On the stability of non-autonomous automatic control systems in the neighborhood of a program manifold

The stability of non-autonomous automatic control systems with respect to the vector function ω in a neighborhood of a program manifold is studied. First, we consider the case where the Erugin function matrix and the Lyapunov matrix are constant, while the control matrix and the feedback matrix are variable. The nonlinear control function is assumed to satisfy a general local quadratic constraint. By constructing a Lyapunov function in quadratic form with a constant matrix, sufficient conditions for the absolute stability of the control system in a neighborhood of the program manifold are obtained. We then consider a control system in which the nonlinearity satisfies a transformed local quadratic constraint. For this system, a Lyapunov function of the form “quadratic form with a constant matrix plus an integral of the local quadratic constraint” is constructed, which yields sufficient conditions for absolute stability of the control system near the program manifold. Next, we address the case where both the Erugin function matrix and the Lyapunov matrix are variable. By constructing a Lyapunov function in quadratic form with a variable matrix, we obtain sufficient conditions for the asymptotic stability of a non-autonomous system without control and the absolute stability of a non-autonomous control system, expressed in terms of generalized Sylvester inequalities. An illustrative example is provided.