Guta, Berhanu(PhD)Aklilu, Kibremarkos2018-07-162023-11-042018-07-162023-11-042017-06http://etd.aau.edu.et/handle/123456789/8799Control theory is an area of applied mathematics that deals with principles,laws, and desire of dynamic systems. Optimal control problems are generalized form of variation problems. A very important tool in variational calculus is the notion of Gateaux-di_erentiabilty.It is the basis of the development of necessary optimality conditions. Euler lagrange di_erential equation(ELDE) is a necessary optimality condition to solve variational problems. The solution of Euler lagrange di_erential equation is an extremal function of a variational problem. Characterizing theorem of convex optimization is the necessary and su_cient condition of many convex problems. i.e Let (P) be given, S is convex set, f is convex function and x0 2 S. Then x02 M(f,S) if and only if f0(x0,x-x0)_ 0 , 8 x2 S. In an optimal control problem our aim is to _nd the optimal state function x_(t) and the optimal control function u_(t) which optimize the objective functional in t2[a,b] by using necessary and su_cient optimality conditions. The necessary optimality conditions for (x_; u_) to be extremal solutions of optimal control problem is the validity of :- Pontryagin minimum principle, of ELDE with TR,and ODE conditions. To determine whether the extremals are optimal solutions of OCP or not; su_cient optimality conditions are required; (e.g checking the convexity of the objective functional and the convexity of the feasible set). Quadratic optimal control problem is a non linear optimization where the cost function is quadratic but the di_erential equation is linear.In quadratic control problem since the objective function is convex then the extremals are the optimal solution of the problem. Linear-Quadratic optimal control problem is an important type of quadratic control problem that simpli_es the work of feed back control system. Optimal control problem can be solved by di_erent methods depending on the type of the problem.This paper mainly considers solving quadratic optimal control problem by using the method of lagrange multiplier. Key words : Variational problem with _xed end points,Variational problem with free right end points,Euler Lagrange Di_erential Equation,optimal control problem,Quadratic control problemenVariational Problem With _Xed end PointsVariational Problem With Free Right end PointsEuler Lagrange Di_Erential EquationOptimal Control ProblemQuadratic Control ProblemThesis