M. ADERIBIGBE Department of Mathematical Sciences, Ekiti State University, Ado Ekiti, Nigeria. Abstract: In this paper, we constructed a control operator sequel to an earlier constructed control operator in one of our papers which enables an Extended Conjugate Gradient Method (ECGM) to be employed in solving discrete time linear quadratic regulator problems with delay parameter in the state variable. The construction of the control operator places scalar linear delay problems of the type within the class of problems that can be solved with the ECGM and it is aimed at reducing the rigours faced in using the classical methods in solving this class of problem. More so, the authors of this paper desire that, the application of this control operator will further improve the results of the ECGM as well as increasing the variant approaches used in solving the said class of optimal control problem. Keywords: Control Operator, Optimal Control, Discrete-Time Linear Regulator Problem, Extended Conjugate Gradient Method and Differential Delay State. I. Introduction In [1] we have considered the construction of a control operator for continuous-time linear regulator problems with delay parameter. This serves as a spring board and motivated the construction of a similar control operator for discrete-time linear quadratic regulator problems with delay parameter. The continuous-time linear quadratic regulator performance measure to be minimized considered by [6] and the Bolza problem of [4] and [3] as: Problem (P1):              (  )(  )   ∫ {      }     1.1 Subject to the differential delay state equation ̇                  1.2        1.3 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, the delay parameter,      is a given piecewise continuous function which is of exponential order on  . According to [2] and with similar report by [1], the controlled differential–delay constraint (1.2) constitutes an important model which has been used variously. Sequel to this, equation (1.1) can be rewritten as:   ∫ {        (  )(  )}         ∫ {      }     1.4   ∫ {  ̇}         ∫ {      }     1.5  ∫ {  ̇            }      1.6 As customary with penalty function techniques, constrained problem equations (1.2) and (1.6) may be put into the following equivalent form:       ∫ {  ̇                  ‖            ̇‖  } 1.7        1.8 where  is the penalty parameter and ‖            ̇‖  is the penalty term. Let us denote by   the product space    [    ]  [    ]     1.9 of the 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: