ABSTRACT
The easiest result of extremum theory is the necessary condition of extremum for smooth functions of one variable. This is the equality to zero of its derivative at the point of extremum. Minimization problems for functionals are more difficult. The relevant stationary condition includes the derivative of the given functional. This result can be generalized to minimization problems on subspaces or on affine varieties. The variational inequality is the general necessary condition of minimum for functionals on convex sets. These results are the basis of the optimal control theory. The minimized functional depends on the state function that depends on the control here. The analysis of the minimization problems for functionals will be continued in the next chapter. We shall consider the sufficiency of extremum conditions, the existence of optimization problems, and some others there.
