On the absolute stability  of automatic control systems in the vicinity of a program manifold

The problem of constructing automatic control systems according to a given program manifold is considered. First, a general method for constructing ordinary differential equations is given, which reduced to choosing some function that provides the stability of a given manifold. Then the absolute stability of automatic control systems in the vicinity of given program manifold is investigated. Nonlinearities satisfy to the conditions of local quadratic connections and they are differentiable in all variables. Sufficient conditions of the absolute stability of the program manifold with respect to the given vector functions are obtained by constructing the Lyapunov function, in the type of “quadratic form plus an integral from nonlinearity”. Frequency conditions of absolute stability for the program manifold of control systems are established. Also sufficient conditions of the absolute stability automatic control systems with non-stationary nonlinearities are received.