Necessary conditions for a minimum of a polynomial of integral functionals

This paper investigates the conditions for a minimum of a “polynomial” functional. Gateaux gradient and necessary conditions for a minimum are obtained for the “polynomial” functional. The necessary minimum conditions are used in the description of the steepest descent method for the considered problem. Further the problem of constrained minimizing of the “polynomial” functional is investigated. Using the theory of exact penalty functions, this problem under constraints reduces to the problem of unconstrained minimization. The resulting minimum conditions allow us to describe the method of hypodifferential descent for the considered problem. Numerical examples of the described methods are included. The problem of minimizing the product of powers of the integrals is widely used in aerodynamics. Some examples of integral equations and the problem of the control theory are given, which can be reduced to the problem of minimizing a “polynomial” functional. Bibliogr. 14. Table 1.