Point Symmetries of Controlled Systems and Their Applications

A problem of finding point symmetries of controlled systems is discussed, basic the-orems and algorithms are formulated. The application to some problems of flight dynamics is suggested. It is well known that the concept of point symmetries of a system of ordinary differential equations can be formulated in two ways: 1) the symmetry transfers solutions of system (1) into ones, 2) the symmetry preserves the form of system (1): ẋi = f i(t, x), i = 1, n. (1) In contrast to the latter, dynamic systems with control are characterized by presence of some free functions in right-hand part of differential equations of the form (2), control actions u: ẋi = f i(t, x, u(·)), i = 1, n. (2) The methodological problem in the development of group analysis of these systems is in the necessity of ”a priori ” choosing the principle of control, i.e., the form of the function u. For example, if u = u(t), we speak about program control, at u = u(t, x) about feedback control, at u = u(t, x, ẋ) about derivative control, etc. Accordingly, the representations of solutions of system (2) are different. The present research is based on the concept of feedback control. Therefore we proceed from system (2) to the equivalent equation in first-order partial derivatives F = X0S = St + f i(t, x, u)Sxi = 0, (3) where X0 = ∂t+ f i∂xi is a vector field associated with system (2), and solutions of (3) are formed by the set of pairs of functions (u(t, x), S(t, x)). We shall consider only controlled systems, when for equation (3) there exist no such solutions S(t, x) which satisfy equation (3) for all functions u(t, x): 6 ∃ S(t, x) : X0S = 0 ∀ u(t, x). (4) The equation (3) has special properties