Guta, Berhanu (PhD)Mamo, Asrat2022-11-282023-11-042022-11-282023-11-042022-11-25http://etd.aau.edu.et/handle/123456789/32668Control 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-differentiability. It is the basis of the development of necessary optimality conditions. ELDE is a necessary optimality condition to solve variational problems. The solution of ELDE is an extremal function of a variational problem. Characterizing theorem of convex optimization is the necessary and sufficient condition of many convex problems. i.e Let (P) be given, S is convex set, f is convex function and x0 ∈ S. Then x0 ∈ M(f, S) if and only if f′(x0, x−x0) ≥ 0, ∀x ∈ S. In an OCP our aim is to find the optimal state function x∗(t) and the optimal control function u∗(t) which optimize the objective functional in t ∈ [a, b] by using necessary and sufficient optimality conditions. The necessary optimality conditions for (x∗(t), u∗(t)) 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; sufficient optimality conditions are required; (e.g checking the convexity of the objective functional and the convexity of the feasible set). QOCP is a non linear optimization where the cost function is quadratic but the differential equation is linear. In quadratic control problem since the objective function is convex then the extremals are the optimal solution of the problem. Linear QOCP is an important type of quadratic control problem that simples the work of feed back control system. OCP can be solved by different methods depending on the type of the problem. This paper mainly considers solving QOCP by using the method of lagrange multiplier and shooting method.enQuadratic OptimalControl ProblemsShooting MethodThesis