This thesis is an extension (and generalization) of the work done by Sergio Blanes(\\cite{40}) . In an optimal control problem, a set of differential equations describes path of control variables to minimize cost function (performance index or cost criterion). The linear quadratic (LQ) optimal control refers to problems that a linear systems must be controlled in a way that maximize or minimize a quadratic cost function. Linear quadratic (LQ) optimal control problems appear in many different fields in engineering as well as quantum mechanics , electrical engineering . These problem are also appeared in the modeling of phenomena in aerospace, aircraft flight control, stochastic differential equation etc. The numerical integration of linear quadratic optimal control problems needs the solution of boundary value problems (BVPs), a non-autonomous matrix Riccati differential equations (RDE) with final conditions coupled with the state vector equation with initial conditions. In these problems, solution have qualitative properties which are relevant to the theoretical study. For example, solution of Riccati differential equation, which appears in the linear quadratic optimal control problems should be positive definite. It seems natural to look for numerical methods (schemes or integrators) that preserve these relevant qualitative properties in order to get trustworthy and accurate results. It is important that one step and multistep methods of order greater than one cannot guarantee that the solution stays positive definite. On the other hand, Riccati differential equations has an associated Hamiltonian system which can be solved using appropriate symplectic integrators and analyzed splitting method as symplectic integrators to solve the Riccati differential equation coupled with the state vector equation both for the autonomous and non-autonomous case. In the case of non-autonomous duplicating the time as two new coordinates and using appropriate splitting methods, high order methods preserving the desired property can be obtained. The methods make successive computations and do not require the storage of intermediate results, so the storage requirements are minimal. Numerical methods considered for solving linear quadratic optimal control problems, are high order, explicit and structure preserving methods. Among this geometric numerical integrations can be referred to the splitting methods and composition methods. The proposed methods are also adapted for solving linear quadratic N-player differential games. The performance of the splitting methods can be considerably improved if the systems is a perturbation (near-integrable) of an exactly solvable problem and the system is properly split. If the equations correspond to a near-integrable system, tailored splitting methods for perturbed systems provide a further improvement. Similar ideas could be used for solving non-linear optimal control problems This theses organized as followed, some important methods and techniques such as splitting methods, composition methods and some properties such as symmetry , symplecity, reversibility in geometric numerical integration are presented in Chapter 2. In the Chapter 3 , the introduced structure preserving methods are used for finding the solution of linear quadratic N-player non-cooperative differential game as a special case of the linear quadratic optimal control problems. Some numerical examples illustrate the performance of the proposed methods.