Abstract

The Noether's Integrals in Optimal Problems of Hypersonic Aerodynamics

N.Bilchenko (Department of Special Mathematics, Kazan State Technical University, Kazan, Russia)

e-mail: Grigory.Bilchenko@ksu.ru

To solve an actual problem of atmospheric entry aerospace planes surfaces heat protection the taking into account of physical and chemical processes (dissociation and ionization) and their flowing finite velocity (non equilibrium flow conditions) is necessary. The heat transfer on the hypersonic aircraft surfaces can be investigated in the limits of boundary layer theory. Majority of works on the theory of optimal controlled boundary layer is connected with the control of incompressible liquid movement, the problems of compressible viscous gas optimal control are considered in a small number of publications. The group-theoretic approach suggested by K.G.Garaev for optimization of distributed parameters systems based on the Lie-Ovsyannikov\'s infinitesimal apparatus and the Noether-Ibragimov's theory of invariant variation problems allows to construct the Noether's integrals for the Euler-Lagrange-Ostrogradsky's systems of equations for various cases of boundary layer control. From the computational standpoint the knowledge of the divergent forms (the first integrals) in the problems of controlled processes optimization simplifies sufficiently the problem of optimal controls search. In the paper the Noether's integrals obtained in three optimal problems of hypersonic aerodynamics are considered (for one problem of ionized gas optimal control and two problems of non equilibrium dissociated gas optimal control).

Section : 12