### Abstract:

Control 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 problem