Control theory is an area of applied mathematics that deals with principles,laws, and desirernof dynamic systems.rnOptimal control problems are generalized form of variation problems. A very importantrntool in variational calculus is the notion of Gateaux-di_erentiabilty.It is the basis of therndevelopment of necessary optimality conditions.rnEuler lagrange di_erential equation(ELDE) is a necessary optimality condition to solvernvariational problems. The solution of Euler lagrange di_erential equation is an extremalrnfunction of a variational problem.rnCharacterizing theorem of convex optimization is the necessary and su_cient condition ofrnmany convex problems. i.ernLet (P) be given, S is convex set, f is convex function and x0 2 S. Then x02 M(f,S) if andrnonly if f0(x0,x-x0)_ 0 , 8 x2 S.rnIn an optimal control problem our aim is to _nd the optimal state function x_(t) and thernoptimal control function u_(t) which optimize the objective functional in t2[a,b] by usingrnnecessary and su_cient optimality conditions.rnThe necessary optimality conditions for (x_; u_) to be extremal solutions of optimal controlrnproblem is the validity of :- Pontryagin minimum principle, of ELDE with TR,and ODErnconditions.rnTo determine whether the extremals are optimal solutions of OCP or not; su_cient optimalityrnconditions are required; (e.g checking the convexity of the objective functional andrnthe convexity of the feasible set).rnQuadratic optimal control problem is a non linear optimization where the cost functionrnis quadratic but the di_erential equation is linear.In quadratic control problem since thernobjective function is convex then the extremals are the optimal solution of the problem.rnLinear-Quadratic optimal control problem is an important type of quadratic control problemrnthat simpli_es the work of feed back control system. Optimal control problem can bernsolved by di_erent methods depending on the type of the problem.This paper mainly considersrnsolving quadratic optimal control problem by using the method of lagrange multiplier.rnKey words : Variational problem with _xed end points,Variational problem with freernright end points,Euler Lagrange Di_erential Equation,optimal control problem,Quadraticrncontrol problem